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- 1. Financial Risk Management 2010-11 TopicsT1 Stock index futures Duration, Convexity, ImmunizationT2 Repo and reverse repo Futures on T-bills Futures on T-bonds Delta, Gamma, Vega hedgingT3 Portfolio insurance Implied volatility and volatility smilesT4 Modelling stock prices using GBM Interest rate derivatives (Bond options, Caps, Floors, Swaptions)T5 Value at RiskT6 Value at Risk: statistical issues Monte Carlo Simulations Principal Component Analysis Other VaR measuresT7 Parametric volatility models (GARCH type models) Non-parametric volatility models (Range and high frequency models) Multivariate volatility models (Dynamic Conditional Correlation DCC models)T8 Credit Risk Measures (credit metrics, KMV, Credit Risk Plus, CPV)T9 Credit derivatives (credit options, total return swaps, credit default swaps) Asset Backed Securitization Collateralized Debt Obligations (CDO)* This file provides you an indication of the range of topics that is planned to be covered in themodule. However, please note that the topic plans might be subject to change.
- 2. Topics Financial Risk Management Futures Contract: Speculation, arbitrage, and hedging Topic 1 Stock Index Futures Contract: Managing risk using Futures Reading: CN(2001) chapter 3 Hedging (minimum variance hedge ratio) Hedging market risks Futures ContractAgreement to buy or sell “something” in the future ata price agreed today. (It provides Leverage.)Speculation with Futures: Buy low, sell high Futures (unlike Forwards) can be closed anytime by taking an opposite position Speculation with FuturesArbitrage with Futures: Spot and Futures are linkedby actions of arbitragers. So they move one for one.Hedging with Futures: Example: In January, a farmerwants to lock in the sale price of his hogs which willbe “fat and pretty” in September. Sell live hog Futures contract in Jan with maturity in Sept
- 3. Speculation with Futures Speculation with FuturesPurchase at F0 = 100Hope to sell at higher price later F1 = 110 Profit/Loss per contractClose-out position before delivery date. Long futureObtain Leverage (i.e. initial margin is ‘low’) $10Example:Example: Nick Leeson: Feb 1995 F1 = 90 0 F0 = 100Long 61,000 Nikkei-225 index futures (underlying F1 = 110 Futures pricevalue = $7bn). -$10Nikkei fell and he lost money (lots of it) - he was supposed to be doing riskless ‘index Short futurearbitrage’ not speculating Speculation with FuturesProfit payoff (direction vectors) F increase F increase then profit increases then profit decreaseProfit/Loss Profit/Loss Arbitrage with Futures -1 +1 Underlying,S +1 or Futures, F -1 Long Futures Short Futures or, Long Spot or, Short Spot
- 4. Arbitrage with Futures Arbitrage with FuturesAt expiry (T), FT = ST . Else we can make riskless General formula for non-income paying security:profit (Arbitrage). F0 = S0erT or F0 = S0(1+r)TForward price approaches spot price at maturity Futures price = spot price + cost of carryForward price, F Forward price ‘at a premium’ when : F > S (contango) For stock paying dividends, we reduce the ‘cost of carry’ by amount of dividend payments (d) F0 = S0e(r-d)T 0 Stock price, St T At T, ST = FT For commodity futures, storage costs (v or V) is negative income Forward price ‘at a discount’, when : F < S (backwardation) F0 = S0e(r+v)T or F0 = (S0+V)erT Arbitrage with Futures Arbitrage with FuturesFor currency futures, the ‘cost of carry’ will be Arbitrage at t<T for a non-income paying security:reduced by the riskless rate of the foreign currency If F0 > S0erT then buy the asset and short the futures contract(rf) If F0 < S0erT then short the asset and buy the futures F0 = S0e(r-rf)T contractFor stock index futures, the cost of carry will be Example of ‘Cash and Carry’ arbitrage: S=£100,reduced by the dividend yield r=4%p.a., F=£102 for delivery in 3 months. 0.04×0.25 F0 = S0e(r-d)T We see F = 100 × e ɶ = 101 £ Since Futures is over priced, time = Now time = in 3 months •Sell Futures contract at £102 •Pay loan back (£101) •Borrow £100 for 3 months and buy stock •Deliver stock and get agreed price of £102
- 5. Hedging with Futures F and S are positively correlated To hedge, we need a negative correlation. So we long one and short the other. Hedging with Futures Hedge = long underlying + short Futures Hedging with Futures Hedging with FuturesSimple Hedging Example: F1 value would have been different if r had changed. You long a stock and you fear falling prices over the This is Basis Risk (b1 = S1 – F1) next 2 months, when you want to sell. Today (say January), you observe S0=£100 and F0=£101 for Final Value = S1 + (F0 - F1 ) = £100.7 April delivery. = (S1 - F1 ) + F0 so r is 4% = b1 + F0 Today: you sell one futures contract In March: say prices fell to £90 (S1=£90). So where “Final basis” b1 = S1 - F1 F1=S1e0.04x(1/12)=£90.3. You close out on Futures. At maturity of the futures contract the basis is zero Profit on Futures: 101 – 90.3 = £10.7 (since S1 = F1 ). In general, when contract is closed Loss on stock value: 100 – 90 =£10 out prior to maturity b1 = S1 - F1 may not be zero. Net Position is +0.7 profit. Value of hedged portfolio However, b1 will usually be small in relation to F0. = S1+ (F0 - F1) = 90 + 10.7 = 100.7
- 6. Stock Index Futures Contract Stock Index Futures contract can be used to eliminate market risk from a portfolio of stocks F0 = S0 × e( r − d )T If this equality does not hold then index arbitrage Stock Index Futures Contract (program trading) would generate riskless profits. Risk free rate is usually greater than dividend yield Hedging with SIFs (r>d) so F>SHedging with Stock Index Futures Hedging with Stock Index FuturesExample: A portfolio manager wishes to hedge her The required number of Stock Index Futures contractportfolio of $1.4m held in diversified equity and to short will be 3S&P500 index TVS 0 $1, 400, 000 Total value of spot position, TVS0=$1.4m NF = − = − = − 3.73S0 = 1400 index point FVF0 $375, 000 Number of stocks, Ns = TVS0/S0 = $1.4m/1400 In the above example, we have assumed that S and=1000 units F have correlation +1 (i.e. ∆ S = ∆ F )We want to hedge Δ(TVSt)= Ns . Δ(St) In reality this is not the case and so we needUse Stock Index Futures, F0=1500 index point, z= minimum variance hedge ratiocontract multiplier = $250FVF0 = z F0 = $250 ( 1500 ) = $375,000
- 7. Hedging with Stock Index Futures Hedging with Stock Index FuturesMinimum Variance Hedge Ratio To obtain minimum, we differentiate with respect to Nf 2∆V = change in spot market position + change in Index Futures position (∂σ V / ∂N f = 0 ) and set to zero = Ns . (S1-S0) + Nf . (F1 - F0) z = Ns S0. ∆S /S0 + ∆ Nf F0. (∆F /F0) z N f ( F V F0 ) 2 σ ∆ F / F 2 = −TVS 0 ⋅ F V F ⋅ σ ∆ S / S ,∆ F / F 0 = TVS0 . ∆S /S0 + ∆ Nf . FVF0 . (∆F /F0) TVS0 where, z = contract multiple for futures ($250 for S&P 500 Futures); ∆S = N f = − ( σ ∆ S / S ,∆ F / F 2 σ ∆F / F ) S1 - S0, ∆F = F1 - F0 F V F0 TVS0 =− β ∆ S / S ,∆ F / FThe variance of the hedged portfolio is 2 2 2 2σ V = (TVS 0 ) σ ∆S / S + ( N f ) ( FVF ) σ ∆F / F 2 2 F V F0 0 where Ns = TVS0/S0 and beta is regression coefficient of the + 2N TVS 0 FVF0 . σ ∆S / S , ∆F / F regression f (∆S / S ) = α 0 + β ∆S / ∆F ( ∆ F / F ) + ε Hedging with Stock Index Futures Hedging with Stock Index FuturesSUMMARY 2 Application: Changing beta of your portfolio: “Market ∂σV / ∂N = 0 implies Timing Strategy” f TVS Nf = 0 .( β h − β p ) TVS 0 FVF0 Nf = − .β p Example: βp (=say 0.8) is your current ‘spot/cash’ portfolio of stocks FVF0 But Value of Spot Position = − FaceValue of futures at t = 0 βp • You are more optimistic about ‘bull market’ and desire a higher exposure of βh (=say, 1.3) • It’s ‘expensive’ to sell low-beta shares and purchase high-beta sharesIf correlation = 1, the beta will be 1 and we just have • Instead ‘go long’ more Nf Stock Index Futures contracts TVS0 Nf = − Note: If βh= 0, then Nf = - (TVS0 / FVF0) βp FVF0
- 8. Hedging with Stock Index Futures Hedging with Stock Index Futures Application: Stock Picking and hedging market risk If you hold stock portfolio, selling futures will place a You hold (or purchase) 1000 undervalued shares of Sven plchedge and reduce the beta of your stock portfolio. If you want to increase your portfolio beta, go long V(Sven) = $110 (e.g. Using Gordon Growth model)futures. P(Sven) = $100 (say) Example: Suppose β = 0.8 and Nf = -6 contracts wouldmake β = 0. Sven plc are underpriced by 10%. If you short 3 (-3) contracts instead, then β = 0.4 Therefore you believe Sven will rise 10% more than the market over the next 3 months. If you long 3 (+3) contracts instead, then β = 0.8+0.4 = 1.2 But you also think that the market as a whole may fall by 3%. The beta of Sven plc (when regressed with the market return) is 2.0 Hedging with Stock Index Futures Hedging with Stock Index Futures Can you ‘protect’ yourself against the general fall in the market and hence any Application: Future stock purchase and hedging market ‘knock on’ effect on Sven plc ? risk Yes . Sell Nf index futures, using: You want to purchase 1000 stocks of takeover target with βp = 2, in 1 month’s time when you will have the cash. TVS N f = − 0 .β p You fear a general rise in stock prices. FVF 0 Go long Stock Index Futures (SIF) contracts, so that gain on the futures will offset the higher cost of these particular shares in 1 month’s time. If the market falls 3% then TVS N f = 0 .β p Sven plc will only change by about 10% - (2x3%) = +4% FVF 0 SIF will protect you from market risk (ie. General rise in prices) but not from But the profit from the short position in Nf index futures, will give you an specific risk. For example if the information that you are trying to takeover additional return of around 6%, making your total return around 10%. the firm ‘leaks out’ , then price of ‘takeover target’ will move more than that given by its ‘beta’ (i.e. the futures only hedges market risk)
- 9. Topics Financial Risk Management Duration, immunization, convexity Repo (Sale and Repurchase agreement) Topic 2 and Reverse Repo Managing interest rate risks Reference: Hull(2009), Luenberger (1997), and CN(2001) Hedging using interest rate Futures Futures on T-bills Futures on T-bonds ReadingsBooksHull(2009) chapters 6CN(2001) chapters 5, 6Luenberger (1997) chapters 3Journal Article Hedging Interest rate risks: DurationFooladi, I and Roberts, G (2000) “Risk Management with Duration Analysis”Managerial Finance,Vol 25, no. 3
- 10. Duration Duration (also called Macaulay Duration) Duration measures sensitivity of price changes (volatility) with Duration of the bond is a measure that summarizes changes in interest rates approximate response of bond prices to change in yields. 1 Lower the coupons A better approximation could be convexity of the bond . T for a given time to n PB = ∑ C t t + ParValueT maturity, greater B = ∑ c i e − y ti (1+ r ) weight t =1 (1+ r ) T change in price to i =1 change in interest n rates ∑ t i ⋅ c i e − y ti n c e − y ti T 2 Greater the time to D = i =1 = ∑ ti i PB = ∑ C t t + ParValueT maturity with a given B i =1 B coupon, greater t =1 (1+ r ) (1+ r ) T change in price to Duration is weighted average of the times when payments change in interest are made. The weight is equal to proportion of bond’s total rates present value received in cash flow at time ti.3 For a given percentage change in yield, the actual price increase is Duration is “how long” bondholder has to wait for cash flows greater than a price decrease Macaulay Duration Modified Duration and Dollar Duration For a small change in yields ∆ y / d y For Macaulay Duration, y is expressed in continuous compounding. dB ∆B = ∆y When we have discrete compounding, we have Modified dy Duration (with these small modifications) Evaluating d B : n If y is expressed as compounding m times a year, we divide D d y ∆ B = − ∑ t i c i e − y ti ∆ y by (1+y/m) ∆B = − B ⋅ D i =1 ⋅ ∆y (1 + y / m) = −B ⋅ D ⋅∆y ∆B = − B ⋅ D* ⋅ ∆y ∆B = −D ⋅∆y B Dollar Duration, D$ = B.D D measures sensitivity of percentage change in bond That is, D$ = Bond Price x Duration (Macaulay or Modified) prices to (small) changes in yields ∆B = − D$ ⋅ ∆y ∆B Note negative relationship between Price (B) So D$ is like Options Delta D$ = − ∆y and yields (Y)
- 11. Duration Duration -exampleExample: Consider a trader who has $1 million in Example: Consider a 7% bond with 3 years to maturity. Assume that the bond is selling at 8% yield.bond with modified duration of 5. This means for A B C D Eevery 1 bp (i.e. 0.01%) change in yield, the value ofthe bond portfolio will change by $500. Present value Weight = Year Payment Discount A× E ∆B = − ( $1, 000, 000 × 5 ) ⋅ 0.01% = −$500 =B× C D/Price factor 8%A zero coupon bond with maturity of n years has a 0.5 3.5 0.962 3.365 0.035 0.017Duration = n 1.0 3.5 0.925 3.236 0.033 0.033A coupon-bearing bond with maturity of n years will 1.5 3.5 0.889 3.111 0.032 0.048have Duration < n 2.0 3.5 0.855 2.992 0.031 0.061 2.5 3.5 0.822 2.877 0.030 0.074Duration of a bond portfolio is weighted average of 3.0 103.5 0.79 81.798 0.840 2.520the durations of individual bonds Sum Price = 97.379 Duration = 2.753 D p o r tfo lio = ∑ (B i i / B )⋅ D i Here, yield to maturity = 0.08, m = 2, y = 0.04, n = 6, Face value = 100.Qualitative properties of duration Properties of durationDuration of bonds with 5% yield as a function ofmaturity and coupon rate. 1. Duration of a coupon paying bond is always less than its maturity. Duration decreases with the increase Coupon rate of coupon rate. Duration equals bond maturity for non- Years to 1% 2% 5% 10% coupon paying bond. maturity 1 0.997 0.995 0.988 0.977 2. As the time to maturity increases to infinity, the 2 1.984 1.969 1.928 1.868 5 4.875 4.763 4.485 4.156 duration do not increase to infinity but tend to a finite 10 9.416 8.950 7.989 7.107 limit independent of the coupon rate. 25 20.164 17.715 14.536 12.754 50 26.666 22.284 18.765 17.384 1+ m λ Actually, D → where λ is the yield to maturity 100 22.572 21.200 20.363 20.067 λ Infinity 20.500 20.500 20.500 20.500 per annum, and m is the number of coupon payments per year.
- 12. Properties of Duration Changing Portfolio Duration3. Durations are not quite sensitive to increase in Changing Duration of your portfolio: coupon rate (for bonds with fixed yield). They don’t If prices are rising (yields are falling), a bond vary huge amount since yield is held constant and trader might want to switch from shorter it cancels out the influence of coupons. duration bonds to longer duration bonds as4. When the coupon rate is lower than the yield, the longer duration bonds have larger price duration first increases with maturity to some changes. maximum value then decreases to the asymptotic limit value. Alternatively, you can leverage shorter maturities. Effective portfolio duration =5. Very long durations can be achieved by bonds with ordinary duration x leverage ratio. very long maturities and very low coupons. Immunization (or Duration matching) Immunization This is widely implemented by Fixed Income Practitioners. Matching present values (PV) of portfolio and obligations This means that you will meet your obligations with the cash time 0 time 1 time 2 time 3 from the portfolio. If yields don’t change, then you are fine. 0 pay $ pay $ pay $ If yields change, then the portfolio value and PV will both change You want to safeguard against interest rate increases. by varied amounts. So we match also Duration (interest rate risk) A few ideas: PV1 + PV2 = PVobligation1. Buy zero coupon bond with maturities matching timing of Matching duration cash flows (*Not available) [Rolling hedge has reinv. risk] Here both portfolio and obligations have the same sensitivity to interest rate changes.2. Keep portfolio of assets and sell parts of it when cash is needed & reinvest in more assets when surplus (* difficult as If yields increase then PV of portfolio will decrease (so will the PV of the obligation streams) Δ value of in portfolio and Δ value of obligations will not identical) If yields decrease then PV of portfolio will increase (so will the PV of the obligation streams)3. Immunization - matching duration and present values D1 PV1 + D 2 PV2 = Dobligation PVobligation of portfolio and obligations (*YES)
- 13. Immunization Immunization Example Suppose only the following bonds are available for its choice. coupon rate maturity price yield duration Suppose Company A has an obligation to Bond 1 6% 30 yr 69.04 9% 11.44 pay $1 million in 10 years. How to invest Bond 2 11% 10 yr 113.01 9% 6.54 in bonds now so as to meet the future Bond 3 9% 20 yr 100.00 9% 9.61 obligation? • Present value of obligation at 9% yield is $414,642.86. • An obvious solution is the purchase of a • Since Bonds 2 and 3 have durations shorter than 10 years, it is not simple zero-coupon bond with maturity 10 possible to attain a portfolio with duration 10 years using these two bonds. years. Suppose we use Bond 1 and Bond 2 of amounts V1 & V2,* This example is from Leunberger (1998) page 64-65. The numbers V1 + V2 = PVare rounded up by the author so replication would give different P1V1 + D2V2 = 10 × PVnumbers. giving V1 = $292,788.64, V2 = $121,854.78. Immunization Immunization Yield 9.0 8.0 10.0 Bond 1 Difficulties with immunization procedure Price 69.04 77.38 62.14 1. It is necessary to rebalance or re-immunize the Shares 4241 4241 4241 portfolio from time to time since the duration depends Value 292798.64 328168.58 263535.74 on yield. Bond 2 2. The immunization method assumes that all yields Price 113.01 120.39 106.23 are equal (not quite realistic to have bonds with Shares 1078 1078 1078 different maturities to have the same yield). Value 121824.78 129780.42 114515.94 Obligation 3. When the prevailing interest rate changes, it is value 414642.86 456386.95 376889.48 unlikely that the yields on all bonds change by the Surplus -19.44 1562.05 1162.20 same amount.Observation: At different yields (8% and 10%), the value of theportfolio almost agrees with that of the obligation.
- 14. Duration for term structure Duration for term structureWe want to measure sensitivity to parallel shifts in the spotrate curve Consider parallel shift in term structure: sti changes to sti + ∆y ( ) Then PV becomesFor continuous compounding, duration is called Fisher-Weil Fisher- n ( ) P ( ∆y ) = − sti + ∆ y ⋅tiduration.duration ∑x i=0 ti ⋅eIf x0, x1,…, xn is cash flow sequence and spot curve is st wheret = t0,…,tn then present value of cash flow is Taking differential w.r.t ∆y in the point ∆y=0 we get n dP ( ∆ y ) n ∑x − sti ⋅ti | ∆ y = 0 = − ∑ t i x t i ⋅ e ti i − s ⋅t PV = ⋅e d ∆y ti i=0 i=0The Fisher-Weil duration is So we find relative price sensitivity is given by DFW n 1 1 dP (0) ∑t − sti ⋅ti D FW = ⋅ x ti ⋅ e ⋅ = − D FW PV i=0 i P (0) d ∆ y Convexity ConvexityDuration applies to only small changes in y Convexity for a bond is nTwo bonds with same duration can have different 1 d 2B ∑ t i2 ⋅ c i e − y t i n c e − y ti change in value of their portfolio (for large changes C = B dy 2 = i =1 B = ∑ t i2 i in yields) i =1 B Convexity is the weighted average of the ‘times squared’ when payments are made. From Taylor series expansion dB 1 d 2B ∆ B = ∆ y + (∆ ) 2 y dy 2 dy 2 ∆ B 1 = − D ⋅ ∆ y + C ⋅ (∆ ) 2 y B 2First order approximation cannot capture this, so we So Dollar convexity is like Gamma measure intake second order approximation (convexity) options.
- 15. Short term risk management using Repo Repo is where a security is sold with agreement to buy it back at a later date (at the price agreed now) Difference in prices is the interest earned (called repo rate rate) It is form of collateralized short term borrowing (mostly overnight) Example: a trader buys a bond and repo it overnight. The REPO and REVERSE REPO money from repo is used to pay for the bond. The cost of this deal is repo rate but trader may earn increase in bond prices and any coupon payments on the bond. There is credit risk of the borrower. Lender may ask for margin costs (called haircut) to provide default protection. Example: A 1% haircut would mean only 99% of the value of collateral is lend in cash. Additional ‘margin calls’ are made if market value of collateral falls below some level.Short term risk management using Repo Hedge funds usually speculate on bond price differentialsusing REPO and REVERSE REPO Example: Assume two bonds A and B with different prices (say price(A)<price(B)) but similar characteristics. Hedge Fund (HF) would like to buy A and sell B simultaneously.This can be financed with repo as follows: (Long position) Buy Bond A and repo it. The cash obtained is used to pay for Interest Rate Futures the bond. At repo termination date, sell the bond and with the cash buy bond back (simultaneously). HF would benefit from the price increase in bond and low repo rate (Futures on T-Bills) (short position) Enter into reverse repo by borrowing the Bond B (as collateral for money lend) and simultaneously sell Bond B in the market. At repo termination date, buy bond back and get your loan back (+ repo rate). HF would benefit from the high repo rate and a decrease in price of the bond.
- 16. Interest Rate Futures Interest Rate FuturesIn this section we will look at how Futures contract written on a Treasury Bill (T-Bill) help in hedging interest rate risks So what is a 3-month T-Bill Futures contract? At expiry, (T), which may be in say 2 months timeReview - What is T-Bill? the (long) futures delivers a T-Bill which matures at T-Bills are issued by government, and quoted at a discount T+90 days, with face value M=$100. Prices are quoted using a discount rate (interest earned as % of face value) As we shall see, this allows you to ‘lock in’ at t=0, the forward Example: 90-day T-Bill is quoted at 0.08 This means annualized 0.08. rate, f12 return is 8% of FV. So we can work out the price, as we know FV. T-Bill Futures prices are quoted in terms of quoted index, Q d 90 (unlike discount rate for underlying) P = F V 1 − 100 360 Q = $100 – futures discount rate (df) Day Counts convention (in US) So we can work out the price as 1. Actual/Actual (for treasury bonds) d f 90 2. 30/360 (for corporate and municipal bonds) F = F V 1 − 3. Actual/360 (for other instruments such as LIBOR) 100 360 Hedge decisions Cross Hedge: US T-Bill Futures Example:When do we use these futures contract to hedge? Today is May. Funds of $1m will be available in August toExamples: invest for further 6 months in bank deposit (or commercial bills) 1) You hold 3m T-Bills to sell in 1-month’s time ~ fear price fall ~ spot asset is a 6-month interest rate ~ sell/short T-Bill futures Fear a fall in spot interest rates before August, so today BUY T- bill futures 2) You will receive $10m in 3m time and wish to place it on a Eurodollar bank deposit for 90 days ~ fear a fall in interest rates Assume parallel shift in the yield curve. (Hence all interest rates ~ go long a Eurodollar futures contract move by the same amount.) ~ BUT the futures price will move less than the price of the 3) Have to issue $100m of 180-day Commercial Paper in 3 months time (I.e. commercial bill - this is duration at work! higher the maturity, more borrow money) ~ fear a rise in interest rates sensitive are changes in ~ sell/short a T-bill futures contract as there is no commercial bill futures prices to interest rates contract (cross hedge) Use Sept ‘3m T-bill’ Futures, ‘nearby’ contract ~ underlying this futures contract is a 3-month interest rate
- 17. Cross Hedge: US T-Bill Futures Cross Hedge: US T-Bill Futures Question: How many T-bill futures contract should I purchase? 3 month Desired investment/protection exposure period period = 6-months We should take into account the fact that: 1. to hedge exposure of 3 months, we have used T-bill futures with 4 months time-to-maturity May Aug. Sept. Dec. Feb. 2. the Futures and spot prices may not move one-to-one Maturity of ‘Underlying’ We could use the minimum variance hedge ratio: in Futures contract TVS0 Nf = .β p FVF0Purchase T-Bill Known $1m Maturity date of Sept.future with Sept. cash receipts T-Bill futures contract However, we can link price changes to interest ratedelivery date changes using Duration based hedge ratioQuestion: How many T-bill futures contract should I purchase? Duration based hedge ratio Duration based hedge ratio Using duration formulae for spot rates and futures: Expressing Beta in terms of Duration: ∆S ∆F TVS0 = − DS ⋅ ∆ys = − DF ⋅ ∆yF Nf = .β p S F FVF0 We can obtain So we can say volatility is proportional to Duration: ∆S ∆F last term by Cov , ∆S ∆F TVS0 S F regressingσ2 = DS ⋅ σ ( ∆ys ) σ2 = DF ⋅ σ ( ∆yF ) = 2 2 2 2 ∆yS = α0 + βy∆yF + ε S F FVF0 σ 2 ∆F ∆S ∆F F Cov , = Ε ( − DS ⋅ ∆ys )( − DF ⋅ ∆yF ) S F TVS0 Ds σ ( ∆ys ∆yF ) = = DS ⋅ DF ⋅ σ ( ∆ys ∆yF ) FVF0 DF σ 2 ( ∆yF )
- 18. Duration based hedge ratio Cross Hedge: US T-Bill Futures Example Summary: REVISITED 3 month Desired investment/protection TVS0 Ds Nf = . βy exposure period period = 6-months FVF0 DF May Aug. Sept. Dec. Feb.where beta is obtained from the regression of yields ∆yS = α0 + β y ∆yF + ε Maturity of ‘Underlying’ in Futures contract Purchase T-Bill Known $1m Maturity date of Sept. future with Sept. cash receipts T-Bill futures contract delivery date Question: How many T-bill futures contract should I purchase? Cross Hedge: US T-Bill Futures Cross Hedge: US T-Bill Futures Suppose now we are in August: May (Today). Funds of $1m accrue in August to be invested for 6- months 3 month US T-Bill Futures : Sept Maturity in bank deposit or commercial bills( Ds = 6 ) Spot Market(May) CME Index Futures Price, F Face Value of $1m Use Sept ‘3m T-bill’ Futures ‘nearby’ contract ( DF = 3) (T-Bill yields) Quote Qf (per $100) Contract, FVF May y0 (6m) = 11% Qf,0 = 89.2 97.30 $973,000 Cross-hedge. August y1(6m) = 9.6% Qf,1 = 90.3 97.58 $975,750 Here assume parallel shift in the yield curve Change -1.4% 1.10 (110 ticks) 0.28 $2,750 (per contract) Qf = 89.2 (per $100 nominal) hence: Durations are : Ds = 0.5, Df = 0.25 Amount to be hedged = $1m. No. of contracts held = 2 F0 = 100 – (10.8 / 4) = 97.30 F FVF0 = $1m (F0/100) = $973,000 Key figure is F1 = 97.575 (rounded 97.58) Gain on the futures position Nf = (TVS0 / FVF0) (Ds / DF ) = TVS0 (F1 - F0) NF = $1m (0.97575 – 0.973) 2 = $5,500 = ($1m / 973,000) ( 0.5 / 0.25) = 2.05 (=2)
- 19. Cross Hedge: US T-Bill Futures Invest this profit of $5500 for 6 months (Aug-Feb) at y1=9.6%: = $5500 + (0.096/2) = $5764 Loss of interest in 6-month spot market (y0=11%, y1=9.6%) = $1m x [0.11 – 0.096] x (1/2) = $7000 Interest Rate Futures Net Loss on hedged position $7000 - $5764 = $1236 (so the company lost $1236 than $7000 without the hedge) (Futures on T-Bonds)Potential Problems with this hedge:1. Margin calls may be required2. Nearby contracts may be maturing before September. So we may have to roll over the hedge3. Cross hedge instrument may have different driving factors of risk US T-Bond Futures US T-Bond FuturesContract specifications of US T-Bond Futures at CBOT: Conversion Factor (CF): CF adjusts price of actual bond to be (CF): Contract size $100,000 nominal, notional US Treasury bond with 8% coupon delivered by assuming it has a 8% yield (matching the bond to Delivery months March, June, September, December the notional bond specified in the futures contract) Quotation Per $100 nominal Price = (most recent settlement price x CF) + accrued interest Tick size (value) 1/32 ($31.25) Last trading day 7 working days prior to last business day in expiry month Example: Possible bond for delivery is a 10% coupon (semi- Delivery day Any business day in delivery month (seller’s choice) annual) T-bond with maturity 20 years. Settlement Any US Treasury bond maturing at least 15 years from the contract month (or not callable for 15 years) The theoretical price (say, r=8%): 40 5 100Notional is 8% coupon bond. However, Short can choose to P=∑ i + = 119.794deliver any other bond. So Conversion Factor adjusts “delivery i =1 1.04 1.0440price” to reflect type of bond delivered Dividing by Face Value, CF = 119.794/100 = 1.19794 (perT-bond must have at least 15 years time-to-maturity $100 nominal) If Coupon rate > 8% then CF>1Quote ‘98-14’ means 98.(14/32)=$98.4375 per $100 nominal ‘98- If Coupon rate < 8% then CF<1
- 20. US T-Bond Futures Hedging using US T-Bond Futures deliver:Cheapest to deliver: Hedging is the same as in the case of T-bill Futures (exceptIn the maturity month, Short party can choose to deliver any Conversion Factor).bond from the existing bonds with varying coupons andmaturity. So the short party delivers the cheapest one. For long T-bond Futures, duration based hedge ratio is given by:Short receives: TVS0 Ds (most recent settlement price x CF) + accrued interest Nf = . β y ⋅ CFCTDCost of purchasing the bond is: FVF0 DF Quoted bond price + accrued interest where we have an additional term for conversion factor for the cheapest to deliver bond.The cheapest to deliver bond is the one with the smallest:Quoted bond price - (most recent settlement price x CF)
- 21. Financial Risk Management Topic 3a Managing risk using Options Readings: CN(2001) chapters 9, 13; Hull Chapter 17
- 22. TopicsFinancial Engineering with OptionsBlack ScholesDelta, Gamma, Vega HedgingPortfolio Insurance
- 23. Options Contract - ReviewAn option (not an obligation), American and European- Put Premium -
- 24. Financial Engineering with optionsSynthetic call optionPut-Call Parity: P + S = C + CashExample: Pension Fund wants to hedge its stock holdingagainst falling stock prices (over the next 6 months) andwishes to temporarily establish a “floor value” (=K) but alsowants to benefit from any stock price rises.
- 25. Financial Engineering with options Nick Leeson’s short straddleYou are initially credited with the call and put premia C + P (at t=0) but if at expirythere is either a large fall or a large rise in S (relative to the strike price K ) then youwill make a loss (.eg. Leeson’s short straddle: Kobe Earthquake which led to a fall in S(S = “Nikkei-225”) and thus large losses).
- 26. Black ScholesBS formula for price of European Call option d1 − σ d 2 =D2=d1 T − rT c = S 0 N (d 1 ) − K e N (d 2 ) Probability of call option being in-the-money and getting stock Present value of the strike price Probability of exercise and paying strike pricec expected (average) value of receiving the stock in the event of =exercise MINUS cost of paying the strike price in the event of exercise
- 27. Black Scholeswhere S σ2 S σ2 ln 0 +r + T ln 0 +r − T K 2 ;d = K 2 d1 = or d 2 = d1 − σ T σ T σ T 2
- 28. Sensitivity of option pricesSensitivity of option prices (American/European non- non- dividend paying) c = f ( K, S0, r, T, σ ) This however can be negative for - + ++ + dividend paying European options. Example: stock pays dividend in 2 weeks. European call with 1 p = f ( K, S0, r, T, σ ) week to expiration will have more + - - + + value than European call with 3 weeks to maturity. Call premium increases as stock price increases (but less than one-for-one) Put premium falls as stock price increases (but less than one- for-one)
- 29. Sensitivity of option pricesThe Greek Letters Delta, ∆ measures option price change when stock price increase by $1 Gamma, Γ measures change in Delta when stock price increase by $1 Vega, υ measures change in option price when there is an increase in volatility of 1% Theta, Θ measures change in option price when there is a decrease in the time to maturity by 1 day Rho, ρ measures change in option price when there is an increase in interest rate of 1% (100 bp)
- 30. Sensitivity of option prices ∂f ∂2 f ∂f ∂f ∂f ∆ = ;Γ = ;υ = ;Θ = ;ρ = ∂S ∂S 2 ∂σ ∂T ∂r Using Taylor series, 1df ≈ ∆ ⋅dS + Γ ⋅ (d S ) + Θ ⋅dt + ρ ⋅dr + υ ⋅dσ 2 2 Read chapter 12 of McDonald text book “Derivative Markets” for more about Greeks
- 31. DeltaThe rate of change of the option price with respectto the share pricee.g. Delta of a call option is 0.6Stock price changes by a small amount, then the optionprice changes by about 60% of that Option price Slope = ∆ = ∂c/ ∂ S C S Stock price
- 32. Delta ∆ of a stock = 1 ∂C ∆ call = = N ( d1 ) > 0 ∂S (for long positions) ∂P ∆ put = = N ( d1 ) − 1 < 0 ∂S If we have lots of options (on same underlying) then delta of portfolio is ∆ portfolio = ∑ N k ⋅ ∆ k kwhere Nk is the number of options held. Nk > 0 if long Call/Put and Nk < 0 if short Call/Put
- 33. DeltaSo if we use delta hedging for a short call position, wemust keep a long position of N(d1) sharesWhat about put options?The higher the call’s delta, the more likely it is that theoption ends up in the money: Deep out-of-the-money: Δ ≈ 0 At-the-money: Δ ≈ 0.5 In-the-money: Δ≈1Intuition: if the trader had written deep OTM calls, itwould not take so many shares to hedge - unlikely thecalls would end up in-the-money
- 34. ThetaThe rate of change of the value of an option withrespect to timeAlso called the time decay of the optionFor a European call on a non-dividend-paying stock, S0 N (d1 )σ − rT 1 − x2 Θ=− − rKe N (d 2 ) where N ( x) = e 2 2T 2πRelated to the square root of time, so the relationship isnot linear
- 35. ThetaTheta is negative: as maturity approaches, the optiontends to become less valuableThe close to the expiration date, the faster the value ofthe option falls (to its intrinsic value)Theta isn’t the same kind of parameter as deltaThe passage of time is certain, so it doesn’t makeany sense to hedge against it!!!Many traders still see theta as a useful descriptive statisticbecause in a delta-neutral portfolio it can proxy forGamma
- 36. GammaThe rate of change of delta with respect to theshare price: ∂2 f ∂S 2Calculated as Γ = N (d1 ) S0σ TSometimes referred to as an option’s curvatureIf delta changes slowly → gamma small → adjustmentsto keep portfolio delta-neutral not often needed
- 37. GammaIf delta changes quickly → gamma large → risky toleave an originally delta-neutral portfolio unchanged forlong periods: Option price C C C S S Stock price
- 38. GammaMaking a Position Gamma-Neutral Gamma- We must make a portfolio initially gamma-neutral as well as delta-neutral if we want a lasting hedge But a position in the underlying share can’t alter the portfolio gamma since the share has a gamma of zero So we need to take out another position in an option that isn’t linearly dependent on the underlying share If a delta-neutral portfolio starts with gamma Γ, and we buy wT optionseach with gamma ΓT, then the portfolio now has gamma Γ + wT Γ T We want this new gamma to = 0: Γ + wT Γ T = 0 −Γ Rearranging, wT = ΓT
- 39. Delta-Theta-GammaFor any derivative dependent on a non-dividend-paying stock,Δ , θ, and Г are relatedThe standard Black-Scholes differential equation is ∂f ∂f 1 2 2 ∂ 2 f + rS + σ S = rf ∂t ∂S 2 ∂S 2where f is the call price, S is the price of the underlyingshare and r is the risk-free rate ∂fBut Θ = , ∆ = ∂f ∂2 f and Γ = ∂t ∂S ∂S 2 1 2 2So Θ + rS ∆ + Θ S Γ = rf 2So if Θ is large and positive, Γ tends to be large and negative,and vice-versaThis is why you can use Θ as a proxy for Γ in a delta-neutralportfolio
- 40. VegaNOT a letter in the Greek alphabet!Vega measures, the sensitivity of an option’s volatility:price to volatility υ = ∂f ∂σ υ = S0 T N (d1 )High vega → portfolio value very sensitive tosmall changes in volatilityLike in the case of gamma, if we add in a tradedoption we should take a position of – υ/υT tomake the portfolio vega-neutral
- 41. RhoThe rate of change of the value of a portfolio ofoptions with respect to the interest rate ∂f ρ= ρ = KTe− rT N (d 2 ) ∂rRho for European Calls is always positive and Rho forEuropean Puts is always negative (since as interest ratesrise, forward value of stock increases).Not very important to stock options with a life of a fewmonths if for example the interest rate moves by ¼%More relevant for which class of options?
- 42. Delta HedgingValue of portfolio = no of calls x call price + no of stocks xstock priceV = NC C + NS S ∂V ∂C = N C ⋅ + N S ⋅1 = 0 ∂S ∂S ∂C NS = −NC ⋅ ∂S N S = − N C ⋅ ∆ c a llSo if we sold 1 call option then NC = -1. Then no of stocks tobuy will be NS = ∆callSo if ∆call = 0.6368 then buy 0.63 stocks per call option
- 43. Delta HedgingExample: As a trader, you have just sold (written)100 call options to a pension fund (and earned anice little brokerage fee and charged a little morethan Black-Scholes price).You are worried that share prices might RISE hence RISE,the call premium RISE, hence showing a loss on yourposition.Suppose ∆ of the call is 0.4. Since you are short,your ∆ = -0.4 (When S increases by +$1 (e.g. from100 to 101), then C decrease by $0.4 (e.g. from 10to 9.6)).
- 44. Delta HedgingYour 100 written (sold) call option (at C0 = 10 each option)You now buy 40-sharesSuppose S FALLS by $1 over the next monthTHEN fall in C is 0.4 ( = “delta” of the call)So C falls to C1 = 9.6To close out you must now buy back at C1 = 9.6 (a GAIN of $0.4)Loss on 40 shares = $40Gain on calls = 100 (C0 - C1 )= 100(0.4) = $40Delta hedging your 100 written calls with 40 shares means thatthe value of your ‘portfolio is unchanged.
- 45. Delta Hedging Call Premium ∆ = 0.5 B ∆ = 0.4 . A 0 . 100 110 Stock PriceAs S changes then so does ‘delta’ , so you have to rebalance your portfolio.E.g. ‘delta’ = 0.5, then you now have to hold 50 stocks for every written call.This brings us to ‘Dynamic Hedging’, over many periods.Buying and selling shares can be expensive so instead we can maintain thehedge by buying and selling options.
- 46. (Dynamic) Delta HedgingYou’ve written a call option and earned C0 =10.45 (with K=100,σ = 20%, r=5%, T=1)At t = 0: Current price S0 = $100. We calculate ∆ 0 = N(d1)= 0.6368.So we buy ∆0 = 0.6368 shares at S0 = $100 by borrowing debt. Debt, D0 = ∆0 x S0 = $63.68At t = 0.01: stock price rise S1 = $100.1. We calculate ∆ 1 = 0.6381.So buy extra (∆ 1 – ∆ 0) =0.0013 no of shares at $100.1. Debt, D1 = D0 ert + (∆ 1 – ∆ 0) S1 = $63.84So as you rebalance, you either accumulate or reduce debtlevels.
- 47. Delta Hedging At t=T, if option ends up well “in the money” Say ST = 163.3499. Then ∆ T = 1 (hold 1 share for 1 call). Our final debt amount DT = 111.29 (copied from Textbook page 247) The option is exercised. We get strike $100 for the share. Our Net Cost: NCT = DT – K = 111.29 – 100 = $11.29How have we done with this hedging? At t = 0, we received $10.45 and at t = T we owe $11.29 0 % Net cost of hedge, % NCT = [ (DT – K )-C0 ] / C0 = 8% (8% is close to 5% riskless rate)
- 48. Delta HedgingOne way to view the hedge:The delta hedge is supposed to be riskless (i.e. no change in value of portfolio of“One written call + holding ∆ shares” , over any very small time interval )Hence for a perfect hedge we require: dV = NS dS + (NC ) dC ≈ NS dS + (-1) [ ∆ dS ] ≈ 0If we choose NS = ∆ then we will obtain a near perfect hedge(ie. for only small changes in S, or equivalently over small time intervals)
- 49. Delta HedgingAnother way to view the hedge:The delta hedge is supposed to be riskless, so any money we borrow (receive)at t=0 which is delta hedged over t to T , should have a cost of rHence: For a perfect hedge we expect: NDT / C0 = erT so, NDT e-r T - C0 ≈ 0If we repeat the delta hedge a large number of times then: % Hedge Performance, HP = stdv( NDT e-r T - C0) / C0HP will be smaller the more frequently we rebalance the portfolio (i.e. buy or sellstocks) although frequent rebalancing leads to higher ‘transactions costs’ (Kurieland Roncalli (1998))
- 50. Gamma and Vega Hedging ∂2 f ∂f Γ = υ = ∂S 2 ∂σLong Call/Put have positive Γ and υShort Call/Put have negative Γ and υGamma /Vega Neutral: Stocks and futures have Γ ,υ = 0So to change Gamma/Vega of an existing optionsportfolio, we have to take positions in further (new)options.
- 51. Delta-Gamma NeutralExample: Suppose we have an existing portfolio of options, with a value ofΓ = - 300 (and a ∆ = 0)Note: Γ = Σi ( Ni Γi )Can we remove the risk to changes in S (for even large changes in S ? )Create a “Gamma-Neutral” PortfolioLet ΓZ = gamma of some “new” option (same ‘underlying’) For Γport = NZ ΓZ + Γ = 0we require: NZ = - Γ / ΓZ “new” options
- 52. Delta-Gamma NeutralSuppose a Call option “Z” with the same underlying (e.g. stock) has a delta =0.62 and gamma of 1.5How can you use Z to make the overall portfolio gamma and delta neutral?We require: Nz Γz + Γ = 0 Nz = - Γ / Γz = -(-300)/1.5 = 200implies 200 long contracts in Z (ie buy 200 Z-options)The delta of this ‘new’ portfolio is now ∆ = Nz.∆z = 200(0.62) = 124Hence to maintain delta neutrality you must short 124 units of the underlying -this will not change the ‘gamma’ of your portfolio (since gamma of stock iszero).
- 53. Delta-Gamma-Vega NeutralExample:You hold a portfolio with ∆ port = − 500, Γ port = − 5000, υ port = − 4000We need at least 2 options to achieve Gamma and Vega neutrality. Thenwe rebalance to achieve Delta neutrality of the ‘new’ Gamma-Veganeutral portfolio.Suppose there is available 2 types of options: Option Z with ∆ Z = 0.5, Γ Z = 1.5, υ Z = 0.8 Option Y with ∆ Y = 0.6, Γ Y = 0.3, υ Y = 0.4We need N Z υ Z + N Y υ Y + υ port = 0 N Z Γ Z + N Y Γ Y + Γ port = 0
- 54. Delta-Gamma-Vega Neutral So N Z ( 0.8 ) + N Y ( 0.4 ) − 4000 = 0 N Z (1.5 ) + N Y ( 0.3 ) − 5000 = 0 Solution: N Z = 2222.2 N Y = 5555.5 Go long 2222.2 units of option Z and long 5555.5 units of option Y toattain Gamma-Vega neutrality. New portfolio Delta will be: 2222.2 × ∆ Z + 5555.5 × ∆ Y + ∆ port = 3944.4 Therefore go short 3944 units of stock to attain Delta neutrality
- 55. Portfolio Insurance
- 56. Portfolio InsuranceYou hold a portfolio and want insurance againstmarket declines. Answer: Buy Put optionsFrom put-call parity: Stocks + Puts = Calls + T-bills Stock+Put = {+1, +1} + {-1, 0} = {0, +1} = ‘Call payoff’This is called Static Portfolio Insurance.Alternatively replicate ‘Stocks+Puts’ portfolio price movementswith‘Stocks+T-bills’ or‘Stocks+Futures’. [called Dynamic Portfolio Insurance] Why replicate? Because it’s cheaper!
- 57. Dynamic Portfolio InsuranceStock+Put (i.e. the position you wish to replicate) N0 = V0 /(S0 +P0) (hold 1 Put for 1 Stock)N0 is fixed throughout the hedge:At t > 0 ‘Stock+Put’ portfolio: Vs,p = N0 (S + P)Hence, change in value: ∂Vs, p ∂ P ∂S ∂ S = N0 (1+ ∆ p ) = N0 1+ This is what we wish to replicate
- 58. Dynamic Portfolio InsuranceReplicate with (N0*) Stocks + (Nf) Futures:N0* = V0 / S0 (# of index units held in shares)N0* is also held fixed throughout the hedge.Note: position in futures costs nothing (ignore interest cost on margin funds.)At t > 0: VS,F = N0* S + Nf (F zf) F = S ⋅ e r (T − t ) ∂ F F r (T − t ) ∂ VS , F =eHence: ∂ F ∂S = N0 + z f N f * ∂S ∂S Equating dV of (Stock+Put) with dV(Stock+Futures) to get Nf : = [N (1 + ∆ ) − N ] * e − r (T − t ) Nft 0 p t 0 zf
- 59. Dynamic Portfolio InsuranceReplicate with ‘Stock+T-Bill’ VS,B = NS S + NB B ∂ VS , B = Ns ∂S (V s , p ) t − ( N S ) t S t NB,t = BtEquate dV of (Stock+Put) with dV(Stock+T-bill) ( N s ) t = N 0 (1 + ∆ p ) t = N 0 (∆ c ) t
- 60. Dynamic Portfolio InsuranceExample:Value of stock portfolio V0 = $560,000 S&P500 index S0 = 280 Maturity of Derivatives T - t = 0.10 Risk free rate r = 0.10 p.a. (10%)CompoundDiscount Factor er (T – t) = 1.01Standard deviation S&P σ = 0.12 Put Premium P0 = 2.97 (index units) Strike Price K = 280 Put delta ∆p = -0.38888 (Call delta) (∆c = 1 + ∆p = 0.6112) Futures Price (t=0) F0 = S0 er(T – t ) = 282.814 Price of T-Bill B = Me-rT = 99.0
- 61. Dynamic Portfolio InsuranceHedge Positions Number of units of the index held in stocks = V0 /S0 = 2,000 index unitsStock-Put Insurance N0 = V0 / (S0 + P0) = 1979 index unitsStock-Futures Insurance Nf = [(1979) (0.6112) - 2,000] (0.99/500) = - 1.56 (short futures)Stock+T-Bill Insurance No. stocks = N0 ∆c = 1979 (0.612) = 1,209.6 (index units) NB = 2,235.3 (T-bills)
- 62. Dynamic Portfolio Insurance1) Stock+Put Portfolio Gain on Stocks = N0.dS = 1979 ( -1) = -1,979 Gain on Puts = N0 dP = 1979 ( 0.388) = 790.3 Net Gain = -1,209.62) Stock + Futures: Dynamic ReplicatinGain on Stocks = Ns,o dS = 2000 (-1) = -2,000Gain on Futures = Nf.dF.zf = (-1.56) (-1.01) 500 = +790.3 Net Gain = -1,209.6
- 63. Dynamic Portfolio Insurance 3) Stock + T-Bill: Dynamic Replication Gain on Stocks = Ns dS = 1209.6 (-1) = -1,209.6 Gain on T-Bills = 0 (No change in T-bill price) Net Gain = -1,209.6 The loss on the replication portfolios is very close to that on the stock-put portfolio (over the infinitesimally small time period).Note:We are only “delta replicating” and hence, if there are large changes in S or changes inσ, then our calculations will be inaccurateWhen there are large market falls, liquidity may “dry up” and it may not be possible totrade quickly enough in ‘stocks+futures’ at quoted prices (or at any price ! e.g. 1987 crash).
- 64. Financial Risk Management Topic 3b Option’s Implied Volatility
- 65. TopicsOption’s Implied VolatilityVIXVolatility Smiles
- 66. ReadingsBooks Hull(2009) chapter 18 VIX http://www.cboe.com/micro/vix/vixwhite.pdfJournal Articles Bakshi, Cao and Chen (1997) “Empirical Performance of Alternative Option Pricing Models”, Journal of Finance, 52, 2003-2049.
- 67. Options Implied Volatility
- 68. Estimating VolatilityItō’s Lemma: The Lognormal Property If the stock price S follows a GBM (like in the BS model), ln( then ln(ST/S0) is normally distributed. σ2 ln S T − ln S 0 = ln( S T / S T ) ≈ φ µ − T , σ T 2 2 The volatility is the standard deviation of the continuously compounded rate of return in 1 year The standard deviation of the return in time ∆t is σ ∆t Estimating Volatility: Historical & Implied – How?
- 69. Estimating Volatility from Historical DataTake observations S0, S1, . . . , Sn at intervals of t years(e.g. t = 1/12 for monthly)Calculate the continuously compounded return in eachinterval as: u i = ln( S i / S i −1 )Calculate the standard deviation, s , of the ui´s 1 n s= ∑ n − 1 i =1 (u i − u ) 2The variable s is therefore an estimate for σ ∆tSo: σ = s/ τ ˆ
- 70. Estimating Volatility from Historical Data Price Relative Daily Return Date Close St/St-1 ln(St/St-1) For volatility estimation 03/11/2008 4443.3 (usually) we assume 04/11/2008 05/11/2008 4639.5 4530.7 1.0442 0.9765 0.0432 -0.0237 that there are 252 06/11/2008 4272.4 0.9430 -0.0587 07/11/2008 4365 1.0217 0.0214 trading days within one 10/11/2008 4403.9 1.0089 0.0089 11/11/2008 4246.7 0.9643 -0.0363 year 12/11/2008 4182 0.9848 -0.0154 13/11/2008 4169.2 0.9969 -0.0031 14/11/2008 4233 1.0153 0.0152 mean -0.13% 17/11/2008 4132.2 0.9762 -0.0241 18/11/2008 4208.5 1.0185 0.0183 stdev (s) 3.5% 19/11/2008 4005.7 0.9518 -0.0494 20/11/2008 3875 0.9674 -0.0332 τ 1/252 21/11/2008 3781 0.9757 -0.0246 24/11/2008 4153 1.0984 0.0938 σ(yearly) τ s / sqrt(τ) = 55.56% 25/11/2008 4171.3 1.0044 0.0044 26/11/2008 4152.7 0.9955 -0.0045 27/11/2008 4226.1 1.0177 0.0175 28/11/2008 4288 1.0146 0.0145 01/12/2008 4065.5 0.9481 -0.0533 Back or forward looking 02/12/2008 03/12/2008 4122.9 4170 1.0141 1.0114 0.0140 0.01147 volatility measure? 04/12/2008 4163.6 0.9985 -0.0015 05/12/2008 4049.4 0.9726 -0.0278 08/12/2008 4300.1 1.0619 0.0601
- 71. Implied Volatility BS ParametersObserved Parameters: Unobserved Parameters:S: underlying index value Black and ScholesX: options strike price σ: volatilityT: time to maturityr: risk-free rate • Traders and brokers often quote implied volatilitiesq: dividend yield rather than dollar prices How to estimate it?
- 72. Implied Volatility The implied volatility of an option is the volatility for which the Black-Scholes price equals (=) the market price There is a one-to-one correspondence between prices and implied volatilities (BS price is monotonically increasing in volatility) Implied volatilities are forward looking and price traded options with more accuracy Example: If IV of put option is 22%, this means that pbs = pmkt when a volatility of 22% is used in the Black-Scholes model.9
- 73. Implied Volatility Assume c is the call price, f is an option pricing model/function that depends on volatility σ and other inputs: c = f (S , K , r , T , σ ) Then implied volatility can be extracted by inverting the formula: σ = f −1 (S , K , r , T , c ) mrk where cmrk is the market price for a call option. The BS does not have a closed-form solution for its inverse function, so to extract the implied volatility we use root- finding techniques (iterative algorithms) like Newton- Newton- Raphson method10 f (S , K , r , T , σ ) − c mrk = 0
- 74. Volatility Index - VIX In 1993, CBOE published the first implied volatility index and several more indices later on. VIX: VIX 1-month IV from 30-day options on S&P VXN: VXN 3-month IV from 90-day options on S&P VXD: VXD volatility index of CBOE DJIA VXN: VXN volatility index of NASDAQ100 MVX: MVX Montreal exchange vol index based on iShares of the CDN S&P/TSX 60 Fund VDAX: VDAX German Futures and options exchange vol index based on DAX30 index options11 Others: VXI, VX6, VSMI, VAEX, VBEL, VCAC
- 75. Volatility Smile
- 76. Volatility Smile What is a Volatility Smile? It is the relationship between implied volatility and strike price for options with a certain maturity The volatility smile for European call options should be exactly the same as that for European put options13
- 77. Volatility Smile Put-call parity p +S0e-qT = c +Ke–r T holds for market prices (pmkt and cmkt) and for Black-Scholes prices (pbs and cbs) It follows that the pricing errors for puts and calls are the same: pmkt−pbs=cmkt−cbs When pbs=pmkt, it must be true that cbs=cmkt It follows that the implied volatility calculated from a European call option should be the same as that calculated from a European put option when both have the same strike price and maturity14
- 78. Volatility Term Structure In addition to calculating a volatility smile, traders also calculate a volatility term structure This shows the variation of implied volatility with the time to maturity of the option for a particular strike15
- 79. IV Surface16
- 80. IV Surface17
- 81. IV SurfaceAlso known as:Volatility smirk 18Volatility skew
- 82. Volatility Smile Implied Volatility Surface (Smile) from Empirical Studies (Equity/Index) Bakshi, Cao and Chen (1997) “Empirical Performance of Alternative19 Option Pricing Models ”, Journal of Finance, 52, 2003-2049.
- 83. Volatility Smile Implied vs Lognormal Distribution20
- 84. Volatility Smile In practice, the left tail is heavier and the right tail is less heavy than the lognormal distribution What are the possible causes of the Volatility Smile anomaly? Enormous number of empirical and theoretical papers to answer this …21
- 85. Volatility Smile Possible Causes of Volatility Smile Asset price exhibits jumps rather than continuous changes (e.g. S&P 500 index) Date Open High Low Close Volume Adj Close Return 04/01/2000 1455.22 1455.22 1397.43 1399.42 1.01E+09 1399.42 -3.91% 18/02/2000 1388.26 1388.59 1345.32 1346.09 1.04E+09 1346.09 -3.08% 20/12/2000 1305.6 1305.6 1261.16 1264.74 1.42E+09 1264.74 -3.18% -ve Price jumps 12/03/2001 1233.42 1233.42 1176.78 1180.16 1.23E+09 1180.16 -4.41% 03/04/2001 1145.87 1145.87 1100.19 1106.46 1.39E+09 1106.46 -3.50% 10/09/2001 1085.78 1096.94 1073.15 1092.54 1.28E+09 1092.54 0.62% Trading was 17/09/2001 1092.54 1092.54 1037.46 1038.77 2.33E+09 1038.77 -5.05% suspended 16/03/00 1392.15 1458.47 1392.15 1458.47 1.48E+09 1458.47 4.65% 15/10/02 841.44 881.27 841.44 881.27 1.96E+09 881.27 4.62% 05/04/01 1103.25 1151.47 1103.25 1151.44 1.37E+09 1151.44 4.28% 14/08/02 884.21 920.21 876.2 919.62 1.53E+09 919.62 3.93% +ve price jumps 01/10/02 815.28 847.93 812.82 847.91 1.78E+09 847.91 3.92% 11/10/02 803.92 843.27 803.92 835.32 1.85E+09 835.32 3.83%22 24/09/01 965.8 1008.44 965.8 1003.45 1.75E+09 1003.45 3.82%
- 86. Volatility Smile Possible Causes of Volatility Smile Asset price exhibits jumps rather than continuous changes Volatility for asset price is stochastic In the case of equities volatility is negatively related to stock prices because of the impact of leverage. This is consistent with the skew (i.e., volatility smile) that is observed in practice23
- 87. Volatility Smile24
- 88. Volatility Smile Possible Causes of Volatility Smile Asset price exhibits jumps rather than continuous changes Volatility for asset price is stochastic In the case of equities volatility is negatively related to stock prices because of the impact of leverage. This is consistent with the skew that is observed in practice Combinations of jumps and stochastic volatility25
- 89. Volatility Smile Alternatives to Geometric Brownian Motion Accounting for negative skewness and excess kurtosis by generalizing the GBM Constant Elasticity of Variance Mixed Jump diffusion Stochastic Volatility Stochastic Volatility and Jump Other models (less complex → ad-hoc) The Deterministic Volatility Functions (i.e., practitioners Black and Scholes) (See chapter 26 (sections 26.1, 26.2, 26.3) of Hull for these alternative specifications to Black-Scholes)26
- 90. Topic # 4: Modelling stock prices, Interest rate derivatives Financial Risk Management 2010-11 February 7, 2011FRM c Dennis PHILIP 2011
- 91. 1 Modelling stock prices 21 Modelling stock prices Modelling the evolution of stock prices is about introducing a process that will explain the random movements in prices. This ran- domness is explained in the E¢ cient Market Hypothesis (EMH) that can be summarized in two assumptions: 1. Past history is re‡ected in present price 2. Markets respond immediately to any new information about the asset So we need to model arrival of new infor- mation that a¤ects price (or much more re- turns). If asset price is S. Suppose price changes to S + dS in a small time interval (say dt). Then we can decompose returns dS into de- S terministic/anticipated part and a random part where prices changed due to some ex- ternal unanticipated news. dS = dt + dW SFRM c Dennis PHILIP 2011
- 92. 1 Modelling stock prices 3 The randomness in the random part is ex- plained by a Brownian Motion process and scaled by the volatility of returns. We can introduce time subscripts and re- arrange to get dSt = St dt + St dWt This process is called the Geometric Brown- ian Motion. Why have we used Brownian Motion process to explain randomness? – In practice, we see that stock prices be- have, atleast for long stretches of time, like random walks with small and fre- quent jumps – In statistics, random walk, being the simplest form, have limiting distribu- tions and since BM is a limit of the random walk, we can easily understand the statistics of BM (use of CLT)FRM c Dennis PHILIP 2011
- 93. 1 Modelling stock prices 4 Next we see, what is this W (and in turnwhat is dW)? Brownian motion is a continuous time (rescaled) random walk. Consider the iid sequence "1 ; "2 ; ::: with mean and variance 2 : Consider the rescaled ran- dom walk model 1 X Wn (t) = p "j n 1 j nt The interval length t is divided into nt equal subintervals of length 1=n and the displace- ments / jumps "j ; j = 1; 2; :::; nt in nt steps are mutually independent random variables. Then for large n; according to Central Limit Theorem: W (t) N ( t; 2 t) :FRM c Dennis PHILIP 2011
- 94. 1 Modelling stock prices 5 Special cases: Standard Brownian Motion arises when we have = 0; and = 1. W is a Standard Brownian Motion if 1. W (0) = 0 2. W has stationary (for 0 s t; Wt Ws and Wt s have the same distrib- D ution. That is, Wt Ws = Wt s N (0; t s)) 3. W has independent increments (for s t; Wt Ws is independent of past his- tory of W until time s) 4. Wt N (0; t)FRM c Dennis PHILIP 2011
- 95. 1 Modelling stock prices 6 For a Brownian Motion only the present value of the variable is relevant for predicting the future [also called Markov property]. There- fore BM is a markov process. It does not matter how much you zoom in, it just looks the same. That is, the random- ness does not smooth out when we zoom in. BM …ts the characteristics of the share price. Imagine a heavy particle (share price) that is jarred around by lighter particles (trades). Trades a¤ect the price movement. what is this dW? Consider a small increment in W W (t + t) = W (t) + "(t + t) where "(t + t) iidN (0; t) [Std BM].FRM c Dennis PHILIP 2011
- 96. 1 Modelling stock prices 7 Taking limit as ! 0; the change in W (t) is dWt = lim W (t + dt) W (t) dt!0 = lim "(t + dt) dt!0 iidN (0; dt) So in the di¤erential form, we can write the Standard Brownian motion process as p dWt = et dt where et N (0; 1)FRM c Dennis PHILIP 2011
- 97. 1 Modelling stock prices 8 Stochastic processes used in Finance Arithmetic Brownian Motion for a shareprice A stock price does not generally have a mean zero and atleast would grow on average with the rate of in‡ation. Therefore we can write dSt = (St ; t) dt + (St ; t)dWt = drif t term + dif f usion term = E(dS) + Stddev(dS) When the drift function (St ; t) = and di¤usion function (St ; t) = ; both con- stants, we have the Arithmetic BM. dSt = dt + dWt p = dt + et dt In the case of ABM, S may be positive or negative. Since prices cannot be negative, we generally use the Geometric BM for asset prices and made the drift and volatility as functions of the stock price.FRM c Dennis PHILIP 2011
- 98. 1 Modelling stock prices 9 Geometric Brownian Motion dSt = St dt + St dWt If S starts at a positive value, then it will re- main positive. The solution of the SDE St is an exponential function which is always pos- itive. Also, note that S will be lognormally distributed. GBM is related to ABM according to dSt = dt + dWt St where is the instantaneous share price volatil- ity, and is the expected rate of return The Hull and White (1987) Model uses GBM.FRM c Dennis PHILIP 2011
- 99. 1 Modelling stock prices 10 Ornstein-Uhlenbeck (OU) Process The Arithmetic Ornstein-Uhlenbeck process is given by dSt = ( St ) dt + dWt where is the long run mean and ( > 0) is the rate of mean reversion. The drift term is the mean reversion component, in that the di¤erence between the long run mean and the current price decides the upward or downward movement of the stock price to- wards the long run mean : Over time, the price process drifts towards its mean and the speed of mean reversion is determined by : This is an important process to model in- terest rates that show mean reversion where prices are pulled back to some long-run av- erage level over time. The Vasicek Model uses this kind of process.FRM c Dennis PHILIP 2011
- 100. 1 Modelling stock prices 11 A special case is when the mean is zero. Then we can write the OU process as dSt = St dt + dWt In the OU process, the stock price can be negative. Therefore we can introduce the Geometric OU process The Geometric OU process is given by dSt = ( St ) St dt + St dWt where the asset prices St would always be positive. So we can model asset prices using the Geo- metric OU process and their log returns will then follow an Arithmetic OU process. dSt = ( St ) St dt + St dWt dSt = ( St ) dt + dWt StFRM c Dennis PHILIP 2011
- 101. 1 Modelling stock prices 12 Square Root Process A square root process satis…es the SDE p dSt = St dt + St dWt This type of process generates positive prices and used for asset prices whose volatility does not increase too much when St increases. Cox-Ingersoll-Ross (CIR) process The CIR combines mean reversion and square root process and satis…es the SDE p dSt = ( St ) dt + St dWt This process was introduced in the Hull and White (1988), and Heston (1993) stochas- tic volatility models. This class of mod- els generated strictly non-negative volatility and accounted for the clustering e¤ect and mean reversion observed in volatility.FRM c Dennis PHILIP 2011
- 102. 1 Modelling stock prices 13 Also used to model short rates features pos- itive interest rates, mean reversion, and ab- solute variance of interest rates increases with interest rates itself. Solving the Stochastic Di¤erential Equa-tions Consider the GBM dSt = St dt + St dWt In the integral form ZT ZT ZT dSt = St dt + St dWt 0 0 0 ZT ZT ST = S0 + St dt + St dWt 0 0 = reimann integ + It^ integ o So we have to solve the intergrals to get a closed form solutions to this SDE.FRM c Dennis PHILIP 2011
- 103. 1 Modelling stock prices 14 We use Ito-lemma to solve this problem. Not all SDE’ have closed form solutions. s When there are no solutions, we have to do numerical approximations for these in- tegrals. Examples: Geometric Brownian Motion dSt = St dt + St dWt has the solution 1 St = S0 e( )t+ 2 Wt 2 Ornstein-Uhlenbeck (OU) Process dSt = ( St ) dt + dWt has the solution Zt t t (t s) S t = S0 e + 1 e + e dWs 0FRM c Dennis PHILIP 2011
- 104. 1 Modelling stock prices 15 Consider the following process dSt = St dt + dWt has the solution Zt St = S 0 e t + e (t s) dWs 0 Simulating Geometric Brownian Motion 1 2 We can write St = S0 e( 2 )t+ Wt in dis- crete time intervals and substituting for Wt as 1 2 p St = St 1 e( 2 ) t+ et t where et N (0; 1) So we randomly draw et and …nd the value of StFRM c Dennis PHILIP 2011
- 105. 2 Interest Rate Derivatives 162 Interest Rate Derivatives The payo¤ of interest rate derivatives would depend on the future level of interest rates. The main challenge in valuing these deriv- atives are that interest rates are used both for discounting and for de…ning payo¤s. For valuation, we will need a model to de- scribe the behavior of the entire yield curve. Black’ Model to price European Options s Consider a call option on a variable whose value is V: To calculate expected payo¤, the model ass- sumes: 1. VT has lognormal distribution with 2 V ar(lnVT ) = T 2. E(VT ) = F0FRM c Dennis PHILIP 2011
- 106. 2 Interest Rate Derivatives 17 The payo¤ is max(VT K; 0) at time T . We discount the expected payo¤ at time T using the risk-free rate given by P (0; T ) We will use the key result that you know from Derivatives: If V is lognormally distributed and stan- dard deviation on ln(V ) is s, then E[max ( V K; 0)] = E[ V ] N (d1 ) KN (d2 ) where ln(E[ V ]=K) + s2 =2 d1 = s ln(E[ V ]=K) s2 =2 d2 = s Therefore, value of the call option is given by c = P (0; T ) [F0 N (d1 ) KN (d2 )]FRM c Dennis PHILIP 2011
- 107. 2 Interest Rate Derivatives 18 where 2 ln(F0 =K) + T =2 d1 = p T and ln(F0 =K) 2 T =2 p d2 = p = d1 T T where – F is forward price of V for a contract with maturity T – F0 is value of F at time zero – K is strike of the option – is volatility of forward contract Similarly, for a put option p = P (0; T ) [KN ( d2 ) F0 N ( d1 )]FRM c Dennis PHILIP 2011
- 108. 2 Interest Rate Derivatives 19 European Bond Options Bond option is an option to buy or sell a particular bond by a certain date for a par- ticular price. Callable bonds and Puttable bonds are ex- amples of embedded bond options. The payo¤ is given by max(BT K; 0) for a call option. To price an European Bond Option: – we assume bond price at maturity of option is lognormal – we de…ne such that standard devia- p tion of ln(BT ) = T – F0 can be calculated as B0 I F0 = P (0; T ) where B0 is bond (dirty) price at time zero and I is the present value of coupons that will be paid during the life of op- tionFRM c Dennis PHILIP 2011
- 109. 2 Interest Rate Derivatives 20 – Then using Black’ model we price of a s bond option Interest Rate Caps and Floors An interest rate Cap provides insurance against the rate of interest on a ‡ oating-rate note rising above a certain level (called Cap rate). Example: Principal amount = $10 million Tenor = 3 months (payments made every quarter) Life of Cap = 5 years Cap rate = 8% If the ‡oating-rate exceeds 8 %, then you get cash of the di¤erence. Suppose at a reset date, 3-month LIBOR is 9%, the ‡oating-rate note would have to pay 0:25 0:09 $10million = $225; 000FRM c Dennis PHILIP 2011
- 110. 2 Interest Rate Derivatives 21 and with the Cap rate at 8%, the payment would be 0:25 0:08 $10million = $200; 000 Therefore the Cap provides a payo¤ of $25,000 to the holder. Consider a Cap with total life of Tn ; a Prin- cipal of L, Cap rate of RK based on a refer- ence rate (say, on LIBOR) with a month maturity denoted by R(t) at date t. The contract follows the schedule: t T0 T1 T2 Tn C1 C2 Cn T0 is the starting date. For all j = 1; :::; n, we assume a constant tenor Tj Tj 1 = On each date Tj ; the Cap holder receives a cash ‡ of Cj ow Cj = L max [R(Tj 1 ) RK ; 0] The Cap is a portfolio of n such options and each call option is known as the caplets.FRM c Dennis PHILIP 2011
- 111. 2 Interest Rate Derivatives 22 Lets now consider a Floor with the same characteristics. On each date Tj ; the Floor holder receives a cash ‡ of Fj ow Fj = L max [RK R(Tj 1 ); 0] The Floor is a portfolio of n such options and each put option is known as the ‡oor- lets.FRM c Dennis PHILIP 2011
- 112. 2 Interest Rate Derivatives 23 Interest rate Caps can be regarded as a port- folio of European put options on zero-coupon bonds. Put-Call parity relation: Consider a Cap and Floor with same strike price RK . Consider a Swap to receive ‡ oat- ing and pay a …xed rate of RK , with no ex- change payments on the …rst reset date. The Put-Call parity states: Cap price = Floor price + value of SwapFRM c Dennis PHILIP 2011
- 113. 2 Interest Rate Derivatives 24 Collar A Collar is designed to guarantee that the interest rate on the underlying ‡oating-rate note always lie between two levels. Collar = long position in Cap + short posi- tion in Floor It is usually constructed so that the price of Cap is equal to price of the ‡ oor. Then the cost of entering into a Collar is zero. Valuation of Caps and Floors If the rate R(Tj ) is assumed to be lognormal with volatility j , the value of the caplet today (t) for maturity Tj is given byCaplett = L P (t; Tj ) FTj 1 ;Tj N (d1 ) RK N (d2 ) where ln(FTj 1 ;Tj =RK ) + 2 (Tj j 1 t) =2 d1 = p (Tj 1 t) FRM c Dennis PHILIP 2011
- 114. 2 Interest Rate Derivatives 25 and q d2 = d1 j (Tj 1 t) where FTj 1 ;Tj is the forward rate underlying the Caplet from Tj 1 to Tj : Similarly,F loorlett = L P (t; Tj ) RK N ( d2 ) FTj 1 ;Tj N ( d1 ) FRM c Dennis PHILIP 2011
- 115. Financial Risk Management Lecture 5 Value at Risk Readings: CN(2001) chapters 22,23; Hull_RM chp 8 1
- 116. Topics Value at Risk (VaR) Forecasting volatility Back-testing Risk Grades VaR: Mapping cash flows2
- 117. Value at Risk Example: If at 4.15pm the reported daily VaR is $10m (calculated at 5% tolerance level) then: I expect to lose more than $10m only 1 day in every 20 days ie. (ie. 5% of the time) The VaR of $10m assumes my portfolio of assets is fixed Exactly how much will I lose on any one day? Unknown !!!3
- 118. Value at Risk Statement (how bad can things get?): “We are x% certain that we will not loose more than V dollars in the next N days” V dollars = f(x%, N days) Suppose asset returns is niid, then risk can be measured by variance/S.D. From Normal Distribution critical values table, we can work out the VaR. Example: For 90% certainty, we can expect actual return to be between the range { µ − 1 .6 5 σ , µ + 1 .6 5 σ }4
- 119. Value at Risk Normal Distribution (N(0,σ))Probability Mean = 0 5% of the area 5% of the area -1.65σ 0.0 +1.65σ Return Only 5% of the time will the actual % return R be below: “ R = µ - 1.65 σ1” where µ = Mean (Daily) Return.5 If we assume µ=0, VaR = $V (1.65 σ1)
- 120. VaR for single asset Example: Mean return = 0 %. Let σ1 = 0.02 (per day) Only 5% of the time will the loss be more than 3.3% (=1.65 x 2%) VaR of a single asset (Initial Position V0 =$200m in equities) VaR = V0 (1.65 σ1 ) = 200 ( 0.033) = $6.6m That is “(dollar) VaR is 3.3% of $200m” = $6.6m VaR is reported as a positive number (even though it’s a loss) Are Daily Returns Normally Distributed? - NO • Fat tails (excess kurtosis), peak is higher and narrower, negative skewness, small (positive) autocorrelations, squared returns have strong autocorrelation, ARCH. • But niid is a (reasonable) approx for portfolios of equities, long term6 bonds, spot FX , and futures (but not for short term interest rates or options)
- 121. VaR for portfolio of assets7
- 122. VaR for portfolio of assets8
- 123. VaR for portfolio of assets9
- 124. VaR for portfolio of assets10
- 125. VaR for portfolio of assets Summary: Variance – Covariance method If Vp is the market value of your portfolio of n assets and wi is the proportionate weight in each asset i then VaR p = V p [ zCz ] 1/2 where z = w1 (1.65σ 1 ) , w2 (1.65σ 2 ) ,… , wn (1.65σ n ) 1 ρ12 … ρ1n ρ ⋱ C = 21 ⋮ ρ n1 1 11
- 126. Forecasting12
- 127. Forecasting σ Simple Moving Average ( Assume Mean Return = 0 ) σ2 t+1|t = (1/n) Σi R2t-i Exponentially Weighted Moving Average EWMA σ2 t+1|t = Σi wi R2t-i wi = (1-λ) λi It can be shown that this may be re-written: σ2t+1|t = λ σ2t| t-1 + (1- λ) Rt2 Longer Horizons: T -rule - for iid returns. σΤ = T σ13
- 128. Forecasting σ Exponentially Weighted Moving Average (EWMA) σ2t+1|t = λ σ2t| t-1 + (1- λ) Rt2 How to estimate λ? 1. Use GARCH models to estimate λ 2. Minimize forecast error Σ (Rt+12 – σ2 t+1|t) where the sum is over all assets, and say 100 days 3. λ = 0.94 as by JPMorgan Suppose λ = 0.94 then weights decline as 0.94, 0.88, 0.83,…. and past observations are given less weight than current forecast of variance.14
- 129. Back-testing In back-testing, we compare our (changing) daily forecast of VaR with actual profit or loss over some historic period. Example: For a portfolio of assets, • forecast all the individual VaRi = Vi1.65 σt+1|t , • calculate portfolio VaR for each day: VaRp = [Z C Z’]1/2 • then see if actual portfolio losses exceed this only 5% of the time (over some historic period, e.g. 100 days).15
- 130. Back-testing Daily $m profit/loss = forecast = actual Days Only 6 violations out of16 100 = just ‘OK’
- 131. VaR and Capital Adequacy-Basle Basle uses a more ‘conservative’ measure of VaR than J. P. Morgan Calc VaR for worst 1% of losses over 10 days Use at least 1-year of daily data to estimate σt+1|t VaRi = 2.33 10 σ ( 2.33 = 1% left tail critical value, σ = daily vol ) Internal Models approach Capital Charge KC KC = Max ( Avg. of previous 60-days VaR x M, previous day’s VaR) M = multiplier (min = 3) Pre-commitment approach • KC set equal to max. forecast loss over 20 day horizon = pre- announced $VaR17 • If losses exceed VaR, more than 1 day in 20, then impose a penalty.
- 132. VaR and Coherent Risk Measures Risk measures that satisfy all the following 4 conditions are called as a Coherent Risk Measure. Monotonicity: X 1 ≤ X 2 ⇒ R ( X 1 ) ≤ R ( X 2 ) (higher the riskiness of the portfolio, higher should be risk capital) Translation invariance: R ( X + k ) = R ( X ) − k ∀k ∈ ℝ (if cash k is added to portfolio, risk should go down by k) Homogeneity: R ( λ X ) = λ R ( X ) ∀λ ≥ 0 (if you change portfolio by a factor of λ, risk is proportionally increased) Subadditivity: R ( X + Y ) ≤ R ( X ) + R (Y )18 (diversification leads to less risk)
- 133. VaR and Coherent Risk Measures VaR violates the subadditivity condition and therefore not coherent. VaR cannot capture the benefits of diversification. VaR can actually show negative diversification benefit! VaR only captures the frequency of default but not the size of default. Even if the largest loss is doubled, the VaR figure could remain the same. Other measures such as Expected Shortfall are coherent measures.19
- 134. Risk Grades20
- 135. Risk Grades RG helps to calculate changing forecasts of risks (volatilities) RG quantifies volatility/risk (similar to variance, std. deviation, beta, etc) RG can range from 0 to over 1000, where 100 corresponds to the average risk of a diversified market-cap weighted index of global equities. So if two portfolio’s have RG1 = 100 and RG2 = 400, portfolio 2 is four times riskier than portfolio 1 RG scales all assets to a common scale and so it is able to compare risk across all asset classes.21
- 136. Risk Grades RG of a single asset σi 252 σi 252 RG = σ ×100 = ×100 i 0.20 base σi is the DAILY standard deviation σbase is fixed at 20% per annum (= 5 yr. av. for international portfolio of stocks) Formula looks complex but RG is just a “scaled” daily standard deviation e.g. If RG = 100% then asset has 20% p.a. risk RG of a portfolio of assets RG 2 = ∑ w i2 RG i2 + ∑ ∑ w i w j ρ RG i RG j22 P
- 137. Risk Grades Risk Grades in 2009 www.riskgrades.com23
- 138. Risk Grades Risk Grades in 2009 of indices heating up and cooling off24
- 139. VaR Mapping (VaR for different assets)25
- 140. VaR for different assets PROBLEMS STOCKS : Too many covariances [= n(n-1)/2 ] FOREIGN ASSETS : Need VaR in “home currency” BONDS: Many different coupons paid at different times DERIVATIVES: Options payoffs can be highly non- linear (ie. NOT normally distributed) SOLUTIONS = “Mapping” (RiskMetricsTM produce volatility & correlations for various assets26 across 35 countries and useful for “Mapping”)
- 141. VaR for different assets STOCKS Within each country use “single index model” SIM FOREIGN ASSETS Treat one asset in foreign country as = “local currency risk”+ spot FX risk (like 2-assets, with equal weight) BONDS Consider each bond as a series of “zeros” OTHER ASSETS Forward-FX, FRA’s, Swaps: decompose into ‘constituent parts’27 DERIVATIVES(non-linear)
- 142. Mapping Stocks Consider ‘p’ = portfolio of stocks held in one country with (Rm , σm) (for e.g. S&P500 in US) Problem : Too many covariances to estimate Soln. All n(n-1)/2 covariances “collapse or mapped” into σm and the asset betas (“n” of them) Single Index Model: Ri = ai + bi Rm + εi Rk = ak + bk Rm + εk assume Eεi εk = 0 and cov (Rm , ε ) = 028 All the systematic variation in Ri AND Rk is due to Rm
- 143. Mapping Stocks 1) In a portfolio idiosyncratic risk εi is diversified away = 0 2) Then each return-i depends only on the market return (and beta). Hence ALL variances and covariances also only depend on these 2 factors. It can be shown that σp = bp σm (i.e. Calculation of portfolio beta requires, only n-beta’s and σm ) 3) Also, ρ = 1 because (in a well diversified portfolio) each return moves only with Rm 4) We end up with VaRp = VP 1.65 ( bP σm ) or equivalently VaRp = (Z C Z’ )1/2 where Z = [ VaR1, VaR2 …. } C is the unit matrix29
- 144. Mapping Foreign Assets (Mapping foreign stocks into domestic currencyVaR)30
- 145. Mapping Foreign assets Example: US resident holds a diversified portfolio of German stocks equivalent to German stocks + Euro-USD, FX risk Use SIM to obtain stdv of foreign (German) portfolio returns, σG Then treat ‘foreign portfolio’ as (two) equally weighted assets: = $V in German asset + $V foreign currency position Then use standard VaR formula for 2-assets31
- 146. Mapping Foreign assets US based investor: with €100m in a German stock portfolio σG = βP σDAX Sources of risk: a) Stdv of the German portfolio (‘local currency’ portfolio) b) Stdv of €/$ exchange rate ( σFX ) c) one covariance/correlation coefficient ρ (between DAX and FX rate) e.g. Suppose when German stock market falls then the € also falls - ‘double whammy’ for the US investor, from this positive correlation, so foreign assets are very risky (in terms of their USD ‘payoff’) Let : S = 1.2 $/ € Dollar initial value Vo$ = 100m x 1.2 = $120m Linear dVP = V0$ (RG + RFX)32 above implies wi = Vi / V0$ = 1
- 147. Mapping Foreign assets Dollar-VaRp = Vo$ 1.65 σp σp = ( σG 2 + σ FX 2 + 2 ρσ Gσ FX )1 2 No ‘relative weights’ appear in the formula Matrix Representation: Dollar VaR Let Z = [ V0,$ 1.65 σG , V0,$ 1.65 σFX ] =[ VaR1 , VaR2 ] V0,$ = $120m for both entries in the Z-vector (i.e. equal amounts) Then VaRp = (Z C Z’ )1/233
- 148. Mapping Bonds (Mapping coupon paying bonds)34
- 149. Mapping Coupon paying Bonds Example: Coupons paid at t=5 and t=7 Treat each coupon as a separate zero coupon bond 100 100 P= + (1 + y5 ) (1 + y7 )7 5 P = V5 + V7 P is linear in the ‘price’ of the zeros, V5 and V7 We require two variances of “prices” V5 and V7 and covariance between these prices. Note: σ5(dV5 / V5) = D σ(dy5) but Risk Metrics provides the price volatilities, σ5(dV5 / V5)35
- 150. Mapping Coupon paying Bonds Treat each coupon as a zero Calculate: price of zero, e.g. V5 = 100 / (1+y5)5 VaR5 = V5 (1.65 σ5) VaR7 = V7 (1.65 σ7) VaR (both coupon payments): VaRp = (Z C Z’ )1/2 = [ VaR + VaR + 2 ρ VaR 5VaR 7 ] 2 5 2 7 1/ 2 r = correlation: bond prices at t=5 and t=7 (approx 0.95 - 0.99 )36
- 151. Mapping FRA37
- 152. Mapping FRA To calculate VaR for a FRA, we break down cash flows into equivalent synthetic FRA and use spot rates only (since we do not know the forward volatilities) Example: Consider an FRA on a notional of $1m that involves lending $1m in 6 months time for a future of 6 months. Receipt of $1m + Interest 0 6m 12m Lend $1m Let y6 = 6.39%, y12 = 6.96% and there are 182 days in the first leg and 183 days in the second leg (day count: actual/365). The implied f6,12 = 7.294% and therefore the 12 month investment will give $1,036,572 return (with round off error).38
- 153. Mapping FRA The original FRA Receipt of $1,036,572 0 6m 12m Lend $1m Synthetic FRA Receipt of $1,036,572 Borrow at 6 month rate from 12 month lending 0 6m 12m Repay 6 Lend at 12 month rate month loan of $1m So at time 0, we borrow $969,121 [=1m / 1+(y6*182/365)] and lend this money at a 12 month rate leading to $1,036,572 [=$969,121*(1+y12)]39
- 154. Mapping FRA Suppose the standard deviation of the prices for 6 month asset is 0.1302% and for 12 month asset is 0.2916%. Suppose ρ = 0.7 To calulate the VaR for each of these positions: VaR6 = $969,121 (1.65) (0.1302%) = $2082 VaR12 = $969,121 (1.65) (0.2916%) = $4663 VaR = [ VaR + VaR + 2 ρ ( −VaR 6 ) VaR12 ] 2 6 2 12 1/ 2 = $353440
- 155. Mapping FX Forwards41
- 156. Mapping FX Forwards Consider a US resident who holds a long forward contract to purchase €10million in 1 year. What is the VaR for this contract? We map Forward into two spot rates and one spot FX rate. Then we calculate VaR from the VaR of each individual mapped asset. Mapping a forward contract42
- 157. Mapping FX Forwards43
- 158. Mapping FX Forwards44
- 159. Mapping Options45
- 160. Mapping Options46
- 161. Mapping Options47
- 162. Mapping Options48
- 163. Mapping Options49
- 164. Financial Risk Management Topic 6 Statistical issues in VaR Readings: CN(2001) chapters 24, Hull_RM chp 8, Barone-Adesi et al (2000) RiskMetrics Technical Document (optional) 1
- 165. Topics Value at Risk for options Monte Carlo Simulation Historical Simulation Bootstrapping Principal component analysis Other related VaR measures Marginal VaR, Incremental VaR, ES2
- 166. MCS - VaR of Call option Option premia are non-linear (convex) function of underlying Distribution of gains/losses is not normally distributed Therefore dangerous to use Var-Cov method Assets Held: One call option on stock Here, Black-Scholes is used to price the option during the Monte Carlo Simulation (MCS). Problem Find the VaR over a 5-day horizon3
- 167. MCS - VaR of Call option If V is price of the option (call or put) and P is price of underlying asset in the option contract (stock) V will change from minute to minute as P changes. For an equal change in P of +1 or -1 ,the change in call premia are NOT equal: this gives “non-normality” in distribution of the4 change in call premium
- 168. MCS - VaR of Call option To find VaR over a 5-day horizon: 1) Given P0 calculate the option price, V0 = BS(P0, K, T0 …..) This is fixed throughout the MCS 2) MCS = Simulate the stock price and calculate P5 3) Calculate the new option price, V5= BS(P5, K, T0 – 5/365, …..) 4) Calculate change in option premium ∆V(1) = V5 – V0 5) Repeat steps 2-4, 1000 times and plot a histogram of the change in the call premium. We can then find the 5% lower cut-off point for the change in value of the call (i.e. it’s VaR).5
- 169. MCS - VaR of Call option 20 18 165% of area 14 Frequency 12 10 8 6 4 2 0 -10 -8 -6 -4 -2 0 2 4 6 8 10 More $-VaR is $5m Change in Call Premium6
- 170. MCS - VaR of Call option Simulate stock prices using P(t+1) = P(t) exp[ (µ - σ2/2) ∆t + σ (∆t)1/2 εt+1 ] To generate 5-day prices from daily prices, you can use the root-T rule ∆t = 5/365 (or 5/252) ( ie. five day) Alternatively, we can generate Tt+5 directly. If P0 is today’s known price. Use a ‘do-loop’ over 5 ‘periods’ to get P5 (using 5 - ‘draws’ of εt+1 )7
- 171. MCS - VaR of Call option Stock price paths generated8
- 172. Monte Carlo Simulation for Two Assets (two call options on different underlying assets)9
- 173. Simulating correlated random variables10
- 174. Simulating correlated random variables11
- 175. Simulating correlated random variables12
- 176. MCS and VaR: two asset example13
- 177. MCS and VaR: two asset example14
- 178. MCS and VaR: two asset example15
- 179. MCS and VaR: two asset example16
- 180. MCS and VaR: two asset example17
- 181. MCS and VaR: two asset example18
- 182. MCS and VaR: two asset example19
- 183. Comparing VaR forecasts20
- 184. Comparing VaR forecasts21
- 185. Historical Simulation (Historical simulation + bootstrapping)22
- 186. Historical Simulation (HS) Suppose you currently hold $100 in each of 2 assets Day = 1 2 3 4 5 6 …1000 R1(%) +2 +1 +4 -3 -2 -1 +2 R2(%) +1 +2 0 -1 -5 -6 -5 _____________________________________________________ ∆Vp($) +3 +3 +4 -4 -7 -7 -5 Order ∆Vp in ascending order (of 1000 numbers) e.g. -12, -11, -11 -10, -9, -9, -8, -7, -7, -6 | -5, -4, -4, …. +8, ……. +14 VaR forecast for tomorrow at 1% tail (10th most negative) = -$623 Above is equivalent to the histogram
- 187. Historical Simulation (HS) This is a non parametric method since we do not estimate any variances or covariances or assume normality. We merely use the historic returns, so our VaR estimates encapsulate whatever distribution the returns might embody ( e.g. Student’s t) as well as any autocorrelation in individual returns. Also, the historic data “contain” the correlations between the returns on the different assets, their ‘own volatility’ and their own autocorrelation over time It does rely on ‘tomorrow’ being like ‘the past’.24
- 188. HS + Bootstrapping Problems: Is data >3 years ago useful for forecasting tomorrow? Use most recent data - say last 100 days ? 1% tail: Has only one number in this tail, for the actual data ! Extreme case ! Actual data might have largest negative (for 100 days ago) of minus 50% - this would be your forecast VaR for tomorrow using historic simulation approach. Is this okay or not?25
- 189. HS + Bootstrapping You have “historic” daily data on each of 10 stock returns (i.e. your portfolio ) But only use last 100 days of historic daily returns, So we have a data matrix of 10 x 100. We require VaR at the 1st percentile (1% cut off) We sample “with replacement” from these 100 observations, giving equal probability to each ‘day’ , when we sample. This allows any one day’s returns to be randomly chosen more than once (or not at all). It is as if we are randomly ‘replaying’ the last 100 days of26 history, giving each day equal probability
- 190. HS + Bootstrapping The Bootstrap Draw randomly from a uniform distribution with an equal probability of drawing any number between 1 and 100. If you draw “20” then take the 10-returns in column 20 and revalue the portfolio. Call this $-value, ∆VP(1) Repeat above for 10,000 “trials/runs” (with replacement), obtaining 10,000 possible (alternative) values ∆VP (i) “With replacement” means that in the 10,000 runs you will “choose” some of the 100 columns more than once. Plot a histogram of the 10,000 values of ∆VP(i) - some of which will be negative Read off the “1% cut off” value (=100th most negative value). This is VaRp27
- 191. Filtered Historical Simulation (FHS)28
- 192. FHS HS assumes risk factors are i.i.d however this is usually not the case. HS assumes that distribution of returns are stable and that the past and present moments of the density function of returns are constant and equal. The probability of having a large loss is not equal across different days. There are periods of high volatility and periods of low volatility (volatility clustering). In FHS, historical returns are first standardized by volatility estimated on that particular day (hence the word Filtered). The filtering process yields approximately i.i.d returns (residuals) that are suited for historical simulation.29 read Barone-Adesi et al (2000) paper in DUO on FHS
- 193. Principal Component Analysis (Estimating risk factors using PCA)30
- 194. Estimating risk factors using PCA31
- 195. Estimating risk factors using PCA32
- 196. Estimating risk factors using PCA33
- 197. Estimating risk factors using PCA34
- 198. Estimating risk factors using PCA35
- 199. Estimating risk factors using PCA36
- 200. Estimating risk factors using PCA37
- 201. PCA and risk management38
- 202. PCA and risk management39
- 203. Other related VaR measures40
- 204. Other related VaR measures41
- 205. Other related VaR measures42
- 206. Other related VaR measures43
- 207. Other related VaR measures44
- 208. Other related VaR measures45
- 209. Other related VaR measures46
- 210. Other related VaR measures47
- 211. Topic # 7: Univariate and Multivariate Volatility Estimation Financial Risk Management 2010-11 February 28, 2011FRM c Dennis PHILIP 2011
- 212. 1 Volatility modelling 21 Volatility modelling Volatility refers to the spread of all likely outcomes of an uncertain variable. It can be measured by sample standard de- viation v u u 1 X T ^=t (rt )2 T 1 t=1 where rt is the return on day t, and is the average return over the T day period. But this statistic only measures the spread of a distribution and not the shape of a dis- tribution (except normal and lognormal). Black Scholes model assumes that asset prices are lognormal (which implies that returns are normally distributed). In practice, returns are however non-normal and also the return ‡uctuations are time vary- ing.FRM c Dennis PHILIP 2011
- 213. 1 Volatility modelling 3 Example: daily returns of S&P 100 show features of volatility clustering Therefore Engle (1982) proposed Autoregres- sive Conditional Heteroscedasticity (ARCH) models for modelling volatility Other characteristics documented in litera- tureFRM c Dennis PHILIP 2011
- 214. 1.1 Parametric volatility models 4 – Long memory e¤ect of volatility (au- tocorrelations remain positive for very long lags) – Squared returns proxy volatility – Volatility asymmetry / leverage e¤ect (volatility increases if the previous day returns are negative)1.1 Parametric volatility modelsARCH model ‘Autoregressive’because high/low volatility tends to persist, ‘ Conditional’ means time- varying or with respect to a point in time, and ‘Heteroscedasticity’is a technical jargon for non-constant volatility. Consider previous t day’ squared returns s ("2 1 ; "2 2 ; :::) that proxy volatility. t t It makes sence to give more weight to recent data and less weight to far away observa- tions.FRM c Dennis PHILIP 2011
- 215. 1.1 Parametric volatility models 5 Suppose we are assuming that previous q ob- servations a¤ect today’ returns. So today’ s s volatility can be q X 2 2 t = j "t j j=1 Pq where i < j for i > j and j=1 j =1 Also we can include a long run average vari- ance that should be given some weight as well q X 2 2 t = V0 + j "t j j=1 where V0 is average variance rate and Pq + j=1 j = 1 The weights are unknown and needs to be estimated. This is the ARCH model introduced by En- gle (1982)FRM c Dennis PHILIP 2011
- 216. 1.1 Parametric volatility models 6 ARCH (1) An ARCH(1) model is given by 2 rt = + "t "t N 0; 2 2 t = 0 + 1 "t 1 Since 2 is variance and has to be positive, t we impose the condition 0 0 and 1 0 Generalization: ARCH(q) model 2 2 2 t = 0 + 1 "t 1 + ::: + q "t q where shocks up to q periods ago a¤ect the current volatility of the process.FRM c Dennis PHILIP 2011
- 217. 1.1 Parametric volatility models 7 EWMA Exponentially weighted moving average (EWMA) model is the same as the ARCH models but the weights decrease exponentially as you move back through time. The model can be written as 2 2 t = t 1 + (1 )u2 t 1 where is the constant decay rate, say 0.94. To see that the weights cause an exponential decay, we substitute for 2 1 : t 2 2 t = t 2 + (1 )u2 t 2 + (1 )u2 t 1 2 2 = (1 ) u2 1 t + (1 ) u2 2 t + t 2 With =0.94 2 t = (0:06) u2 t 1 + (0:056) u2 t 2 + (0:883) 2 t 2 2 Substituting for t 2 now:FRM c Dennis PHILIP 2011
- 218. 1.1 Parametric volatility models 8 2 2 t = (1 ) u2 t 1 + (1 ) u2 t 2 + 2 t 3 + (1 )u2 t 3 2 3 2 = (1 ) u2 1 t + (1 ) u2 2 t + (1 )u2 3 t + t 3 With =0.94 2 t = (0:06) u2 1 + (0:056) u2 2 + (0:053) u2 3 + (0:83) t t t 2 t 3 Risk Metrics uses EWMA model estimates for volatility with = 0.94.FRM c Dennis PHILIP 2011
- 219. 1.1 Parametric volatility models 9 Generalized ARCH (GARCH) model This model generalizes the ARCH speci…ca- tion. As one increase the q lags in an ARCH model for capturing the higher order ARCH e¤ects present in data, we loose parsimonity. Bollerslev (1986) proposed GARCH(p; q) q p X X 2 2 2 t = 0 + j "t j + j t j j=1 j=1 where the weights 0 0; j 0 and j 0: Further, for stationarity of this autore- gressive model, we need the condition q p ! X X j + j <1 j=1 j=1 In this model, today’ volatility is explained s by the long run variance rate, the past squared observations, and the past volatility history.FRM c Dennis PHILIP 2011
- 220. 1.1 Parametric volatility models 10 Special case: GARCH(1,1) 2 2 2 t = 0 + 1 "t 1 + 1 t 1 where 0 ( 1 + 1) < 1: To see that the weights cause an exponential decay, consider a GARCH(1,1) process 2 2 2 t = 0 + 1 "t 1 + 1 t 1 2 We substitute for t 1: 2 2 2 2 t = 0 + 1 "t 1 + 1 0 + 1 "t 2 + 1 t 2 2 2 2 2 = 0 + 0 1 + 1 "t 1 + 1 1 "t 2 + 1 t 2 2 Substituting for t 2 now: 2 t = 0 + + 1 "2 1 + 0 1 t 2 1 1 "t 2 + 2 2 2 1 0 + 1 "t 3 + 1 t 3 2 2 2 = 0+ 0 1+ 0 1+ 1 "t 1 + 1 1 "t 2 + 2 2 3 2 1 1 "t 3 + 1 t 3 i 1 So the weight applied to "2 t i is 1 1 : The weights decline at rate :FRM c Dennis PHILIP 2011
- 221. 1.1 Parametric volatility models 11 GARCH is same as EWMA in assigning ex- ponentially declining weights to past obser- vations. However, GARCH also assigns some weight to the long-run average variance rate. When the intercept parameter 0 = 0 and 1 + 1 = 1; then GARCH reduces to a EWMA. A GARCH(1,1) model can be interpreted as an 1 order ARCH. To see this, consider a ARCH(1) 2 2 2 2 t = 0 + 1 "t 1 + 2 "t 2 + ::: + 1 "t 1 Problem of how to estimate this – Use the Koyck transform Assume 1 is declining – E¤ect of lagged residuals falls as time goes by 1 > 2 > 3 > ::: > 1FRM c Dennis PHILIP 2011
- 222. 1.1 Parametric volatility models 12 Also assume a geometric decline in 1 such that k k = a where 0 < <1 Gives conditional variance equation as: 2 t = 0+ a "2 1 + a 2 "2 2 + ::: + a 1 "2 1 t t t 2 2 2 1 2 = 0+a "t 1 + "t 2 + ::: + "t 1 Now consider one lag of the variance equa- tion 2 1 t 2 2 2 1 2 t 1 = 0 +a "2 t 2 + "t 3 + ::: + "t 1 Multiply by through 2 2 2 1 2 t 1 = 0 +a "2 t 2 + "t 3 + ::: + "t 1 2 2 Now derive t t 1 using the above equa- tions 2 2 2 t 2 t 1 = 0 +a "2 t 1 + "t 2 + ::: + 1 "2 1 t 0 +a "t 2 + "t 3 + ::: + 1 "2 1 2 2 2 tFRM c Dennis PHILIP 2011
- 223. 1.1 Parametric volatility models 13 Solving 2 2 t t 1 = ( 0 0 ) + a "2 t 1 2 t = ( 0 0 ) + a "2 t 1 + 2 t 1 GJR model or TARCH model Introduced by Glosten, Jagannathan, Run- kle (1993). Hence called GJR model. Also called Threshold ARCH (TARCH) model. Asymmetries in conditional variances could also be introduced by distinguishing the sign of the shock. We can therefore separate the positive and negative shock and allow for di¤erent coef- …cients in a GARCH framework. q q p X 2 X 2 X + 2 t = 0+ j "+ j + t j "t j + j 2 t j j=1 j=1 j=1 If + = j for j = 1; 2; :::; q then this re- j duces to a GARCH(p,q) model.FRM c Dennis PHILIP 2011
- 224. 1.1 Parametric volatility models 14 GJR(1993) proposed keeping the original GARCH framework and adding an extra component that captures the negative shocks. This is because bad news usually has a greater im- pact on volatility than good news. The GJR(1,1) model is 2 + 2 2 2 t = 0 + 1 "t 1 + T " t 1 Dt 1 + 1 t 1 where 1 for "t 1 <0 Dt 1 = 0 for "t 1 0 Remarks: – For testing symmetry, we consider test- ing H0 : T = 0: The leverage e¤ect is seen in T > 0: + – 1 is the coe¢ cient for positive shocks. + – + T = 1 is the coe¢ cient for 1 negative shocks. – For positivity of the conditional vari- ances, we require 0 > 0; + 0; and 1 + 1 + T 0. Hence, T is allowed to be negative provided + > j T j : 1FRM c Dennis PHILIP 2011
- 225. 1.1 Parametric volatility models 15 Exponential GARCH (EGARCH) Recall that in the case of (G)ARCH models, we provided certain coe¢ cient restrictions in order to ensure 2 (conditional variance of t "t ) is non-negative with probability one. An alternative way of ensuring positivity is specifying an EGARCH framework for 2 : t Another characteristic of the model is that it allows for positive and negative shocks to have di¤erent e¤ects on conditional vari- ances (unlike GARCH). The model re‡ ects the fact that …nancial markets respond asymmetrically to good news and bad news. EGARCH(p,q) model is q q X "t j X "t j 2 ln t = 0 + j + j j=1 t j j=1 t j p X 2 + j ln t j j=1FRM c Dennis PHILIP 2011
- 226. 1.1 Parametric volatility models 16 where if theh error terms "t = i q t N (0; 1) then = E "tt = 2 Remarks: – We can use other fat failed distribu- tions such as student t; or GED in the case of non-normal errors. In this case, will take other forms. – Specifying the model as a logarithm en- sures positivity of 2 : Therefore the lever- t age e¤ect is exponential rather than quadratic. – We divide the errors by the conditional standard deviations, "tt : Therefore we standardize (scale) the shocks. "t – t captures the relative size of the shocks and j captures the sign of the relative shocks – The magnitude is captured by the vari- able that substracts the mean from the absolute value of the scaled shocks. Example:FRM c Dennis PHILIP 2011
- 227. 1.1 Parametric volatility models 17 – Suppose we specify a EGARCH(0,1) 2 ln t = 0 + 1 t 1 + 1 [j t 1j ] where t 1 = "t 1 = t 1: – Consider the estimated component: ^1 t 1 + ^ 1 [j t 1j ] where ^ 1 = 0:3; ^ 1 = 0:6 and = 0:85: – Case 1: impact of positive scaled shock +1:0 0:3 (1) + 0:6 [j1j 0:85] = 0:39 Case 2: impact of negative scaled shock 1:0 0:3 ( 1) + 0:6 [j 1j 0:85] = 0:21 We see that positive shock has a greater im- pact than negative shock for ^ 1 positive. If we have ^ 1 negative, say ^ 1 = 0:3; a +1:0 shock will have an impact of 0:21 and a 1:0 shock will have an impact of 0.39.FRM c Dennis PHILIP 2011
- 228. 1.2 Non-parametric volatility models 18 Thus, ^ 1 allows for the sign of the shock to have an impact on the conditional volatility; over and above the magnitude captured by ^ 1:1.2 Non-parametric volatility mod- elsRange-based estimators Suppose log prices of assets follow a Geo- metric Brownian Motion (GBM). The vari- ous variance estimators have been proposed in literature. Notation: – volatility to be estimated – Ct closing price on date t – Ot opening price on date t – Ht high price on date t – Lt low price on date tFRM c Dennis PHILIP 2011
- 229. 1.2 Non-parametric volatility models 19 – ct = ln Ct ln Ot , the normalized clos- ing price – ot = ln Ot ln Ct 1 , the normalized opening price – ht = ln Ht ln Ot , the normalized high price – lt = ln Lt ln Ot , the normalized low price The classical sample variance estimator of variance 2 is 1 X T 2 2 ^ = [(oi + ci ) (o + c)] T 1 i=1 where 1X T (o + c) = (oi + ci ) T i=1 and T is the total number of days consid- ered. So this is the average volatility over T days.FRM c Dennis PHILIP 2011
- 230. 1.2 Non-parametric volatility models 20 Parkinson (1980) introduced a range estima- tor of daily volatility based on the highest and lowest prices on a particular day. He used the range of log prices to de…ne 1 ^2 = t (ht lt )2 4 ln 2 since it can be shown that E (ht lt )2 = 4 ln(2) 2t Garman and Klass (1980) extended Parkin- son’ estimator where information about open- s ing and closing prices are incorporated as follows: ^ 2 = 0:5 (ht t lt )2 [2 ln 2 1] c2 t Parkinson (1980) and Garman and Klass (1980) assume that the log-price follows a GBM with no drift term. This means that the average return is assumed to be equal to zero. Rogers and Satchell (1991) relaxes this assumption by using daily opening, high- est, lowest, and closing prices into estimat- ing volatility.FRM c Dennis PHILIP 2011
- 231. 1.2 Non-parametric volatility models 21 Rogers and Satchell (1991) estimator is given by ^ 2 = ht (ht ct ) + lt (lt ct ) t This estimator performs better than the es- timators proposed by Parkinson (1980) and Garman and Klass (1980). Yang and Zhang (2000) proposed a re…ne- ment to Rogers and Satchell (1991) estima- tor for the presence of opening price jumps. Due to overnight volatility, the opening price and the previous day closing price are mostly not the same. Estimators that do not incor- porate opening price jumps underestimate volatility. Yang and Zhang (2000) estimator is given by ^ 2 = ^ 2 + k^ 2 + (1 open close k)^ 2 RS where ^ 2 and ^ 2 open close are the classical sam- ple variance estimators with the use of daily opening and closing prices, respectively. ^ 2RS is the average variance estimator introduced by Rogers and Satchell (1991).FRM c Dennis PHILIP 2011
- 232. 1.2 Non-parametric volatility models 22 The constant k is set to be 0:34 k= 1:34 + (T + 1)=(T 1) where T is the number of days. Realized Volatility Realized volatility is referred to volatility es- timates calculated using intraday squared returns at short intervals such as 5 or 15 minutes. For a series that has zero mean and no jumps, the realized volatility converges to the con- tinuous time volatility. Consider a continuous time martigale process for asset prices dpt = t dWt where dWt is a standard brownian motion.FRM c Dennis PHILIP 2011
- 233. 1.2 Non-parametric volatility models 23 Then the conditional variance for one-period returns, rt+1 pt+1 pt is Z t+1 2 s ds t which is called the integrated volatility (or the true volatility) over the period t to t + 1: 2 We don’ know what t t is. So we estimate it. Let m be the sampling frequency such that there are m continuously compounded re- turns in one unit of time (say, one day). The j th return is given by rt+j=m pt+j=m pt+(j 1)=m The realized volatility (in one unit of time) can be de…ned as X 2 RVt+1 = rt+j=m j=1;:::;mFRM c Dennis PHILIP 2011
- 234. 1.2 Non-parametric volatility models 24 Then from the theory of quadratic variation, if sample returns are uncorrelated, Z t+1 ! X 2 2 p lim s ds rt+j=m = 0 m!1 t j=1;:::;m As we increase sampling frequency, we get a consistent estimate of volatility. In the presence of jumps, RV is no longer a consistent estimator of volatility. An extension to this estimator is the stan- dardized Realized Bipower Variation mea- sure de…ned as [a;b] 1 [1 (a+b)=2] X m a b BVt+1 = rt+j=m rt+(j 1)=m m j=1 for a; b > 0: When jumps are large but rare, the simplest case where a = b = 1 captures the jumps well.FRM c Dennis PHILIP 2011
- 235. 1.2 Non-parametric volatility models 25 High frequency returns measured below 5 minutes are a¤ected by market microstruc- ture e¤ects including nonsynchronous trad- ing, discrete price observations, intraday pe- riodic volatility patterns and bid– bounce. askFRM c Dennis PHILIP 2011
- 236. 2 Multivariate Volatility Models 262 Multivariate Volatility Mod- els Multivariate modelling of volatilities enable us to study movements across markets and across assets (co-volatilities). Applications in …nance: asset pricing and portfolio selection, market linkages and in- tegration between markets, hedging and risk management, etc. Consider a n-dimensional process fyt g : If we denote as the …nite vector of parameters, we can write yt = t ( ) + "t where t ( ) is the conditional mean vector and 1=2 "t = H t ( ) zt 1=2 where Ht ( ) is an N N positive de…nite matrix. The N 1 vector zt is such that zt iidD (0; IN )FRM c Dennis PHILIP 2011
- 237. 2 Multivariate Volatility Models 27 where IN is the identity matrix of order N: The matrix Ht is the conditional variance matrix of yt How do we parameterize Ht ? Vech Representation Bollerslev, Engle and Wooldridge (1988) pro- pose a natural multivariate extension of the univariate GARCH(p; q) models where q p X 0 X vech (Ht ) = W + Ai vech "t i "t i + j vech (Ht j ) i=1 j=1 where vech is the vector-half operator, which stacks the lower triangular elements of an N N matrix into a [N (N + 1) =2] 1 vec- tor. The challenge in this parameterization is to ensure Ht is positive de…nite covariance ma- trix. Also, as the number of assets N in- crease, the number of parameters to be esti- mated is very large.FRM c Dennis PHILIP 2011
- 238. 2 Multivariate Volatility Models 28 f"t g is covariance stationary if all the eigen- values of A and B are less than 1 in mod- ulus. Bollerslev, Engle and Wooldridge (1988) pro- posed a "Diagonal vech" representation where Ai and j are diagonal matrices. Example: For N = 2 assets and a single- period lag model (p = q = 1), 2 3 2 3 2 32 3 h11;t w1 a11 "2 1 1;t 4 h21;t 5 = 4 w2 5 + 4 a22 5 4 "2;t 1 ; "1;t 1 5 h22;t w3 a33 "2 1 2;t 2 32 3 b11 h11;t 1 +4 b22 54 h21;t 1 5 b33 h22;t 1 The diagonal restriction reduces the num- ber of parameters but the model is not al- lowed to capture the interactions in vari- ances among assets (copersistence, causality relations, asymmetries) P The diagonal vech is stationary i¤ q aii + Pp i=1 j=1 bjj < 1FRM c Dennis PHILIP 2011
- 239. 2 Multivariate Volatility Models 29 BEKK Representation Engle and Kroner (1995) propose a BEKK representation where q p X 0 X 0 0 Ht = cc + Ai "t i " t i A0i + j Ht j j i=1 j=1 where c is a lower triangular matrix and therefore cc0 will be positive de…nite. Also, by estimating A and B rather than A and B ; we ensure positive de…niteness. In the case of 2 assets:h11;t h12;t a11 a12 "2 1 "1;t 1 ; "2;t 1 = cc0 + 1;th21;t h22;t a21 a22 "2;t 1 ; "1;t 1 "2 1 2;t 0 a11 a12 b11 b12 + a21 a22 b21 b22 0 h11;t 1 h12;t 1 b11 b12 h21;t 1 h22;t 1 b21 b22 To reduce the number of parameters to be estimated, we can impose a "diagonal BEKK" model where Ai and Bj are diagonal.FRM c Dennis PHILIP 2011
- 240. 2 Multivariate Volatility Models 30 Alternatively, we can have Ai and Bj as scalar times a matrix of ones. In this case, we will have a "scalar-BEKK" model. Diagonal BEKK andP Scalar-BEKK are co- P variance stationary if q a2 + p b2 < Pi=1 nn;i P j=1 nn;j 1 8n = 1; 2; :::; N and q a2 + p b2 < 1 i=1 i j=1 j respectively. Constant Conditional Correlation (CCC)Model Bollerslev (1990), assuming conditional cor- relations constant, proposed that conditional covariances (Ht ) can be parameterized as a product of corresponding conditional stan- dard deviations. Ht = Dt RDt p = ij hii;t hjj;t 2 p 3 h11;t 6 .. 7 where Dt = 4 . 5; R = p hN N;tFRM c Dennis PHILIP 2011
- 241. 2 Multivariate Volatility Models 31 2 3 1 12 1N 6 . . 7 6 21 1 . 7 6 . .. 7 4 . . . 5 N1 1 Each conditional standard deviations can be in turn de…ned as any univariate GARCH model such as GARCH(1,1) hii;t = wi + i "2 1 + i hii;t i;t 1 i = 1; 2; :::; N Ht is positive de…nite i¤ all N conditional covariances are positive and R is positive de…nite. In most empirical applications, the condi- tional correlations are not constant. There- fore Engle (2002) and Tse and Tsui(2002) propose a generalization of the CCC model by allowing for conditional correlation ma- trix to be time-varying. This is the DCC model.FRM c Dennis PHILIP 2011
- 242. 2 Multivariate Volatility Models 32 Tests for Costant Correlations Tse (2000) proposes testing the null that p hijt = ij hiit hjjt against the alternative that p hijt = ijt hiit hjjt where the conditional variances, hiit and hjjt are GARCH-type models. The test statistic is an LM statistic which is asymptotically 2 (N (N 1) =2) : Engle and Sheppard (2001) propose another test with the null hypothesis H0 : Rt = R for all t against the alternative H1 : vech (Rt ) = vech R + 1 vech (Rt 1 )+ ::: + p vech (Rt p ) The test statistic employed is again chi-squared distributed.FRM c Dennis PHILIP 2011
- 243. 2 Multivariate Volatility Models 33 Dynamic Conditional Correlation (DCC)model Tse (2000) and Engle and Sheppard (2001) propose tests of constant conditional corre- lation hypothesis. In most applications, we see the hypothe- sis of constant conditional correlation is re- jected. Engle (2002) propose the DCC framework, Ht = Dt Rt Dt where Dt is the matrix of standard devia- tions (as de…ned in the case of CCC), hii;t can be any univariate GARCH model and Rt is the conditional correlation matrix. We then standardize each return by the dy- namic standard deviations to get standard- ized returns. Let h p i 1 ut = "t diag h11t hN N tFRM c Dennis PHILIP 2011
- 244. 2 Multivariate Volatility Models 34 be the vector of standardized residuals of N GARCH models. These variables now have standard deviations of one. We now model the conditional correlations of raw returns ("t ) by modelling conditional covariances of standardized returns (ut ). We de…ne Rt as 1=2 1=2 Rt = diag (Qt ) Qt diag (Qt ) where Qt is an N N symmetric positive de…nite matrix given by q p ! q p X X X X 0 Qt = 1 i j Q+ i ut i ut i + j Qt j i=1 j=1 i=1 j=1 where Pq Pp – i 0; j 0; i=1 i + j=1 j <1 P – Q = T T ut u0t is the standardized 1 t=1 unconditional covariance matrixFRM c Dennis PHILIP 2011
- 245. 2 Multivariate Volatility Models 35 Example: Consider the following model: rt = "t and "t N (0; Ht ) where Ht = var (rt j t 1) We can write Ht as Ht = Dt Rt Dt where 2 p 3 h11;t 6 ... 7 – Dt = 4 5 is di- p hN N;;t agonal matrix of conditional standard deviations. Each hii;t follows a GARCH(1,1) process – u t = Dt 1 " t 2 p 3 1 2 p 3 1 q11 q11 6 ... 7 6 ... 7 Rt =4 5 Qt 4 5 p p qN N t qN N t whereFRM c Dennis PHILIP 2011
- 246. 2 Multivariate Volatility Models 36 h PL PS i P – Qt = 1 l=1 l s=1 s Q+ L l=1 l ut l u0t l + PS s=1 s Qt s P – Qt = Q + L ll=1 ut l u0t l Q + PS s=1 s Qt s Q – For the case of one lags, Qt = Q+ 1 ut 1 u0t 1 Q + 1 Qt 1 Q Remarks: – Qt is a function of: 1. unconditional covariance matrix Q of standardized residuals 2. covariance matrix of standardized residuals u for L lags 3. Qt s i.e s lags of itself 8 > > uncond. component > > > > (of resid) > < addl. persistent component – Q at time t = + > > from past L lags (of resid) > > > > + include past S period addl. > : persistent component (of Q)FRM c Dennis PHILIP 2011
- 247. 2 Multivariate Volatility Models 37 Two-step Estimation of DCC models Under the assumption of normality of inno- vations, Engle and Sheppard show that the DCC can be estimated in 2 steps. Let "t N (0; Ht ) : Let be vector of un- known parameters in matrix Ht : The log-likelihood function is given by 1X T 0l( ) = N log (2 ) + log jHt j + rt Ht 1 rt 2 t=1 1X T 0 = N log (2 ) + log jDt Rt Dt j +rt Dt 1 Rt 1 Dt 1 rt 2 t=1 1X T 0 = N log (2 ) +2 log jDt j + log jRt j +ut Rt 1 ut 2 t=1FRM c Dennis PHILIP 2011
- 248. 2 Multivariate Volatility Models 38where ut = Dt 1 rt is the standardized returns.Adding and substracting u0t ut we get 1X T 0l( ) = (N log (2 ) + 2 log jDt j + rt Dt 1 Dt 1 rt 2 t=1 0 u0t ut + log jRt j + ut Rt 1 ut ) Remarks: – The representation allows us to decom- pose the log-likelihood l ( ) as a sum of the volatility part (lV ( )) contain- ing the parameters in matrix D and the correlation part (lC ( )) containing the parameters in matrix R: – That is, we partition vector of parame- ters into 2 subsets: = f ; g : The log-likelihoods can be written as l ( ) = lV ( ) + lC ( j ) where 1X T 0 lV ( ) = N log (2 ) + log jDt j2 + rt Dt 2 rt 2 t=1FRM c Dennis PHILIP 2011
- 249. 2 Multivariate Volatility Models 39 1X T 0 lC ( j ) = log jRt j + ut Rt 1 ut u0t ut 2 t=1 – The volatility term lV ( ) is apparently the sum of the individual GARCH like- lihoods, which is jointly maximized by seperately maximizing each term. This gives us the parameters : – Then the correlation term lC ( j ) is maximized conditional over the volatil- ity parameters that were estimated be- fore. – In the correlation term, Rt takes the DCC form diag (Qt ) 1=2 Qt diag (Qt ) 1=2 The two-step estimation procedure: 1. Estimate the conditional variances (volatil- ity terms ) using MLE. That is, maximize the likelihood to …nd ^ ^ = arg max flV ( )g 2. Then we compute the standardized returns ut = Dt 1 rt and we estimate the correlationsFRM c Dennis PHILIP 2011
- 250. 2 Multivariate Volatility Models 40 among the returns "t of several assets. Here we maximize the likelihood function of the correlation term ^ n o ^ = arg max lC ^ j The DCC estimation method employs the assumption of Normality in conditional re- turns, which is generally not the case …nan- cial assets. One can use alternative fat-tailed distribu- tions such as student-t, laplace, logistic in order to represent the data.FRM c Dennis PHILIP 2011
- 251. 2 Multivariate Volatility Models 41 Univariate Volatility Models References: Hull (2010) Risk management and …nancial institutions, Chapter 9 Christo¤ersen (2003) Elements of …nancial risk management, Chapter 2 Andersen, T. G. And L. Benzoni (2008), “Re- alized volatility” Chapter in Handbook of , Financial Time Series, Springer Verlag. Additional references: Andersen, T. G., Bollerslev, T., Diebold, F. X. and P. Labys (1999), “(Understand- ing, optimizing, using and forecasting) real- ized volatility and correlation” Manuscript, , Northwester University, Duke University and Pennsylvania University. Published in re- vised form as “Great realizations” in Risk, March 2000, 105-108. Andersen, T., Bollerslev, T., Diebold, F.X. and Ebens, H. (2001), "The distribution of stock return volatility" Journal of Financial Economics, 61, 43-76.FRM c Dennis PHILIP 2011
- 252. 2 Multivariate Volatility Models 42 Andersen, T. G., Bollerslev, T., Diebold, F. X. and P. Labys (2003), “Modelling and forecasting realized volatility” Economet- , rica, 71, 529-626. Andersen, T. G., Bollerslev, T., Diebold, F. X. and J. Wu (2004), “Realized beta: persistence and predictability” Manuscript, , Northwester University, Duke University and Pennsylvania University. Multivariate Volatility Models Readings: Christoggersen (2003) Elements of …nancial risk management, Chapter 3 Engle, R (2002) Dynamic Conditional Cor- relation: A Simple Class of Multivariate Gen- eralized Autoregressive Conditional Heteroskedas- ticity Models Journal of Business & Eco- nomic Statistics. vol. 20, no. 3 Bauwens, L; Laurent, S and Rombouts, J (2006) Multivariate GARCH models: A Sur- vey Journal of Applied Econometrics. vol. 21 pp. 79–109FRM c Dennis PHILIP 2011
- 253. 2 Multivariate Volatility Models 43 Engle, R (2009) Anticipating Correlations: A New Paradigm for Risk Management Prince- ton University PressFRM c Dennis PHILIP 2011
- 254. Financial Risk Management Topic 8a Credit Risk Measures Readings: CN(2001) chapters 25 Hull’s (Risk Management) book chapters 14, 15
- 255. TopicsC-VaR Credit Risk Measures4models Credit Metrics KMV Credit Monitor (and Merton (1974) model) CSFP Credit Risk Plus McKinsey’s Credit Portfolio View, CPV credit rate -> find change VaR
- 256. Credit Metrics Approach
- 257. AAA bond -> AA -> value bond reduce -> find final figure of C-Var Credit Metrics CreditMetrics (J.P. Morgan 1997) measures risk, associated with credit events, for a portfolio of rated exposures (such as corporate bonds) We will cover the following topics: Transition probabilities1 Valuation whats the value chage (V1 -> V2) how much am i 2 losing? Joint migration probabilities bc have 3 4 bonds -> joint Many Obligors: Mapping and MCS not normal
- 258. Transition probabilities Credit ratings measure credit risk. change in rating Practical issue: How do we measure changes in credit risk (i.e. credit ratings)?Problems not much data Lack of historical data to measure proportion of firms that migrate between each credit ratings The distribution of credit returns are highly skewed (see graph on next slide) -> not use standart deviation Therefore ‘change in value’ cannot be explained by std. dev.
- 259. Transition probabilities Frequency Distribution for a 5 year BBB 0.900 BBB bond after 1 year pro prob that BBB jump to b AC-Var can use standartdeviation histogra 0.100 m thickness : 0.075 fal l A BB 0.050 0.025 B AA Default CCC AAA 0.000 50 60 70 80 90 100 110 Revaluation at Risk Horizon Skewed: prob(BBB to Default) is low but change in value is large whereas prob(BBB to A) relatively high but change in value is relatively small this is assumption that can known by lookig at this
- 260. Transition probabilitiesTo calculate Transition Prob: of every rating bonds Suppose we have a sample of 1000 firms and their bond credit ratings from 1980-2000. do it for all others Consider bonds initially rated CCC. default only how rate going to For each year, 1-year marginal mortality rate (MMR1) is ( change historic value of CCC bonds defaulting in 1 year al MMR1 = total value of CCC bonds at beginning of 1 year Transition probability is average MMR: 2000 weigh by number of (tong Wi P (CCC to default ) = ∑ ALL =1) t firms bonds wi MMR1,i 1980 historical for all teh bonds
- 261. Transition probabilitiesWe repeat this exercise for CCC-rated bonds movingto all possible ratings.Also, we repeat this for all other bonds.This gives us the Empirical Transition ProbabilitiesThis model assumes probability of transition in anyyear is independent of probability in earlier years(called first-order Markov process).Survival rates: SRi = 1 − MMRi (use quite much in practice)
- 262. want to do MSC at the question: if BAD case -> whats the max end loss? C-VaR mone y We intend to calculate Credit Value at Risk (C-VaR) Aim is to establish the $-value for 1% tail cut-off, from histogram of all possible values of firm’s bonds at end-year, after credit migrationThis requires: calculating the probability of migration between different credit ratings and the calculation of the expected value of a single bond in different potential credit ratings. deriving the possible values of portfolio of n-bonds at year- end, after all migrations Calculating the probabilities (likelihood) of joint migration of n-bonds,. between credit ratings. under VaR setting
- 263. C-VaR for One Bond First consider calculations for a Single Bond e. g Possible Transitions {A stays at A} or { A to B} or {A to D} for example 3 possible transition Suppose a bank holds “senior unsecured” A-rated bond with 6% coupon and 7-year maturity.want to Credit risk horizon is 1 year ahead (assumed).calculate Transition probabilities, calculated using historical data, are: 1 Transition Matrix (Single Bond) Initial Probability : End-Year Rating (%) Sum Rating A B D A pAA = 92 pAB = 7 pAD = 1 100 %
- 264. C-VaR for One Bond We calculate market value of the bond at the end of 1-year. Consider a set of forward rates want to calculate market value One Year Forward Zero Curves discount 1yr starting from 4yr start from back using different Credit Rating 1y f12 f13 f14 1 … forward A 3.7 4.3 4.9 … A better rate than rate in the (get B 6.0 7.0 8.0 … B market) Notes : f12 = one-year forward rate applicable from the end of year-1 to the end of year-2 etc. Forward rates for end-year A-rated are lower than for B- rated ~ reflects different credit risk If A-rated bond stays A-rated, the value of bond at the end discount of year 1 is: using different $6 $6 $6 $ 1 06 = $6 + + + + ... +valu rates V A, A (1 + f1, 7 ) 6e 2 3 (1 .03 7) (1 .0 43) (1 .04 9) (assume 6% annual coupons paid at end of the year)
- 265. C-VaR for One Bond If A-rated bond migrates to B rating, the value of bond at the end of year 1 is: $6 $6 $6 $106VA, B = $6 + + + + ... + 2 (1.06) (1.07) (1.08) 2 (1 + f1,7 )6 use B rate Suppose calculations yield VA,A = $109 and VA,B = $107 If A-rated bond catastrophically defaults, the value of the bond is the recovery value calculated from the recovery rates given below. VA,D = 51% of face value = $51 D: Recovery Rates After Default (% of par value) default Seniority Class Mean (%) Standard Deviation (%) Senior Secured 53 27 Senior Unsecured 51 25 Senior Subordinated 38 24 Subordinated 33 20 Junior Subordinated 17 11
- 266. C-VaR for One Bond To calculate C-VaR, we have the ingredients: 1 Transition probabilities 2 Bond values associated to different transitionsC-VaR (using standard deviation method): =1.65 σv Mean and Standard Deviation of end-year Value is 3 waighte Vm = ∑ i =1 p iVi = 0.92($109) + 0.07($107) + 0.01($51) = $108.28 d mea n lay cac gia tri tu slide 3 different b/w V & 3 tren σv = ∑ p (V − Vm ) = ∑ pV − (Vm ) = $5.78 mean 2 2 2 i i i i multiply i =1 formular for i =1 second moment - mean to get VaR volatility square(we can calculate σ around the mean Vm or around VA,A)
- 267. C-VaR for One Bond This calculation assumes value of bond in the various states is known with no uncertainty. Usually there is uncertainty in value of the bond and so we associate σ to the three states. not use previous equation (in them std ∑ p (V ) − (V )recommended 3 practice) σv = +σ 2 2 2to use this i i i m i =1 We know the standard deviation associated to recovery rates in the event of default (σdefault = 25%) Another assumption is the distribution of outcomes are normal and so standard deviation is a good measure. Alternatively, one can use a particular percentile value as a measure of C-VaR. historical simulation
- 268. C-VaR for Two Bonds We extend previous calculations to the case of two bonds. Table below summarizes transition probabilities and bond values for an A-rated and B-rated bond: sigma & Probabilities and Bond Value (Initial A-Rated Bond) migration Year End Probability $ValueA & B are Rating % calculatindependent e = 109 A pAA = 92 VAA to make it simpler B pAB = 7 VAB = 107 dont care about D pAD = 1 VAD = 51 Cov Notes : The mean and standard deviation for initial-A rated bond are Vm,A = 108.28, σV,A = 5.78. Probability and Value (Initial B-Rated Bond) historic Year End Rating Probability al $Value A pBA = 3 VBA = 108 use forward B pBB = 90 VBB = 98 rate D pBD = 7 VBC = 51 Notes : The mean and standard deviation for initial-B rated bond are Vm,B = 95.0, σV,B = 12.19.
- 269. C-VaR for Two Bonds There are 9 possible values for the two-bond portfolio at the end of 1-year. Joint year-end values for the two bonds is: Possible Year End Value (2-Bonds) 9 Obligor-1 (initial-A rated) Obligor-2 (initial-B rated) 1. A 2. B 3. D VBA = 108 VBB = 98 VBD = 511. A VAA = 109 217 207 160 all possible value2. B VAB = 107 215 205 1583. D VAD = 51 159 ad 149 102Notes : The values in the ith row and jth column of the central 3x3 matrix are d simply the sum of the values in the appropriate row and column (eg. entry for D,D is 102 = 51 + 51).
- 270. C-VaR for Two BondsWe also need the Joint Likelihoods or probabilities of creditmigration of the two bonds.If we assume independence between credit migrations of thetwo bonds, the calculation is straight forward.Joint likelihood can be calculated from the transition matrix Transition Matrix (percent) Initial End Year Rating Row Sum Rating 1. A 2. B 3. D 1. A 92 7 1 100 2. B 3 90 7 100 3. D 0 0 100 100 % Note: If you start in default you have zero probability of any rating change and 100% probability of staying in default.
- 271. C-VaR for Two BondsJoint Likelihoods are calculated under independence.Example: Prob(‘A remains A’ and ‘B remains B’) = Prob(‘Aremains A’) x Prob(‘B remains B’) = 0.92 x 0.90 = 0.828assume similar to Joint Migration Probabilities : πij(percent)undependent value Obligor-1 (initial-A rated) Obligor-2 (initial-B rated) 1. A 2. B 3. D p21 = pAB = p22 = pBB = p23 = pBD = 3 90 7 1. A p11 = pAA = 2.76 82.8 6.44 92 2. B p12 = pAB = 0.21 6.3 0.49 7 3. D p13 = pAD = 0.03 0.9 0.07 1
- 272. C-VaR for Two BondsC-VaR (using standard deviation method):The formulae for calculating mean and standard deviation isthe same as in the case of one bond. bond 1 and bond 2 3 3 Vm , p = ∑∑π i =1 j =1 ij Vi , j = $203.29 Vi , j − (Vm , p ) = $13.49 3 3 ∑ ∑π 2 σ v, p = ij 2 i =1 j =1 second moment
- 273. similar to historical simulation C-VaR for Two BondsC-VaR (using percentile method):Order VA+B in from lowest to highest sor tThen add up their joint likelihoods until these reach the 1%value. 1% loss VA+B = {$102, $149, $158, $159, …, $217} sort trang 16 πi,j = {0.07, 0.9, 0.49, 0.43, …, 2.76} cumulate -> cut off at the tail$149 is the ‘unexpected loss’, at 1% levelIt is customary to measure the VaR relative to the ‘meanvalue’ rather than relative to the ’initial value’ of thebond/loan portfolio difference b/w expected and unexpectedHence: C-VaR = $54.29 (= mean value – 1% extremevalue = $203.29 - $149)
- 274. C-VaR - summary • C-VaR of a portfolio of corporate bonds depends on the joint migration likelihoods (probabilities), the value of the obligor (bond) in default (based on the seniority class of the bond), and the value of the bond in any new credit rating histor yhistorical data-> Credit Rating Seniority Credit Spread Migration Recovery Rate in Value of Bond Likelihoods Default in new Rating Standard Deviation or Percentile Level for C-VaR
- 275. Measuring Joint Credit Migration
- 276. Measuring Joint Credit MigrationThree ways:1. Historic credit migration data (dont have the data -> limit (problem) - This becomes difficult when one has to measure correlations between many rating possibilities - Lack of data2. Bond spread data difference bw yield y(AAA) - y(BB) = spread - Movements in credit yield spread between two firms reflect changes in credit quality - Use bond pricing model to extract probabilities of credit rating changes from observed changes in credit spreads3. Asset value approach (concentrate on this) looking at asset (stock) -> use it to calculate credit (bond) - Movements in firm’s stock price reflect changes in credit quality and hence signal possible change in credit ratings(we discuss point 3 now…)
- 277. Asset value approach – one stockOne stock example: find the threshold (gioi han) de xac dinh luc nao chuyen Consider initial BB-rated firm. Lower the stock returns, lower it’s credit ratings but there threshol d are range of return values where ratings remain the same. Suppose we assume stock returns are normal and σ is known. nguoc de tinh gia We can ‘Invert’ the normal distribution to obtain cut-off find where the bond tri going to fall -> get points for stock returns (R) corresponding to the known the average probability of default (e.g. 1.06% below) Then from normal distribution bell: Pr(default) = Pr(R<ZDef) = Φ(ZDef/σ) = 1.06%Hence: ZDef = Φ-1(1.06%) σ = -2.30σ(In Excel: =NORMSINV(0.0106) = -2.30)
- 278. Asset value approach – one stockProbability BB BBB want to calculate Probability of default B this off (cut =1.06 level) A CCC AA Def AAA transition pointCut-off level: -2.30 -2.04 -1.23 1.37 2.39 2.93 3.43 <- Calc. ZTransition probability: 1.06 1.00 8.84 80.53 7.73 0.67 0.14 0.03 <- known know n use differnt forward rate Prob ( BB to CCC) = 1.0 We assume (for simplicity) that the mean return for the stock of an initial BB-rated firm is zero
- 279. Asset value approach – one stockIf the stock return falls by more than -2.30σ (%) then weassume transition of the firm from BB to D (and we revaluethe bond using recovery rates in default)Now lets consider transition from BB to CCC rating.Suppose 1% is the observed transition probability. normalPr(CCC) = Pr(ZDef<R<Zccc) =distribution /σ) - Φ(ZDef/σ) = 1.00 Φ(ZCCC cut offHence: Φ(ZCCC/σ) = 1.0 + Φ(ZDef/σ) = 1.0+1.06 = 2.06 leveland ZCCC = Φ-1(2.06/ σ) = -2.04σIf stock return fall by more than -2.04σ (%) then we assumetransition of the firm from BB to CCC (and we revalue thebond using CCC forward rates)
- 280. Asset value approach – one stock In a similar fashion, we obtain cut-off points for all credit rating changes for the initial BB-rated firm, summarized in the table:calculate all cut offlevels
- 281. Asset value approach – two stocksConsider now two stocks: ‘A-rated’ firm along with the‘BB-rated’ firm.Denote R’ as it’s asset returns, σ’ as it’s standard deviation,and Z’Def, Z’CCC, etc as it’s asset return cut-offs/thresholds.Table summarizes the calculations:
- 282. Asset value approach – two stocks Until now we have calculated individual obligor credit rating changes. To calculate two credit rating changes jointly, we assume two asset returns are correlated and bivariate normal with σ2 ρσσ use Σ= cholesky ρσσ σ 2 Suppose we wish to calculate P(BBB to BB and A to BBB). Let’s call this Y%. joint P(BBB to BB and A to BBB) = Pr(ZB <R<ZBB, Z’BB prob Z BB ZBBB <R’<Z’BBB) = ∫ZB ∫ Z BB f ( R, R , Σ) not known -> use dR dR’ = Y% simulate We use the same procedure to calculate all joint rating migration πi,j for the two obligors.
- 283. EXAM THIS Asset value approach – using MCS As number of obligors and number of ratings increase, the dimensions of joint migration likelihood matrix π explode. Instead, we estimate the distribution of portfolio credit values using Monte Carlo Simulation (MCS). MCS We have already found the ‘cut-off points’ Z for each obligor Now simulate the joint stock returns (with a known correlation) and associate these outcomes with a JOINT credit rating Revalue the n-bonds at these new ratings ~ this is the 1st MCS outcome, Vp(1) Repeat above m-times and plot a histogram of Vp(i) Order the Vp(i) from lowest to highest and read off the $-value of the portfolio at the 1% left tail cut-off pointAssumes stock return correlations correctly reflect the economic conditions, that influence credit migration correlations
- 284. KMV’s Credit Monitor
- 285. KMV’s Credit Monitor As in Credit Metrics model, KMV Credit Monitor model also links stock prices to default probabilities. It uses Merton (1974) model to make this link. Unlike Credit Metrics which allows for upgrades and downgrades, KMV’s model is a default only model.Next, we will learn about Merton (1974) model and KMV(see Topic 8b slides)
- 286. CSFP Credit Risk Plus
- 287. CSFP Credit Risk PlusUses Poisson to give default probabilities and mean default rate µcan vary with the economic cycle.Assume bank has 100 loans outstanding and estimated default rate= 3% p.a. implying µ = 3 defaults per year (from the 100 firms).Probability of n-defaults e−µ µ n p (n, defaults ) = n!p(0) = = 0.049, p(1) = 0.149, p(2) = 0.224…p(8) = 0.008 etc ~humped shaped probability distribution (see figure on next slide).Cumulative probabilities:p(0) = 0.049, p(0-1) = 0.198, p(0-2) = 0.422, … p(0-8) = 0.996“p(0-8)” indicates the probability of between zero and eight defaultsTake 8 defaults as approximation to the 99th percentile (1% cut off)
- 288. CSFP Credit Risk Plus Probability Distribution of Losses Probability Expected Loss0.224 Unexpected Loss0.049 99th percentile Economic Capital Loss in $’s $30,000 $80,000
- 289. CSFP Credit Risk PlusAverage loss given default LGD = $10,000 then:Expected loss = (3 defaults) x $10,000 = $30,000Unexpected loss (99th percentile) = 8 defaults x $10,000 = $80,000(Note: Line in text p.716 is incorrect)Capital Requirement = Unexpected loss - Expected Loss = 80,000 - 30,000 = $50,000TFOLIO OF LOANSSuppose the bank also has another 100 loans in a ‘bucket’ with anaverage LGD = $20,000 and with µ = 10% p.a.Repeat the above exercise for this $20,000 ‘bucket’ of loans and deriveits (Poisson) probability distribution.Then ‘add’ the probability distributions of the two buckets (i.e. $10,000and $20,000) to get the probability distribution for the portfolio of 200loans (we ignore correlations across defaults here)
- 290. McKinsey’s Credit Portfolio View, CPV
- 291. McKinsey’s Credit Portfolio View, CPVMark-to-market model with direct link to macro variablesExplicitly model the link between the transition probability (e.g. p(C toD)) and an index of macroeconomic activity, y. pit = f(yt) where i = “C to D” etc.y is assumed to depend on a set of macroeconomic variables Xit (e.g.GDP, unemployment etc.) yt = g (Xit, vt) i = 1, 2, … nXit depend on own past values plus other random errors εit.(sayVAR(1))It follows that: pit = k (Xi,t-1, vt, εit) Each transition probability depends on past values of the macro-variables Xit and the error terms vt, εit. Clearly the pit are correlated.
- 292. CPVMonte Carlo simulation to adjust the empirical (or average) transitionprobabilities estimated from a sample of firms (e.g. as inCreditMetrics).Consider one Monte Carlo ‘draw’ of the error terms vt, εit (whichembody the correlations found in the estimated equations for yt andXit above).This may give rise to a simulated probability pis = 0.25, whereas thehistoric (unconditional) transition probability might be pih = 0.20 . Thisimplies a ratio of ri = pis / pih = 1.25Repeat the above for all initial credit rating states (i.e. i = AAA, AA, …etc.) and obtain a set of r’s.
- 293. CPVThen take the (CreditMetrics type) historic 8 x 8 transition matrixTt and multiply these historic probabilities by the appropriate ri sothat we obtain a new ‘simulated’ transition probability matrix, T.T , now embodies the impact of the macro variables and hencethe correlations between credit migrationsThen revalue our portfolio of bonds using new simulatedprobabilities which reflect one possible state of the economy.This would complete the first Monte Carlo ‘draw’ and give us onenew value for the bond portfolio.Repeating this a large number of times (e.g. 10,000), providesthe whole distribution of gains and losses on the portfolio, fromwhich we can ‘read off’ the portfolio value at the 1st percentile.
- 294. SUMMARY A COMPARISON OF CREDIT MODELS Characteristics J.P.Morgan KMV CSFP McKinsey CreditMetrics Credit Monitor Credit Risk Plus Credit Portfolio ViewMark-to-Market MTM MTM or DM DM MTM or DM(MTM) or DefaultMode (DM)Source of Risk Multivariate normal Multivariate normal Stochastic default Macroeconomic stock returns stock returns rate (Poisson) VariablesCorrelations Stock prices Option prices Correlation between Correlation between Transition Stock price mean default rates macro factors probabilities volatilitySolution Method Analytic or MCS Analytic Analytic MCS
- 295. Topic # 8b: Merton model and KMV Financial Risk Management 2010-11 March 3, 2011FRM c Dennis PHILIP 2011
- 296. 2Merton (1974) model and KMV Assume …rm’ balance sheet looks like this: s Consider a …rm with risky assets A; that follow a GBM. Suppose …rm is …nanced by a simple capital structure, namely one debt obligation (D) and one type of equity (E) A0 = E0 + D0 where (Et )t 0 is a GBM that describes evo- lution of equity of the …rm and (Dt )t 0 is some process that describes market value of debt obligation of the …rm.FRM c Dennis PHILIP 2011
- 297. 3 Suppose debt holders pay capital D0 at time t = 0 and get F (includes principal and in- terest) at time t = T: For debt holders (lending banks), credit risk increase i¤ P [AT < F ] > 0 When default probability > 0, we can con- clude that D0 < F e rt where r is the risk free rate. That is, debt holders would need credit risk premium. In other words, D0 is smaller, greater the riskiness of the …rm. To hedge this credit risk, debt holders go Long a Put contract on A; with strike F and maturity T: This guarantees credit protection against de- fault of payment:FRM c Dennis PHILIP 2011
- 298. 4 Debt holder’ portfolio consist on a loan and s Put contract. At t = 0; D0 + P0 (A0 ; A ; F; T; r) At t = T; portfolio value is F Using non-arbitrage principal, at t = 0 rt D0 + P0 (A0 ; A ; F; T; r) = Fe So the present value of D0 is rt D0 = F e P0 (A0 ; A ; F; T; r) (1)FRM c Dennis PHILIP 2011
- 299. 5 The …gure below summarizes this. Next, we can think about the value of equity E0 in terms of a Call option. Equity holders of the …rm have the right to liquidate the …rm (i.e. paying o¤ the debt and taking over remaining assets) Suppose liquidation happens at maturity date t = T: Two Scenarios:FRM c Dennis PHILIP 2011
- 300. 6 1. AT < F : In this case, there is default and there is not enough to pay the debt holders. Moreover, equity holders have a payo¤ of zero. 2. AT F : In this case, there is a net pro…t for equity holders after paying o¤ the debt (AT F ). So the total payo¤ to equity holders is max(AT F; 0) which is the payo¤ of an European call op- tion on A with strike F and maturity T: The present value of equity is therefore E0 = C0 (A0 ; A ; F; T; r) (2) Combining equation 1 and 2, we can obtain A0 = E0 + D0 rt = C0 (A0 ; A ; F; T; r) + F e P0 (A0 ; A ; F; T; r) rt A0 + P0 (A0 ; A ; F; T; r) = C0 (A0 ; A ; F; T; r) + F e which is nothing but the put-call parity.FRM c Dennis PHILIP 2011
- 301. 7 The above also shows that equity and debt holders have contrary risk preferences. By choosing risky investment in some asset A, with higher volatility A ; equity holders will increase the option premium for both call and put. This is good for equity holders as they are long a call and this is bad for debt holders as they have a short position. Equity holders have limited downside risk but unlimited upside potential. The main application of this Merton’ op-s tion pricing framework is to estimate default probabilities. This is implemented in the KMV credit monitor system. KMV applies Black-Scholes formula in re- verse. Conventionally, we observe price of the un- derlying, strike price, etc and we calculateFRM c Dennis PHILIP 2011
- 302. 8 the value of the derivative. In this applca- tion, we begin with value of the derivative (value of equity E0 as a call option) and given strike F ; and calculate the unobserved value of A0 : Another complication: we need the value of A but since we do not observe value of A, we have to infer A from volatility of returns on equity E : The Black-Scholes gives the value of equity today as rT E0 = A0 N (d1 ) Fe N (d2 ) (3) where ln(A0 =F ) + (r + A =2)T d1 = p A T and p d2 = d1 A T The value of debt today is D0 = A0 E0FRM c Dennis PHILIP 2011
- 303. 9 To calculate above we need A0 and A (un- knowns). Using Ito lemma, we yield the relationship between the volatilities at t = 0 @E E E0 = A A0 (4) @A Here @E=@A is the delta of the equity. So @E=@A = N (d1 ) So we have two equations (3 and 4) and two unknowns (A0 and A ). We can use Excel Solver to obtain the solu- tion to both these equations. Now we can calculate the default probabili- ties (called Expected Default frequency, EDF) as pi = P [AT < F ] This can be shown to be pi = P [Zi < DD]FRM c Dennis PHILIP 2011
- 304. 10 where Zi N (0; 1) and DD is the ‘ distance to default’ (i.e. number of standard devia- tions the …rm’ assets are away from default) s ln(A0 =F ) + ( A A =2)T DD = p A T where A is the mean return/growth rate of the assets.FRM c Dennis PHILIP 2011
- 305. 11FRM c Dennis PHILIP 2011

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