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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

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