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### Lecture 2

1. 1. RISK MANAGEMENTLECTURE 2 a. Fundamentals of financial instruments b. Futures and forwards c. Options 1
2. 2. Part 1FUNDAMENTALS OFFINANCIALINSTRUMENTS a. Present and future value b. Maturity c. Duration d. Convexity 2
3. 3. 1. Present, future value PV Present Value FV Future Value FV PV  T,i P i interest rate (discount rate) (1  i )T T number of periods FV  PV (1  i )T T=f(Years,m(per per year)Part 2. Fundamentals of statistics 𝐵𝑒𝑐𝑎𝑢𝑠𝑒 𝑇 = 𝑚 ∗ 𝑌 Compound i mY period FV  PV (1  ) m 𝐷𝑒𝑓𝑖𝑛𝑒 𝑥 = 𝑚 𝑖 i xiY FV  PV (1  ) x 1 x iY Aggregation FV  PV [lim x (1  ) ] x (Present) Value of any investmentLecture 2 is the sum total of all future FV  PV [e]iY financial benefits T CashFlow P 1 (1  i )T 3
4. 4. 2. Price sensitivity There are four measures of bond price sensitivity Maturity Macaulay Duration (effective maturity) Modified Duration Convexity. Maturity The time left to maturity on a bondPart 2. Fundamentals of statistics The longer the time to maturity, the more sensitive a particular bond is to changes in the rate of return. FV PV  i (1  ) mY m • Bond A matures in 10 years and has a required rate of return of 10%.Lecture 2 10 year steaper 5 year • Bond B has a maturity of 5 Steaper years and also has a required rate of return of 10% 4
5. 5. 2. Price sensitivity Duration (Macaulay) But duration is a better measure of term than maturity Relationship price - maturity is affected when considered non-zero coupon bonds.: many of the cash flows occur before the actual maturity of the bond and the relative timing of these cash flows will affect the pricing ofPart 2. Fundamentals of statistics the bond. T Dm   t  wt  Period * Payments t 1 PV(CFt ) CFt /(1  y )t wt   PV( Bond ) P qLecture 2 w t 1 t 1 5
6. 6. 2. Price sensitivity Duration (Macaulay) We find the weighted values This bond is 6 YPart 2. Fundamentals of statistics to maturity The semi-annual duration for this bond is 10.014 semianual. Annual: 5Lecture 2 We find the value of the bond by discounting each Duration is a of the flows function of cash flows Summation of cash flows 6
7. 7. 2. Price sensitivity Modified Duration More direct measure of the relationship between changes in interest rates and changes in bond prices Modified Duration, D, is defined as dP: change of price 1 𝜕𝑃 dY: change of rate of ret. 𝐷=− ∗ 𝑃 𝜕𝑌 First derivative of thePart 2. Fundamentals of statistics bond price WR discount factor 𝑇 𝜕𝑃 1 𝐶𝑡 =− ∗ 𝑡 𝜕𝑌 1+ 𝑦 (1 + 𝑦) 𝑡 1 𝑇 𝐶𝑡 1 (1 + 𝑦) 𝑡 𝐷= ∗ 𝑡 1+ 𝑦 𝑃 1 𝑇 𝐶𝑡 𝑇 𝐶𝑡 (1 + 𝑦) 𝑡 1 (1 + 𝑦) 𝑡 𝐷= 𝑡 𝑀𝑜𝑑. 𝐷 = ∗ 𝑡Lecture 2 𝑃 1+ 𝑦 𝑃 1 1 7
8. 8. 2. Price sensitivity Modified Duration So, at the end we have • Macaulay Duration is an average or effective maturity.Part 2. Fundamentals of statistics • Modified Duration really measures how small changes in the yield to maturity affect the price of the bond. From the definition of Modified Duration we can write % Change in bond price = - Mod. Duration times the change in yield to maturity 𝛻𝑃 = −𝐷 ∗ 𝛻𝑦 𝑃 How much in % will the price change when the yield changes?Lecture 2 8
9. 9. 2. Price sensitivity Convexity Second derivative of price with respect to yield to maturity • Measures how much a bond’s price-yield curve deviates from a straight line Notice the convex shape of price-yield relationshipPart 2. Fundamentals of statistics Bond 1 Price Bond 2 YieldLecture 2 Bond 1 is more convex than Bond 2 Price falls at a slower rate as yield increases 9
10. 10. 2. Price sensitivity Convexity Second derivative of the 1 P2 bond price WR discount Convexity  P 2 y factorPart 2. Fundamentals of statistics 2P 1 N  Ct    2 y (1  y ) 2   (1  y)t t 1  t (t  1)   Ct  N   1 (1  y ) t This seems the w 2  Convex  t (t  1)  (1  y ) t 1  P     Lecture 2 10
11. 11. Lecture 2 Part 2. Fundamentals of statistics Convexity 2. Price sensitivity Period11
12. 12. 2. Price sensitivity Convexity Recall approximation using only duration: P   Dm  y * P The predicted percentage price change accounting for convexity is:Part 2. Fundamentals of statistics P  1     Dm  y    Convexity  (y ) 2  * P 2  Adding the convexity adjustment corrects for the fact that Modified Duration understates the true bond price. This is a really good approximation btw. change of yield and its effect on priceLecture 2 12
13. 13. Part 2FUTURES ANDFORWARDS a. Basics b. The futures contract c. Determinants of prices d. Future prices V expected spot prices 13
14. 14. 1. Basics Definition Financial contract obligating the buyer to purchase an asset (or the seller to sell an asset) at a predetermined future date and price. • Obligation !! • Commitment today to make a transaction in the future Practical example Farmer Mill • Sell a product • Buy a product. Only that product • No diversification. (single product) • It is worried about the future price Forward Contract Agreed price no real transaction (money) • Deliver a product • Deliver the moneyPart 2. Futures • Get the money for sure • Get the product for sure It is a zero sum gameLecture 2 Each long position has a short position Futures do not affect the market price Can be seen as a Risk Management Technique 14
15. 15. 1. Basics The formalization of the forward contract is the futures market Futures Forward • Standardization • Contracts more liquid No money changes until delivery • Margin to market. Daily settling up of gains and losses • Margin account Long position on a future Short position on a future Buy a contract: Commitment to Sell a contract: Commitment to purchase a product in the future deliver a product in the future Price of the future Price of the future Profit Profit Value of the forwardPart 2. FuturesLecture 2 Profits can be <0 Profit=Spot-F0 Loss=F0- Spot Profit=F0-Spot Price Price 15
16. 16. 1. Basics Existing contracts Mechanisms of Futures • Agricultural Commodities • Organized exchanges • Metals and Minerals • Cash delivery instead product • Foreign currencies • Standardization: specific contracts • Financial: index and single and maturities stocks • Clearing House: trading partner for each trade (credibility) • Marking to market: put positions at market price • Margins Money MoneyPart 2. Futures Long Short Clearing House position positionLecture 2 Commodity Commodity 16
17. 17. 2.The futures contract Mechanisms of Futures Marking to market process • At the beginning of the trade, each trader establishes a margin account • Can be cash or near cash assets • Both parties must give the margin • 5% - 15% total value of contract • Instead of waiting until the maturity date, the clearing house requires traders to realize gains and losses in a daily basis • The daily settling is called Marking to Market (MtM) • When margin account falls below a maintenance margin, the trader receives a margin call to give more money or close the operation Convergence property (avoid arbitrage) Futures price on delivery date and Silver is traded Today MtMPart 2. Futures spot price must converge at maturity 14,10 per 5000 ounces There are two sources of a commodity 1 14,20 0,10 500 2 14,25 0,05 250 : futures and spot, and both must beLecture 2 3 14,18 -0,07 -350 the same 4 14,18 0,00 0 Difference of values times 5000 ounces 5 14,21 0,03 150 0,11 550 550 17
18. 18. 3. Determination of future prices Spot-Futures Parity Theorem Futures can be used to hedge changes in the value A perfect hedged portfolio should provide the risk free rate to avoid arbitrage • SPX500 at 1500 • An investor has a position in an SPX500 indexed portfolio Long • Future price of SPX500 is 1550 • The investor wants to hedge the market risk HOW? Short sell a future contract of SPX500 @ 1550 Final value of P 1510 1530 1550 1570 1590 Profit=F0-Spot Payoff of short 40 20 0 -20 -40 Convergence!! Dividend 25 25 25 25 25 TOTAL 1575 1575 1575 1575 1575Part 2. Futures • Any increase in the value of the indexed portfolio is offset by an equal decrease in the payoff of the position. 𝐹0 + 𝐷 − 𝑆0Lecture 2 𝑟𝑓 = • The final value is independent of the market price 𝑆0 • The rf (risk free rate) is 5% (1575-1500)/1500 𝐹0 = 𝑆0 (1 + 𝑟 𝑓 ) − 𝐷 𝑇 𝐹0 = 𝑆0 (1 + 𝑟 𝑓 − 𝑑)18 Any deviation from parity would give rise to arbitrage
19. 19. 3. Determination of future prices Spot-Futures Parity Theorem Example • rt becomes 4% • F0=1535 • But the actual future price is 1550 What could be the strategy? Short overpriced futures Will get Buy the under-priced stock using money at 4% the future price and dividend Initial Cash Cash Flow in Flow 1 year Borrow 1500 and repay with interest in 1 year 1500 -1560 Buy stock -1500 St+25Part 2. Futures Enter short future position 0 1550-St 0 15Lecture 2 • Net initial investment is 0 • Cash flow in 1 year is 15 no matter the price of the stock (riskless) • When misprice, the market will equilibrate prices 19
20. 20. 4. Future prices VS Expected spot prices How well future price forecast the REAL spot price? Three basic theories Expectations Hypothesis • Futures price equals the expected value of the future spot price of asset 𝐹0 = 𝐸(𝑃 𝑇 ) • Expected profit = 0 • Prices of goods at all future dates are known • Resembles a market with no uncertainties • Ignores risk premiums Normal Backwardation • Hedgers (Farmers) must give an expected profit to speculators to attract their investments 𝐹 < 𝐸(𝑃 ) 0 𝑇 • Expected profit: 𝐸 𝑃 𝑇 − 𝐹0Part 2. Futures Contango • Purchasers of commodities need the productLecture 2 𝐹0 > 𝐸(𝑃 𝑇 ) 𝐹0 − 𝐸 𝑃 𝑇 20
21. 21. 4. Future prices VS Expected spot prices How well future price forecast the REAL spot price? Basic theories F0 Contango 𝐹0 > 𝐸(𝑃 𝑇 ) 𝐹0 = 𝐸(𝑃 𝑇 ) Expectations Hypothesis 𝐸(𝑃 𝑇 )Part 2. Futures Backwardation Today’s price should be cheaper to 𝐹0 < 𝐸(𝑃 𝑇 )Lecture 2 attract buyers 21
22. 22. Part 3OPTIONS a. Definition b. Values at expiration c. Option strategies d. Put-Call parity relation e. Option valuation: f. Exotic options 22
23. 23. 1.The option contract Definition Are financial instruments that give to the holder the • RIGHT (not an obligation) to buy or sell an asset • at an specific time (depending on the option) • at some specific price • can be purchased or sold Call option Gives its holder the right to PURCHASE an asset for a specific price (strike or exercise price) Market Or on before some specific date Strike • When it is not profitable, it expires The value of the option is (St-K) Stock price – Strike pricePart 3. Options Market C December call option @ 30Lecture 2 Holder can buy C at a price of 30 if market is > 30 Holder does not have the obligation to exercise the call, So he/she will exercise it only when it is profitable 23
24. 24. 1.The option contract Definition Are financial instruments that give to the holder the • RIGHT (not an obligation) to buy or sell an asset • at an specific time • at some specific price • can be purchased or sold Put option Gives its holder the right to SELL an asset for a specific price (strike or exercise price) Or on before some specific date Market Strike When it is not profitable, it expires The value of the option is (K-St) Strike price - Stock pricePart 3. Options Market 3M January put option @ 85 Holder can sell 3M at a price of 85 if market is < 85Lecture 2 Holder does not have the obligation to exercise the put, So he/she will exercise it only when it is profitable 24
25. 25. 1.The option contract Additional language Option “in the money”: it is profitable Option “out of the money”: it is unprofitable Option “at the money”: S=K American option: the right to buy(sell) at any time before expiration European option: the right to buy(sell) at expiration American option are more expensive than European options Option on assets other than stocks are traded. • Index options: • Future options • Foreign currency options: buy/sell a quantity of foreign currency for a specific amount of local currency. (this is different that a currency future contract)Part 3. Options • Interest rate options The premium: (cost of the option)Lecture 2 • Upfront payment (unlike forwards) • Purchase price of the option. Represents the compensation to have the right to exercise the option • This cost has to be included 25
26. 26. 2.Values of options at expiration Call option Right to buy an asset St-K St>K Payoff to call holder 0 otherwise Profit Price Premium K (Strike Price) Limited riskPart 3. OptionsLecture 2 Risk Management technique 26
27. 27. 2.Values of options at expiration Put option Right to sell an asset 0 St>K Payoff to put holder K-St St<K Profit Price Premium K (Strike Price)Part 3. Options Limited riskLecture 2 27
28. 28. 2.Values of options at expiration Call option (writer) Exposes the writer to losses when market falls The writer will receive a call and will be -(St-K) St>K obligated to deliver a stock worth St Payoff to call writer 0 otherwise Profit Income Price K (Strike Price)Part 3. Options The income is Unlimited risk given by the premium, butLecture 2 there is unlimited risk 28
29. 29. 2.Values of options at expiration Bullish strategy Bearish strategy Bearish strategy Bullish strategyPart 3. OptionsLecture 2 29
30. 30. 2.Values of options at expiration What is the difference among some portfolios? We have \$10.000 to spend Portfolio A: Only stocks Portfolio B: Only calls. K=100 Portfolio C: T+ 10% callsPart 3. Options 1. While purchasing shares I can afford 100 units, by using calls I can haveLecture 2 access to 1000 shares 2. Option offers leverage!! 3. Options’ return is 0% because I spent all the money in the premium 4. Consider combinations of financial assets 30
31. 31. Very conservative investment strategy 3. Option strategies St ≤ K St > K I have an obligation to give stocks, but I am long Covered call St St Payoff of stock 0 -(St-K) Stock St K XYZ is trading at \$17. Sell someone the right to purchase your K XYZ stock for \$17.50 for a premium of \$2. Write a Call (sell someone the right to buy)Part 3. OptionsLecture 2 Covered call 31
32. 32. 3. Option strategies Covered call Payoff of stock Stock K If you are long, (very long) and Write a you have a target price to take Call (sell profits, you might want to write someone a call. the right to buy) Pension funds are always long,Part 3. Options but using a covered call, they can hedge some positions.Lecture 2 Covered call Also you can get some cash from premiums 32
33. 33. 3. Option strategies Product of the combination of options • Only stock seems risky Protective Put Payoff of stock St ≤ K St > K St St + Stock K-St 0 K K St • You bought 500 shares of stock XYZ Long at \$50, and it rises to \$70. But, price put could drops to \$65…\$60. Hmm. ATM • When the price rises to \$70, I can buy 5 puts (each put contract represents 100Part 3. Options shares of stock) at \$2 per contract with \$65 strike price. Commissions are \$8.20 New price : \$50Lecture 2 Protecti ve put Exercise the put and sell at \$65 when price is \$50. Fees 33
34. 34. 3. Option strategies Product of the combination of options Protective Put Payoff of stock Stock K Long put ATM This is clearly a protectivePart 3. Options portfolio strategyLecture 2 Protecti ve put Fees 34
35. 35. • Call and put with same exercise 3. Option strategies price and same expiration day • Ideal when prices will move a lot in price, no matter the direction Straddle (Long) • Are bets on volatility Payoff of stock • In no volatility, both premiums are lost St < K St ≥ K Call 0 St-K 0 K-St 0 -C K K-St St-K Put XYZ stock is trading at \$40. Long straddle: buy put for \$200 and a call for \$200. Cost, 400 0 If XYZ = \$50 -P Put will expirePart 3. Options Call in the money. \$1000. Profit \$600.Lecture 2 Covered If XYX= \$40, both expire worthless call and there is a loss of \$400 35
36. 36. 3. Option strategies Straddle (Long) Payoff of stock Call 0 -C K The point here is the cost of the Put TWO premiums. This could be expensive 0 -PPart 3. Options Investors who sell straddles are betting on stability.Lecture 2 Covered Nick Leeson is famous for that call 36
37. 37. • Same date 3. Option strategies • Same stock Spreads: Product of the combination of options, puts or calls Bullish spread • Different exercise prices Payoff of stock • Three outcomes • Holders profit when price increases St ≤ K1 K1 < St ≤ K2 St ≤ K2 Long call 0 St-K1 St-K1 K1 0 0 -(St-K2) 0 St-K1 K2-K1 Short XYZ at \$42 could rally. call Buying call for \$300 at \$40 Writing call for \$100. at \$45 Investment \$200. K2 XYZ rise and closes at \$46 on expiration date.Part 3. Options Long call at 40: + \$600 Short call at 45: - \$100 Net: \$500.Lecture 2 Covered Net profit : \$500-200=\$300 call XYZ declined to \$38, trader lose his entire investment of \$200, which is also his maximum possible loss. 37
38. 38. 4. Put-call parity relation Definition: • Establish a relationship between the prices of an European put and call options of the same class • Combinations of options can create positions that are the same as holding the stock itself First portfolio: • Call option • Risk free investment with face value = exercise price o a call • Same expiration datePart 3. OptionsLecture 2 St ≤ K St > K St ≤ K St > K St ≤ K St > K + K K K St 0 St-K 38
39. 39. 4. Put-call parity relation Second portfolio: • Put option • Long stock • Must produce the same scenario Therefore, the call+bond must cost the same than the put + stock to establishPart 3. Options Initial payoff must be the price of the assetLecture 2 Put – call parity theorem c + PV(x) = p + s 𝑐 + 𝐾𝑒 −𝑟𝑡 = 𝑝 + 𝑆0 39
40. 40. 5. Option valuation Determinants (in call option case) • Stock price: direct relation • Exercise price: inverse relation • Volatility: direct relation • K=30 • S1 has a volatility btw \$10 and 50. EV: 6 • S2 has a volatility btw \$20 and 40 EV:3 S1 10 20 30 40 50 Payoff 0 0 0 10 20 S2 20 25 30 35 40 Payoff 0 0 5 10 Each price has prob= 0.2Part 3. Options • Time to expiration: direct relation • Interest rate: direct relation because the more r, the less PV of KLecture 2 • Dividend rate of stock: inverse relation. When stocks pay out their dividends, the share price adjusts downward to compensate for the pay-out 40
41. 41. 5. Option valuation • An additional relation: time VS price • If S<K, the option is worthless? • At expiration date: YES • Before expiration date, always there is a chance that the option becomes profitable Payoff of stock Time Value Most of an option’s time value typically is a type of volatility value Long call High volatility near K1 K1 Intrinsic ValuePart 3. Options Payoff by immediate exercise Time ValueLecture 2 The sensitivity of the option value to the amount of time to expiry is known as the options theta. 41
42. 42. 5. Option valuation Restrictions on the value of options • European option < American option • Price cannot be negative • Call price low bound 𝑐 + 𝐾𝑒 −𝑟𝑡 = 𝑝 + 𝑆0 Put option 𝑐 − 𝑝 = 𝑆0 − +𝐾𝑒 −𝑟𝑡 𝑐 ≥ 𝑆0 − 𝐾𝑒 −𝑟𝑡 𝑝 ≥ 𝐾𝑒 −𝑟𝑡 - 𝑆0 • Call upper bound, is the stock price 𝑐 ≤ 𝑆0 𝑐 ≤ 𝐾0Part 3. OptionsLecture 2 42
43. 43. 5. Option valuation Binomial option pricing model • Proposed by Cox, Ross and Rubinstein. “Option Pricing: A Simplified Approach”, Journal of Financial Economics, 1979, 7, 229-263. • Replication principle: Two portfolios producing the exact same future payoffs must have the same value. • Otherwise, there will be opportunities for riskless arbitrage. • Use this model to price European call options.Part 3. Options Main idea: construct a synthetic portfolio that replicate option’s payoffs usingLecture 2 rF and stocks . These portfolios SHOULD have the same return to avoid arbitrage Find the value of that portfolio. That must be the price of the call 43
44. 44. 5. Option valuation Binomial option pricing model S0=110 C1= 10 S0=100 K=100 S0= 90 C2= 0 • If the investors borrow money, the interest rate=6% for one year. • What is the price of the European call option? We can replicate the payment of the call by a suitable portfolio : f ( underlying asset + risk free ) Stock BondPart 3. Options C1 Payoff 10  N 110  B  (1  0.06) Two eq = Two unk C2 Payoff 0  N  90  B  (1  0.06)Lecture 2 N  0.5 B  42.4528 44
45. 45. 5. Option valuation Binomial option pricing model S=100, it will move to either 110 or 90 in one year X=100, r=6% Form a synthetic portfolio: short position in a bond (sell a bond to borrow money) at \$42.4528 and long position in ½ share of stock after 1 year ST=110 ST=90 Synthetic portfolio stock 55 45 bond -45 -45 Net payoff 10 0Part 3. OptionsLecture 2 Call Call 10 0 45
46. 46. 5. Option valuation Binomial option pricing model Since the payoff (value) for the synthetic portfolio is exactly the same as that for the Call option in all circumstances, the price (initial value) of the portfolio must be the same as that of the Call. C0  N  S0  B  0.5 100  42.4528  7.5472  7.55Part 3. OptionsLecture 2 46
47. 47. 5. Option valuation Black – Scholes valuation • Assumes that the price follow a Geometric Brownian Motion (GBM) with constant drift and volatility. 𝑑𝑆 = 𝜇𝑑𝑡 + 𝜎𝑑𝑧 𝑆 • The model incorporates the • constant price variation of the stock • the time value of money • the options strike price • the time to the options expiry. Assumptions 1. Stock pays no dividendsPart 3. Options 2. Option can only be exercised upon expiration (European) 3. Market direction cannot be predicted, hence "Random Walk." 4. No commissions, taxes are charged in the transaction.Lecture 2 5. Short sales allowed 5. Interest rates remain constant. 6. Stock returns are normally distributed, thus volatility is constant over time. 47
48. 48. 5. Option valuation Black – Scholes valuation Co call option value So current stock price N(d) cumulative distribution function of the standard normal distribution T-t time to maturity r risk free rate 𝜎 is the volatility of the underlying asset Scenario 1: N(d) = 1Part 3. Options High probability the option will be exercised Intrinsic valueLecture 2 Scenario 2: N(d) = 0 No probability the option will be exercised Scenario 3: N(d) = btw 0 and 1 Value depends on the call potential value (PV) 48
49. 49. 6. Exotic Options Asian Options Payoff depends on the average price of the underlying asset over a certain period of time as opposed to at maturity Barrier Options A type of option whose payoff depends on whether or not the underlying asset has reached or exceeded a predetermined price. Lookback option Payoffs that depend in part on the minimum or maximum price of the underlying asset during the life of the option. • Payoff could be against the max or min instead of the final pricePart 3. OptionsLecture 2 49