Review of Derivatives I. Matti Suominen, Aalto

Transcription

1 Review of Derivatives I Matti Suominen, Aalto

2 25 SOME STATISTICS: World Financial Markets (trillion USD) Securitized loans Corporate bonds Financial institutions' bonds Public debt Equity market Source: Deutche Bank

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4 Global&Derivative&Markets&(Source&BIS)&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Notional&amounts&outstanding&at&yearend,&in&trillions&of&USD& &!! 1998& & 26& & 214& Exchangetraded&instruments& 14& & 69& & 75& & Futures& 8,4& & 26& & 27&!!!Interest!rate!!!!!!!!!!!!!!!!!! 8,!! 24!! 25!!!!Currency!!!!!!!!!!!!!!!!!!!!!!,!!,2!!,2!!!!Equity!index!!!!!!!!!!!!!!!!!!!!,3!! 1!! 1,6! Options& 5,6& & 44& & 38&!!!Interest!rate!!!!!!!!!!!!!!!!!! 4,6!! 38!! 31!!!!Currency!!!!!!!!!!!!!!!!!!!!!!,!!,1!!,1!!!!Equity!index!!!!!!!!!!!!!!!!!!!!,9!! 6!! 5,6! Turnover&of&financial&Derivatives&traded&on&organized&exchanges& Notional&amounts&in&trillions&of&US&dollars&!! &!!!! Total&turnover& 388!! 186!! 1936!! Interest!rate!futures! 296!! 1169!! 1267! Interest!rate!options! 56!! 446!! 334! Currency!futures! 2,6!! 17!! 29! Currency!options!,5!! 1,1!! 3! Equity!index!!futures! 18,9!! 74!! 155! Equity!index!options! 14,3!! 99!! 148!!!!!!!

5 DAILY TURNOVER USD BILLIONS FX TURNOVER INTEREST RATE DERIVATIVES TURNOVER

6 Forwards and Futures In a forward contract you make a firm commitment to buy or sell a certain quantity of, say, currency on some future date (3, 6 or 12 months ahead) at a price, the forward price, which is fixed already today. The forward price is set so that it costs nothing to enter a forward contract.

7 Forward Pricing Example: Price of GM share today $5 Risk-free interest rate 3% p.a. 12 months forward price of GM share F 12 (price such that it costs nothing to enter a forward contract) Expected dividends over the next 12 months. The forward price can be understood by looking at a buyer s situation: Buying GM today costs: Buying GM using a forward contract costs:

8 Forward Pricing The cost of these two alternatives must be the same: PV(F 12 )$5. a F 12 $5 x (1+.3) More generally: The fair forward price is: F T t (S t -PV[dividends]) x (1+r A ) (T-t) where r A is the simple risk-free annual interest rate.

9 Forward prices versus expected future prices Note that all we have to know in order to determine the forward price is: The current price of the underlying. The interest rate. The length of time until delivery. In our example, the forward price is different from the expected stock price in 12 months time (if investors are risk averse) which is: E( S ) S (1 For instance, if: Market risk premium π 6% β of GM stock 1 E(S 12 ) $5 x ( ) rf + bp) Does the fact that today s forward price for GM stock is below the expected future price of GM mean that you should purchase GM forward contracts today? {Note: To get the expected stock price, we use CAPM: E(R GM )r f +βπ for the expected return and multiply S x (1+E(R GM ))}

10 Exception In commodity markets the relationship between futures and spot prices is not tight due to: Cost of storage (like negative dividend) Convenience yield for holding commodity inventory (like dividend) Hence the future s price can be above the spot (contango) or below the spot price (backwardation) depending on the marginal trader s cost of storage and convenience yield

11 Options Call (Put): Right, but no obligation, to buy (or sell) stock at expiration date at price EX. Can be bought or written (give someone else an option). Payoff Payoff EX S T EX S T Payoff Payoff EX S T EX S T EX Strike price S T Price of the underlying at expiration date

12 Put-Call Parity: Look at payoffs at expiration. Two portfolios that have same payoff at expiration: EX EX S S 1: Stock + put with exercise price EX Payoff at expiration S + max(ex-s;) max (S;EX) 2: BOND paying EX + call with exercise price EX Payoff at expiration EX + max(s-ex;) max (S;EX) Costs today: S + P Costs today: PV(EX) + C Must be equally expensive today, hence S + P PV(EX) + C P PV(EX) S + C

13 Binomial Option Pricing Model To be able to derive the price of an option, we need a model of how the underlying behaves. Here, we model stock prices using a binomial tree: u 2 S S us ds uds d 2 S Each period the stock price can go up by a factor u or down by a factor d. T

14 Two-Period Binomial Tree Model S5, u1.1, d.9, (1+r f )1.5, EX5, T2 Stock price tree: S uu x (1.1) 2 S Call price tree: C S u S d C u C d S ud 49.5 S dd 4.5 C uu max(1.5;) C ud C dd

15 Two-Period Binomial Tree Model Start at t2 and work backwards, solving one-period trees, using either Replicating portfolio method, or Risk-neutral pricing. Using replicating portfolio method First, at S u node (time t1) (replicating portfolio: h u shares, borrow B u ) h B C u u u C S h h uu uu u u Sud - C (1 + r ) S - C -S u ud f ud - B u ud

16 Two-Period Binomial Tree Model At S d node (time t1) (replicating portfolio: h d shares, borrow B d ) hd,bd,cd At S node (time t) (replicating portfolio: h shares, borrow B ) h B C C S h Sd - C (1 + r ) h u u - C -S S d f d - B d

17 Alternatively: Using risk-neutral pricing Recall that the risk-neutral probability is: p 1+ rf - d u - d C C u C uu 1.5 C ud C d C C u d pc uu + (1 - p)c (1 + r ) f ud ; C dd C pc + (1 - p)c (1 + r ) f u d 5.36

18 Warrants u w s s u C n n n W ø ö ç ç è æ + W? d w s s d C n n n W ø ö ç ç è æ + The payoffs of the warrant are proportional to those of the call. Hence: w s s C n n n W ø ö ç ç è æ +

19 Allowing for more Frequent Price Changes Keep T fixed and reduce t (fraction of year during which prices change once). S t us ds u 2 S uds d 2 S How to choose u and d? Take u e s Dt T and d 1 u where σ is the empirically observed annual volatility of the stock returns.

20 Final Stock Price Distribution As t this approaches a continuous model of stock prices, where the stock returns (between time t and tt) are distributed normally. Probability Density Standard deviation σ Return (T)

21 Are Real Stock Returns Normally Distributed? Monthly Stock Returns of Boeing ( ) Number of samples % -36% -34% -32% -3% -28% -26% -24% -22% -2% -18% -16% -14% -12% -1% -8% -6% -4% -2% % 2% 4% 6% Stock Returns 8% 1% 12% 14% 16% 18% 2% 22% 24% 26% 28% 3% 32% 34% 36% 38%

22 Black-Scholes Option Pricing Formula Define: σ annual volatility of stock returns r interest rate risk-free, annual (continuously compounded) T time to expiration EX exercise price N(d) Pr{z d}, where z is distributed according to a standard normal distribution. As t " the replicating portfolio at time approaches: C h S B Where: h N(d 1 ) and B N(d 2 )PV (EX) d 1 ln(s/ EX) + rt s T + s T 2 d 2 d 1 - s T PV (EX) EXe -rt

23 DIVIDENDS Option holders are not entitled to the dividend. To take into account the effect of dividends we typically assume that 1) Dividends are known with certainty 2) Assume that price drops by the amount of dividends when dividends are paid. For European Options we either - Build a binomial tree model that includes dividends - Adjust the Black-Scholes model

24 DIVIDENDS Binomial tree approach when there are dividends S Su Sd Su 2 (1 - d ) S (1 - d ) Sd 2 (1 - d ) Su 3 (1 - d ) Su (1 - d ) Sd (1 - d ) To price a European option with dividends you proceed just as before: Calculate the option payoffs at maturity and work your way backwards either using the replicating portfolio or the risk neutral pricing method. Sd 3 (1 - d ) Ex - dividend date Fig. 1: Tree when stock pays a known dividend yield, d, at one particular time.

25 DIVIDENDS Binomial tree approach when there are dividends S Su Sd Su 2 -D S-D Sd 2 -D Ex-dividend date To Price American Options (options that you can exercise also before the maturity date) you may want to exercise a call early to get the dividend. In this case, we have to build an option value tree and at each node compare whether we want to exercise the option early or not. Fig. 2: Tree when dollar amount of dividend, D, is assumed known.

26 COMMON METHOD: Adjusting the Black & Scholes model Example of Black & Scholes model with dividends: Call option on a dividend paying stock S 5 EX 45 s 18% T.25 r c.583 Stock pays a dividend of $4 in 2 month s time. ð PV(dividend) 4 x e-.583 (2/12) 3.96 Replace S in Black-Scholes formula with: S - PV(dividend) Value of call (with dividend) $2.6 Value of call (no dividend) $5.83

27 Arbitrage : One-period binomial stock price model again Let s start with a one-period model, i.e., at maturity of the option the stock can have either gone up or down: S 5, u1.1, d What are the payoffs of a call option, expiring at t1, with X5, when r f 5%?

28 Option Pricing by No-Arbitrage Given our model of stock prices, we price options by constructing a replicating portfolio. Replicating portfolio: buy Δ stocks borrow B such that: 55 Δ 1.5 B 5 45 Δ 1.5 B

29 Option Pricing by No-Arbitrage This leads to two equations in two unknowns, Δ and B Example In General 55 Δ 1.5 B 5 Δ us B (1+r f ) C u 45 Δ 1.5 B Δ ds B (1+r f ) C d Δ.5 B [Option Delta or Hedge Ratio] D h By no-arbitrage, the current value of the option must equal the current value of the replicating portfolio: C.5 x C Δ S B B Cu us DdS - C (1 + r ) f d - Cd - ds

30 How to take advantage of mispricing? What if the call option is trading at 3.8 in the market? Buy low: Sell high: t t 1 up state down state Buy ½ share Borrow write call (1+r)21.43 Net +.23

31 Real options Typically the Underlying S the Present Value of the Cash Flow from the project (after some initial investment or adjustment cost) EX some initial investment or adjustment cost EXAMPLE: MW Petroleum

32 Review questions for lecture 3: Q1: Bullmart Inc. s common stock is currently trading at 33p, and can go up by a factor of 1.2 or down by a factor of 1/1.2 each period. The periodic riskfree rate is 5%. What is the value of a two-period European put on Bullmart with a strike price of 3p? (Bullmart does not pay any dividends.)

33 Q1: To price the put option, we first determine the terminal payoffs. Then, because all we need to do is to price the option, the quickest method is to use risk neutral pricing, working backwards from t2. The two-period stock price tree for Bullmart is:

34 The risk-neutral probability is: 1.5-1/ / Put pay-off at various nodes: max (3-475,) max (3-33,) max (3-229,) 71

35 Thus, the put price tree is: Put pay-off: max (3-475,) 59. x +. 41x max (3-33,) 59. x +. 41x max (3-229,) 71 The price of the two-period European put on Bullmart is thus 1.75p.

36 Review Questions Q1: Valuing a two-period convertible bond with coupon payments. Goldenbust corporation s assets are currently worth $4,. Each period the value of Goldenbust s assets (those remaining after servicing their debt) either increase or decrease by 5%. They have 15 shares outstanding. They have, in addition, issued 1 bonds. Each bond has a face value of $1 and matures in 2 periods. receives $1 per period coupon. can be converted into.5 shares at any time. The company pays no dividends, the t coupon has just been paid, the risk free rate is 8% and it is constant. a) What would be the value of their debt without the conversion feature (straight debt)? b) What is the value of the convertible debt?

37 Review Questions: Answers Q1: Step 1: Two period asset value tree after the periodic coupon has been paid (ex-coupon): t t1 t2 A 2 (59x1.5) A 1 (4x1.5) A 4 A 2 (59x.5) A 2 (19x1.5) A 1 (4x.5) A 2 (19x.5) Step 2: u 1.5, d.5 ð r d p u - d Also: g NC N + N S C

38 Review Questions: Answers Value of straight bond after the periodic coupon has been paid (ex coupon) (Aasset value, B bond value): t t1 t2 A B 2 1 A 1 59 B A 4 A B 2 1 B A B 2 1 A 1 19 B A 2 85 B 2 85 The value of straight debt after the current period s coupon has been paid is D Straight 11.3.

39 Value of the convertible bond: Review Questions: Answers t t1 t2 A if convert:.25 x if don t convert: 1 A 1 59 if convert:.25 x A 4 if convert:.25 x 4 1 market value: A if convert:.25 x if don t convert: 1 market value: A if convert:.25 x if don t convert: 1 A 1 19 if convert:.25 x market value: So the value of the convertible bond is: D Convertible A 2 85 if convert:.25 x if don t convert: 85

40 Value of the convertible bond: Review Questions: Answers t t1 t2 A if convert:.25 x if don t convert: 1 A 1 59 if convert:.25 x A 4 if convert:.25 x 4 1 market value: If bond is also callable at 115 at t1: - If called investors select max(147.5;115) Company calls if < MV A if convert:.25 x if don t convert: 1 market value: A if convert:.25 x if don t convert: 1 A 1 19 if convert:.25 x market value: So the value of the convertible bond is: D Convertible A 2 85 if convert:.25 x if don t convert: 85