Basics of Derivative Pricing

Transcription

1 Basics o Derivative Pricing 1/ 25

2 Introduction Derivative securities have cash ows that derive rom another underlying variable, such as an asset price, interest rate, or exchange rate. The absence o arbitrage opportunities places restrictions on the derivative s value relative to that o its underlying asset. For orward contracts, no-arbitrage considerations alone may lead to an exact pricing ormula. For options, no-arbitrage restrictions cannot determine an exact price, but only bounds on the option s price. An exact option pricing ormula requires additional assumptions on the probability distribution o the underlying asset s returns (e.g., binomial). 2/ 25

3 Forward Contracts on Assets Paying Dividends Let F 0 be the date 0 orward price or exchanging one share o an underlying asset periods in the uture. This price is agreed to at date 0 but paid at date > 0 or delivery at date o the asset. Hence, the date > 0 payo to the long (short) party in this orward contract is S F 0, ( F 0 S ) where S is the date spot price o one share o the underlying asset. The parties set F 0 to make the date 0 contract s value equal 0 (no payment at date 0). Let R > 1 be the per-period risk-ree return or borrowing or lending over the period rom date 0 to date, and let D be the date 0 present value o dividends paid by the underlying asset over the period rom date 0 to date. 3/ 25

4 Forward Contract Cash Flows Consider a long orward contract and the trades that would exactly replicate its date payo s: Date 0 Trade Date 0 Cash ow Date Cash ow Long Forward Contract 0 S F 0 Replicating Trades 1) Buy Asset and Sell Dividends S 0 + D S 2) Borrow R F 0 F 0 Net Cash ow S 0 + D + R F 0 S F 0 In the absence o arbitrage, the cost o the replicating trades equals the zero cost o the long position: or S 0 D R F 0 = 0 (1) F 0 = (S 0 D) R (2) 4/ 25

5 Forward Contract Replication I the contract had been initiated at a previous date, say date 1, at the orward price F 1 = X, then the date 0 value (replacement cost) o the long party s payo, say 0, would still be the cost o replicating the two cash ows: 0 = S 0 D R X (3) The orward price in equation (2) did not require an assumption regarding the random distribution o the underlying asset price, S, because it was a static replication strategy. Replicating option payo s will entail, in general, a dynamic replication strategy requiring distributional assumptions. 5/ 25

6 Basic Characteristics o Option Prices The owner o a call option has the right to buy an asset in the uture at a pre-agreed price, called the exercise or strike price. Since the option owner s payo is always non-negative, this buyer must make an initial payment to the seller. A European option can be exercised only at the maturity o the option contract. Let S 0 and S be the current and maturity date prices per share o the underlying asset, X be the exercise price, and c t and p t be the date t prices o European call and put options, respectively. Then the maturity values o European call and put options are c = max [S X ; 0] (4) p = max [X S ; 0] (5) 6/ 25

7 Lower Bounds on European Option Values Recall that the long (short) party s payo o a orward contract is S F 0 (F 0 S ). I F 0 is like an option s strike, X, then assuming X = F 0 implies the payo o a call (put) option weakly dominates that o a long (short) orward. Because equation (3) is the current value o a long orward position contract, the European call s value must satisy c 0 S 0 D R X (6) Furthermore, combining c 0 0 with (6) implies c 0 max S 0 D R X ; 0 (7) By a similar argument, p 0 max R X + D S 0 ; 0 (8) 7/ 25

8 Put-Call Parity Put-call parity links options written on the same underlying, with the same maturity date, and exercise price. c 0 + R X + D = p 0 + S 0 (9) Consider orming the ollowing two portolios at date 0: 1 Portolio A = a put option having value p 0 and a share o the underlying asset having value S 0 2 Portolio B = a call option having value c 0 and a bond with initial value o R X + D Then at date, these two portolios are worth: Portolio A = max [X S ; 0] + S + DR = max [X ; S ] + DR Portolio B = max [0; S X ] + X + DR = max [X ; S ] + DR 8/ 25

9 American Options 9/ 25 An American option is at least as valuable as its corresponding European option because o its early exercise right. Hence i C 0 and P 0, the current values o American options, then C 0 c 0 and P 0 p 0. Some American options early exercise eature has no value. Consider a European call option on a non-dividend-paying asset, and recall that c 0 S 0 R X. An American call option on the same asset exercised early is worth C 0 = S 0 X < S 0 R X < c 0, a contradiction. For an American put option, selling the asset immediately and receiving $X now may be better than receiving $X at date (which has a present value o R X ). At exercise P 0 = X S 0 may exceed R X + D S 0 i remaining dividends are small.

10 Binomial Option Pricing The no-arbitrage assumption alone cannot determine an exact option price as a unction o the underlying asset. However, particular distributional assumptions or the underlying asset can allow the option s payo to be replicated by trading in the underlying asset and a risk-ree asset. Cox, Ross, and Rubinstein (1979) developed a binomial model to value a European option on a non-dividend-paying stock. The model assumes that the current stock price, S, either moves up by a proportion u, or down by a proportion d, each period. The probability o an up move is. 10/ 25

11 Binomial Option Pricing cont d S % & us with probability ds with probability 1 Let R be one plus the risk-ree rate or the period, where in the absence o arbitrage d < R < u. (10) Let c equal the current value o a European call option written on the stock and having a strike price o X, so that its payo at maturity equals max[0; S X ]. Thus, one period prior to maturity: 11/ 25

12 Binomial Option Pricing cont d c u max [0; us X ] with probability c % & c d max [0; ds X ] with probability 1 (11) To value c, consider a portolio containing shares o stock and $B o bonds so that its current value is S + B. This portolio s value evolves over the period as S + B % & us + R B with probability ds + R B with probability 1 (12) 12/ 25

13 Binomial Option Pricing cont d With two securities (bond and stock) and two states (up or down), and B can be chosen to replicate the option s payo s: us + R B = c u (13) ds + R B = c d (14) Solving or and B that satisy these two equations: = c u c d (u d) S (15) B = uc d dc u (16) (u d) R Hence, a portolio o shares o stock and $B o bonds produces the same cash ow as the call option. 13/ 25

14 Binomial Option Pricing Example Thereore, the absence o arbitrage implies c = S + B (17) where is the option s hedge ratio and B is the debt nancing that are positive/negative (negative/positive) or calls (puts). Example: I S = $50, u = 2, d = :5, R = 1:25, and X = $50, then us = $100; ds = $25; c u = $50; c d = $0. Thereore, = 50 0 (2 :5) 50 = / 25

15 Binomial Option Pricing cont d so that B = 0 25 (2 :5) 1:25 = 40 3 c = S + B = 2 3 (50) 40 3 = 60 3 = $20 This option pricing ormula can be rewritten: c = S + B = c u c d (u d) + uc d dc u (18) (u d) R h i R d u d max [0; us X ] + u R u d max [0; ds X ] = which does not depend on the stock s up/down probability,. R 15/ 25

16 Binomial Option Pricing cont d Since the stock s expected rate o return equals u + d(1 ) 1, it need not be known or estimated to solve or the no-arbitrage value o the option, c. However, we do need to know u and d, the size o the stock s movements per period which determine its volatility. Note also that we can rewrite c as d c = 1 R [bc u + (1 b) c d ] (19) where b R u d is the risk-neutral probability o the up state. b = i individuals are risk-neutral since which implies that [u + d (1 )] S = R S (20) 16/ 25

17 Binomial Option Pricing cont d = R d = b (21) u d so that b does equal under risk neutrality. Thus, (19) can be expressed as c t = 1 E b [ct+1 ] (22) R where b E [] denotes the expectation operator evaluated using the risk-neutral probabilities b rather than the true, or physical, probabilities. 17/ 25

18 Multiperiod Binomial Option Pricing Next, consider the option s value with two periods prior to maturity. The stock price process is us % & u 2 S S % & dus (23) ds % & d 2 S so that the option price process is 18/ 25

19 Multiperiod Binomial Option Pricing cont d 19/ 25 c % & c u % & c d % & c uu max 0; u 2 S X c du max [0; dus X ] c dd max 0; d 2 S We know how to solve one-period problems: X (24) c u = bc uu + (1 b) c du (25) R c d = bc du + (1 b) c dd (26) R

20 Multiperiod Binomial Option Pricing cont d With two periods to maturity, the next period cash ows o c u and c d are replicated by a portolio o = cu c d shares o dc u (u d )S stock and B = uc d (u d )R o bonds. No arbitrage implies c = S + B = 1 R [bc u + (1 b) c d ] (27) which, as beore says that c t = 1 R b E [ct+1 ]. The market is complete over both the last period and second-to-last periods. Substituting in or c u and c d, we have c = 1 R 2 hb 2 c uu + 2b (1 b) c ud + (1 b) 2 c dd i 20/ 25

21 Multiperiod Binomial Option Pricing cont d = 1 b 2 R 2 max 0; u 2 S X + 2b (1 b) max [0; dus X ] + 1 h(1 R 2 b) 2 max 0; d 2 S X i which says c t = 1 be [c R 2 t+2 ]. Note when a market is complete each period, it becomes dynamically complete. By appropriate trading in just two assets, payo s in three states o nature can be replicated. Repeating this analysis or any period prior to maturity, we always obtain c = S + B = 1 R [bc u + (1 b) c d ] (28) 21/ 25

22 Multiperiod Binomial Option Pricing cont d Repeated substitution or c u, c d, c uu, c ud, c dd, c uuu, and so on, we obtain the ormula, with n periods prior to maturity: 2 3 c = 1 nx 4 n! R n b j (1 b) n j max 0; u j d n j S X 5 j! (n j)! or c t = 1 R n j=0 (29) be [c t+n ]. De ne a as the minimum number o upward jumps o S or it to exceed X. Then or all j < a (out o the money): while or all j > a (in the money): max 0; u j d n j S X = 0 (30) max 0; u j d n j S X = u j d n j S X (31) 22/ 25

23 Multiperiod Binomial Option Pricing cont d Thus, the ormula or c can be simpli ed: c = 1 Xn n! R n b j (1 b) n j u j d n j S X j=a j! (n j)! (32) Breaking up (32) into two terms, we have Xn n! c = S b j (1 b) n j u j d n j j=a j! (n j)! R n Xn XR n n! b j (1 b) n j (33) j=a j! (n j)! The terms in brackets are complementary binomial distribution unctions, so that (33) can be written 23/ 25

24 Multiperiod Binomial Option Pricing cont d where b 0 c = S[a; n; b 0 ] XR n [a; n; b] (34) u b and [a; n; b] is the probability that the R sum o n random variables that equal 1 with probability b and 0 with probability 1 b is a. For time to maturity and per-unit variance 2 (depending on u and d), as the number o periods n! 1, but the length o each period n! 0, this ormula converges to: where z c = SN (z) h ln S XR XR N z p (35) i = ( p ) and N () is the cumulative standard normal distribution unction. 24/ 25

25 Summary Forward contract payo s can be replicated using a static trading strategy. Option contract payo s require a dynamic trading strategy. A dynamically complete market allows us to use risk-neutral valuation. Dynamically complete markets imply replication o payo s in all uture states, but we may need to execute many trades to do so. 25/ 25