2 The binomial pricing model

Transcription

1 2 The binomial pricing model 2. Options and other derivatives A derivative security is a financial contract whose value depends on some underlying asset like stock, commodity (gold, oil) or currency. The underlying asset has a market value S t which changes randomly with time t. Most of the derivatives are fixed-term contracts, which expire at certain time T called maturity. Next are some basic examples. A forward contract is an agreement between two parties whereby one contracts to buy from the other a given asset (stock) at specified price (forward price) K at delivery date T. A (European) call option is a contract which confers the holder the right, but not the obligation, to buy a given asset (stock) at prescribed price, called strike price K at time T. The party issuing the contract ( writer of the option, typically a bank) is obliged to sell the asset should the holder decide to exercise her right. A (European) put option is a contract which confers the holder the right, but not the obligation, to sell a given asset at prescribed price, called strike price K at time T. The party issuing the contract ( writer of the option) is obliged to buy the asset should the holder decide to exercise her right. Speaking of European type of option we mean that the option can be exercised only at the maturity, but not earlier. For instance, on 25/09/3 you possess a call option for buying a share of ABZ for 220p on 25/09/204. Suppose the market price of ABZ share on 25/09/204 will be 250p. In this case you can exercise the option, that is buy a share for 220p, then sell the share for 250p, thus gaining a profit of 30p. In the options exchange, the contract will be settled by just paying you out 30p. If on 25/09/204 the market price of a share is 200p, you will not exercise, as it makes not financial sense and you are not obliged to exercise. On 25/09/3 the value of stock could be higher (the option is in the money ) or lower ( out of the money ) than the strike 220p; but in any case the option has some positive time value due to the chance that at the maturity the option will be in the money. The payoff at time T for holder of each of the above three types of contracts depends on the stock price S = S T at time T in a simple way: it is S K for forward, (S K) + for call option, and (K S) + for put option. Throughout we shall use the notation x + for max(x, 0), called positive part of x (e.g. 7 + = 7, ( 3) + = 0). Exercise 2.. Graph the payoffs as functions of S. Exercise 2.2. You buy a call option with strike K = 2 and write a call option with strike K 2 = 3. Both options are issued on the same stock, and have the same maturity time T. Graph the payoff of your portfolio as a function of the stock price S T at maturity. At any time from the start of the contract to maturity T the contract has some value, which may be negative for forwards. The holder of the security (buyer) has a long position, while the counter-party (seller, writer) is said to have a short position.

2 Puts and calls are known as plain vanilla options. More complex options may have payoff also depending on the stock price before the maturity (exotic options), and can be exercised before they expire (American options), 2.2 Risk-free investments Most of the market model include a risk-free investment like bonds or saving account in a bank. Normally we shall assume that the rate of returns for such investment is the same for both borrowing and investing, and in simple case that the rate is a constant r > 0. For instance, let r be the annual interest rate. The interest can be compounded simply (one time a year), multiply over discrete time periods (e.g. monthly, quarterly) or continuously. If the interest is compounded k times per year, then investing amount P (the principal ) returns in a year P (+r/k) k. If the interest is compounded continuously, then investing amount P returns P e rt over time t. Example 2.3. On your credit card you borrow 90 pounds at interest rate 8% compounded monthly. After one year you owe pounds ( ) 2 = Example 2.4. On your credit card you borrow 90 pounds at interest rate 8% compounded continuously. After five months you owe pounds 90 e 0.8 5/2 = 97 Exercise 2.5. Suppose the annual interest rate on saving account is r. Let R k be the return from investing P pounds for a year with k-times compounded interest, and let R be the analogous return from investing P pounds with continuously compounded interest. Argue that R k converges to R as k. What is the return on such investment for T years (in multiply compounded, respectively continuously compounded ineterest)? It is useful to go back and forth in time with interest rate computations. The present (time 0) value of capital P at time t is the riskless investment at time 0 which returns P at time t. The present value of a pound at time is e r for continuous compounding, and ( + r) for simple compounding. More generally, an asset worth a pound at time t has present value e rt (for t 0) for continuous compounding, respectively ( + r) t (for t = 0,, 2,... ) for simple compounding. 2.3 The no arbitrage principle A fundamental problem of the financial mathematics is the pricing problem for derivatives. The question is how to determine a reasonable price of a derivative any time before the maturity. The pricing methodology based on the no arbitrage principle has become a cornestone of the theory. 2

3 Definition 2.6. An arbitrage opportunity is a trading strategy which requires zero investment, incurs no losses but may yield a positive profit (with some probability). The arbitrage pricing paradigm sticks with the assumption that the market has no arbitrage opportunities. Example 2.7. (Price of a forward) Let the risk-free investment rate be r, with interest compounded continuously. Consider a forward contract on a share of a stock, with delivery price K and maturity T. The payoff of this contract is F T = S T K at time T. Consider also a portfolio comprised of one share of the stock, which is priced S 0 at time 0, and a cash position Ke rt (negative amount means money borrowed from bank). So the price of the portfolio at time 0 is S 0 Ke rt. At time T the value of the portfolio will be S T (Ke rt )e rt = S T K. We see that at time T the portfolio pays the same as the forward contract. The arbitrage pricing requires that the price F 0 of forward at time 0 must be F 0 = S 0 Ke rt, for otherwise there could be an arbitrage opportunity. The investor who sells a forward contract for F 0 can hedge the risks by investing the sale proceeds into one share of the stock and Ke rt in bonds. Suppose the contrary, that the forward is traded at time 0 on the market at some price F 0 > S 0 Ke rt. In that case you can write (or borrow) a forward contract (so assume obligations to be fulfilled in time T ), receive F 0 in cash and invest S 0 Ke rt in the stockand-bond portfolio. At time 0 you pocket positive amount A = F 0 (S 0 Ke rt ) > 0, while at time T you will have S T K to pay back your obligations, keeping positive amount Ae rt in your pocket whichever the stock price S T occurs to be. Example 2.8. (The put-call parity) Observe the identity (S T K) + + K = (K S T ) + + S T. Let S t, P t, C t be the price at time t (for 0 t T ) of stock, put option and call option, respectively. We know that P T = (K S T ) +, C T = (S T K) + are the payoffs of put and call options, so rearranging C T + K = P T + S T. The portfolio one call option, Ke r(t t) pounds in bonds is worth some C t +Ke r(t t) pounds at time t, and will be valued at C T + K = P T + S T at time T, like the portfolio one put option, one share of stock. The arbitrage pricing implies that the portfolios should be equally valued at time t, hence the put-call parity formula C t + Ke r(t t) = P t + S t for all t [0, T ]. Whichever, the put and call prices P t, C t happen to be, they must satisfy the formula. Exercise 2.9. Suppose the prices of stock, call and put satisfy S t Ke r(t t) < C t P t for some time t < T. Find an arbitrage opportunity. 3

4 2.4 One-period binomial model We have been successful with solving promptly the pricing problem for forwards because these are relatively simple derivatives. For calls, puts and more complex instruments we need a model for the price development of the underlying (stock). The simplest model of the kind is the market with one stock and one bond (money), and the one-period evolution with two possible moves of the stock. Suppose the market evolves over one period, from time 0 to time, with no intermediate valuations. At time zero we have a stock with some known price S 0 > 0, while at time the price depends on chance. We model the chance as a binary choice from sample space Ω = {H, T } (here: T stands for tail, not for maturity ), thus S = S (ω) is a random variable depending on ω Ω. We may think of tossing (perhaps, biased) coin, so that if the coin lands heads the price is S (H) = us 0, and if tails S (T ) = ds 0. The up factor u and down factor d are some given positive numbers with d < u. The stock might move up or down with certain true or market probabilities p and q. However, these are of secondary interest for us, provided both events H, T are possible. For the riskless money market (bonds) we assume (simply compounded) interest rate r, so pound invested at time 0 will yield + r pounds at time. To exclude the arbitrage we must assume d < r + < u. Exercise 2.0. Suppose r + / (d, u). Find an arbitrage opportunity. To apply the no arbitrage principle for pricing an option, we need to find a hedging strategy in the stock+money market. Example 2.. Suppose S 0 = 4, u = 2, d = /2, r = /4. Then S (H) = 8 and S (T ) = 2. Consider a European call with maturity time and strike K = 5. To hedge the option we may start with capital X 0 =.2, buying 0 = shares at time 0, and investing 2 X 0 0 S 0 = 0.8 in bonds (cash position). At time the cash position will be ( + r)(x 0 0 S 0 ) =, and the stock position will be either S 2 0(H) = 4 or S 2 0(T ) =, leaving us with the wealth either X (H) = 2 S (H) + ( + r)(x 0 0 S 0 ) = 3 or X (T ) = 2 S (T ) + ( + r)(x 0 0 S 0 ) = 0. On the other hand, the payoff of call is (S (H) K) + = 8 5 = 3 if ω = H; and (S (T ) K) + = (2 5) + = 0 if ω = T. We see that a stock-bond portfolio of worth X 0 =.2 hedges (replicates) the call. By the no arbitrage principle the time-0 value of the call option must be X 0 =.2. The example suggests a general way of pricing derivatives in the one-period model. Suppose an option has payoff V (ω) depending on ω {H, T }. We are looking for a portfolio of some worth X 0, with 0 shares of stock and X 0 0 S 0 invested in bonds, 4

5 such that at time the portfolio worth is V (ω) whichever ω. The value of portfolio at time is X = 0 S + ( + r)(x 0 0 S 0 ) = ( + r)x (S ( + r)s 0 ). We want to choose X 0, 0 to have X (ω) = V (ω) for ω = H, T, that is ( ) X r + S (H) S 0 ( ) X r + S (T ) S 0 = r + V (H) () = r + V (T ). (2) A smart way to solve (),(2) is to multiply them by yet indefinite factors p, respectively, q = p, then add them up to get ( ) X r + [ ps (H) + qs (T )] S 0 = r + [ pv (H) + qv (T )]. Suppose we have chosen p to achieve then From (3) easily p = + r d u d and subtracting (2) from () we get S 0 = r + [ ps (H) + qs (T )], (3) X 0 = r + [ px (H) + qv (T )]. (4), q = u r u d, (5) 0 = V (H) V (T ) S (H) S (T ). (6) Formula (6) is called the delta hedging formula. A portfolio worth (4) with 0 shares will hedge short position in the option. The no arbitrage principle prescribes that the time-0 value of the option be V 0 = r + [ pv (H) + qv (T )]. The numbers p, q are positive and add to. We can therefore interpret them as probabilities for H and T (upward/downward moves). These are the risk-neutral probabilities, which are characterised by the condition (3), which says that the the discounted expected stock price at time is equal to S 0. In other words, under the risk-neutral probability the stock moves, on the average, exactly like the riskless bond. Warning: the risk-neutral probabilities should not be confused with the market probabilities p, q. Under the market probabilities the stock should perform, on the average, better than a bond, for otherwise the investors had no incentive to invest in stocks. 5

6 2.5 Multiperiod binomial model The one-period model can be readily extended to multiple periods, with two possible stock price moves in every situation. When the move is up, the price is multiplied by factor u, down by factor d. For N-period model, we can think of N coin tosses, and adopt for the sample space Ω = {H, T } N, which is the set of 2 N sequences ω = ω... ω N with ω j {H, T }. The stock price at time 0 is a fixed value S 0 and in time n =,..., N the stock price S n depends on chance ω through the first n coin tosses ω... ω n, so does not depend on further coin-tosses ω n... ω N. We can write therefore S n (ω... ω n ) instead of S n (ω) with ω = ω... ω N. For instance S 3 (HT H) = u 2 ds 0, S 4 (HT T T ) = ud 3 S 0, etc. Consider an option (derivative security) which pays V N at maturity N, where V N = V N (ω... ω N ) depends on N coin-tosses. This notation includes the possibility that the option is path dependent (aka exotic ), with a payoff V N depending not only on S N but also on stock prices before maturity N. In the multiperiod market, a trading strategy may adjust the portfolio with every new stock price tick at times n =,..., n Let (X n, n ) for n = 0,,..., N be a stock-bond portfolio process with total wealth X n and n shares of stock at time n. The portfolio process is self-financing if it satisfies the recursion X n+ = n S n+ + ( + r)(x n n S n ), n = 0,,..., N, (7) meaning that the wealth of portfolio at time n + comes from the investment at time n. No dividends, no consumption. Introduce p = + r d, q = u r u d u d, and define recursively backward in time V N, V N 2,..., V 0 by V n (ω... ω n ) = r + [ pv n+(ω... ω n H) + qv n+ (ω... ω n T ), (8) so each V n depends on ω... ω n only. Further define n (ω... ω n ) = V n+(ω... ω n H) V n+ (ω... ω n T ) S n+ (ω... ω n H) S n+ (ω... ω n T ). (9) Theorem 2.2. The portfolio process defined by equations (7), (8), (9) replicates the option, that is X N (ω... ω N ) = V N (ω... ω N ) for all ω... ω N. To construct a hedge at time N, when the tosses ω... ω N are fixed, we just argue like in the -period model, finding N and X N. This will give X N (ω... ω N ) = r + [ pv N(ω... ω N H) + qv N (ω... ω N T )]. 6

7 A full proof is by induction, showing that X n (ω... ω n ) = V n (ω... ω n ) for all ω... ω n and n =,..., N. Following the no arbitrage principle we must set the option price at time n equal to V n, for all n = 0,..., N. Exercise 2.3. In three-period binomial model, consider a path-dependent option which pays V 3 = max 0 n 3 S n S 3 at the maturity time N = 3. Suppose u =.5, d = 0.7, r = 0., S 0 = 5. Calculate the price of the option at time Pricing calls and puts Call and put options are not path dependent. Their payoff is determined only by the price of underlying stock at the maturity time. This feature simplifies formulas, because at any time n only S n should be taken into account, and not all previous stock price moves. Within the framework of N-period model let V N = v N (S N ), where v N is some given function of the stock price. As can be argued by backward induction (and will be shown later), the option price V n depends on S n, hence can be written as V n = v n (S n ) for some function v n, n N. The recursion for the time-n value of the option simplifies as v n (s) = r + [ pv n+(us) + qv n+ (ds)], n = 0,..., N. (0) Moreover the time-0 price of the option becomes V 0 = (r + ) N N j=0 ( ) N ( p j q N j v N S0 u j d N j). () j Exercise 2.4. Write analogous formula for v n (s). Show that this formula agrees with the recursion (0). Consider the coin-tossing space Ω = {H, T } N with the risk-neutral probability measure #tails(ω) P(ω) = p #heads(ω) q (e.g. #heads(hhht HT ) = 6 #tails(hhht HT ) = 4). Note that #heads has Binomial(N, p) distribution. We can write () as the discounted expected payoff (or present value of the expected payoff) V 0 = ( + r) N Ẽv(S N). 7