A1: American Options in the Binomial Model

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1 Appendix 1 A1: American Options in the Binomial Model So far we were dealing with options which can be excercised only at a fixed time, at their maturity date T. These are european options. In a complete model, lie the Binomial or the Blac-Scholes model, these options can be replicated exactly. Now we are considering options which can be excercised at an arbitrary time t [0, T. These are called american options. Thus, if a ban is selling an american option, it has to be prepared to pay the payoff to the customer not only at t = T, but at an arbitrary t [0, T. Therefore it would be desirable to replicate the option for all t [0, T, not only at t = T. However, this is not possible. But it is possible to set up a selffinancing strategy whose portfolio value is always greater or equal to the option s payoff and this leads to an arbitrage free fair price for the american option contract. We first consider the situation in discrete time, in the Binomial model. Let Π S = S 0 X j (1.1 be the price process for the Binomial model. That is, the X j are identically independently distributed with { U = 1 + u with probability p X j = (1.2 D = 1 + d with probability q = 1 p The discounted price process s = R S, R = 1 + r, is given by where the x j = X j /R have the distribution Π s = s 0 x j (1.3 db({x j } = N Π db(x j (1.4 db(x = { p δ(x U/R + (1 p δ(x D/R } dx (

2 192 A1 Let C be the payoff of some (exotic or not american option. That is, excercising the option at time {1, 2,..., N} gives a payment of C = C ( S 0,..., S (1.6 For a simple call or put (1.6 reduces to C = C(S. Let c = R C(S 0,..., S (1.7 be the discounted payoff at time. We are looing for some selffinancing strategy (δ whose discounted portfolio value v = R V at time is always bigger than c, v = v 0 + δ j 1 (s j s j 1! c (1.8 At time = N, we should have v N = c N. At time = N 1, v N 1 should be such that we can guarantie v N = c N at = N and it must be at least c N 1. In the first lecture on the Binomial model and in lecture 5 we saw that we can guarantie v N = c N at = N if we define v N 1 = p v N (s 0,..., s N 1, s N 1 U/R + (1 p v N (s 0,..., s N 1, s N 1 D/R = E B[v N s N 1 (1.9 where p = R D U D is the probability of the equivalent martingale measure d B. That is, if we define a sequence of discounted portfolio values (u 0 N inductively by then we must have u N = c N (1.10 u 1 = max { c 1, E B[u s 1 }, = N, N 1,..., 1 (1.11 v u = 0,..., N (1.12 Observe that we cannot write equal in (1.12 since it is not clear that the portfolio values u defined in (1.10,1.11 can be generated by some selffinancing strategy. In fact, in general they cannot. Thus we have to find the smallest selffinancing strategy, which has the lowest price v 0 and satisfies (1.12. It is obtained as follows. Theorem 8.1: Let (u 0 N be the Snell envelope of the discounted american claim (c 0 N defined by (1.10,1.11. For = 1,..., N, let δ 1 (s 0,..., s 1 = u (s 0,..., s 1, s 1 U/R u (s 0,..., s 1, s 1 D/R s 1 U/R s 1 D/R (1.13

3 A1 193 and let v = u 0 + δ j 1 (s j s j 1 (1.14 be the discounted portfolio value of the selffinancing strategy defined by (1.13 with initial price u 0. Then any selffinancing strategy ṽ = ṽ 0 + δ j 1 (s j s j 1 with ṽ c for all satisfies ṽ u for all. In particular, each such strategy has a larger price than (1.14, ṽ 0 u 0. The relation between (u and (v is given by v = u + (max { } u (1.15 where ũ 1 := E B[u s 1. Proof: We first show (1.15. Recall from Theorem 1.1 that if δ 1 is given by (1.13 and if ũ 1 is given by ũ 1 = p u (s 0,..., s 1, s 1 U/R + (1 p u (s 0,..., s 1, s 1 D/R then there is the relation Hence, = E B[u s 1 (1.16 ũ 1 + δ 1 (s s 1 = u (1.17 v = u 0 + (1.17 = u 0 + = u + (1.11 = u + δ j 1 (s j s j 1 ( uj ũ j 1 ( uj 1 ũ j 1 (max { } c j 1, ũ 1 ũ 1 (1.18 which proves (1.15. The inequalities ṽ u for all can be obtained by induction. For = N, ṽ N c N = u N. Suppose ṽ u holds for. Then ṽ 1 = E B[ṽ s 1 E B[u s 1 = ũ 1 (1.19

4 194 A1 Since also by assumption ṽ 1 c 1 we have ṽ 1 max { } c 1, ũ 1 = u 1 which completes the induction. Theorem 8.1 solves the hedging problem for american options in the Binomial model and, by approximation with small t, also for the Blac-Scholes model. One has to compute the Snell envelope (u which can be easily done on an excel-sheet using the definition (1.10,1.11 and then the delta s for a selffinancing strategy are given by (1.13. Another characterization of the sequence (u, which is probably of more theoretical interest, can be given in terms of stopping times. It reads as follows. Theorem 8.2: Let T = { τ : {s j } 0 j N τ({s j } {, + 1,..., N}, τ stopping time } be the set of all stopping times bigger or equal than. Let c = R C(S 0,..., S be the discounted payoff of some american option and let (u be the Snell envelope defined by (1.10,1.11. Then where τ u=c is the stopping time defined by u = sup E B[ c τ s (1.20 τ T = E B[ cτ s u=c (1.21 τ u=c = min{j u j = c j } (1.22 Proof: From (1.15 we have and for m u = v (max { } (1.23 In particular, u m = u + (u m u = u + v m v u τ u=c = u + v τ u=c v = u + v τ u=c v m (max { } τ u=c τ u=c (max { c j 1, ũ j 1 } ũj 1 (ũ j 1 ũ j 1 (1.24 = u + v τ u=c v (1.25

5 A1 195 Thus, E B[ cτ s u=c = E B[ uτ s u=c = u + E B[ vτ s u=c v Lemma 8.3 = u (1.26 which proves (1.21. Furthermore, for every stopping time τ T, c τ u τ τ = u + v τ v (max { } (1.27 and therefore, using Lemma 8.3 below again, E B[ c τ s u + E B[ v τ v s }{{} =0 u E B [ (max { cj 1, ũ j 1 } ũj 1 χ(j τ s } {{ } 0 (1.28 This implies also sup τ T E B[ c τ s u and, because of (1.26, the equal sign follows. Lemma 8.3 (Stopping Theorem: Let v = v 0 + δ j 1(s j s j 1 be a selffinancing strategy and let τ be a stopping time. Let Ẽ denote the expectation with respect to the martingale measure. Then Ẽ[v τ = v 0, and, for τ, Ẽ[v τ s = v (1.29 Proof: For τ we have Ẽ[v τ s = v + = v + = v Ẽ [ δ j 1 (s j s j 1 χ(j τ s Ẽ [ δ j 1 (s j s j 1 s N Ẽ [ δ j 1 (s j s j 1 χ(j > τ s Ẽ [ δ j 1 (s j s j 1 χ(j > τ s (1.30 Since τ is a stopping time, we have τ = τ({s m } 0 m τ. Thus, for τ < j, τ is a function of at most s 0, s 1,..., s j 1. But then we can move the integrals over s n,..., s j directly to the

6 196 A1 s j in the last line of (1.30 to obtain Ẽ[v τ s = v Ẽ [ (Ẽ[sj δ j 1 s j 1 s j 1 χ(j > τ s }{{} =0 = v (1.31 since (s j is a martingale with respect to Ẽ. The next lemma states that if the option value of some european option is always bigger than the option s payoff, V (S C(S, then the Snell envelope coincides with the discounted V. Lemma 8.4: Let V be the portfolio value of a replicating strategy for some european option with payoff C(S 0,..., S N, that is, V is the option value at time. Then, if V (S 0,..., S C(S 0,..., S = N, N 1,..., m (1.32 the undiscounted Snell envelope U = R u satisfies In particular, if for c = R C(S then v c and u = v for all. U = V = N, N 1,..., m. (1.33 E B[c s 1 c 1 (1.34 Proof: We prove (1.33 by induction on. For = N, u N = c N = v N. Suppose (1.33 holds for. Observe that since v = v 1 + δ 1 (s s 1 we have v 1 = E B[v s 1. Then u 1 = max { c 1, E B[u s 1 } = max { c 1, E B[v s 1 } = max { c 1, v 1 } = v 1 (1.35 if v 1 c 1. This proves (1.33. (1.34 follows similar. By induction, starting at = N with v N = c N, v 1 = E B[v s 1 E B[c s 1 c 1 (1.36

7 A1 197 and the lemma is proven Condition (1.34 holds for a call option, but not for a put. Namely, for a call c = R (S K + = (s K/R + = (s 1 x K/R + (1.37 Since f(x = (s 1 x K/R + is a convex function, we can apply Jensen s inequality, f(x dp (x f ( x dp (x for any probability measure dp and convex function f, to obtain E B[c s 1 = (s 1x K/R d b(x + ( s 1 x d b(x K/R + = (s 1 K/R + R 1 (s 1 K/R 1 + = c 1 (1.38 If one tries the same computation for a put, one ends up with (K/R E B[c s 1 = s 1x d b(x + ( K/R s 1 x d b(x + = (K/R s 1 + (1.39 but this cannot be estimated against (K/R 1 s 1 + for R 1. Thus we can summarize Corollary 8.5: Let the discounting factor R 1. Then the values of american and european calls coincide but the values of american and european puts differ. The value of an american put at time is given by the undiscounted Snell envelope U = R u defined by (1.10,1.11.