手写代写-CHAPTER 1
时间:2021-05-19
CHAPTER 1 Economical and financial background 1 CHAPTER 2 Probability background 3 CHAPTER 3 Stochastic processes in discrete time 5 CHAPTER 4 Mathematical finance in discrete time In this chapter we introduce mathematical models for financial markets in dis- crete time. Throughout this chapter we use a set Ω with a probability P and a finite time horizon filtration (Fn)n=0,...,N where we assume that we have no information at time 0, i.e. F0 = {∅,Ω}. We use the notation I = {0, . . . , N} for the set of time points. 1. The model We model a financial market with d+ 1 assets: a risk-less asset (bank account) S0, and d risky assets S1, . . . , Sd. We assume a constant interest rate i and S0n := (1 + i) n as the value for our bank account at time n, i.e. the amount that is accumulated with 1 unit of money from our starting time 0 (note that S00 = 1). For j ≥ 1, the price of the jth risky asset at time n is denoted by Sjn and we assume that (Sjn)n=0,...,N is an adapted process, i.e. the price S j n is known at time n. We also denote the corresponding discounted prices by S˜jn := Sjn S0n = Sjn (1 + i)n for j = 0, . . . , d and n = 0, . . . , N . We always assume that the processes Sj are adapted. 2. Trading strategies Definition 4.1. A trading strategy H is a family H = (H0, . . . ,Hd) of pre- dictable stochastic processes, i.e. for any j = 0, . . . , d, Hj is predictable, where Hj = (Hjn)n=0,...,N , and for j ≥ 1, Hjn denotes the number of assets we hold in the jth risky asset at time n, H0nS 0 n is the amount of money you have in the bank account at time n. The value of the trading strategy at time n is defined by Vn(H) := d∑ j=0 HjnS j n. Definition 4.2. A trading strategy H is self-financing if 0 = ∆Hn+1Sn := d∑ j=0 ∆Hjn+1S j n, ∀n = 0, . . . , N − 1 where Sn := (S 0 n, . . . , S d n), and ∆H j n+1 := H j n+1 −Hjn. Remark 4.3. Let H be a trading strategy. 7 8 4. MATHEMATICAL FINANCE IN DISCRETE TIME (1) At time n the prices are equal to Sn, if we readjust our portfolio from Hn to Hn+1, before the prices change to Sn+1, then this will cost us d∑ j=0 (Hjn+1 −Hjn)Sjn = ∆Hn+1Sn. The trading strategy is self-financing if ∆Hn+1Sn = 0. Thus, self- financing simply means that we have no portfolio readjustment costs, money is handled only inside the portfolio. Observe that ∆Hn+1Sn = 0 i.e. d∑ j=0 Hjn+1S j n = d∑ j=0 HjnS j n. We deduce from the last equality that Vn+1(H)− Vn(H) = d∑ j=0 Hjn+1(S j n+1 − Sjn) = Hn+1∆Sn+1. (2) The value of the portfolio at time n equals its initial value plus the trading gains plus the readjustment gains, i.e. Vn(H) = V0(H) + n∑ k=1 Hk∆Sk + n∑ k=1 ∆HkSk−1. The trading strategy is self-financing if and only if for any n = 1, . . . N , Vn(H) = V0(H) + n∑ k=1 Hk∆Sk = V0(H) + (H • S)n. (3) One can make similar notions with discounted values. That is V˜n(H) := Vn(H)/S 0 n = ∑d j=0H j nS˜ j n is the discounted value process and one has V˜n(H) = V˜0(H) + n∑ k=1 Hk∆S˜k + n∑ k=1 ∆HkS˜k−1, note that V˜0(H) = V0(H). The trading strategy is self-financing if and only if for any n = 1, . . . N , V˜n(H) = V˜0(H) + n∑ k=1 Hk∆S˜k = V0(H) + (H • S˜)n. Definition 4.4. A trading strategy H is an arbitrage if it is bounded and self-financing such that (1) (no initial position) H0 = 0 , (2) (positivity) Vn(H) ≥ 0 for any n = 0, . . . , N , (3) (chance to have a gain) P (VN (H) > 0) > 0. The market S = (S0, . . . , Sd) is called free of arbitrage (NA) if there is no arbitrage. The markets without arbitrage are the sensible markets while those with arbitrage allow market participants to make money out of nothing. For this reason, we need a tool that allows us to decide whether a market allows for arbitrage or not. This will be done in Theorem 4.8. Definition 4.5. A claim is a random variable C which is FN -measurable. A duplication strategy, replication strategy or hedge is a bounded self-financing trading strategy H such that VN (H) = C. The market S = (S0, . . . , Sd) is called complete if for any claim C there is a duplication strategy. 3. ABSENCE OF ARBITRAGE AND COMPLETENESS 9 3. Absence of arbitrage and completeness In this section we characterize absence of arbitrage and completeness. We first summarize the main results. For these we need one definition. (Recall Definition 2.12. from Chapter 2 for the notion Q ∼ P .) Definition 4.6. An equivalent martingale measure (EMM) is a measure Q ∼ P under which S˜1, . . . , S˜d are martingales. (Some people write P ∗ instead of Q.) Remark 4.7. S˜1, . . . , S˜d are martingales under Q means that for any j = 1, . . . , d, EQ[S˜ j n|Fn−1] = S˜jn−1, ∀n ≥ 1. The previous definition essentially states that an EMM is a probability measure under which all discounted price process are martingales and that this probability is equivalent to the original probability. In case that the probability space is finite Ω = {ω1, . . . , ωN} with all scenarios having positive probability P ({ωj}) > 0 for any j = 1, . . . , N equivalence to P means that all scenarios have positive probability under Q as well, in other words, P and Q are equivalent if they agree on which scenarios are impossible. We can now characterize (NA) markets. The following statement says that (NA) holds if and only if there is at least one EMM. Theorem 4.8 (Fundamental theorem of asset pricing (FTAP)). The following statements are equivalent: (1) The market is free of arbitrage. (2) There is at least one EMM. The less important (but sometimes very useful) concept of completeness allows to be characterized in terms of EMMs as well. Theorem 4.9 (Completeness theorem). Assume that the market satisfies (NA). The following statements are equivalent: (1) The market is complete. (2) There is exactly one EMM. Proof of Theorem 4.8. We prove that (2)⇒ (1) (the proof of (1)⇒ (2) is out of the scope of this lecture). We assume that there is an EMM Q. Let H be a bounded self-financing trading strategy such that H0 = 0 and Vn(H) ≥ 0 for n = 1, . . . , N (These are the only candidates to be an arbitrage). Since H is self-financing we have, by Remark 4.3 (3), that for n = 1, . . . , N V˜n(H) = V˜0(H) + (H • S˜)n = (H • S˜)n, where we have the last equality because V˜0(H) = 0 because H0 = 0. Theorem 3.13. in Chapter 3 yields that (V˜n(H))n∈I is a Q-martingale because for any j = 1, . . . , d, S˜j is a Q-martingale. Thus, we have 0 = V˜0(H) = EQ[V˜N (H)] ≥ 0. Consequently, EQ[V˜N (H)] = 0 which implies (because V˜N (H) ≥ 0) that V˜N (H) = 0 Q-a.s. so VN (H) = 0 Q-a.s. Since Q ∼ P we have VN (H) = 0 P -a.s. This implies P (VN (H) > 0) = 0. Hence, H is not an arbitrage. (What is written in blue is clarified in the part 2 of the slides of Chapter 4). Remark 4.10. From a practical point the FTAP is important because it pro- vides a method to construct markets with (NA) and it tells you that this is the only such method; also it connects an important meaningful economic statement with 10 4. MATHEMATICAL FINANCE IN DISCRETE TIME a structural mathematical statement. The completeness theorem is more about (1)⇒ (2) because this makes such models very tractable. Example 4.11. (1) Consider a market with time horizon N = 1, d = 2 risky assets, zero interest rate r = 0, Ω = {a, b, c}, with P ({a}) = 12 , P ({b}) = P ({c}) = 14 . The initial prices of the risky assets are S10 = 100 = S20 and their final prices at time 1 are S 1 1 and S 1 2 where for any ω ∈ Ω, S11(ω) =  130 if ω = a, 110 if ω = b, 90 if ω = c, S12(ω) =  90 if ω = a, 90 if ω = b, 110 if ω = c. Find an arbitrage. Answer: Starting with no initial position, H0 = (0, 0, 0). By the self-financing condition we know that money is handled only inside the portfolio. In order to buy 1 of each asset we need 100 + 100 = 200 which must be taken from the bank account. Thus H1 = (−200, 1, 1). We have V0(H) = 0, V1(H) =  −200 + 130 + 90 = 20 if ω = a, −200 + 110 + 90 = 0 if ω = b, −200 + 90 + 110 = 0 if ω = c. So, V1(H) ≥ 0, and P (V1(H) > 0) = P ({a}) = 12 > 0. It is an arbitrage. (2) In a market model with d = 1 risky asset, time horizon N = 1, interest rate i = 10%, Ω = {1, 2} with P ({1}) = P ({2}) = 12 . The risky asset has an initial price S10 = 105, and a final price S 1 1 given by S11(ω) = { 135 if ω = 1, 95 if ω = 2. Let (Fn)n=0,1 be the filtration given by F0 = {∅,Ω}, and F1 = σ(S11). Find the equivalent martingale measure(s). Answer: A probability Q is an equivalent martingale measure if: Q ∼ P and S˜1 is a Q-martingale. Q ∼ P it means that P and Q they agree on which scenarios are im- possible, i.e. q1 := Q({1}) > 0 and q2 := Q({2}) > 0 (because P ({1}) > 0 and P ({2}) > 0). S˜1 is a Q-martingale, i.e. EQ[S˜11 |F0] = S˜10 , where S˜10 = S10 = 105, and S˜11 = S11 1+i . We have EQ[S˜ 1 1 |F0] = EQ[S˜11 ] because F0 is the trivial σ-algebra. Therefore EQ[S˜11 |F0] = S˜10 ⇔ EQ[S˜11 ] = 105 ⇔ EQ[ S 1 1 1+i ] = 105 ⇔ EQ[S11 ] = 105(1 + i) ⇔ EQ[S11 ] = 105(1.1) ⇔ EQ[S11 ] = 115.5 ⇔ 135q1 + 95q2 = 115.5 On the other hand Q is a probability, i.e. q1 + q2 = 1. We deduce that Q is an equivalent martingale measure if and only if:{ 135q1 + 95q2 = 115.5, q1 + q2 = 1, where q1 and q2 should be strictly positive i.e. q1 > 0 and q2 > 0. We solve the system, then we find the following solution: q2 = 0.4875 and q1 = 0.5125. (Observe that the solution satisfies the required condition: 3. ABSENCE OF ARBITRAGE AND COMPLETENESS 11 q1 > 0 and q2 > 0). Finally, the EMM (equivalent martingale measure) is the probability Q where Q({1}) = 0.5125 and Q({2}) = 0.4875. The two theorems have strong implications. One is regarding possible prices for claims. Theorem 4.12 (Risk-Neutral Valuation Formula). (1) Let C be a claim and Q be an EMM. Then Cn := S 0 nEQ [ C S0N |Fn ] is a fair price for C at time n (i.e. a price for which C may be traded at time n without introducing arbitrage). (2) Also, if C is a claim and Cn a fair price for it at time n, then there is an EMM Q such that the above formula holds. Theorem 4.13 (Risk-Neutral Valuation Formula). Let C be a claim, assume that the market S = (S0, . . . , Sd) satisfies (NA) and let Q be an EMM. (1) If C can be duplicated, then the only prices Cn for which C may be traded at time n without introducing arbitrage is given by Cn := S 0 nEQ [ C S0N |Fn ] . (2) If C cannot be duplicated, then any element of( inf R EMM ER [ C S0N ] , sup R EMM ER [ C S0N ]) is a price for C at time 0 which does not introduce arbitrage. (The interval is open and non-empty) (3) If we define Cn := S 0 nEQ [ C S0N |Fn ] , then the market (S0n, . . . , S d n, Cn)n=0,...,N satisfies (NA). If the market is complete, then every claim can be duplicated, thus (1) can always be applied for claims in complete markets. In other words, in a complete market every claim has a unique fair price at each point of time. Proof of Theorem 4.13. We give only the proof of (3). Define Cn := S 0 nEQ[ C S0N |Fn] for any n. Observe that C˜n = EQ[ C S0N |Fn], then, by Proposition 3.12. in Chapter 3, (C˜n)n=0,...,N is a Q-martingale and, hence, Q is an EMM for the market (S0n, . . . , S d n, Cn)n=0,...,N . Theorem 4.8 yields that this market is free of arbitrage. Remark 4.14. Theorem 4.13 provides price formulas within the (NA) (as you can see in (1) and (2)) framework and a method to extend markets with a price process belonging to a claim (as you can see in (3)). Theorem 4.15. Let C be a claim and assume (NA). Then the following are equivalent (1) There is a hedge for C. (2) EQ[C] results in the same number across all EMMs Q. 12 4. MATHEMATICAL FINANCE IN DISCRETE TIME Remark 4.16. This theorem gives an idea about how one can verify if a claim C can be hedged or not under the assumption of (NA)! Exercise 1. Consider the model from Example 4.11 (2). What is the price at time 0 of a European call option (on the risky asset S1) with strike K = 110 and maturity T = 1? Definition 4.17. A pricing measure P ∗ is an EMM such that we assume that any bounded claim C is traded at time n at the price Cn := S 0 nEP∗ [ C S0N |Fn]. Remark 4.18. If a market is given, then there is either one or no pricing measure at all. (The latter can happen if some claims are not liquidly traded and as a consequence have no fixed price). In a complete arbitrage free market there is a pricing measure which is the only EMM. Proposition 4.19. Assume that the market satisfies (NA). Let H and H˜ be two bounded self-financing trading strategies. If at time N the value of H is equal to that of H˜ i.e. VN (H) = VN (H˜), then Vn(H) = Vn(H˜) for any n ∈ {0, . . . , N}. Proposition 4.20. Let C be a claim, assume that the market satisfies (NA) and let Q be an EMM. Suppose that C can be duplicated, and let H be a duplication strategy for C, then Cn = Vn(H) for any n ∈ {0, . . . , N}, where Cn is given in Theorem 4.13 (1), and Vn(H) is the value of H at time n. Proof. Cn := S 0 nEQ[ C S0N |Fn] = S0nEQ[ VN (H) S0N |Fn] = S0nEQ[V˜N (H)|Fn] = S0nEQ[V˜0(H) + (H • S˜)N |Fn] = S0n(V˜0(H) + (H • S˜)n) = S0nV˜n(H) = Vn(H), where we have the first equality because H is a duplication for C, the third equality by Remark 4.3 (3), the fourth one because, thanks to Theorem 3.13. in Chapter 3, ((H•S˜)n)n=0,...,N is a Q-martingale (because for j = 1, . . . , d, S˜j is a Q-martingale). Exercise 2. Consider the same model as in Exercise 1. Consider the claim C := (S11 − 110)+, and let H be a duplication strategy for C. What is V0(H)? 4. Binomial model In this section we introduce the most simple market model which still has inter- esting mathematical features. It is called Cox-Ross-Rubinstein model or binomial model. Definition 4.21. A market S is called binomial model if there is only one risky asset ’d = 1’, there are numbers −1 < d < u and S10 > 0 with S1n+1 = Tn+1S 1 n, for n = 0, . . . , N − 1, where (1) T1, . . . , TN are independent and identical distributed, 4. BINOMIAL MODEL 13 (2) T1 = { 1 + u with probability p, 1 + d with probability 1− p, where p ∈ (0, 1) is a fixed probability, Fn := σ(S10 , . . . , S1n), F := FN . (In a more advanced course, we can prove that for n ≥ 1, σ(S10 , . . . , S1n) = σ(T1, . . . , Tn)) From the definition of the binomial model we can see that for n = 0, . . . , N − 1 S1n+1 = { S1n(1 + u) with probability p, S1n(1 + d) with probability 1− p, and that for instance S11 ∈ {S10(1 + u), S10(1 + d)}, and S12 ∈ {S10(1 + u)2, S10(1 + u)(1 + d), S10(1 + d) 2}. The main statement is Theorem 4.22. A binomial model S is complete. The binomial model S is arbitrage free if and only if d < i < u, where i is the interest rate. If S is arbitrage free , then the EMM Q satisfies: (1) T1, . . . , TN are Q-independent and Q-identical distributed, (2) Q(T1 = 1 + u) = i−d u−d =: q and Q(T1 = 1 + d) = u−i u−d = 1− q. Proof. The completeness of the binomial model S is proved in Theorem 4.23 below. In this proof we show only that if d < i < u then the model is arbitrage free. To this end, let d < i < u and define the probability measure Q as indicated in (2). This measure is equivalent to P (because q > 0, and 1−q > 0) and we admit that Q satisfies (1). As it is mentioned above, Fn := σ(S10 , . . . , S1n) (since S10 is a constant, then we admit that F0 = σ(S10) = {∅,Ω}), and for n ≥ 1, Fn = σ(T1, . . . , Tn) (we admit also this last result). We have for n ≥ 1, EQ[S˜ 1 n|Fn−1] = EQ[ S1n−1Tn (1 + i)n |Fn−1] = S1n−1 (1 + i)n EQ[Tn|Fn−1] = S˜1n−1 1 + i EQ[Tn|Fn−1] = S˜1n−1 1 + i EQ[Tn] = S˜1n−1 1 + i (q(1 + u) + (1− q)(1 + d)) = S˜1n−1 1 + i (1 + i) = S˜1n−1, where we have the second equality because S1n−1 is Fn−1-measurable (known fac- tor), and we have the fourth equality because T1, . . . , TN are independent under Q (we admit this result). Thus, S˜1 is a Q-martingale, and it is proved above that Q ∼ P , thus Q is an EMM. We want to prove now that the binomial model is complete. In the following theorem there is a statement on how replicating strategies look like. 14 4. MATHEMATICAL FINANCE IN DISCRETE TIME Theorem 4.23. Let C be a claim in a binomial model S, define C˜N := C S0N , and define for any n = 1, . . . , N, H1n(ω) := E[C˜n|S10 , . . . , S1n−1, Tn = 1 + u](ω)− E[C˜n|S10 , . . . , S1n−1, Tn = 1 + d](ω) (u−d) 1+i S˜ 1 n−1 , (where in the last line ω ∈ Ω), C˜n−1 := C˜n −H1n∆S˜1n, H10 := H 1 1 , and define for any n = 1, . . . , N, H0n := C˜n −H1nS˜1n, H00 := H 0 1 . Then H is a duplication strategy for C and V˜n(H) = C˜n for any n = 0, . . . , N . Proof of theorem 4.23. Let C be a claim and (Hn)n=0,...,N , (C˜n)n=0,...,N be as described. First, the process H is bounded (this is obvious if Ω is a fi- nite set, we admit the result if Ω is infinite). Let us prove that H1 is pre- dictable, indeed, define An(ω) := E[C˜n|S10 , . . . , S1n−1, Tn = 1 +u](ω), and Bn(ω) := E[C˜n|S10 , . . . , S1n−1, Tn = 1 + d](ω), and A0 := 0 =: B0. A, B are predictable pro- cesses, because A0 and B0 are constant, and for n ≥ 1, An and Bn are functions of S10 , . . . , S 1 n−1. Observe that H1n = An −Bn u−d 1+i S˜ 1 n−1 which is Fn−1-measurable because it is the quotient of Fn−1 measurable functions, and observe that H10 := H 1 1 so H 1 0 is F0-measurable. Thus H1 is predictable. Let us prove now that H0 is predictable. For n ≥ 1, H0n = C˜n−H1nS˜1n, and we have C˜n is Fn-measurable, thus E[C˜n|Fn] = C˜n. Therefore, for any ω ∈ Ω, C˜n(ω) = E[C˜n|σ(S10 , . . . , S1n−1, S1n)](ω) = E[C˜n|σ(S10 , . . . , S1n−1, S1n−1Tn)](ω) = { E[C˜n|S10 , . . . , S1n−1, Tn = 1 + u](ω) if Tn(ω) = 1 + u, E[C˜n|S10 , . . . , S1n−1, Tn = 1 + d](ω) if Tn(ω) = 1 + d, = { An(ω) if Tn(ω) = 1 + u, Bn(ω) if Tn(ω) = 1 + d. We deduce that, for n ≥ 1 H0n(ω) = C˜n(ω)− (H1nS˜1n)(ω) = { An(ω)−H1n(ω)S˜1n−1(ω) 1+u1+i if Tn(ω) = 1 + u, Bn(ω)−H1n(ω)S˜1n−1(ω) 1+d1+i if Tn(ω) = 1 + d, = { An(ω)− (1 + u)An(ω)−Bn(ω)u−d if Tn(ω) = 1 + u, Bn(ω)− (1 + d)An(ω)−Bn(ω)u−d if Tn(ω) = 1 + d, = (1 + d)Bn(ω)− (1 + u)An(ω) u− d . Consequently, for n ≥ 1, H0n = (1+d)Bn−(1+u)Anu−d which is Fn−1-measurable, and observe that H00 := H 0 1 so H 0 0 is F0-measurable, thus H0 is predictable. Finally, we proved that H0 and H1 are predictable processes, thus H is a trading strategy. 4. BINOMIAL MODEL 15 We prove now that H is self-financing. Indeed, for n = 1, ∆H0nS˜ 0 n−1 + ∆H1nS˜ 1 n−1 = 0 because H 0 1 = H 0 0 and H 1 1 = H 1 0 , for n ≥ 1, ∆H0nS˜ 0 n−1 + ∆H 1 nS˜ 1 n−1 = ∆H 0 n + ∆H 1 nS˜ 1 n−1 = H 0 n −H0n−1 + ∆H1nS˜1n−1 = ∆C˜n − (H1nS˜1n −H1n−1S˜1n−1) + ∆H1nS˜1n−1 = ∆C˜n − (H1nS˜1n −H1n−1S˜1n−1) + (H1n −H1n−1)S˜1n−1 = ∆C˜n +H 1 nS˜ 1 n−1 −H1nS˜1n = ∆C˜n −H1n(S˜1n − S˜1n−1) = H1n∆S˜ 1 n −H1n∆S˜1n = 0. Thus, H is a self-financing strategy. It remains to prove that VN (H) = C. Observe that C˜n = H 0 n +H 1 nS˜ 1 n for n = 0, . . . , N , i.e . V˜n(H) = C˜n for n = 0, . . . , N . So, for n = N , V˜N (H) = C˜N = C S0N , i.e. VN (H) = C. Finally, we deduce that H is a duplication for C. For n = 0, . . . , N , a price Cn is Fn-measurable, thus thanks to Proposition 3.5. in Chapter 3 there is a function fn such that Cn = fn(S 1 0 , . . . , S 1 n). For the sake of simplicity, fn is denoted Cn, and consequently a fair price for C at time n is Cn(S 1 0 , . . . , S 1 n). The claim C is FN -measurable, so there is a function f such that C = f(S10 , . . . , S 1 N ), for the sake of simplicity, f is denoted C, and consequently the claim is equal to C(S10 , . . . , S 1 N ). Proposition 4.24. Let S be a binomial model with (NA) (i.e. d < i < u where i is the interest rate), and denote its unique EMM by Q. Let C be a claim. (1) The fair prices for the claim at each time point n are CN (S 1 0 , . . . , S 1 N ) = C(S 1 0 , . . . , S 1 N ), Cn(S 1 0 , . . . , S 1 n) = (1 + i) −1[q Cn+1(S10 , . . . , S1n, S1n(1 + u)) + (1− q)Cn+1(S10 , . . . , S1n, S1n(1 + d)) ] , for n = 0, . . . , N − 1, where q = i−du−d . (2) There is a duplication strategy H for C where for n = 1, . . . , N , H1n(S 1 0 , . . . , S 1 n−1) = Cn(S 1 0 , . . . , S 1 n−1, S 1 n−1(1 + u))− Cn(S10 , . . . , S1n−1, S1n−1(1 + d)) S1n−1(1 + u)− S1n−1(1 + d) , H10 (S 1 0) = H 1 1 (S 1 0). H0n(S 1 0 , . . . , S 1 n−1) = C˜n(S 1 0 , . . . , S 1 n)−H1n(S10 , . . . , S1n−1)S˜1n, for n = 1, . . . , N, and H00 (S 1 0) = H 0 1 (S 1 0). (3) For n = 0, . . . , N , V˜n(H) = C˜n. Corollary 4.25. Consider a continuous function f : R→ R and the European claim C with payoff function f , i.e. C = f(S1N ). Define the function c(x) := EQ [ f(xΠNj=1Tj) S0N ] = N∑ j=0 ( N j ) qj(1− q)N−j f(x(1 + u) j(1 + d)N−j) S0N . Then the price for the claim at time 0 is given by C0 = c(S 1 0). 16 4. MATHEMATICAL FINANCE IN DISCRETE TIME 5. American options In this section we consider American put and call options in an arbitrage-free market. We start with one general observation. The value of an American option depends on the point of view. If you have an American option, then you would try to maximize its payout. As we compare later values with earlier values via expectations under EMMs we would say the value of an American option is given by Abuy0 := max τ stopping time EQ [ f(S1τ )) S0τ ] where f is the payoff function (f(x) = (x − K)+ for calls, f(x) = (K − x)+ for puts). On the other hand, if we sell an American option we cannot know when the other side exercises the option. They might exercise the option at a point of time we find suboptimal but they like for other reasons. We would like to have a portfolio whose value is larger than or equal to the exercise value of the American option. Definition 4.26. A superhedge for a stochastic process Z is a trading strategy H such that Vn(H) ≥ Zn, n ∈ I A cheapest superhedge for a stochastic process Z is a superhedge H such that for any other superhedge K one has V0(H) ≤ V0(K). There is of course no reason to use more expensive superhedges. This leads to the following price for the seller: Asell0 := min H self-financing:∀n∈I,f(S1n)≤Vn(H) V0(H). Lemma 4.27. Let S be any market model. Then Abuy0 ≤ Asell0 . Proof. Let H be a self financing-trading strategy such that V0(H) = A sell 0 and ∀n ∈ I : f(S1n) ≤ Vn(H). Then, we have for any stopping time τ EQ [ f(S1τ )) S0τ ] ≤ EQ [ V˜τ (H) ] = V0(H), where we have the last equality thanks to Theorem 3.15 in Chapter 3. Theorem 4.28. Let S be any market model, i ≥ 0 and K > 0. Assume that the European call option with strike K and maturity N is traded at the price C0 at time 0. Then the price (both buy and sell) for the American call option with strike K and maturity N satisfies Abuy0 = A sell 0 = C0. If instead i ≤ 0, then the price (both buy and sell) for the American put coin- cides with the price for the European put with the same strike and maturity. Proof. Define f(x) := (x−K)+, Zn := f(S 1 n) S0n and let U be the Snell envelope of Z. If a ∈ (0, 1] we have f(ax) = (ax−K)+ = a(x−K/a)+ ≤ a(x−K)+ = af(x). We have UN = ZN = C˜N and for n = 0, . . . , N − 1 with Un+1 = C˜n+1 we have Un = max{Zn,EQ[Un+1|Fn]} = max{Zn, C˜n}. We also have Zn = f(S1n) S0n = f(S0n/S 0 NEQ[S 1 N |Fn]) S0n ≤ f(EQ[S 1 N |Fn]) S0N ≤ EQ[f(S 1 N )|Fn] S0N = C˜n 5. AMERICAN OPTIONS 17 and, hence, Un = C˜n. Since, C˜ is a martingale so is U . Thus, the stopping time T = N is optimal (because UT = ZT and U T is a martingale). Consequently, we have Abuy0 = EQ[UT ] = U0 = C˜0. Investing into one European call option yields a trading strategy H such that V0(H) = C0 and Vn(H) = Cn = S 0 nC˜n = S 0 nUn ≥ S0nZn = f(S1n). Thus, Asell0 ≤ V0(H) = C0 = Abuy0 ≤ Asell0 which yields the claim in case of non-negative interest rates. If we have negative interest rate, then we use with f(x) := (K−x)+ and a ≥ 1 that f(ax) = a(K/a− x)+ ≤ a(K − x)+ = af(x) instead. This will allow to make the same conclusion for American put in case of non-positive interest rate. For the remainder of this section we concentrate on the binomial model. To this end we fix the interest rate i > 0, the initial stock price S10 and the parameters −1 < d < i < u as well as the time horizon N ∈ I and the probability p := P (Tn = 1 + u). Recall that the risk neutral probability, i.e. the EMM Q, is given by q := Q(Tn = 1 + u) = i−d u−d . In this model European options have unique fair prices. As we assume positive interest rate we know that the American call option has the same price as the European call option but we cannot make the same conclusion for the American put option. We fix a strike K > 0 for an American put on the risky asset and briefly describe how to price it in the binomial model. The underlying idea is to calculate the Snell envelope U of the discounted payoff Zn := (K−S1n)+ S0n . We have UN = ZN and proceed backward. Theorem 4.29. Assume that the market model is complete (which is the case of the binomial model). Then the price of the American put option is the same for buyer and seller: Abuy0 = max τ stopping time EQ [ (K − S1τ )+ S0τ ] = U0 = min H self-financing: ∀n∈I,(K−S1n)+≤Vn(H) V0(H) = A sell 0 , where U is the Snell envelope of Z, where Zn := (K−S1n)+ S0n , for n = 0, . . . , N . Proof. Let Un = U0 + Bn + Mn be the Doob-decomposition of U . U is a supermartingale and, hence, B is decreasing. Thus Un ≤ U0 + Mn. Since the market is complete there is a duplication strategy H for (U0 + MN )S 0 N . Then, V˜N (H) = U0 + MN and V˜ (H) and U0 + M are martingales. Consequently, we have U0 + Mn = V˜n(H) for any n. Also, we have Zn ≤ Un ≤ U0 + Mn = V˜n(H) and, hence, H is a trading strategy with (K − S1n)+ ≤ Vn(H). The stopping time T := min{n ≥ 0 : Un = Zn} satisfies UT is a martingale and Abuy0 = EQ [ (K − S1T )+ S0T ] = max τ stopping time EQ [ (K − S1τ )+ S0τ ] = U0. 18 4. MATHEMATICAL FINANCE IN DISCRETE TIME We get Asell0 ≤ V0(H) = V˜0(H) = U0 = Abuy0 ≤ Asell0 . Finally, from a practical point one calculates the Snell envelope along the bi- nomial tree simply from UN = ZN and Un = max{Zn,EQ[Un+1|Fn]}. We know how to find the price of the American put option in the binomial model and that buyer and seller prices are the same. Moreover, we have a method to find the optimal stopping time. However, we have not yet a method to find the cheapest superhedge. We start by observing that complete models have a cheapest superhedge. A statement which is in fact true for any model! Lemma 4.30. Let Z be an adapted process. Then there is a cheapest superhedge. In the binomial model we can find the superhedge explicitly! Proposition 4.31. Let Zn = (K−S1n)+ S0n and U be its Snell envelope. Let U = U0 + M + A be the Doob-decomposition of U where M is the Q-martingale part. The hedge H = (H0, H1) for the claim C := S0N (U0 +MN ) is a cheapest superhedge for the American put, where H1 satisfies H1n = EQ[Un|S10 , . . . , S1n−1, Tn = 1 + u]− EQ[Un|S10 , . . . , S1n−1, Tn = 1 + d] EQ[S˜1n|S10 , . . . , S1n−1, Tn = 1 + u]− EQ[S˜1n|S10 , . . . , S1n−1, Tn = 1 + d] = EQ[Un|S10 , . . . , S1n−1, Tn = 1 + u]− EQ[Un|S10 , . . . , S1n−1, Tn = 1 + d] (u−d) 1+i S˜ 1 n−1 , for n ≥ 1, H10 = H11 , and H0 satisfies H00 = U0 −H10S10 , H0n = H 0 n−1 − S˜1n−1(H1n −H1n−1), for n ≥ 1.






















































































































































































































































































































































































































































































































































































































































































































































































































































































































































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