1 MATH3075 Mathematical Finance (Normal) Due by 4 pm on Thursday, September 17, 2015 1. [10 marks] Elementary market model. Consider a single-period two-state market model M = (B, S) with the two dates: 0 and 1. Assume that the stock price S0 at time 0 is equal to $27 per share, and that the price per share will rise to either $28 or $31 at the end of a period, that 3 is, at time 1, with probabilities 4 and 1 respectively. Assume that the one-period 4 simple interest rate r equals 10%. We consider call and put options written on the stock S with the strike price K = $28.5 and the expiry date T = 1. (a) Construct unique replicating strategies for these options as vectors (0 , 1 ) R2 such that V1 (0 , 1 ) = 0 B1 + 1 S1 . Note that V1 (0 , 1 ) = V1 (x, ) where x = 0 + 1 S0 and = 1 . (b) Compute the arbitrage prices of call and put options through replicating strategies. (c) Check that the put-call parity relationship holds. (d) Find the unique risk-neutral probability P for the market model M and recompute the arbitrage prices of call and put options using the risk-neutral valuation formula. (e) How will the replicating portfolios and arbitrage prices of the call and put options change if we assume that the interest rate r equals 5%? 2. [10 marks] Single-period market model. Consider a single-period security market model M = (B, S) on a nite sample space = {1 , 2 , 3 }. Assume that the savings account B equals B0 = B1 = 1 and the stock price S satises S0 = 6 and S1 = (S1 (1 ), S1 (2 ), S1 (3 )) = (3, 6, 8). The real-world probability P is such that P(i ) = pi > 0 for i = 1, 2, 3. (a) Show that the model M is arbitrage-free by checking that no arbitrage opportunity exists. (b) Find the class M of all risk-neutral probability measures for the model M. Is this market model complete? (c) Find the class A of all attainable contingent claims. (d) Check that the contingent claim X = (X(1 ), X(2 ), X(3 )) = (2, 4, 7) is attainable and compute its arbitrage price 0 (X) using the replicating strategy for X. (e) Consider again the contingent claim X = (2, 4, 7). Show that the expected value ( ) X EQ B1 does not depend on the choice of a risk-neutral probability Q M. ASSIGNMENT 1 MATH3975 Mathematical Finance (Advanced) Due by 4 pm on Thursday, September 17, 2015 1. [10 marks] Elementary market model. Consider a single-period two-state market model M = (B, S) with parameters S0 > 0 and 0 < d < 1 + r < u. (a) Find the probability measure P such that ( ) B1 B0 . EP = S1 S0 (b) Compute the Radon-Nikodym density of P with respect to the risk-neutral probability P. (c) Let X be any contingent claim. Show that the price 0 (X) satises ( ) X . 0 (X) = S0 EP S1 (d) Let X = C1 = (S1 K)+ for some K > 0. Show that the price C0 equals C0 = S0 P(S1 > K) K(1 + r)1 P(S1 > K). (e) Express the arbitrage price P0 of the put option in terms of probability measures P and P and derive the put-call parity relationship using part (d). 2. [10 marks] Single-period market model. Consider a single-period security market model M = (B, S) on a nite sample space = {1 , 2 , 3 }. We assume that the savings account B equals B0 = 1 and B1 = 4. The stock price S satises S0 = 2.5 and S1 = (S1 (1 ), S1 (2 ), S1 (3 )) = (18, 10, 2). The real-world probability P is such that P(i ) = pi > 0 for i = 1, 2, 3. (a) Find the class M of all martingale measures for the model M. Is the market model M arbitrage free? Is this model complete? (b) Find the replicating strategy for the claim X = (X(1 ), X(2 ), X(3 )) = (5, 1, 3) and compute the arbitrage price 0 (X) at time 0. (c) Compute the arbitrage price 0 (X) using the risk-neutral valuation formula with an arbitrary martingale measure Q from M. (d) Show that the contingent claim Y = (Y (1 ), Y (2 ), Y (3 )) = (10, 8, 2) is not attainable and nd the range of arbitrage prices 0 (Y ). (e) For the contingent claim Z = (20, 16, 4), nd the minimal initial endowment x for which there exists a portfolio (x, ) with V0 (x, ) = x and such that the inequality V1 (x, )(i ) Z(i ) holds for i = 1, 2, 3. ASSIGNMENT 2 MATH3075 Mathematical Finance (Normal) Due by 4 pm on Thursday, 22 October 2015 1. [10 marks] The CRR model: European claim. Consider the CRR model of stock price S with T periods and parameters d < 1 + r < u, where r is the one-period interest rate. We denote by P the unique S martingale measure for the discounted stock price S = B . (a) Consider the digital put and call options with expiry date T and the payoffs given by 1 if ST K, CT (K) := 1[K,) (ST ) = 0 if ST < K, and PT (K) := 1[0,K) (ST ) = 0 1 if ST K, if ST < K, respectively. Establish the put-call parity relationship for digital options for a xed strike K > 0. (b) Consider the digital call option with expiry date T and the payoff CT (K) = 1[K,) (ST ) where the strike K = S0 (1 + r)T . Using the risk-neutral valuation formula 1 t (X) = Bt EP (XBT | Ft ), compute the arbitrage price C0 (K) for this option at time t = 0. (c) Let X and Y be two European claims with maturity T in the CRR model. Assume that the equality U (X) = U (Y ) holds for some date U such that 0 < U < T . Does the equality t (X) = t (Y ) holds for every 0 t < U ? (d) Let X and Y be two European claims with maturity T in the CRR model. Assume that the equality U (X) = U (Y ) holds for some date U such that 0 < U < T . Does this assumption imply that t (X) = t (Y ) for every U < t T? 2. [10 marks] The CRR model: American claim. u d Let M = (B, S) be the CRR model with T = 2 and S0 = 100, S1 = 120, S1 = 90. Assume that the interest rate r = 0. Consider the American put option with the reward process g(t, St ) = (Kt St )+ where K0 = 105, K1 = 116, K2 = 111. (a) Compute S2 and nd the risk-neutral probability measure P for the model. (b) Compute the price process P a of this option using the recursive relationship a Pta = max g(t, St ), (1 + r)1 EP Pt+1 | Ft a with the terminal condition P2 = g(2, S2 ). (c) Find the rational exercise time 0 for the holder of this option. (d) Find the replicating strategy for the issuer of the option up to the rational exercise time 0 and check that Vt () = Pta (X) for all t 0 . ASSIGNMENT 2 MATH3975 Mathematical Finance (Advanced) Due by 4 pm on Thursday, 22 October 2015 1. [10 marks] Complete model: European claim. Consider a discrete time, complete, arbitrage-free model M = (B, S) where Bt = (1 + r)t for every t = 0, 1, . . . , T and S is a nonnegative stock price process dened on a nite probability space (, F, P). Let X = g(ST ) be a path-independent European claim with maturity T where the payoff function g : R+ R is twice b continuously differentiable and a |g (z)| dz < + for all 0 a < b. (a) Using the integration by parts formula, show that for every x, y R+ , y g(x) = g(y) + g (y)(x y) + 0 + (z x) g (z) dz + y (x z)+ g (z) dz. (b) Let C0 (K) and P0 (K) are prices at time 0 of European call and put options with strike K. Using the risk-neutral valuation formula, show that the arbitrage price 0 (X) of the claim X = g(ST ) satises for any xed y R+ 0 (X) = y 1 BT g(y) + g (y)(C0 y P0 ) + y P0 (K)g (K) dK + 0 C0 (K)g (K) dK. y (c) Let us take g(x) = ax + b for some real numbers a and b. Using the formula derived in (b) with some y > 0 and the put-call parity relationship, show 1 that the price of X = g(ST ) at time 0 equals aS0 + bBT . 2 (d) Apply the formula derived in (b) to the claim X = ST and y = 0 to compute the price 0 (X) in terms of prices of call options with all strikes K > 0. (e) Assume that the stock price S is strictly positive. Consider the claim X = 1 ln(ST /F0 ) where F0 = S0 BT . Express the price 0 (X) in terms of prices of put and call options by taking y = F0 . 2. [10 marks] The CRR model: American claim. Let M = (B, S) be the CRR model with the horizon date T = 2 with r = 0 and u d S0 = 8, S1 = 11, S1 = 7. Consider an American claim with maturity date T = 2 u d and the reward process g(t, St ) given by: g(0, S0 ) = g(1, S1 ) = g(1, S1 ) = 12 and uu ud du dd g(2, S2 ) = 14, g(2, S2 ) = g(2, S2 ) = 10, g(2, S2 ) = 18. (a) Compute the stock price S2 and nd the unique martingale measure P. (b) Compute the arbitrage price (X a ) for this claim using the recursive relationship: 2 (X a ) = g(2, S2 ) and t (X a ) = max g(t, St ), EP (1 + r)1 t+1 (X a ) | Ft . (c) Check whether the arbitrage price (t (X a ); t = 0, 1, 2) is a martingale under P with respect to the ltration F. (d) Find the rational exercise time 0 for the holder of the American claim. (e) Find the replicating strategy for the claim up to time 0 and check that Vt () = t (X a ) for all t 0 . MATH3075/3975 Financial Mathematics Week 10: Solutions Exercise 1 Consider the CRR model M = (B, S) with the horizon date T = 2, the risk-free rate r = 0.1, and the following values of the stock price S at times t = 0 and t = 1: S0 = 10, u S1 = 13.2, d S1 = 9.9. Let X be a European contingent claim with the maturity date T = 2 and the payo at maturity given by the formula ( )+ X = min(S1 , S2 ) 10 . (a) The stock price and the martingale measure are given by the following diagram uu S2 7 = 17.424 o 1/3 oooo oo oo ooo P(1 ) = 1/9 u S1 = 13.2 y A 1/3 yyy yy2/3 yyy yyy ' ud S2 = 13.068 P(2 ) = 2/9 du S2 7 = 13.068 P(3 ) = 2/9 S0 = Y 10 Y YY YY YY YY 2/3 YY YY YY YY Y\u001d o 1/3 oooo o ooo ooo d S1 = 9.9 y yyy 2/3 yyy yyy yy' dd S2 = 9.801 1 P(4 ) = 4/9 (b) To show that X is a path-dependent contingent claim, it suces to note that X = (X(1 ), X(2 ), X(3 ), X(4 )) = (3.2, 3.068, 0, 0). Hence for the same value of the stock price S2 (2 ) = 13.068 = S2 (3 ) we have two possible values of the payo, depending on the level of the stock price at time 1 (note that S1 (2 ) = 13.2 = 9.9 = S1 (3 )). We thus see that no function f : R R such that X = f (ST ) exists. We conclude that X is a path-dependent claim (an example of a lookback option). (c) Our next goal is to compute the arbitrage price of X using the riskneutral valuation formula, for t = 0, 1, 2, ( ) X t (X) = Bt EP Ft . BT The price process t (X) of the European claim X is given by uu 2 (X) = 3.2 k5 1/3 kkkkk kk kkk kkk 1 u 1 (X) = 2.8291 2/3 ) ; xx xx xx xx 1/3 xxx xx xx xx x xx xx x ud 2 (X) = 3.068 2 du 5 2 (X) = 0 3 0 (X) = 0.8573 p pp pp pp pp pp2/3 pp pp pp pp pp p# k 1/3 kkkkk kk kkk kkk d 1 (X) = 0 2/3 ) dd 2 (X) = 0 4 (d) We start by considering the replicating strategy (0 , 1 ) at time t = 1. 1 1 - Assume rst that the stock price has risen during the rst period. Then we need to solve 1.10 + 17.4241 = 3.2, 1 1 1.10 + 13.0681 = 3.068. 1 1 2 Hence (0 , 1 ) = (2.4291, 0.0303) if the stock price has risen du1 1 ring the rst period, that is, for {1 , 2 }. We check that u u V1u () = 0 + 1 S1 = 2.4291 + 0.0303 13.2 = 2.8291 = 1 (X). 1 1 - Let us now assume that the stock price has fallen during the rst period. Then we need to solve 1.10 + 13.0681 = 0, 1 1 1.10 + 9.8011 = 0. 1 1 Hence (0 , 1 ) = (0, 0) if the stock price has fallen during the rst 1 1 d period, that is, for {3 , 4 }. Obviously, V1d () = 0 = 1 (X). In the second step, we search for the replicating portfolio at time t = 0. We need to solve 1.10 + 13.21 = 2.8291, 0 0 1.10 + 9.91 = 0. 0 0 Hence (0 , 1 ) = (7.7157, 0.8573) for all s. We check that 0 0 V0 () = 0 + 1 S0 = 7.7157 + 0.8573 10 = 0.8573 = 0 (X). 0 0 The evolution of the portfolio is given by the following diagram V uu () = 3.2 42 iiii iiii iiii iiii (0 , 1 ) = (2.4291, 0.0303) 1 1 7 ppp ppp p ppp ppp p ppp ppp p ppp ppp pp * V2ud () = 3.068 (0 , 1 ) = (7.7157, 0.8573) 0 0 xxx xxx xxx xxx xxx xxx xxx xxx xxx xx' ii4 iiii iiii ii iiii V2du () = 0 (0 , 1 ) = (0, 0) 1 1 * 3 V2dd () = 0 Exercise 2 (MATH3975) We apply the CRR call option pricing formula T T (T ) (T ) K k T k pk (1 p)T k C0 = S0 p (1 p) k k (1 + r)T k=k k=k where k is the smallest integer k such that ) ( (u) K k log . > log d S0 dT We consider the European call option with strike price K = 10 and maturity date T = 5 years. We assume that the initial stock price S0 = 9, the risk-free simple interest rate is r = 0.01 and the stock price volatility equals = 0.1 per annum. The CRR parametrization for u and d with t = 1 gives 1 u = e t = e = 1.105171, d = = 0.904837. u Consequently, 1+rd pu p= = 0.524938, p = = 0.574402. ud 1+r (a) We rst compute the call price at time 0. It is easy to check that k = k(S0 , T ) = 4 and thus 5 ( ) 5 (5) 10 5 k k 5k p (1 p)5k = 0.552247. C0 = 9 p (1 p) (1.01)5 k=4 k k k=4 Hence the price of the option at time 0 equals C0 = 0.552247. (b) We will now compute the prices of the option at time t = 1. The time to maturity is now equal to T 1 = 4. If S1 = uS0 = 9.9465 then we obtain the new value of k(uS0 , T 1) = 3 and thus in that case 4 4 ( ) (4) 10 4 k u k 4k C1 = 9.9465 p (1p) p (1p)4k = 0.920649. k (1.01)4 k=3 k k=3 If S1 = dS0 = 8.1435 then we obtain the value of k(dS0 , T 1) = 4 and thus 4 4 ( ) (4) 10 4 k k 4k d p (1 p) p (1 p)4k C1 = 8.1435 k (1.01)4 k=4 k k=4 = 8.1435 (0.574402)4 10 (0.524938)4 = 0.156793. 4 (1.01) In Week 11, the same problem will be solved using the backward induction method. 4 MATH3075/3975 Financial Mathematics Week 11: Solutions Exercise 1 Assume the CRR model M = (B, S) with T = 3, the stock u d price S0 = 100, S1 = 120, S1 = 90, and the risk-free interest rate r = 0.1. We consider the American put option on the stock S with the maturity date T = 3 and the constant strike price K = 121. The option has the following reward process, for t = 0, 1, 2, 3, g(St , t) = (K St )+ = (121 St )+ . (a) We rst compute the arbitrage price Pta of this option for t = 0, 1, 2, 3. We start by noting that the unique risk-neutral probability measure P satises p= d (1 + r)S0 S1 2 1+rd 110 90 = = . = u d ud 120 90 3 S1 S1 The stock price process St is given by 172.8 v; vv vv vv vv 144 rr x< rr xx rr x rr xx x r# x 120pp 129.6 v; x< pp vv xx pp x vv pp xx vv vv xx " 100pp 108 rr < rr pp xx rr pp xx rr pp xx r# " xx 90 pp 97.2 ; vv pp vv pp v pp vv " vv 81 rr rr rr rr r# 72.9 1 The price process Pta is represented by the following diagram <0 yy yy y yy yy v; 0 iii vv ii vv v ii vv ii vv " 3.9394 rr ; y< 0 y rr vv yy rr vv yy rr vv y v r# yy vv 21 rr ; 13 ii rr vv ii rr vv ii v rr ii vv r# v " v 31 rr 23.8 < rr yy rr yy rr yy r# yy 40 ii ii ii ii " 48.1 and the rational exercise decisions are given by (1 = exercise) 1 @ @0a aaa aa a\u001e 1 aa 1 @ aa aa a\u001e 1 aa aa aa a\u001e 1 @ aa aa aa a\u001e @1 aa aa aa a\u001e 1 @ 1 aa aa aa a\u001e 1 2 (b) The rational exercise times for the holder of the American put option are: 0 = 0, 1 () = 2 for {1 , 2 , 3 , 4 }, 1 () = 1 for {5 , 6 , 7 , 8 }, 2 = 2 and 3 = 3. a (c) Suppose that the option was sold for the price P0 = 21 and it was not immediately exercised by its holder. The issuer may establish the replicating portfolio for the European claim X = (3.9394, 31) with maturity 1 by solving the following equations 1.1 0 + 120 1 = 3.9394, 0 0 0 1 1.1 0 + 90 0 = 31. We nd that (0 , 0 ) = (101.9835, 0.90202) and thus the initial we0 1 alth needed to establish this portfolio equals V0 () = 101.9835 0.90202 100 = 11.7815. The dierence 21 11.7815 is the net prot of the issuer at time 0. This argument can be extended to any date t. Exercise 2 We consider the CRR binomial model with the risk-free rate r = 0 and the following values of the stock price S at times t = 0 and t = 1: S0 = 100, u S1 = 120, d S1 = 90. We examine the American call option with maturity date T = 3 and the following reward process g(St , t) = (St Kt )+ , where the variable strike Kt satises K0 = K1 = 100, K2 = 105, K3 = 110. (a) We rst compute the arbitrage price Xta of this option for t = 0, 1, 2, 3. The unique risk-neutral probability measure P satises d 1+rd (1 + r)S0 S1 100 90 1 p= = = = . u d ud 120 90 3 S1 S1 3 The stock price process St is given by 172.8 ; vv vv v vv vv 144 rr x< rr xx rr x rr xx x r# x 120pp 129.6 v; x< pp vv xx pp x vv pp xx vv xx " vv 100pp 108 rr x< rr pp xx rr pp x rr pp xx x r# " x 90 pp 97.2 v; pp vv pp vv pp vv " vv 81 rr rr rr rr r# 72.9 The price process Xta of the call option is given by 62.8 t: tt tt tt tt 9 39 tt rr tt tt rrr tt rrr t$ rr r 20 vv 19.6 9 t: vv rr tt vv rr tt vv rr tt vv rr tt % rr 8.1185v 6.5333t vv t rr9 tt vvv tt rrr vvv tt rrr v% tt rr $ 2.1778v :0 vv tt t vvv tt vvv tt vv tttt % 0 ttt tt tt tt tt $ 0 4 The rational exercise decisions of the holder are given by 1 @ @1a aaa aa a\u001e 0 aa 0 @ aa aa a\u001e 0 aa aa aa a\u001e 1 @ aa aa aa a\u001e @1 aa aa aa a\u001e 1 @ 1 aa aa aa a\u001e 1 (b) The rational holder should exercise the American option at time t = 1 whenever the stock price rises during the rst period. Otherwise, he should not exercise the option till time 2. Hence the rational exercise time 0 is a stopping time 0 : {0, 1, 2} given by 0 () = 1 for {1 , 2 , 3 , 4 }, 0 () = 2 for {7 , 8 }, 0 () = 3 for {5 , 6 }. (c) We now take the position of the issuer of the option: - At t = 0, we need to solve 0 + 120 1 = 20, 0 0 0 + 90 1 = 2.1778. 0 0 Hence (0 , 1 ) = (51.2888, 0.5941) for all s. 0 0 - If the stock price has risen during the rst period, the option is exercised by its holder. Hence we do not need to compute the strategy at time 1 for {1 , 2 }. 5 - If the stock price has fallen during the rst period, we need to solve 0 + 108 1 = 6.5333, 1 1 0 + 81 1 = 0. 1 1 Hence (0 , 1 ) = (19.599, 0.2420). 1 1 - If the stock price has fallen twice then the option is exercised and has value 0. - If the stock price has fallen during the rst period and has risen during the second period then we need to solve 0 + 129.6 1 = 19.6, 2 2 0 1 2 + 97.2 2 = 0. Hence (0 , 1 ) = (58.8, 0.6050). 1 1 We conclude that the replicating strategy = (0 , 1 ) is represented by the following diagram V1u () = 20 nn7 nnn nn nnn nnn n nnn nnn nn nnn (0 , 1 ) = (51.29, 0.59) 0 0 www www www www www www www www www & (0 , 1 ) = (58.8, 0.61) 2 2 hhh4 hhhh hhhh hhh hhhh (0 , 1 ) = (19.60, 0.24) 1 1 * 6 V2dd () = 0 Exercise 3 (MATH3975) We consider the European call option with strike price K = 10 and maturity date T = 5 years. We assume that the initial stock price S0 = 9, the risk-free interest rate is r = 0.01 and the stock price volatility equals = 0.1 per annum. We use the CRR parametrization for u and d with t = 1, that is, we set u = e t = 1.105171, d= 1 = 0.904837. u Consequently, 1+rd = 0.524938. ud Using the backward induction method, we obtain the following results. p= (a) The price at 0 of the European call option equals C0 = 0.5522. The detailed computations are summarised on the next page. (b) The price at 0 of the European put option equals P0 = 1.0669. For details, see the next page. (c) The put-call parity at time t = 0 reads C0 P0 = 0.5522 1.0669 = 0.5147 = 9 K 10 = S0 . 5 (1.01) (1 + r)T a (d) The price at time 0 of the American put option equals P0 = 1.2112. The option should not be exercised at time 0, but it should be exercised by its holder at time 1 if the stock price falls during the rst period. For the full description of the rational exercise decisions of the holder, see the next page. 7 Week 11 Volatility Stock at 0 Strike price Interest 0 .01 0.1 9 10 Exercise 3 Maturity 5 tilde p 1 - tilde p up down 1.105171 0 .904837 0 .524938 0 .475062 2 3 4 1 0 9 9.946538 10.99262 12.14873 l3.42642 8.143537 9 9.946538 10.99262 7.368577 8.143537 9 6.667364 7.368577 6.03288 Year Stock price 5 14.83849 12.14873 9 .946538 8.143537 6.667364 5.458776 European call price 0.552247 0.920649 1.498351 2.357597 3.525432 4.838491 0 .156793 0 .301676 0 .580436 1.116781 2.148729 0 0 0 0 0 0 0 0 0 0 European put price 1.066904 0 .583914 0.211627 0.011828 0 0 1.62306 1.007577 0.436858 0 .025146 0 2.337325 1.659424 0.053462 0 .90099 3.l35597 2.5324l3 1.856463 3.86811 3.332636 4.541224 Put-call parity American put price Exercise decision -0.51466 1.211179 0 .650276 0 .233532 0.011828 0 0 1.856463 1.124461 0.483428 0 .025146 0 2.631423 l.856463 1 0.053462 3.332636 2.631423 l.856463 3.96712 3.332636 4.541224 No No Yes No No Yes No No Yes Yes Yes No Yes Yes Yes Yes Yes Yes Yes Yes Yes Week 11 Exercise 4 Interest 0.05 up down 1.1 Year Stock price Stock at 0 Strike K 25 27 tilde p 1 - tilde p 0.9 0.75 0.25 alpha 0.02 0 25 1 27.5 22.5 2 30.25 24.75 20.25 3 33.275 27.225 22.275 18.225 4 36.6025 29.9475 24.5025 20.0475 16.4025 5 40.26275 32.94225 26.95275 22.05225 18.04275 14.76225 6 44.28903 36.23648 29.64803 24.25748 19.84703 16.23848 13.28603 Upper payoff H_t 2.02 0.02 4.52 0.02 2.27 6.77 0.02 0.02 4.745 8.795 0.02 0.02 2.5175 6.9725 10.6175 0.02 0.02 0.02 0.02 0.06725 0.02 4.96775 2.762525 8.97725 7.172975 12.25775 10.78153 13.73398 Lower payoff L_t 2 0 4.5 0 2.25 6.75 0 0 4.725 8.775 0 0 2.4975 6.9525 10.5975 0 0 0 0 0.04725 0 4.94775 2.742525 8.95725 7.152975 12.23775 10.76153 13.71398 Game option price X^g_t Exercise decision 2 L 0.02 0.009022 0.005964 0.001683 0.000408 6.43E-05 4.5 2.25 0.02 0.02 0.005847 0.001519 6.75 4.725 2.4975 0.06725 0.02 8.775 6.9525 4.94775 2.742525 10.5975 8.95725 7.152975 12.23775 10.76153 13.71398 H L N L L N H L L N H L L L N N H L L L N N H L L L L 7 48.71793 39.86012 32.61283 26.68322 21.83173 17.86232 14.61463 11.95742 8 53.58972 43.84613 35.87411 29.35154 24.0149 19.64855 16.07609 13.15316 10.76168 9 58.94869 48.23075 39.46152 32.2867 26.41639 21.61341 17.6837 14.46848 11.83785 9.685512 10 64.84356 53.05382 43.40767 35.51537 29.05803 23.77475 19.45207 15.91533 13.02163 10.65406 8.716961 11 71.32792 58.35921 47.74844 39.06691 31.96383 26.15223 21.39728 17.50686 14.3238 11.71947 9.588657 7.845265 12 78.46071 64.19513 52.52328 42.9736 35.16022 28.76745 23.537 19.25755 15.75618 12.89142 10.54752 8.629791 7.060738 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.336777 0.02 0.02 0.02 5.188272 3.0051 0.60361 0.02 9.157677 7.371445 5.40659 3.245249 12.40537 10.94391 9.336301 7.567931 15.06258 13.86684 12.55152 11.10467 16.25832 15.18215 13.99837 17.33449 16.36594 18.30304 0.02 0.02 0.02 0.02 0.02 0.867774 5.622724 9.513138 12.6962 15.30053 17.43134 19.17474 0.02 0.02 0.02 0.02 0.02 0.02 3.482996 7.762451 11.26382 14.12858 16.47248 18.39021 19.95926 0 0 0 0 0 0 0 0 0 0 0 0 0.316777 0 0 0 5.168272 2.9851 0.58361 0 9.137677 7.351445 5.38659 3.225249 12.38537 10.92391 9.316301 7.547931 15.04258 13.84684 12.53152 11.08467 16.23832 15.16215 13.97837 17.31449 16.34594 18.28304 0 0 0 0 0 0.847774 5.602724 9.493138 12.6762 15.28053 17.41134 19.15474 0 0 0 0 0 0 3.462996 7.742451 11.24382 14.10858 16.45248 18.37021 19.93926 0 0.00027 0.005572 0.336777 5.168272 9.137677 12.38537 15.04258 L N N H L L L L 0 0 0.001134 0.02 2.9851 7.351445 10.92391 13.84684 16.23832 L L N H L L L L L 0 0 0 0.004762 0.60361 5.38659 9.316301 12.53152 15.16215 17.31449 L L L N H L L L L L 0 0 0 0 0.02 3.225249 7.547931 11.08467 13.97837 16.34594 18.28304 L L L L H L L L L L L 0 0 0 0 0 0.847774 5.602724 9.493138 12.6762 15.28053 17.41134 19.15474 L L L L L L L L L L L L 0 0 0 0 0 0 3.462996 7.742451 11.24382 14.10858 16.45248 18.37021 19.93926 L L L L L L L L L L L L L MATH3075/3975 Financial Mathematics School of Mathematics and Statistics University of Sydney Semester 2, 2015 Tutorial sheet: Week 10 Background: Section 4.2 - European Options in the CRR Model. Exercise 1 Consider the CRR model M = (B, S) with the horizon date T = 2, the risk-free rate r = 0.1, and the following values of the stock price S at times t = 0 and t = 1: S0 = 10, u S1 = 13.2, d S1 = 9.9. Let X be a European contingent claim with the maturity date T = 2 and the payo at maturity given by the formula ( )+ X = min(S1 , S2 ) 10 . (a) Find the martingale measure P for the market model M = (B, S). (b) Show explicitly that X is a path-dependent contingent claim. (c) Let Ft = FtS = (S0 , . . . , St ) for t = 0, 1, 2. Compute the arbitrage price of X using the risk-neutral valuation formula, for t = 0, 1, 2, ( ) X t (X) = Bt EP Ft . BT (d) Find the replicating strategy t , t = 0, 1, for the claim X and check that the wealth process V () of the unique replicating strategy for X coincides with the price process (X) computed in part (c). Exercise 2 (MATH3975) We take for granted the CRR call option pricing formula T T (T ) (T ) K k T k C0 = S0 p (1 p) pk (1 p)T k k k (1 + r)T k=k k=k where k is the smallest integer k such that ( ) (u) K > log . k log d S0 dT 1 (a) Compute the arbitrage price C0 of the European call option with strike price K = 10 and maturity date T = 5 years. Assume that the initial stock price equals S0 = 9, the risk-free interest rate is r = 0.01 and the stock price volatility equals = 0.1 per annum. Use the CRR parametrization for the parameters u and d, that is, set u = e t , d= 1 , u with the time increment t = 1. u d (b) Compute the prices C1 and C1 at time t = 1 for the same option using the CRR call option pricing formula. 2 MATH3075/3975 Financial Mathematics School of Mathematics and Statistics University of Sydney Semester 2, 2015 Tutorial sheet: Week 11 Background: Section 4.4 - American Options in the CRR Model. Exercise 1 Assume the CRR model M = (B, S) with T = 3, the stock u d price S0 = 100, S1 = 120, S1 = 90, and the risk-free interest rate r = 0.1. Consider the American put option on the stock S with the maturity date T = 3 and the constant strike price K = 121. (a) Find the arbitrage price Pta of the American put option for t = 0, 1, 2, 3. (b) Find the rational exercise times t , t = 0, 1, 2, 3 for the holder of the American put option. (c) Show that there exists an arbitrage opportunity for the issuer if the option is not rationally exercised by its holder. Exercise 2 Assume the CRR model M = (B, S) with T = 3, the stock u d price S0 = 100, S1 = 120, S1 = 90, and the risk-free interest rate r = 0. Consider the American call option with the expiration date T = 3 and the running payo g(St , t) = (St Kt )+ , where the variable strike price equals K0 = K1 = 100, K2 = 105 and K3 = 110. (a) Find the arbitrage price Xta of the American call option for t = 0, 1, 2, 3. (b) Find the holder's rational exercise times 0 for the American call option. (c) Find the issuer's replicating strategy for the American call option up to the rational exercise time 0 Exercise 3 Consider the CRR binomial lattice model M = (B, S) with the initial stock price S0 = 9, the interest rate r = 0.01 and the volatility equals = 0.1 per annum. Use the CRR parametrization for u and d, that is, u = e t , with the time increment t = 1. 1 d= 1 , u We consider call and put options with the expiration date T = 5 years and strike K = 10. (a) Compute the price process Ct , t = 0, 1, . . . , 5 of the European call option using the binomial lattice method. (b) Compute the price process Pt , t = 0, 1, . . . , 5 for the European put option. (c) Does the put-call parity relationship hold for t = 0? (d) Compute the price process Pta , t = 0, 1, . . . , 5 for the American put option. Will the American put option be exercised before the expiration date T = 5 by its rational holder? Exercise 4 (MATH3975) Consider the game option (See Section 4.5) with the expiration date T = 12 and the payo functions h(St ) and (St ) where Ht = h(St ) = (K St )+ + and Lt = (St ) = (K St )+ where = 0.02 and K = 27. Assume the CRR model with d = 0.9, u = 1.1, r = 0.05 and S0 = 25. (a) Compute the arbitrage price process (Xtg )T for the game option using t=0 the recursive formula, for t = 0, 1, . . . , T 1, { [ ( gu )]} gd Xtg = min h(St ), max (St ), (1 + r)1 pXt+1 + (1 p)Xt+1 with T (X g ) = (ST ). (b) Find the optimal exercise times 0 and 0 for the holder and the issuer of the game option. Recall that { } 0 = inf t {0, 1, . . . , T } | Xtg = (St ) and { } 0 = inf t {0, 1, . . . , T } | Xtg = h(St ) . 2