ASSIGNMENT 1: SOLUTIONS MATH3075 Mathematical Finance (Normal) 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 $54 per share, and that the price per share will rise to either $56 or $62 at the end of a period, that is, at time 1, with equal probability. Assume that the one-period simple interest rate r equals 10%. We consider call and put options written on the stock S, with the strike price K = $59 and the expiry date T = 1. (a) Construct unique replicating strategies for these options as vectors (П•0 , П•1 ) в€€ R2 such that V1 (П•0 , П•1 ) = П•0 B1 + П•1 S1 . Note that V1 (П•0 , П•1 ) = V1 (x, П•) where x = П•0 + П•1 S0 and П• = П•1 . (b) Compute 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 15%. 2. [10 marks] Single-period market model. Consider a single-period security market model M = (B, S) on a finite space в„¦ = {П‰1 , П‰2 , П‰3 }. Assume that the savings account B equals B0 = B1 = 1 and the stock price S satisfies S0 = 7 and S1 = (S1 (П‰1 ), S1 (П‰2 ), S1 (П‰3 )) = (4, 6, 9). The real-world probability P is such that P(П‰i ) = pi > 0 for i = 1, 2, 3. (a) Check directly that the model M is arbitrage-free. (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(П‰1 ), X(П‰2 ), X(П‰3 )) = (2, 4, 7) is attainable and compute its arbitrage price ПЂ0 (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 в€€ M. Solution to Question 1 Note that П•0 is the amount invested in the bank account. The initial wealth of our portfolio is thus given by x = V0 (x, П•) = П•0 + П•1 S0 and the value of the portfolio at time t = 1 is given by V1 (x, П•) = (x в€’ П•1 S0 )(1 + r) + П•1 S1 = П•0 (1 + r) + П•1 S1 . (a) For the call option with the payoff C1 = (S1 в€’ 59)+ , the unique replicating strategy (П•0 , П•1 ) can be found by solving the following equations: ( ) ( 0) ( ) 1.1 62 3 П• = =в‡’ (П•0 , П•1 ) = (в€’28/1.1, 1/2). 1 1.1 56 П• 0 For the put option with the payoff P1 = (59 в€’ S1 )+ , the unique replicating strategy (П•0 , П•1 )is computed through the following system of linear equations: ( ) ( 0) ( ) 1.1 62 П• 0 = =в‡’ (П•0 , П•1 ) = (31/1.1, в€’1/2). 1 1.1 56 П• 3 (b) Using part (a), we can compute the price of the put and the call. We know that the initial endowment x of the replicating portfolio is given by x = П•0 + П•1 S0 . Hence the prices of the put and call options are equal to C0 = в€’28/1.1 + 27 = 1.5454, P0 = 31/1.1 в€’ 27 = 1.1818. (c) On the one hand, we have C0 в€’ P0 = 1.5454 в€’ 1.1818 = 0.3636. On the other hand, we obtain S0 в€’ K/(1 + r) = 54 в€’ 59/1.1 = 0.3636. Hence the put call-parity relationship holds. (d) Recall that P(П‰1 ) = 1+rв€’d , uв€’d P(П‰2 ) = 1 в€’ P(П‰1 ). Hence the unique risk-neutral probability measure is P(П‰1 ) = 0.56667 and P(П‰2 ) = 0.43333. The arbitrage prices can be computed using the risk-neutral valuation formula and, of course, the prices should also be 10/9 for both options. For instance, the price of the call satisfies ( ) ( ) C1 1 1 C0 = EP = EP (S1 в€’ K)+ = Г— 0.5667 Г— 3 = 1.5454. 1+r 1+r 1.1 (e) The new replicating strategy (П•0 , П•1 ) for the call option is given by ( ) ( 0) ( ) 3 1.05 62 П• = =в‡’ (П•0 , П•1 ) = (в€’28/1.05, 1/2). 1 П• 0 1.05 56 For the put option, we obtain ( ) ( 0) ( ) 0 1.05 62 П• = =в‡’ (П•0 , П•1 ) = (31/1.05, в€’1/2). 1 П• 3 1.05 56 The new price of the call is C0 = 0.3333 and price of the put is P0 = 2.5238. Hence C0 в€’ P0 = в€’2.1904 and S0 в€’ K/(1 + r) = в€’2.1904. Solution to Question 2 (a) If x = 0 then the wealth at time 1 satisfies, for every П•1 в€€ R, V1 (0, П•1 ) = П•1 (S1 в€’ (1 + r)S0 ) = П•1 ((4, 6, 9) в€’ (7, 7, 7)) = (в€’3П•1 , в€’П•1 , 2П•1 ). It is not possible that all three numbers are non-negative, unless П•1 = 0. In that case, V1 (0, П•1 ) = (0, 0, 0). Hence no arbitrage opportunity exists in the model M = (B, S) and thus M is arbitrage-free. (b) We need to solve q1 + q2 + q3 = 1, 0 < qi < 1 and (since r = 0) EQ (S1 ) = EQ (S1 ) = 4q1 + 6q2 + 9q3 = S0 = 7. The equation above is equivalent to EQ (S1 в€’ S0 ) = в€’3q1 в€’ q2 + 2q3 = 0. Let q3 = О±. Then q1 = 3О±в€’1 , q2 = 3в€’5О± and thus the inequalities 0 < q1 < 1 and 2 2 0 < q2 < 1 hold when 1/3 < О± < 3/5. Consequently, the class M of all risk-neutral probability measures is given by { } 3О± в€’ 1 3 в€’ 5О± M = (q1 , q2 , q3 ) q1 = , q2 = , q3 = О±, О± в€€ (1/3, 3/5) . 2 2 Since the risk-neutral probability measure is not unique, we conclude market model M is not complete. (c) The class A of all attainable claims is the two-dimensional linear subspace of R3 spanned by the vectors 1 = (1, 1, 1) and S1 = (4, 6, 9). Formally, it equals A = {X в€€ R3 | X = a1 1 + a2 S1 for some a1 , a2 в€€ R} or, equivalently A = {X в€€ R3 | X = c1 B1 + c2 S1 for some c1 , c2 в€€ R}. Any other equivalent representation for the plane A is fine as well. (d) We need to solve equations П•0 + 4П•1 = 2, П•0 + 6П•1 = 4, П•0 + 9П•1 = 7. The strategy (П•0 , П•1 ) = (в€’2, 1) replicates X and thus X is attainable. The price of X at time t = 0 thus equals ПЂ0 (X) = V0 (П•) = в€’2B0 + S0 = 5. (e) For every 1/3 < О± < 3/5, we obtain (recall that B1 = 1) ( ) X 3О± в€’ 1 3 в€’ 5О± EQ =2 +4 + 7О± = 5, B1 2 2 which shows that the expected value does not depend on the choice of a risk-neutral probability measure Q from the class M.

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