MATH3075/3975 FINANCIAL MATHEMATICS TUTORIAL PROBLEMS

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1 MATH307/37 FINANCIAL MATHEMATICS TUTORIAL PROBLEMS School of Mathematics and Statistics Semester, 04

2 Tutorial problems should be used to test your mathematical skills and understanding of the lecture material. Some of these problems are similar to those appearing on the assignments and final examination. The weekly onehour tutorial sessions give you an opportunity to work on these problems with the help of a tutor and to ask any questions you may have. Needless to say that there is a big difference between understanding someone else s answer to a question and being able to produce your own solution. It is very tempting to just read the question and then immediately view the answer, but that is not the best way to use these problems. You are strongly encouraged to first try to solve the problems by yourself and then use the answers to check your work.

3 MATH307/37 Financial Mathematics School of Mathematical Sciences Semester, 04 Tutorial sheet: Week Background: Chapter 6 Probability Review. Exercise Assume that joint probability distribution of the two-dimensional random variable (X, Y ), that is, the set of probabilities is given by: P(X = i, Y = j) = p i,j for i, j =,, 3, p, = /, p, = /, p,3 = 0, p, = /3, p, = 0, p,3 = /6, p 3, = /, p 3, = /8, p 3,3 = /. (a) Compute E P (X Y ), that is, E P (X Y = j) for j =,, 3. (b) Check that the equality E P (X) = E P [ E P (X Y )] holds. (c) Are the random variables X and Y independent? Exercise The joint probability density function f (X,Y ) of random variables X and Y is given by f (X,Y ) (x, y) = y e x/y e y, (x, y) R +, and f (X,Y ) (x, y) = 0 otherwise. (a) Check that f (X,Y ) is a two-dimensional probability density function. (b) Show that E P (X Y = y) = y for all y R +. Exercise 3 Let X be a random variable uniformly distributed over (0, ). Compute the conditional expectation E P (X X < /). Exercise 4 Let X be an exponentially distributed random variable with parameter λ > 0, that is, with the probability density function f X (x) = λ e x λ for all x > 0. Compute the conditional expectation E P (X X > ). Exercise We assume that P(X = ±) = /4, P(X = ±) = /4 and we set Y = X. Check whether the random variables X and Y are correlated and/or dependent. Exercise 6 (MATH37) Let U and V have the same probability distribution and let X = U + V and Y = U V. Examine the correlation and independence of the random variables X and Y (provide relevant examples).

4 MATH307/37 Financial Mathematics School of Mathematics and Statistics Semester, 04 Tutorial sheet: Week 3 Background: Section. Elementary Market Model. Exercise What is the price at time 0 of a contingent claim represented by the payoff h(s ) = S? Give at least two explanations. Exercise Give a proof of the put-call parity relationship. Exercise 3 Compute the hedging strategies for the European call and the European put in Examples.. and... Exercise 4 Consider the elementary market model with the following parameters: r =, S 4 0 =, u = 3, d =, p = 4. Compute the price of the digital 3 call option with strike price K and the payoff function given by {, if S K, h(s ) = 0, otherwise. Exercise Prove that the condition d < + r < u implies that there is no arbitrage in the elementary market model. Exercise 6 (MATH36) Under the assumptions of Section., show that there exists a random variable Z such that the price x of a claim h(s ) can be computed using the formula x = E P (Zh(S )) where the expectation is taken under the original probability measure P. A random variable Z is then called a pricing kernel (note that it does not depend on h). Hint: You may use the fact that the probability measures P and P are equivalent.

5 MATH307/37 Financial Mathematics School of Mathematics and Statistics Semester, 04 Tutorial sheet: Week 4 Background: Section. Single-Period Market Models. Exercise Verify the equality (see Section.) V t := V ( t = x B t n ) ϕ j S j 0 + j= n ϕ j Ŝ j t () for t {0, } with B 0 = and B = + r, and derive the equality j= Ĝ (x, ϕ) = n ϕ j Ŝj = j= n ϕ j (Ŝj Ŝj 0) () j= where Ĝ (x, ϕ) := V (x, ϕ) V 0 (x, ϕ). Exercise We consider the market model M = (B, S, S ) introduced in Example.., but with k = 4 and the stock prices in state ω 4 given by S(ω 4 ) = 0 and S (ω 4 ) = 0. The interest rate equals r =. Stock prices at time t = 0 are given by S0 = and S0 = 0, respectively. Random stock prices at time t = are given in the following table S S ω ω ω 3 ω (a) Compute explicitly the random variables V (x, ϕ), G (x, ϕ), V (x, ϕ) and Ĝ(x, ϕ). (b) Does G (x, ϕ) (or Ĝ(x, ϕ)) depend on the initial endowment x?

6 Exercise 3 Consider the market model M = (B, S, S ) introduced in Exercise. (a) Give an explicit representation for the linear space W R 4. (b) Find explicitly the linear space W R 4. (c) Is the market model M = (B, S, S ) arbitrage free? (d) Find the class M of all risk-neutral probability measures for M using the equality M = W P +. Exercise 4 Consider the market model M = (B, S) with k = 3, n =, r =, S 0 = and the random stock price S given by the table S 60 ω ω ω 3 Find the class M of all risk-neutral probability measures for this market model by making use of Definition..4. Exercise (MATH37) Give a proof of Proposition

7 MATH307/37 Financial Mathematics School of Mathematics and Statistics Semester, 04 Tutorial sheet: Week Background: Section. Single-Period Market Models. Exercise Consider the market model M = (B, S) introduced in Exercise 4 (Week 4). We have k = 3, r =, S 0 =. Moreover, the stock price S is given by the following table S 60 ω ω ω 3 Are there any values for K such that the call option (S K) + represents an attainable contingent claim? Exercise Consider the stochastic volatility model M = (B, S) introduced in Example..3 and assume that 0 r < h. (a) Characterise the class of all attainable contingent claims in M and check whether the model M is complete. (b) Describe the class M of all risk-neutral probability measures for M. (c) Describe the set of all arbitrage prices for the call option (S K) + where the strike K satisfies S 0 ( + l) < K < S 0 ( + h). (d) (MATH37) Assume that r = 0. Check directly whether the call option with strike K such that S 0 ( + l) < K < S 0 ( + h) is attainable and find the range of values of its arbitrage price

8 MATH307/37 Financial Mathematics School of Mathematics and Statistics Semester, 04 Tutorial sheet: Week 6 Background: Section. Single-Period Market Models. Exercise Consider a single-period three-state market model M = (S, B) with the two dates: 0 and. We assume that there are two assets: the savings account B with the initial value B 0 = and a risky stock with the initial price S 0 = 4. The risk-free simple interest rate r equals 0%. Assume that the stock price S satisfies (S (ω ), S (ω ), S (ω 3 )) = (8,, 3) and the real-world probability measure P equals: P(ω ) = 0.3, P(ω ) = 0.3, P(ω 3 ) = 0.4. (a) Show directly that the model M = (B, S) is arbitrage free, that is, no arbitrage opportunities exist in this model. Do not use here the FTAP (Theorem..), but refer instead to Definition..3 in Course Notes. (b) Consider the call option with the expiry date T = and strike price K = 4. Examine the existence of a replicating strategy for this option. (c) Find explicitly the class of all attainable contingent claims. (d) Find the class M of all risk-neutral probability measures Q = (q, q, q 3 ) on the space Ω = (ω, ω, ω 3 ) for the model M. (e) Find all expected values ( ) (S 4) + E Q + r where Q ranges over the class M of all risk-neutral probability measures for the model M. (f) (MATH37) Find the minimal initial endowment x for which there exists a portfolio (x, ϕ) such that the inequality holds for every ω Ω. V (x, ϕ)(ω) (S (ω) 4) +

9 MATH307/37 Financial Mathematics School of Mathematics and Statistics Semester, 04 Tutorial sheet: Week 7 Background: Chapter 3 Multi-Period Market Models. Exercise We consider the conditional expectation E P (X G) where G is generated by a finite partition (A i ) i I of the sample space Ω = {ω,..., ω k }. Specifically, let k = and A = {ω, ω }, A = {ω 3 }, A 3 = {ω 4, ω }. Let the probability measure P be given by P(ω ) = P(ω ) = 0., P(ω 3 ) = 0.3, P(ω 4 ) = 0., P(ω ) = 0.3. Consider the random variable X : Ω R be given by X(ω i ) = i for i =,...,. (a) Find the probability distribution of the random variable X. (b) Compute the conditional expectation E P (X G). (c) Find the probability distribution of the random variable Y := E P (X G). (d) Show that E P (X) = E P (E P (X G)). Exercise (MATH37) We consider the conditional expectation E P (X G) where the σ-field G is generated by a finite partition (A i ) i I of the sample space Ω = {ω,..., ω k }. (a) Show that the conditional expectation E P (X G) satisfies equation (3.) in the course notes, that is, E P (X G)(ω)P(ω), G G. ω G X(ω)P(ω) = ω G (b) Deduce from this equality that E P (X) = E P (E P (X G)).

10 Exercise 3 Consider the two-period market model M = (B, S) with the savings account B given by B 0 =, B = + r, B = ( + r) with r = 0. and the stock price S evolving according to the following diagram S = 0 ω S = S = 6 ω S 0 = S = 4 ω 3 S = 3 3 S = ω 4 (a) Compute the probabilities of the states ω, ω, ω 3, ω 4. (b) Compute the conditional expectation E P (S F S ): (b) using the formula E P (S F S ) = m i= Ai P(A i ) ω A i S (ω)p(ω), (b) using directly the conditional probabilities. (c) Compute E P (S ) directly and using the equality E P (S ) = E P (E P (S F S )).

11 (d) Check whether the following trading strategies ϕ = (ϕ 0, ϕ ) are F S - adapted and self-financing: I t = 0 t = ω ω ω 3 ω 4 ϕ 0 t ϕ t II t = 0 t = ω ω ω 3 ω 4 ϕ 0 t ϕ t III t = 0 t = ω ω ω 3 ω 4 ϕ 0 t ϕ t

12 MATH307/37 Financial Mathematics School of Mathematics and Statistics Semester, 04 Tutorial sheet: Week 8 Background: Chapter 3 Multi-Period Market Models. Exercise We consider the two-period market model M = (B, S) with the savings account B given by B 0 =, B = + r, B = ( + r), where the interest rate r = 0.. The stock price process S is represented by the following diagram S = 0 ω 3 S = 7 S = 6 ω S 0 = 3 S = 4 ω 3 S = 3 3 S = ω 4

13 (a) Find the risk-neutral probability measure Q for the model M = (B, S). (b) Compute the replicating strategy for the digital call option with strike K = 8 and maturity T =, that is, {, if S 8, X = h(s ) = 0, otherwise. Describe the arbitrage price process for the digital call option. (c) Compute the arbitrage price process for the Asian option with the payoff at maturity T = given by the following formula Y = ( ) ( ) + S0 + S + S 4. 3 Exercise (MATH37) We consider a discrete-time stochastic process X = (X t, t = 0,,... ) defined on a probability space (Ω, F, P) endowed with a filtration F = (F t ) t 0. It is assumed throughout that a process X is adapted to the filtration F. (a) Assume that X has independent increments with respect to F, meaning that for any t = 0,,... the increment X t+ X t is independent of the σ-field F t. Show that the process Y given by the formula Y t := X t E P (X t ), t = 0,,..., is a martingale with respect to the filtration F. (b) Let A 0 = 0 and for t = 0,,... A t+ A t = E P (X t+ X t F t ). () (b) Verify that the process Ỹ given by the equality Ỹt := X t A t for t = 0,,... is a martingale. (b) We assume that the process Ŷt := X t Ât for t = 0,,... is a martingale where the process  satisfies:  0 = 0 and Ât+ is F t - measurable for every t = 0,,... (we then say that the process  is F-predictable). Show that  = A where the process A is given by formula () with A 0 = 0.

14 (c) Assume that a process X = (X t, t = 0,,..., T ) represents a gamble, meaning here that if the game is played at time t then the (positive or negative) reward at time t + per one unit of the bet equals X t+ X t. The random amount of the size of a bet is given by an arbitrary F- adapted process H called a gambling strategy. The profits/losses after t rounds of the game when a gambling strategy H is followed are given by the following equality (by convention, G 0 = 0) t G t := H u (X u+ X u ). u=0 Note that one does not pay any fee for the right to play the game X. By definition, we say that the game X is fair if there is no gambling strategy H such that E P (G t ) 0 for some t T. (c) Show that the game is fair if and only if X is a martingale with respect to the filtration F. (c) Consider a general F-adapted process X. Argue that the game will become a fair game if the player is required to pay at time t the fee A t+ A t per one unit of the bet where A is an F-predictable process satisfying equality (). 3

15 MATH307/37 Financial Mathematics School of Mathematics and Statistics Semester, 04 Tutorial sheet: Week 0 Background: Section 4. European Options in the CRR Model. Exercise Consider the CRR model M = (B, S) with the horizon date T =, the risk-free rate r = 0., and the following values of the stock price S at times t = 0 and t = : S 0 = 0, S u = 3., S d =.. Let X be a European contingent claim with the maturity date T = and the payoff at maturity given by the formula X = ( min(s, S ) 0 ) +. (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 F t = Ft S = σ(s 0,..., S t ) for t = 0,,. Compute the arbitrage price of X using the risk-neutral valuation formula, for t = 0,,, ( X ) π t (X) = B t E P Ft. B T (d) Find the replicating strategy φ t, t = 0,, for the claim X and check that the wealth process V (φ) of the unique replicating strategy for X coincides with the price process π(x) computed in part (c). Exercise (MATH37) We take for granted the CRR call option pricing formula T ( ) T C 0 = S 0 p k ( p) T k K T ( ) T p k ( p) T k k ( + r) T k k= k where k is the smallest integer k such that ( u k log > log d) ( K S 0 d T k= k ).

16 (a) Compute the arbitrage price C 0 of the European call option with strike price K = 0 and maturity date T = years. Assume that the initial stock price equals S 0 =, the risk-free interest rate is r = 0.0 and the stock price volatility equals σ = 0. per annum. Use the CRR parametrization for the parameters u and d, that is, set with the time increment t =. u = e σ t, d = u, (b) Compute the prices C u and C d at time t = for the same option using the CRR call option pricing formula.

17 MATH307/37 Financial Mathematics School of Mathematics and Statistics Semester, 04 Tutorial sheet: Week Background: Section 4.4 American Options in the CRR Model. Exercise Assume the CRR model M = (B, S) with T = 3, the stock price S 0 = 00, S u = 0, S d = 0, and the risk-free interest rate r = 0.. Consider the American put option on the stock S with the maturity date T = 3 and the constant strike price K =. (a) Find the arbitrage price P a t of the American put option for t = 0,,, 3. (b) Find the rational exercise times τ t, t = 0,,, 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 Assume the CRR model M = (B, S) with T = 3, the stock price S 0 = 00, S u = 0, S d = 0, and the risk-free interest rate r = 0. Consider the American call option with the expiration date T = 3 and the running payoff g(s t, t) = (S t K t ) +, where the variable strike price equals K 0 = K = 00, K = 0 and K 3 = 0. (a) Find the arbitrage price X a t of the American call option for t = 0,,, 3. (b) Find the holder s rational exercise times τ 0 for the American call option. (c) Find the issuer s replicating strategy for the American call option up to the rational exercise time τ 0 Exercise 3 Consider the CRR binomial lattice model M = (B, S) with the initial stock price S 0 =, the interest rate r = 0.0 and the volatility equals σ = 0. per annum. Use the CRR parametrization for u and d, that is, with the time increment t =. u = e σ t, d = u,

18 We consider call and put options with the expiration date T = years and strike K = 0. (a) Compute the price process C t, t = 0,,..., of the European call option using the binomial lattice method. (b) Compute the price process P t, t = 0,,..., for the European put option. (c) Does the put-call parity relationship hold for t = 0? (d) Compute the price process P a t, t = 0,,..., for the American put option. Will the American put option be exercised before the expiration date T = by its rational holder? Exercise 4 (MATH37) Consider the game option (See Section 4.) with the expiration date T = and the payoff functions h(s t ) and l(s t ) where and H t = h(s t ) = (K S t ) + + α L t = l(s t ) = (K S t ) + where α = 0.0 and K = 7. Assume the CRR model with d = 0., u =., r = 0.0 and S 0 =. (a) Compute the arbitrage price process (X g t ) T t=0 for the game option using the recursive formula, for t = 0,,..., T, { [ ) X g gu t = min h(s t ), max l(s t ), ( + r) ( px ]} t+ + ( p)x gd t+ with π T (X g ) = l(s T ). (b) Find the optimal exercise times τ 0 and σ 0 for the holder and the issuer of the game option. Recall that τ 0 = inf { t {0,,..., T } X g t = l(s t ) } and σ 0 = inf { t {0,,..., T } X g t = h(s t ) }.

19 MATH307/37 Financial Mathematics School of Mathematics and Statistics Semester, 04 Tutorial sheet: Week Background: Chapter The Black-Scholes Model. Exercise Consider the Black-Scholes model M = (B, S) with the initial stock price S 0 =, the continuously compounded interest rate r = 0.0 per annum and the stock price volatility equals σ = 0. per annum. (a) Using the Black-Scholes call option pricing formula C 0 = S 0 N ( d + (S 0, T ) ) Ke rt N ( d (S 0, T ) ) compute the price C 0 of the European call option with strike price K = 0 and maturity T = years. (b) Using the Black-Scholes put option pricing formula P 0 = Ke rt N ( d (S 0, T ) ) S 0 N ( d + (S 0, T ) ) compute the price P 0 for the European put option with strike price K = 0 and maturity T = years. (c) Does the put-call parity relationship hold? C 0 P 0 = S 0 Ke rt (d) Recompute the prices of call and put options for modified maturities T = months and T = days. (e) Explain the observed pattern of call and put prices when the time to maturity goes to zero.

20 Exercise Assume that the stock price S is governed under the martingale measure P by the Black-Scholes stochastic differential equation ds t = S t ( r dt + σ dwt ) where σ > 0 is a constant volatility and r is a constant short-term interest rate. Let 0 < L < K be real numbers. Consider the contingent claim with the payoff X at maturity date T > 0 given as X = min ( S T K, L ). (a) Sketch the profile of the payoff X as the function of the stock price S T at maturity date T and find the decomposition of the payoff X in terms of the payoffs of standard call and put options with different strikes. (b) Compute the arbitrage price π t (X) at any date t [0, T ]. Take for granted the Black-Scholes pricing formulae for European call and put options. (c) Find the limits of the arbitrage price lim L 0 π 0 (X) and lim L π 0 (X). (d) Find the limit of the arbitrage price lim σ π 0 (X). Exercise 3 (MATH37) Consider the stock price process S under the Black and Scholes assumption, that is, ( ( S t = S 0 exp r ) ) σ t + σw t where W is a the Wiener process under the martingale measure P. (a) Show that Ŝt := e rt S t is a martingale under P with respect to the filtration F = (F t ) t 0 generated by the stock price process S. Hint: Use the property that S t S s is independent of F s for 0 s < t. (b) Compute the expectation E P(S t ) and the variance Var P(S t ) of the stock price under the martingale measure P using the martingale property of Ŝ under P. Exercise 4 (MATH37) We consider the call option pricing function, that is, the function v : R + [0, T ] R such that C t = v(s t, t) for all t [0, T ] where C t is the Black-Scholes price of the call option. (a) Check directly that v satisfies the Black-Scholes PDE. You may use the partial derivatives given in Section. of the course notes. (b) Show that v satisfies the terminal condition v(s, T ) = (s K) + in the sense that lim t T v(s, t) = (s K) +.