Capítulo 1 Basic Concepts Exercise 1.1 Let B(t, T ) denote the cost at time t of a risk-free 1 euro bond, at time T. (i) Assume that the interest rate is a deterministic function. Show that the absence of arbitrage requires that B(0, 1)B(1, 2) = B(0, 2). (ii) Now suppose that B(0, 1) and B(0, 2) are known at time 0 but B(1, 2) will not be known until time 1. What goes wrong in the previous argument? Exercise 1.2 The present price of a stock is 50. The price of an European call option with strike price 47.5 and maturity 180 days is 4.375. The cost of a risk-free 1 euro bond 180 days is 0.98. (i) Consider an European put option with price 1.025 (same strike price and maturity as the call option). Show that this is inconsistent with put-call parity. (ii) Describe how you can take advantage of this situation, creating an arbitrage possibility example. Exercise 1.3 For each of the following portfolios, draw the expiry payoff diagram and explain waht view of the market holding this position expresses: (i) One long call and one long put options, both with the same strike (straddle strategy). (ii) One short forward and two long calls, all with the same strike price. (iii) One long call and two long puts, all with the same strike price (strip strategy). (iv) One long put and two long calls, all with the same strike price (strap strategy). (v) One long call with strike price K 2 and one long put, with strike price K 1. (vi) One long call with strike price K 1, one long call with strike price K 2 and two short calls, both with strike price (K 1 + K 2 )/2 (buttterfly strategy). Exercise 1.4 Today s price of a share is 94 euros. A European call option (involving 10 shares) with maturity 3 months and strike price equal to 95 is worth 4.70 euros. An investor strongly believes that the price of these shares will increase. He hesitates between buying today 100 shares or buying 2000 call options. Which advice would you give him? 1
Exercise 1.5 Consider the following product: upon maturity, the owner gets 1000 euros from a bond plus an additional value, function of the price of the crude upon maturity of the bond. This additional value is equal to 60 times the excess (if positive) of the crude price p compared with 60 euros, i.e., 60(p 60) +. Furthermore this additional value cannot be larger than 2550 Euros. Argue that this derivative is a combination of a bond, with a call and a put. 2
Capítulo 2 Discrete Time Models Exercise 2.1 You suspect that company XP T O will merge with company XP T A in the coming month. Presently, XP T O stock is trading at 0.85 Euros. Assume a 2-period binomial tree, where there is a 60% chance that the merge will occur, in which case the stock will be worth 1.20 Euros; if not, the stock will downward plunge to 0.30 Euros. If r = 2% per annum, what is the value of a call with strike price K = 1 Euro and maturity one month? Exercise 2.2 A European put option with strike price 45 Euros matures in one year. The current price is 50 Euros, and the risk-free rate is 5 % per annum. Divide the one-year interval in two six-month intervals, consider a 2-binomial tree model, with u = 1.17 and d = 0.88. (i) Determine the put price. (ii) Specify the hedging portfolio of this conting claim. Exercise 2.3 The current price is 100 Euros. The risk-free interest rate is 5% per annum. Consider a one-year European call option on this price, with strike 100. (i) Divide the one-year period into six-month intervals, and derive the recombinant tree with u = 1.25 and d = 0.82. Calculate the risk-neutral probabilities. What is the option value? (ii) Suppose that the market price of this option is 16 Euros. Assuming that the market is truly described by the tree of (a), then that means that there is an arbitrage possibility. Explain how can you use this opportunity to earn a risk-free profit. Exercise 2.4 A special kind of one-year put option is written on the forward price. The current forward price is 40 Euros, and the current strike price is 40. At month 6, if the price is below 35, the strike price is lowered to 35; otherwise it remains unchanged. The risk-free interest rate is 5 % per annum. Assume a 4-period binomial tree with u = 1.21 and d = 0.81 to value the option. Explain why this option is difficult to value. Exercise 2.5 The current stock price is 40 Euros. Over each of the next 3-month period it is expected to go up by 10% or to go down by 10%. The risk-free interest rate is 12% per annum. (i) Assuming a 2-period binomial model, determine the value of a 6-month European call option with strike price of 42 Euros. (ii) The same, but now assume that it is an American call option. 3
Exercise 2.6 Consider a 2-year European put, with a strike price equal to 52 Euros on a stock whose current price is 50 Euros. We assume that there are two time steps of one year, and in each time step the stock price either goes up by 20% or moves down by 20%. We assume also that the risk-free interest rate in 5% per annum. Give the price of this put, as well as the price of the American put. Should it be exercised early? Exercise 2.7 A stock price is currently 30 Euros. During each 2-month period for the next 4 months it will increase by 8% or reduce by 10%. The risk-free interest rate is 5%. Use a 2-step binomial tree to calculate the value of a derivative that pays off max[(30 S T ), 0] 2, where S T is the stock price in 4 months. If the derivative is American-style, should it be exercised early? Exercise 2.8 Prove that if B t denotes the risk free investment at time t, then: [ ] Π(0; X) Π(1; X) = E Q. B 0 B 1 Exercise 2.9 For w Ω, let: Λ(w) = 1 q(w) 1 + R p(w) where q(w) = Q(w) and Q is the martingale measure. Prove that the arbitrage free price of any contingent claim X is given by: Π(0; X) = E P [Λ.X]. 4
Capítulo 3 Introduction to Stochastic Calculus Exercise 3.1 Show that a sum of martingales is still an martingale. Exercise 3.2 Let M = {M t, t IN 0 } be a martingale, with E[Mt 2 ] <, t, defined on the filtred probability space (Ω, F, F, P), where F = {F, IN } is the filtration. Let η = {η t, t IN 0 } be a stochastic process defined on the same probability space with the following property: η t is F -measurable, such that E[ηt 2 ] <, t. Show that the process N = {N t, t IN 0 } defined by t N t = N 0 + η k (M k M k 1 ) k=1 is a martingale as long as N 0 is F -measurable. Exercise 3.3 Let X be a random variable such that E P [ X ] <, where (Ω, F, F, P) is its filtrered probability space. Show that the stochastic process M = {M t, t IN 0 }, where M t = E P [X F ], is a martingale. Exercise 3.4 Let W 1 e W 2 be two Brownian motions, with drift µ 1 e µ 2 and variances σ1 2 e σ2 2, respectively. What process follows W 1 + W 2 if: a) W 1 and W 2 are not correlated? b) The correlation between W 1 and W 2 is ρ 0? Exercise 3.5 Let W be a Brownian motion. Denote by τ a the first time W crosses a, i.e.: τ a = inf{t : W t a} i) Verify the reflection principle: P (W t x τ a t) = P (W t 2a x τ a t). ii) Using the reflection principle, find the density function of τ a and verify that E[τ a ] =. Exercise 3.6 The evolution of a stock price, modelled by the stochastic process S = {S t, t 0}, where S(t) = e µt+σw (t) (W is a brownian motion), with S 0 = 1. 5
(i) Derive an expression for P (S t x). (ii) Derive the median and expected value of S t. (iii) Determine an expression for the condicional expectation E[S t F s ], where s < t and {F s, s 0} is the filtration associated with the process S. (iv) Find conditions on µ and σ under which the process {S t, t 0} is a martingale. (v) State. with reasons, whether or not the stock would be a good long term investment in this case. Exercise 3.7 Let S = {S t, t 0} be the process of prices of a stock. Explain the difference in the following models and should what is, in your opinion, the more appropriade one: ds = µsdt + σs dt; ds = µsdt + σ dt; ds = µdt + σ dt ds = µdt + σs dt where µ and σ are constant. Exercise 3.8 Data suggests that the short-term interest rate, r = {r(t), t 0}, follows the following stochastic process: dr = a(b r)dt + rcdw where a, b abd c are positive constants and W is a Brownian motion. Describe the nature of such process. Exercise 3.9 Suppose that S follows a geometric Brownian motion, with drift µ and volatility σ: ds = µsdt + σsdw (where W is the Brownian motion). Show that S n also follows a geometric Brownian motion. Exercise 3.10 Let X and Y be two processes adapted to the same Brownian motion W, such that: dx = µ X dt + σ X dw ; dy = µ Y dt + σ Y dw. Apply Itô s formula to 1 2 ((X + Y )2 X 2 Y 2 ) = XY do derive d(xy ). Now suppose that X and Y are adpated to two independent Brownian motions. Then prove that in this case the chain rule for the Itô s calculus is the same as in the deterministic chain rule. Exercise 3.11 Show that if B is a zero-volatility process and X is any stochastic process, then d(bx) = BdX + XdB. Exercise 3.12 Use Itô s formula to compute E[W 4 (t)], where W is a Brownian motion. Exercise 3.13 Compute the stochastic differential of X when: a) X(t) = t 0 g(s)dw (s) (g is a stochastic process adapted to W ) 6
b) X(t) = eαw (t) c) X(t) = e αy (t) where Y follows the following SDE: dy = µdt + σdw d) X(t) = Y 2 (t), where Y follows the following SDE: dy = αy dt + σy dw Exercise 3.14 Let g be a deterministic function and W a Brownian motion; define X as follows: Prove that for a IR: and therefore X(t) N(0, t 0 g2 (s)ds). X(t) = t 0 E[e iyx(t) = e y2 2 g(s)dw (s). t 0 g2 (s)ds Exercise 3.15 Suppose that {W t } and {V t } are two independent Brownian processes. Z t = Wt 2 + V t 2, the distance from the origin of a Brownian particle to the plane. Show that Z 3 = W3 2 + V 3 2 E[Z 4 V 3, Z 3 ] 2 + Z3 2. Let 7