ASSIGNMENTS FINANCIAL DERIVATIVES AND STOCHAS- TIC ANALYSIS, TMA285

Transcription

1 ASSIGNMENTS FINANCIAL DERIVATIVES AND STOCHAS- TIC ANALYSIS, TMA285 Week 1 1. Probability Suppose X : Ω (, ) is a bounded random variable. a) Show that E[X log(x)] E[X] log(e[x]) b) Compute E[X log(x)] and E[X] log(e[x]) when X has a uniform distribution in the interval (, 1). 2. Probability: Let W be a Brownian motion. Find and 3. Itô isometry: Set (t) = t E[e W (1)+W (2) ] sin(t + u)dw (u), t 1 (t)dw (t). Find E[ 2 (t)] and E[X 2 ]. 4. Martingales: Determine the constants a, b, c, d so that W 3 (t) + (at + b)w 2 (t) + (ct + d)w (t) is a martingale 5. Stochastic integral: a) Suppose 3W ( 1 3 ) W (1 2 ) + 2W (3 4 ). Find a function f(t), t 1 such that f(t)dw (t). b) Compute V ar(x). 6. Measurability: a) Show that all open sets are in the Borel σ-algebra F b) Show that if the sets A 1, A 2,... F, then i=1 A i F Week 2 1. Itô formula: a) State (without proof) the two dimensional Itô formula, and use it to prove the Itô product rule (Corollary 4.6.3). b) Take two Brownian motions W 1, W 2, and define X 1 (t) = W 1 (t), for t [, T ]. Calculate d(x 1 (t)x 2 (t)). 1 X 2 (t) = ρw 1 (t) + 1 ρ 2 W 2 (t)

2 2 2. Martingales: Let X(t) = µt + σw (t), where µ, σ are non-zero constants, and W (t) is a Brownian motion. Let f be a (at least) two times differentiable function. For which such functions is f(x(t)) a martingale? 3. How good is the simple function approximation of the stochastic integral? Let f be a real-valued continuous function on the unit interval [, 1] and set f(t)dw (t) n k=1 where n is a fixed positive integer. Prove that V ar(x) f( k 1 n )(W ( k n ) W (k 1 n )) max (f(s) f(t)) 2 s t 1 n 4. Martingales: Show that the process e t 2 cosh W (t) is a martingale. Week 3 1. Maximum of Brownian motion: Suppose T > and M(T ) = max t T W (t). Find E[e M(T ) ]. 2. Stochastic differential equations: a) Solve the equation dx(t) = tx(t)dt + dw (t), t where X() = 1. (Hint: d(e t2 2 X(t))) b) Determine the covariance function of X(t), i.e. find Cov(X(s), X(t)). (Hint: First assume s t and then t s) 3. Stochastic differential equations: a) Solve the equation dx(t) = X(t)/(T t)dt + dw (t), t T where X() =. (Hint: d(e log(t t) X(t))) b) Determine the variance of X(t), 4. Convexity or Put-Call parity Show that c(t, S(t), K, T ) max(s(t) K, ) using both Jensen s inequality and the Put-Call parity. 5. Probability: Let Y be an integrable random variable on (Ω, F, P), and let the σ-algebra G F. Define Err = Y E[Y G]. Let X be another G-measurable random variable with finite variance. When the existence of the conditional expectation is established, one uses a theorem on orthogonal projection which states that there exists a random variable E[Y G] such that V ar(err) V ar(y X), and E[X(Y E[Y G])] = for all X. The inequality above means that the estimate E[Y G] minimizes the variance of the error among all estimates based on the information in G. Show that this inequality holds as long as E[X(Y E[Y G])] =, which it always true for the conditional expectation. (Hint: Let µ = E[Y X], and calculate the variance of Y X after introducing E[Y G] in an appropriate way...)

3 3 Week 4 1. Simulated stock prices: Consider a stock price process σ2 (µ S(t) = S()e 2 )t+σw (t). Plot 5 realizations of the stock price process if S() = 1, µ =.5, σ =.5, and T = Hedging a call option: Assume the same model as in the previous problem. Suppose h = T/N and t n = nh, n =,..., N. Define cash(t ) by the equation c(t, S(t ), K, T ) = c (t )S(t ) + cash(t ) where c (t) = c s(t, S(t), K, T ). Set Π 1 = c (t )S(t 1 ) + cash(t )e rh where r is the interest rate. Next define cash(t 1 ) by the equation and Π 1 = c (t 1 )S(t 1 ) + cash(t 1 ) Π 2 = c (t 1 )S(t 2 ) + cash(t 1 )e rh and continue the process until maturity T. Plot the theoretic call prices c(t i, S(t i ), K, T ) and the portfolio values Π i, i = 1,..., N, in the same figure when S(t ) = 1, K = 15, T =.5, µ =.5, r =.3, σ =.35, and N = 13. Find Π N max(s(t ) K, ) for 1 realizations. What happens if you hedge using parameters that are different than the correct ones? Different T, σ, r, K? 3. Hedging error: It is known that, with notation as in the previous assignment, the normalized hedging error N(Π c)(t) =d T 2 t c ss(t)σ 2 S(t) 2 dw (t), where = d denotes equality in distribution, and where W is a Brownian motion which is independent of the stock price! Analyze the hedging error in the previous assignment. Multiply the error with N. Does this normalizing sequence seem appropriate, and if so, in what sense? For how large N does the normalized hedging error start to look like a stochastic integral w r t Brownian motion? When is the error large? When is it small? How is it affected if you increase N? Compare with the central limit theorem. Fix a stock price trajectory for a large N. Does the error seem to converge to another stochastic process X as N increases or does it keep moving around? Why is this, you think? 4. Monte Carlo pricing and hedging: Calculate the price and hedge of a call option using the Monte Carlo method. Compare the numerical performance of the standard and centred approximations of the hedging ratio, respectively. Further, compare the numerical accuracy when you use the same sequence of random outcomes to simulate both c(t, x + h) and

4 4 c(t, x h), with the accuracy when you generate two different random sequences to simulate c(t, x + h) and c(t, x h). What do you see? Finally, calculate the price of an Exotic option of your choosing (barrier options, lookback options, Asian options,...). Argue, using the Strong Law of Large Numbers, that the simulated price using the Monte Carlo method must converge to the true price almost surely. 5. Black-Scholes model: Assume that < L < K. A (European) derivative pays Y = min(l, S(T ) K ) at time T. Describe how this derivative can be hedged in the time interval [, T ). (Hint: Describe the payoff of the contract as a portfolio of calls and puts, which you know how to both price and hedge, right?) Week 5 1. Black-Scholes model: We know how to price anything that depends only on the price of the underlying stock at time T, i.e. anything that has a payoff function h(x) at time T. Find the prices of the contracts paying max(log S(T ) K, ) and S2 (T ), respectively. 2. Black-Scholes model: Suppose S() < B, T >, and M(T ) = max u T S(u). Find the price at time zero of a barrier option of European type paying the amount Y = 1 [M(T )<B] to its owner at time of maturity T. 3. Girsanov s theorem: State Girsanov s theorem. Write up the stochastic differentials D(t)S(t) and D(t)X(t), where D(t) = e rt, S(t) is a stock in the Black-Scholes model, and X(t) is the portfolio process defined in Sec Apply Girsanov s theorem on these processes in such a way that both are martingales under the new measure. Finally, what does the change of measure consist of? 4. Black-Scholes model: Derive the Black-Scholes equation where σ = σ(x, t) is a deterministic function of the underlying x and time t. 5. Existence Itô integral: Show that the Itô integral exists for bounded and adapted. You may assume that L 2 is complete. Week 6 1. Black-Scholes formula: Derive the pricing formula and the hedge of a European call option starting from the Black-Scholes equation, arguing that you are aloud to use the Feynman-Kac formula. 2. Black-Scholes model II - Local volatility: We assume that σ = σ(x, t) is a deterministic function of the underlying x and time t. Show that the hedging ratio is still c x. Derive the pricing formula, and the hedge, of a call option if the underlying stock has the dynamics. ds(t) = αdt + σdb(t), for α, σ >. Start from the corresponding Black-Scholes type equation, and argue that you are aloud to use the Feynman-Kac formula.

5 3. General pricing formula: Write up the general pricing formula at time t for any F T -measurable payoff V (T ), and show that it can be used to find the price of for the European call option in the Black-Scholes model. Argue that the Feynman-Kac pricing formula is actually the expected value of a payoff h(s(t )) under the risk-neutral probability measure. 4. The implied distribution: Calculate the implied distribution at the time of expire T for some underlying asset of your choosing, using the option prices from that asset. (Hint: Exercise 5.9). 5