[SOLVED] 编程代写 FINANCE 2021–22 STAT0013

STOCHASTIC METHODS IN FINANCE 2021–22 STAT0013

Exercises 9: Risk-neutral pricing
For all questions, assume that the risk-free rate is continuously compounded, is constant and is r.

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1. A European digital call option1 has an “all or nothing payoff” function, so that at expiration date the option pays off £1 if ST ≥ K, and nothing if ST < K, where K is the strike price. Assume that the underlying stock follows geometric Brownian motion. Use the risk- neutral valuation approach to find the value of this digital call option with strike K.2. Find the Delta of the European digital call option from question 1.3. A stock follows a price process of geometric Brownian motion with volatility σ. A derivative based on this stock will provide a payoff at expiration time T of £H1 if ST ≤ K1 and £H2 if ST ≥ K2, where H1,H2,K1 and K2 are positive constants with K2 > K1. The payoff is zero if K1 < ST < K2, and early exercise is not allowed. Use the risk-neutral valuation approach to find a formula for the price of the derivative.4. Suppose you are given a vector of n independent, pseudo-random num- bers generated from a uniform [0,1] distribution, i.e. ui ∼ U[0,1] for i = 1, 2, …, n. Determine an expression based on the Monte-Carlo sim- ulation technique that uses these numbers to estimate the price of a European style derivative based on an underlying stock whose price process St follows the stochastic differential equationdSt = μStdt + σStdBt,and whose payoff at expiration time T is ST2 . Assume the risk-free rateis r and that the current stock price is S0.1Sometimes also called a binary call option 5. If a stock price in a risk-neutral world follows the SDE dSt = rStdt + σStdBtthen use Itoˆ’s lemma to show that the price process relative to the riskless bond, say Rt, (i.e., the process St/Rt), has zero drift in the risk-neutral world2. 2Having zero drift is a property of a “fair game” process. A fair game is a stochastic process Xt that satisfies the result E[XT |Xt] = Xt for T > t, so that the expectation of any future value is the current value, given that we know the current value. If the process is an Ito process it will have zero drift. A “fair game” is also referred to as a martingale

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