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MATH5816 Continuous Time Financial Modelling
发布时间:2021-11-05
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MATH5816 Continuous Time Financial Modelling
Assignment 2, 2021
Exercise 1. Optimal Stopping. Let where µ is a constant and W is a standard Brownian motion.
1. Find all the values for µ such that (St) is a strict submartingale. In this case, show that
where the supremum is over all bounded stopping times.
Exercise 2. American option. We consider the standard Black-Scholes model of the stock price. The discounted reward process an American put option with a constant strike K and an expiration date T > 0 is defined by the equality
1. Show that
is a submartingale under
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2. Argue that the price of an American call option is the same as an European call option.
Exercise 3. Forward-start option. We assume the Black-Scholes model of the stock price. The forward-start call option has the payoff at expiry where T is the expiry date and 0 ≤ U ≤ T is the strike determination date.
1. Show that the price at time 0 of the forward-start option can be represented as follows:
where
is the exercise event and the probability measures
and
are given on
by
where the process
follows a martingale under
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2. Find the arbitrage price for the forward-start option at time 0.