STATS 723 - Stochastic methods in finance - 2022 Assignment 3
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STATS 723 - Stochastic methods in finance - 2022
Assignment 3
1. Let (Wt )0≤t≤T be a Brownian motion under a probability measure P , and define another probability measure Q by
= exp │ │T - W Wt2 dt、、 .
(a) Show that under Q, (Wt ) is an Ornstein-Uhlenbeck process for 0 s t s T, and find an explicit expression (in terms of an integral) for its internal Brownian motion. ))/≠( This a Cameron-Martin-Girsanov change of probability.
(b) The expression for suggests that outcomes where Wt stays close to zero for 0 s t s T should be more likely under Q than under P . With this in mind, consider the event A = ≥}Wt } s a for 0 s t s T{, and show that
Q(A) ≤ e(T —a2 T —a2 )/2P (A).
(c) The tendency of Brownian motion to wander suggests that for large T, P (A) should be small. Use the result of (b) to establish that for T ≤ ,
P ╱ }Wt } s for 0 s t s T← s e —T/4 .
2. The price of a barrel of oil is $100 today; in one year’s time it could be either $70 or $140. A floor contract at $90 is one that compensates the holder for any fall in price below $90: if the oil price ends up being $70, it will pay the holder $20, otherwise it has no value. (A useful contract to have, if one happens to be an oil producer looking for some certainty of revenue.) The risk-free interest rate is zero.
(a) Find a replicating strategy for the floor contract.
(b) If I want to negotiate a floor contract today, how much should I expect to pay for it?
(c) Suppose we are able to negotiate the sale of floor contracts to a client for an up-front payment of $12. Explain how to ensure that we profit from this transaction, without exposing ourselves to oil price risk.
(d) This model of the oil price has an equivalent martingale measure; find this measure.
3. The share prices of two companies, Alpha Corp. and Betaco, over a 24-week period are modelled by the following discrete-time process. There are four equally likely scenarios ω 1 ,ω2 , ω3 , and ω4 .
|
now |
12 weeks hence ω 1 , ω2 ω3 , ω4 |
24 weeks hence ω 1 ω2 ω3 |
(Alpha) S1 (t) (Betaco) S2 (t) |
12 6.00 |
15 6.60 |
11.00 5.80 |
A risk-free bank account, which pays no interest, is also available.
(a) Draw a scenario tree illustrating the model.
(b) Find a self-financing trading strategy which is an arbitrage opportunity.
(c) Find a self-financing trading strategy φ with Vφ (0) = 100, P (Vφ (2) ≤ 95) = 1, and P (Vφ (2) > 110) > 0. ))/≠( There are plenty of solutions; intelligent guesswork will get you a long way towards finding one of them.
(d) Show that arbitrage opportunities must exist, by demonstrating the non-existence of an equivalent mar- tingale measure.
(e) We can make the model arbitrage-free by changing the final Alpha price under scenario ω2 from 12.50 to another value. Find that value, and the resulting equivalent martingale measure.
2022-04-25