STAT0013 – Stochastic Methods in Finance (2022)
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STAT0013 – Stochastic Methods in Finance
Examination Paper 2021/2022, Level 6
Section A
A1 A short forward contract that was negotiated some time ago will expire in six months and has a delivery price of £35. The current forward price for six-month forward contracts is £40. The risk-free interest rate (with continuous compounding) is 5%.
(a) What is the current value of the short forward contract? [4]
(b) Three months later, the price of the stock is £50 and the risk-free interest rate is still 5%.
What is the value of the short forward contract at this time? What is the price of a contract entered into at this time with the same expiration date as the first short forward contract? [5]
A2 The process (Bt, t < 0) is a standard Brownian motion. Assume that the stock price St
follows the SDE:
dSt = µtStdt + σtdBt ,
with S0 = 1, and:
µt = 4, σt = 0, for 0 ● t ● 5
(
What is the distribution of S8?
A3 A stock price is currently £60. Over each of the next two four-month periods it is expected to
go up by 10% or down by 5%. The risk-free rate is 7% per annum with continuous compounding.
(a) Calculate the value today of an eight-month European put option with a strike price of £70. [5]
(b) When, if ever, would it be worth exercising an eight-month American put option with a strike price of £70? [5]
(c) What is the value today of an eight-month European call option with a strike price of £70? [4]
A4 European call options and European put options with the same strike price of £20 and expiry
date 1 year from now, based on the same underlying stock, are traded at £7 and £4 respectively. The risk-free interest rate is 2%.
(a) Assuming that no-arbitrage holds, find the current value of the stock.
Hint: Use put-call parity. [4]
(b) Use as current stock price the one you found in part (a). Assume that the put option is instead traded at £8. Construct a portfolio that creates an arbitrage opportunity in this
scenario. Find the risk-free profit generated by this portfolio. [7]
Section B
B1 Assume a stock has current price £75. Assume there are two European call options on the
stock, with exercise price £75 and £85, respectively. The risk free interest rate is 7% per annum, the time to maturity is 2 months and the volatility is 15% per annum.
(a) You are given that the Delta and Gamma of the first call with exercise price『75 are
equal to 0.588 and 0.085, respectively. Find the Delta and Gamma for each of the second
European call. [5]
(b) Constuct a portofolio that contains 500 shares together with approriate number of derivatives, so that it is a delta-gamma neutral portfolio. [6]
B2 (a) In a year’s time a stock will either be valued at $10 or $20. A certain derivative will be
worth $5 or $2 in each case respectively. Assume the risk-free rate is 5%. Find a replicating portfolio for this derivative today stating clearly whether each portfolio component is a long or a short position. [6]
(b) Let Bt be a standard Brownian motion. Assume that dXt = µdt + σ ^XtdBt . Find the
SDE satisfied by Yt = ^Xt . [5]
B3 (a) Assume in this question that all the derivatives have the same maturity date and the same underlying asset. Write explicitly the payoff function at maturity and then draw the payoff diagram (payoff at date of derivative maturity versus price of underlying asset) for a portfolio consisting of a short position in two European put options with exercise price
£5 and a long position in three European call options with exercise price £2. [7]
(b) Using forwards and/or put and call options with appropriate strike prices, create a portfolio with the following payoff at maturity T:
Payoff = _4~ST _ 10~ + 3(ST _ 5) [5]
B4 A stock price St follows Geometric Brownian motion with drift and volatility parameters µ = 4 and σ = 2, respectively. Assume that S0 =『35 and that the risk-free interest rate is 8% with continuous compounding. Suppose that ST is the stock price at the end of five months. Apply the risk-neutral evaluation formula to determine the current price of a derivative that provides a payoff of ST(3) at the end of five months.
Hint: The moment generating function of Z > N(0, 1) is equal to E [ etz ] = et /22 . [6]
B5 Assume that (Bt, t < 0) is a standard Brownian motion.
Consider the two Geometric Brownian motions:
Xt = e(u1 − 
)t+o1Bt
Yt = e(u2 − 
)t+o2Bt
so that Xt has drift parameter µ 1 and variance parameter σ 1 , whereas the corresponding parameters for Yt are µ2 and σ2 .
(a) Let Rt = Xt + Yt . Show that when µ 1 = µ2 = µ and σ 1 = σ2 = σ > 0, then Rt is also
a Geometric Brownian motion. Give the initial condition, the drift parameter and the
variance parameter for Rt . [4]
(b) Let St = Xt . Yt . Show that St is a Geometric Brownian motion under any choice of µ 1 , µ2 and any choice of σ 1 > 0, σ2 > 0. Give the initial condition, the drift parameter and
the variance parameter for St . [6]
B6 Assume that (Bt, t < 0) is a standard Brownian motion.
(a) Let Xt = ^tB1 . Is Xt a Brownian motion? Explain briefly your answer. [4]
(b) Let Yt = (1 + t)B
_ tB1 . Compute the expectation and covariance of Yt . Is Yt a
Brownian motion? Explain your reasoning. [6]
2023-05-20