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STAT0013: Stochastic Methods in Finance I

Examination Paper 2019/2020

Q1 (a) Draw the profit/loss diagram for a long European Put option and a long European Call option.               [4]

(b) A butterfly spread is made up of 4 European Call options on the same asset with

(i) Long call, Strike price K1 = 戈25 and cost C1 = 戈2.5.

(ii) Two short calls each with Strike price K2 = 戈27 and cost C2 = 戈1.5.

(iii) Long call, Strike price K3 = 戈29 and cost C3 = 戈1.2

Draw the profit/loss diagram for the butterfly spread.  In the process demonstrate under what conditions a profit is made.  [8]

(c) At time t = 0, Mr X plans to invest 戈10, 000 over the next two years. His bank offers him three investment strategies with the following payments over the 2 years:

A: One payment of 12, 100 after two years.

B: A payment of 戈3, 000 after 1 year and a final payment of 戈8, 800 after two years.

C: A payment of 戈10, 000 after 1 year and a final payment of 戈981 after two years.     Assume that after 1 year, i.e.  at time t = 1, Mr X invests any payment in a risk- free 1-year bond with interest rate r per annum, where r is unknown at time t = 0. Assume that interest rate is discretely compounded.

Calculate the accumulated capital after 2 years for each strategy as a function of r. For each strategy (A,B,C) determine the range of values of r, for which each of the respective strategies gives the largest payment.        [8]

(d) A British football team signs a contract with a promising player from an European team to buy him from his current team after one year.  The price the British team has to pay after 1 year is 1 million EUR. The football team takes out a forward contract with a bank to supply this for a set amount of sterling. What should be the agreed price if the interest rate in EUR is 3%, the interest rate in pounds is 5% and the current exchange rate is 戈1 = 1.2 EUR? Describe the arbitrage opportunities if the forward contract strike price is incorrectly set as:

(i)  戈900, 000

(ii)  戈700, 000 [10]

Q2 (a) Derive the Put-Call parity relationship

P + S = C + Ee −r(T −t)

which relates European Put (P) and European Call(C) options.  Both options have the same Strike price (E) and time to expiry (T) and are based on the same share (S). The present time is t.        [5]

(b) Through using Put-Call Parity relationship, determine the value of a European Call option

(C), expressing it in its simplest form, given that the value of a European Put option (P) is,

P = Ee −r(T −t)N ( ·d2 ) · SN ( ·d1 )

where

ln (S/E) + (r + σ 2 )(T · t)

σ(T · t)1/2                    ,

ln (S/E) + (r · σ 2 )(T · t)

σ(T · t)1/2                    ,

and each of the remaining symbols have their usual meaning.                                     [5]

(c) Consider the binomial tree of pricing an option.  In each step, the share price (S) may either increase by a multiple u(>  1) or decrease by d(< 1). Assuming that p is the probability of an up move, construct a binomial tree with δt = 1 month to calculate the price of a 3 month American put option if, S(t = 0) = 戈80, E = 戈86, r = 8% and σ = 25%. For the calculation you may assume

u   =   er6t ┌ 1 + ^eσ2 6t · 1] ,

d = er6t ┌ 1 · ^eσ2 6t · 1] ,

er6t · d

u · d  .

[10]

Q3 (a) Explain what delta decay, rho and theta of a portfolio of derivatives are, and provide a  mathematical expression for them.      [4]

(b) Assume a stock has current price £50. Assume there are two European call options on the stock, with exercise price £48 and £55, respectively. The risk free interest rate is 5% per annum, the time to maturity is 4 months and the volatility is 15% per annum. How can we make a 100 shares portfolio delta-gamma neutral?                     [10]

(c)  Assume that (Bt, t , 0) and (Wt, t , 0) are independent standard Brownian motions. Show that

Wt + 2Bt

5

(d) Carefully explain the differences between a forward and a future contract.             [3]

Q4 Assume that the stock price St satisfies a standard Brownian motion.

(a) Provide two reasons why geometric Brownian motion is a more suitable model for stock prices rather than generalised Brownian motion.             [4]

(b) Assume that the current stock price is £110, the riskless interest rate is 3% per annum and the volatility is 7% per annum.  Let S2  be the stock price after two years.  What is the value of a financial product that pays S2 · 120 if S2  > 120, 100 · S2 if S2  < 100 and zero otherwise? Assume that all black-Scholes assumptions hold. [10]

(c) Assume that Xt satisfies:

Xt = X0 exp(4St · 2t)       where  X0 = 1.

Find the SDE satisfied by Xt . [5]

(d) We know that Yt satisfies the SDE

dYt = dt · dSt

Find Yt explicitly. [6]