MTH771P / MTH771U: Foundations of Mathematical Modelling in Finance 2021

Question 1  [30 marks].

(a) Give a short definition or explanation of each of the following terms, using your own words:

(ii) An American option.                                                                                              [2]

(iii) The Law of One Price.                                                                                            [2]

(b) A continuous random variable X has the probability density function

fX ←xJ =『 ψ(Ce) _8北     

where β and M are fixed parameters (both strictly positive), and C is the normalisation constant.

(ii) Next, determine the cumulative distribution function FX ←xJ.                                    [3]

(iii) Using the previous result, calculate the constant C, expressed in terms of β and M.   [3]

(iv) Finally, determine the expectation of X .                                                                  [4]

(c) Consider a European put option with strike price K and expiry time T. The underlying

share does not pay any dividends during the lifetime of the option.

Suppose that, at some earlier time t, the market price of the option is Pt , and the market price of the share is St  > ψ . The continuously-compounded risk-free interest rate is r.   Show that Pt must satisfy each of the following conditions, otherwise there will be       arbitrage opportunities:

(i)  Pt  > ψ .

(ii)  Pt  < Ke_r(T_t) .

(iii)  Pt  > Ke_r(T_t) - St .

[4] [4] [4]

Hint: For each case separately, identify an arbitrage opportunity if the condition does not hold.

Question 2  [20 marks].

(including a bank account or other risk-free asset) and a finite number of outcomes,

state the First and Second Fundamental Theorems of Asset Pricing.                                 [4]

Now consider the following specific one-period model, with two assets (A and B) and three possible outcomes:

There is also a bank account paying an interest rate gψ淬 per period.

(b) Verify that there is no arbitrage in this market, and that the market is complete.

Note: Show every step of your proof explicitly. Clearly state any theorems that you use,       at the points in your proof when you use them.                                                                [7]

Consider a so-called basket call option that has payoff at time { given by

V1  三 max ╱  - K, ψ、,

where A1 and B1 are the prices at time { of assets A and B respectively, and K 三 ψ .5ψ is the pre-agreed strike price.

(c) Determine the fair price V0 of this option at time ψ .                                                          [4]

(d) Suppose that you buy {ψψψ of these options at time ψ .

Calculate how many shares (units) of assets A and B you would need to buy or sell (also at time ψ), in order to make your overall position risk-free.

Note: Show detailed working for your solution.                                                               [5]

Question 3  [15 marks].   Consider the three-period binomial model, with market parameters S0  三 5ψ , u 三 {.g, d 三 ψ .9 and R 三 ψ .{5. (S0 is the initial price of the stock, u and d denote   the multiplicative jump sizes of the stock price at each time step, and R is the risk-free interest rate per period.) Denote the stock price process as S 三 ←S0 , S1 , S2 , S3 J.

A particular type of option has payoff at time N 三 3 given by

V3  三 max ╱K -  , ψ、,

where K 三 55 is the pre-agreed strike price.

(a) Calculate the fair price V0 of this option at time 0.                                                          [12]

(b) Suppose that you buy such an option at time ψ . How many shares of the underlying

stock would you need to buy or sell, also at time ψ, such that your overall position is

risk-free during the first time step?                                                                                  [3]

Question 4  [15 marks].

(a)    (i) Give a formal definition of the Wiener process.                                                        [2]

(ii) What is the relationship between geometric Brownian motion and the Wiener               process?                                                                                                                 [1]

(iii) Explain briefly (in your own words) why geometric Brownian motion is often              considered to be a reasonable model for the behaviour of stock prices in the real           world.                                                                                                                    [2]

(b) Now consider a European call option, with expiry time T 三 g years, and strike price    K 三 {ψ5. The current price of the underlying stock is S0  三 {ψψ , and the stock price is assumed to follow geometric Brownian motion with annualised drift µ 三 ψ .{4 and       volatility σ 三 ψ .3. You may assume that the underlying stock pays no dividends during the lifetime of the option. The continuously-compounded annualised risk-free interest  rate is r 三 ψ .ψg.

Using the discrete-time binomial model with N 三 d periods, find an approximate value

Question 5  [20 marks].   Consider a derivative whose payoff at expiry time T is given by

CT (ST ) = max ╱log ╱ 、 ,  0、,

where K > 0 is the strike price and ST is the price of the underlying share at expiry. This type of derivative is known as a “log option” . Derivatives of this type have to be cash-settled.

(a) Carefully sketch the graph of CT as a function of ST .

(b) We showed in the lectures that the Black-Scholes price V0 at time 0 of a vanilla European option with payoff function VT (ST ) is given by

[2]

V0  =

o

VT  ╱S0 exp 尸(r - σ 2 /2)T + σ ^T x┌、e_α2 /2dx,

_o