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MTH771P / MTH771U Foundations of Mathematical Modelling in Finance 2021

发布时间：2021-12-25

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MTH771P / MTH771U:

Foundations of Mathematical Modelling in Finance

Question 1 [30 marks].

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

(i) An Asian option.

(ii) An American option.

(iii) The Law of One Price.

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

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

(i) First, carefully sketch the function fX (x).

(ii) Next, determine the cumulative distribution function FX (x).

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

(iv) Finally, determine the expectation of X.

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 > 0. 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 > 0.

(ii) Pt < Ke、r(T、t) .

(iii) Pt > Ke、r(T、t) - St .

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

Question 2 [20 marks].

(a) In the context of a general one-period market model with a ﬁnite number of assets

(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.

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 20% 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.

Consider a so-called basket call option that has payoff at time 1 given by

where A1 and B1 are the prices at time 1 of assets A and B respectively, and K = 0.50 is the pre-agreed strike price.

(d) Suppose that you buy 1000 of these options at time 0.

Calculate how many shares (units) of assets A and B you would need to buy or sell (also at time 0), in order to make your overall position risk-free.

Note: Show detailed working for your solution.

Question 3 [15 marks]. Consider the three-period binomial model, with market parameters S0 = 50, u = 1.2, d = 0.9 and R = 0.15. (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 = (S0 , S1 , S2 , S3 ).

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

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

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

(b) Suppose that you buy such an option at time 0. How many shares of the underlying

stock would you need to buy or sell, also at time 0, such that your overall position is

risk-free during the ﬁrst time step?

Question 4 [15 marks].

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

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

(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.

(b) Now consider a European call option, with expiry time T = 2 years, and strike price K = 105. The current price of the underlying stock is S0 = 100, and the stock price is assumed to follow geometric Brownian motion with annualised drift µ = 0.14 and volatility σ = 0.3. 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 = 0.02.

Using the discrete-time binomial model with N = 8 periods, ﬁnd an approximate value for the fair price of this option at time 0.

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

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

assuming that the underlying share pays no dividends during the lifetime of the option. The various symbols here have their usual meanings, and you may assume that S0 , σ and T all are strictly positive.

By using this result, or otherwise, prove that the Black-Scholes price C0 at time 0 of the log option described above is given by

where

and Φ(x) and φ(x) are the cumulative distribution function and probability density function respectively of the standard normal distribution.

(c) By (partially) differentiating the expression for C0 with respect to S0 , ﬁnd the formula for the delta ∆ of this log option.

(d) Finally, by differentiating once again, ﬁnd the formula for the gamma Γ .

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