代写MATH40082 (Computational Finance)帮做R语言程序

MATH40082 (Computational Finance)

Main Assignment: Simulation Methods

1 Background

1.1 Stock Options

The trader has calibrated a specialised risk neutral process for some underlying stock price. Given the current stock is S0, market prices indicate the risk-neutral distribution of the stock price at time T is given by:

for some calibrated functions f and v2.

Consider a financial contract C(S, t) written on the underlying asset S with payo↵ on expiry

C(S, T) = g(S).

Then for any given payo↵, the analytic solution may be found by carrying out the numerical integration

To carry out a Monte Carlo valuation of the financial contract, we may use samples from a standard random normal distribution

to write the equation

Then we can simply average out the discounted payoff over n to get our approximation to the value of the financial contract:

where g(S) is the desired payo↵ of the contract.

1.2 Path Dependent Options

Now assume that the risk neutral stochastic process follows the SDE

dS = f(S, t)dt + v(S, t)dW.

The path dependent options you will be pricing depends on S(tk) which are the share prices at K + 1 equally spaced sampling times t0, t1,..., tk = k∆t, ..., tK with t0 = 0, tK = T and

Use an Euler type scheme to write

for k = 1, 2, ..., K to estimate the underlying asset values at each time. Here φi,k is a random draw from a Normal distribution.

For path dependent options, the payo↵ function can be written g(S(t0), ..., S(tk), ..., S(tK)) and so the value of the path dependent option can be approximated by

Asian Option

Assume that a discretely sampled Asian option has a payo↵ depending on the discretely sampled average given by

Then we can write

g(S(t0), ..., S(tK)) = G(S(tK), A),

where G(S, A) is the payo↵ function depending the type of option.

There are di↵erent classes of Asian option, resulting in different payoff conditions. In this coursework we look at simple European style. call or put options. A fixed strike call option will have the payoff

G(S, A) = max(A − X, 0)

where X is the strike price and a floating strike call option would be

G(S, A) = max(S − A, 0).

where A is sometimes called the average strike price.

A fixed strike put option will have the payo↵

G(S, A) = max(X − A, 0)

where X is the strike price and a floating strike put option would be

G(S, A) = max(A − S, 0).

where A is the strike price.

Lookback Option

The discretely sampled Lookback option has a payo↵ depending on the discretely sampled maximum or minimum given by

or

Then we can write

g(S(t0), ..., S(tK)) = G(S(tK), A),

where G(S, A) is the payo↵ function depending the type of option.

There are different classes of Lookback option, resulting in di↵erent payo↵ conditions. In this coursework we look at simple European style. call or put options. We can either have a floating strike S or a fixed strike X. For example a floating strike Lookback call option would give

G(S, A) = max(S − A, 0)

where A must be the minimum, and a floating strike Lookback put option would be

G(S, A) = max(A − S, 0).

where A must be the maximum.

A fixed strike call option will have the payo↵

G(S, A) = max(A − X, 0)

where X is the strike price and A must be the maximum. and a fixed strike put option will have the payo↵

G(S, A) = max(X − A, 0)

where X is the strike price and A must be the minimum.

Barrier Options

The discretely sampled knock-out barrier option will be knocked out (and return a value of zero) if the a barrier asset price B is crossed before the maturity date.

The option will be an “up” option if the knock out condition is on S>B, or a “down” option if the condition is on S<B.

Let the variable A be a binary variable such that

Then we can write

g(S(t0), ..., S(tK)) = A · G(S(tK)),

where G(S) is the payo↵ function depending the type of option.

So for example an up-and-out knockout barrier call option has the payo↵

G(S) = max(S − X, 0)

where