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