QF4102 Assignment 2 : AY24/25 Semester 1 – Financial Modelling and Computation, NUS, Singapore

1. (i) Implement the explicit difference scheme (II), that is, the backward-time scheme with Vinin place of V in+1, to estimate the value of a European vanilla call option.

function v = FD_eds_call(S0,X,r,T,sigma,q,N,I)

Use linear interpolation if the current input values do not lie exactly on the grid.

(ii) Use your Matlab function to estimate the value of the option with S0 = X = 1,
and a time to maturity of 0.5 year. The current risk free rate is 2%, the volatility of the asset is 0.5, and the dividend yield is 3%. Obtain the estimate with ∆t = 0.01 and h = 0.05. Compare your estimate with the exact Black-Scholes price.

(iii) Determine a lower bound for N = T/∆t so that all coefficients in the finite
difference equation are nonnegative.

(iv) Use the lower bound in (iii) to obtain a new estimate using the explicit difference scheme. Compare your estimate with the Black-Scholes price.

v) Gradually reduce the value of N, and through multiple calls to the above function, determine the cut-off value for N where the option estimates becomes meaningless. Comment on the value and estimates obtained.

2. (i) Write down the algorithm to estimate a fixed-strike arithmetic Asian call option using a two-state-variable forward shooting grid method. Assume that the option is newly issued.

(ii) Implement the above algorithm in a Matlab function. Use the following header for your function:

function v = fsg_fixArithAsianCallNew(S0,X,r,T,sigma,q,N,L)

and save the function in its own file.

(iii) Test your implementation with S0 = X = 100, r = 0.03, T = 1, N = 4 and L = 2,
to verify the correctness of your implementation. Assume that the underlying
asset pays no dividends, and that the volatility of the asset is 0.22.
Note: In your implementation, you need not store option values for all time points with a 3-dimensional array. Instead, for the backward time iterations, you only need to store values with two 2-dimensional arrays for the two successive time points.

(iv) Modify your algorithm and implementation in parts (i) and (ii) to estimate the value of a arithmetic Asian option that is not newly issued, with a running average runavg based on Nhist time periods. Use the following header for your function:

function v = fsg_fixArithAsianCall(S0,X,r,T,sigma,q,N,L,runavg,Nhist )

(v) Using the above function, obtain estimates of the value of a arithmetic average Asian call option, which was issued 2 months ago, with 1/2 year to expiry. The current underlier price is $95, and the strike price of the option is $90. The risk free rate is 4%, the volatility of the underlier is 30%, the historical average of the underlier is $93, and the underlying asset pays no dividends. Use N = 60, 120, 180, 240 time periods in your implementation, and for each value of N, obtain option value estimates for ρ = 1, 1 2 , 1 4 . Tabulate your numerical results, and comment on the values obtained. Also obtain the runtimes for each value of N and ρ, and tabulate the results, and comment.

(vi) Plot the runtimes versus N. Comment on the plot obtained, and making reference to the two-state-variable binomial tree method in the previous assignment, comment on the computational efficiency of the forward shooting grid method.