The partial differential equation (PDE) for an option price f = f (x, t) where the option
price is expressed as a function of the log stock price, x = lnS, and time t is given as
The grid for the log stock price is given as
xvec = [xmin, xmin+x, xmin+2x, . . . , xmax−x, xmax]
however, we cannot set xmin = ln0 as this is −, so we set xmin = lnS0−2×pwhere
= T −t is the time-to-maturity of the option. Similarly, set xmax = lnS0 +2×p.
Assume the initial stock price is S0 = 100, an interest rate of r = 5% and a volatility of
Derive the coefficients on the explicit finite difference algorithm by discretising the above
PDE and solving for fi, j−1 in terms of fi+1, j, fi, j and fi−1, j.
Consider an optionwith the following payoff: f (ST ,T)=max_K −100×__ST
Assume the time-to-maturity of the option is T = 3/12 years, the strike price is K = 110,
and the continuously compounded risk-free rate is r = 5%.
This option is not path dependent hence it can be shown that the price of this option follows
the same partial differential equation (PDE) as a vanilla European call or put option but
with different terminal and boundary conditions.
• Use the explicit finite difference (EFD) method to price a European version of this slightly
exotic put option.
• Report the option price for the initial stock price of S0 = 100 and provide a plot of the time
0 option price versus the vector of stock prices.
• Price the American version of this exotic put option using the explicit finite difference
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