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Finance and matlab

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

follows:

The grid for the log stock price is given as

xvec = [xmin, xmin+x, xmin+2x, . . . , xmax−x, xmax]

however, we cannot set xmin = ln0 as this is −, so we set xmin = lnS0−2×pwhere

= 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

= 20%.

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

S0 _1/T_,0_.

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

method.

Habilidades: Finanças, Matlab and Mathematica

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