# Time independent Black-Scholes with jumps

Alex Townsend, 28th October 2011

## Contents

Chebfun example ode/BlackScholes.m

The Black-Scholes equation is a partial differential equation for modelling the price of an European option in terms of underlying equity prices [1]. In this example we consider the time independent Black-Scholes equation which is a one-dimensional ODE.

## Good investment?

Let's suppose you buy an European option for £50 that depends on the share value of Apple Inc. and the risk-free interest rate is 3%. At the time you decide to sell the option an incremental tax applies so that you pay 20% of the price of the share rounded down to the nearest multiple of 10. If the underlying share is worth £1, you lose all your investment and when its worth £50 you will be able to sell your option for £150.

r = 1.03; % Risk-free interest rate vol = 1; % Volality tax = 0.2; % Rate of tax taxpts = 10:10:40; N = chebop(@(s,V) .5*vol*s.^2.*diff(V,2) + r*s.*diff(V) - r*V,[1,50]); N.lbc = @(V) V+50; N.rbc = @(V) V-150; N.bc = @(V) jump(V,taxpts)+tax*feval(V,taxpts); y=N\0; plot(y), hold on; title('Profit/loss versus underlying share price','FontSize',16); xlabel('Share Price in pounds'); ylabel('Profit');

## Break-even point and double your money

As a shrewd investor, you would like to know the underlying share price when you break-even and when you double your money. This can be computed by the roots command.

fprintf('Break-even point = £%1.2f\n',roots(y)); fprintf('Double your money = £%1.2f\n',roots(y-100));

Break-even point = £2.15 Double your money = £25.43 Double your money = £30.00 Double your money = £30.46

Note that you do not double your money when the underlying share price is £30 this is just where the solution jumps across the £100 profit mark.

Don't lose all your money!