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));
```Break-even point = £2.15