# Perpetual Option Price under Black Scholes model

Would like to ask you, how would you price an Option which has its starting underlying price S0 = 70 dollars, with no dividends, and that pays 0.5 dollars each time the underlying price hits a barrier of 30 dollars.

Normally the initial Black Scholes PDE will be converted into a simpler ODE, has time decay = 0 (due to the perpetuity of the option).

How would you solve it\price it? (risk free rate is assumed to be constant)

I am mainly interested on the final pricing equation.

Thanks!

• It's fairly straight-forward if $r$ is constant (you may need a positive dividend yield for perpetual options to have finite values). You also need to specify some model for the stochastic interest rates (Hull White? Cox-Ingersoll-Ross? etc.) Out of curiosity, what is your motivation? Where does this problem arise? – Kevin Jul 20 at 18:42
• That may be dumb, but I would say to run Monte Carlo simulation into like 1000 years into the future, count the times the barrier was hit and average the results. The GBM process tends to infinity when r-0.5vol^2 is positive otherwise it tends to zero a.s. By the time the process reaches 1000 years it will be of order 1e10 or 0, so it doesn't matter anymore. quant.stackexchange.com/questions/24825/… – emot Jul 20 at 19:39
• Why isn’t the ‘option’ worth 70? Proof: buy the option for 70, sell the underlying for 70. All upfront and dividend cash flows match in perpetuity. – dm63 Jul 21 at 11:57
• I would rather solve the analytical solution, by defining the proper ODE, with the respective boundaries, and arrive at a final value (which will be a function of risk free rate r) - all starting from the original the Black Scholes PDE – Joquim Jul 21 at 13:07
• @Joquim the Black Scholes PDE/ODE won't hold if rates are stochastic. Your (perpetual) option value is a function of $S$ and $r$. To obtain the right differential equation, you need to specify the distribution (SDE) of the short rate. If $r$ is deterministic though, you obtain a simple ODE that you can solve in closed-form – Kevin Jul 21 at 15:18