# Effect of massive volatility on BS formula

I am experimenting with very high volatility on the standard Black-Scholes formula. I set risk free to zero, time to expiry to 1, volatility to 1 (=100%), and underlying to 1. Then I simulate the profit and loss on (i) the delta hedge account and (2) the option account between t=1 and t=0 and compute compare the expected return on both. There is a persistent bias in the result – the expected return on the option account is consistently higher.

I’m wondering if this is caused by the difference between volatility of the log returns and actual returns. There is little difference at low volatility, but it is huge at such massive vols. It could be calculation error of course, but I don’t think so because everything works fine at typical volatilities.

• what measure are you simulating? how are you discretizing? how many steps are you using? – Mark Joshi Mar 27 '15 at 23:59
• Thanks. Drawing z from a standard normal distribution, simulating lognormal = exp(z). With S and K set to 1, this gives delta = N(.5) and price of option = delta-N(-.5). – quis est ille Mar 29 '15 at 8:21

## 1 Answer

i would guess it is the difference between

$$\exp(-0.5\sigma^2 T + \sqrt{T}\sigma Z)$$

and

$$\exp(\sqrt{T}\sigma Z)$$

The first is correct, the second is wrong.

• Why is the second wrong? – quis est ille Mar 30 '15 at 10:36
• Ito's lemma... look up the change of coordinates from geometric Brownian motion when taking logs. This is in any financial maths text book including mine. – Mark Joshi Mar 30 '15 at 21:30