Quantitative Question – BLACK SCHOLES Consider a call option on a stock. Assume that Black-Scholes prices the option correctly if all of the assumptions of Black-Scholes hold true. Assume in addition that these assumptions really are true save one: there is a chance that the stock could instantly plunge to zero sometime before expiry. Does Black-Scholes then misprice the option? Too high or too low? Why?
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2 Answers
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Black Scholes assumes the stock price follows Geometric Brownian Motion, which can approach but never hit zero. If it is possible that the stock price hits zero, then that is a very adverse scenario that will make the call worthless that is not taken into account by Black-Scholes. So the BS formula is clearly overstating the value of the call.
You can think of the true value of the call as being some convex combination of the BS value and 0, where the weights reflect the likelihood (in some probability measure) of the two scenarios:
$$C_{TRUE} = (1-p)*C_{BS} + p*0$$
the likelier the "go to zero" scenario the bigger p is.
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$\begingroup$ (P.S. I am assuming that the $\sigma$ used in $C_{BS}$ is the usual diffusion parameter that describes c.t. fluctuations in price and that the sudden drop to zero is exogenous to this diffusion and unforseen). $\endgroup$– nbbo2Commented Feb 8, 2017 at 20:28
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If the probability of stock going to 0 is reasonably high the BS implied vol you would pay for the option will reflect that. BS equation does not decide the market price of the option.
Strictly speaking such names would be priced as a credit instrument with appropriate default model.
