Why Black-Scholes Overprices Calls When the Stock Can Default to Zero
Summary
The document considers a call option when the underlying stock can abruptly fall to zero before expiry, while the other Black-Scholes assumptions are retained. Standard Black-Scholes geometric Brownian motion does not reach zero, so its formula omits this default-like outcome. Since a zero stock value makes the call worthless, ignoring that possibility can overstate the option’s value relative to a model that includes it.
One answer illustrates the adjustment as a probability-weighted combination of the Black-Scholes value and zero, with the weight representing the zero-price event under an appropriate probability measure. A second answer notes that market prices can reflect default risk through implied volatility, and suggests treating such securities with a credit instrument framework and a suitable default model. The simple mixture is intuition rather than a complete pricing method: the document does not specify event timing, recovery, or how default risk interacts with the remaining assumptions.
Key ideas
- Black-Scholes geometric Brownian motion does not model a stock price jumping to zero.
- A call is worthless if the underlying reaches zero, so omitting that event can overstate its value.
- A probability-weighted mixture illustrates how a zero-price event could reduce call value.
- Defaultable stocks may require a credit model, and observed option prices can reflect default risk.
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# Answer by nbbo2 (score 2, accepted)
# When a stock's price could suddenly drop to zero before expire. does black-scholes misprice the option? Too high or Too low?
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?
## Answer by nbbo2 (score 2, accepted)
https://quant.stackexchange.com/a/32294
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.
## Answer by Ashkar (score 0)
https://quant.stackexchange.com/a/32321
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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.