Deriving an Implied Volatility Smile from Bimodal Outcomes

Summary

The note describes how a two-outcome stock-price distribution can produce a volatility smile. In the example, a biotech company’s value either jumps upward after regulatory approval or falls sharply after denial. The method prices a call at each strike by taking the risk-neutral expected payoff across those two possible outcomes, using the probability and jump levels specified in the cited model.

For each strike, that expected payoff is then matched to the price from the Black–Scholes–Merton model, and the volatility that gives the match is the implied volatility. The answer reports that this procedure produces a plot similar to the referenced example, but it does not provide the plotted data or a full derivation. The shape depends on the assumed jump outcomes, risk-neutral probability, and pricing setup; it should not be treated as a general description of all biotech event distributions or observed option markets.

Key ideas

• A bimodal outcome can be represented by a risk-neutral distribution with two possible terminal values.
• Compute each call’s value as the probability-weighted expected payoff at its strike.
• Invert the Black–Scholes–Merton pricing formula to obtain implied volatility for each strike.
• The resulting smile depends on the assumed outcomes and risk-neutral probabilities.

Tags

# The shape of the volatility smile for bimodal outcome


# The shape of the volatility smile for bimodal outcome












Let's say that we have a biotech company that awaits FDA approval. In the case of approval the company gets a cash injection and in the case of denial it is pretty much bankrupt. Clearly, this is a very bimodal outcome. According to the this website the volatility smile looks as follows:

Can someone explain to me why it would look as above? There is no clear explanation given.

## Answer by Kermittfrog (score 3, accepted)

https://quant.stackexchange.com/a/59621

According to the blog post you cited above, all you have to do is simply back out Black Scholes Implied Volatilities from the prices in the first part of the website.

For a given strike $X$, risk-free rate of zero, and jump-level $H$, the present value according to the blog post is:

$$PV(H,X)=\mathbb{E}_\mathbf{Q}\left((S-X)^+\right)=p_\mathbf{Q}(H-X)^+ + (1-p_\mathbf{Q})(\frac{1}{H}-X)^+$$

In his example, $p_\mathbf{Q}=\frac{1}{1+H}$.

Having thus obtained the PV, you invert the classical Black-Scholes-Merton option pricing equation to find the implied vol:

$$ \sigma_{implied}: Call(S_0=1,X,r=0,y=0,\tau=1,\sigma_{implied})\stackrel{!}{=} PV(H,X)$$

Doing this, and using his $H$-parameter of $H=1.2$, I come up with a quite similar plot:

HTH?

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.