# 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.

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?

• thanks for your help and time. Any idea why his tails are flat and yours are upwards sloping? Dec 3 '20 at 9:58
• No, unfortunately not - Maybe if you ask him he could say whether he flattened the wings or maybe there's still some more magic going on. Good luck! Dec 3 '20 at 10:15
• I think he makes the assumption that outcomes <1/H and >H are not possible and therefore the IV is zero for these strikes Dec 3 '20 at 10:42