Smile Implied Probability Distribution

We can actually make some progress without any knowledge of the true dynamic causing the IV smile. 


Suppose we are interested in a particular expiry T because we have a European contract only depending on the spot at T, and not on the path the spot takes between now and T.

Let say this contract has payout function A(ST) so that to price it we want to calculate:

Smile implied distribution1

where pT is the probability density function for the spot level at time T under the risk-neutral measure. 


The IV smile tells us the price of vanilla call options at all strikes, and therefore we know:    Smile implied distri 2

where B is the BS formula but without the discount factor. 


It is easy to check that differentiating the call option payouf with respect to strike gives a step function: 

Smile implied distri 3


The expectation of a step function is logically equal to the cumulative probability distribution, which we denote: P(ST > K): 

Smile implied distri 4


Then, we can differentiate the cumulative probability to get the probability density function: 

Smile implied distri 5


When we know the IV smile at a given expiry, we can therefore deduce the risk-neutral probability density function. Then we can go back to our first equation of this section and use this density function to price any European contract. 


Using vanilla prices in this way to determine the probability density is known as the Breeden and Litzenberger approach (1978). 






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