Inferring Stock Price Distribution from Option Quotes

It is no secret that options and the underlying instruments are very intimately related and movements in options markets are highly influenced by the movements in the underlying market. In particular, we know that options are forward looking, as they are essentially bets on the future prices of the underlying. Therefore, it seems natural to think that option quotes give some notion of what the market thinks the price of the underlying is going to be at expiry. In particular, it would be useful to know what the consensus expectation of the underlying price at expiry is, based on how the options are priced, as that could result in several arbitrage trades or trades that compare the consensus price and our forecasted price.

The Pricing of Options

A call option is defined as the right (but not obligation) to buy the underlying instrument at a predefined price (known as the strike price) at a future date (known as the expiry date). Similarly, a put option is defined as the right (but not obligation) to sell the underlying at a strike price at expiry.

Logically, the intrinsic value of an option would depend on what the price of the underlying is going to be at expiry. A call (put) option would therefore only payout if the underlying price is higher (lower) than the strike at expiry. This naturally leads us to define a few useful terms: an In the Money (ITM) option is an option whose strike such that if I were to exercise now, I would have a net profit. An Out…