Is the Call Option Price a Probability Measure?
Rahul Bhattacharya June 28, 2011
Is the Call Option price a probability measure? If the price of a call option on a particular stock trading at 100, is say, $4.50, then can we say that this price represent a probability measure? Talk to any trader and he will say, of course not. He would say that the call price can always be expressed as a percentage of the spot price but in no way can it symbolize a probability measure.
Actually, he would be wrong to say that. The Call option price is indeed a probability measure, as shown by Dilip Madan of University of Maryland, recently. But first let’s tackle a few mathematical preliminaries.
What is a probability measure?
Mathematically speaking, a “measure” relates to the size of an object (or an event). For example, the length of a line, the area of a two dimensional surface or the volume of a three dimensional objects such as a cube represents the size of the object. Also, areas of two surfaces can be added up, just like volume or lengths. When we talk about the “size” of an object, we in essence, assign a nonnegative real number to the object which defines or characterizes that size. Therefore, “length”, “area” or “volume” is known as “measures” (on some real valued space).
Just as for objects, so it is for events. When we talk about the “probability” of an event happening, we are, in essence, talking about the size of that event and we try to measure the size of that event by assigning a probability to that event. In that respect, the “probability” is also a measure, just like “area” or “volume”. However, there is one big difference. Probability of an event assigns a value of 1 (one) to the entire space of events (known as the probability space). All probabilities must add up to one. The sum of the probability of an event happening and not happening is equal to one.
A digital option price is a probability measure. A digital call price is equal to the probability of an option finishing in the money and a digital put price is equal to the probability of the option finishing out of the money. The sum of the probability of the option finishing in the money (i.e. the spot at maturity is equal to or greater than the strike) and the probability of the option finishing out of the money (i.e. the spot at maturity is less than the strike) is equal to one.
But how can we say that a vanilla call or a put option is a probability measure? In a seminal article in 2008, Dilip Madan, at the University of Maryland, has shown that a call price can be represented as a probability measure. In fact, generally speaking, Madan has shown that a call price is the probability that after the maturity of the option the stock price remains below the strike forever.
Actually, it can be shown quite easily and very elegantly, as Carr and Madan have done in their paper, that a call or a put option price can symbolize a probability measure under a different measure.
If is the price of the asset at maturity and is the strike price then the (undiscounted) price of a call option is given by:
In terms of the expectation operator we can write the above price as:
Where, the above expectation is under a risk neutral probability measure, .
Now, the value (price) of the asset today, , can be represented under a suitable probability measure as:
Therefore, since is constant, the normalized call price can be written as
Here, the modified call price is under a different probability measure . This is actually known as the share measure.
The buyer of a call option can choose to get paid in shares instead of US Dollars and in that case the claim – the payoff from the call option – gets valued using the share price tilted measure. The two measures are equivalent, as in the prices of the claims – call option – paid out in US Dollars can be seen as equivalent price measures where the claim is paid out in shares.
Secondly, the function has a minimum value of zero and a maximum value of 1 (one).
For various values of the normalized call price will assign a value of 1 (one) to the entire space of payoffs. Hence, the normalized call price, is indeed a probability measure.
Reference: Saddle Point Methods of Option Pricing, Peter Carr and Dilip Madan, The Journal of Computational Finance, 2009.
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