## Apr 1, 2011

### The Probability And Expected Return, Part 1

Many option traders look at delta as indicator of probability of potential payoff, however are not aware of that payoff's magnitude. They make qualitative judgments and use vague rules for decision making. For example a seller of OTM options will tell you that the odds of going in the money are only 10%, so he’ll make money approximately (adjusting by premium and trading costs) 90% of the time. Such simplistic thinking however does not take into account what exactly will happen if option goes in the money. The concept that I'm explaining here is not some high-level math, but rather a simple and fast rule of thumb that can be easily used in practical trading.

With the usual Black-Scholes assumptions:
Option price = E[Discount * Max(S-K,0)], where S is the stock price as expiration, E is the expectation operator, P is the probability, Discount is exp(-rT) . Max(S-K,0) is a payoff of a call option at expiration; similar analysis will hold for puts. In simple words, the options price today is how much you would expect it to be worth at expiration, discounted by interest rate. For simplicity I'll ignore the discount factor.

Option price = E[Max(S-K,0)] =
E[Max(S-K,0)|S>K] * P[S>K]
+ E[Max(S - K,0)|S<=K] * P[S<=K].

It means that the options price is equal to its expected payoff in the money multiplied by the probability of being in the money, and its expected payoff out of the money multiplied by the probability of being out of the money. Because option is not worth anything when it is out of the money, the second term is zero.

Option price = E[Max(S-K,0)] = E[Max(S-K,0) |S>K] * P[S>K]

The last term is the delta of the option. Now we can invert the equation, obtaining

E[Max(S-K,0) |S>K] = E[Max(S-K,0)] / P[S>K] = E [Max(S-K,0)] /Δ

The option payoff conditioned on it finishing in the money is option price divided by delta.

Example:

The current price of an asset is \$100, strike is \$130, interest rate is 10%, and time to maturity is ¼ year. The option trades at \$0.80, or at 35% annual volatility. Delta of the option is 10%. While the option will let the seller collect premium 90% of the time, the seller faces a potential risk of losing the premium and more if the option goes in the money. This value is \$0.80/0.1 = \$8.00

 Probability Result Expected Payoff 90% Collect premium \$0.80 10% Lose premium, or more \$0.80-\$8.00=-\$7.20

Selling this option can be thought of as making a binary bet that will pay \$0.80 90% of the time, and lose \$7.20 10% of the time. Simple calculation 0.8*0.9 – 7.2*0.1 = 0 confirms that it is a fair bet.