No expected return in Black Scholes formula: But how about the gamma?

Question

A lot has been written about the fact that the expected return of the underlying asset is not part of the Black Scholes formula. I understand the argument that the performance of the underlying asset can be hedged out via delta hedging and that the dealer who has sold the option is therefore indifferent to the performance of the underlying asset.

But I have some doubts about how the expected return could influence the expected realised gamma of the dealer and, in consequence, the fair value of the option. Here is an example:

Scenario 1: Standard Black Scholes:

• Say the dealer sells a 2 year call option on a stock with current price 100 dollars, option strike price 120 dollars and implied volatility of 10%. Assume the risk free rate is zero.
• Under the standard Black Scholes formula, assuming the drift is the risk free rate, which is zero here, it would require a 2 standard deviation move of the stock for the option to become at-the-money.
• The dealer's losses from being short gamma (and realised via constant delta hedging) are expected to be rather low because the delta of the call option is low and the same is the case for the gamma. Only those few Brownian motion return paths where the stock performs extraordinarly well will be in an area where delta and gamma are high.

Scenario 2: When using an expected return

• Though if the expected return were to be let's say 10% then over the 2 year period the drift would be 10% * 100 dollars * 2 = 20 dollars so the center of the probability distribution at the option maturity would now be at-the-money (120 dollars).
• I'm well aware of the difficulties of measuring (or even defining) the expected return of an asset. But for the sake of this example can we please assume that 10% is the true expected return.
• This means the delta of the option would be much higher. The dealer's short gamma position would be much higher as well (in absolute terms) and the dealer would expect to lose much more from being short gamma because now a much larger proportion of the possible Brownian motion paths of the stock will be in a price region where delta and gamma are high.

If we consider that the option is priced by the dealer and the dealer sells the option at least at the expected value of his PnL from dynamic delta hedging across all Brownian motion return paths, then shouldn't the option price differ between Scenario 1 and Scenario 2?

Wouldn't the dealer expect a higher option premium if he knew that the expected return was 10% rather than the risk free rate of zero?

Answer 1

In the reduced variables, BS PDE becomes "theta equals gamma".

Answer 2

Generally speaking theoretical expected losses from hedging error should be equal to expected profits.

If you think about your second example it isn't much different from asking the same for a one year expiry and $110 strike. All that matters in pricing is the cost of hedging, I don't see how either of your examples could result in significantly higher hedging costs.

If you want to examine your question properly then you should write some code and run some hedging procedure simulations with different expected returns then plot resulting PnL histograms.

Answer 3

1. "Gamma should be higher, resulting in higher PnL so price should be higher" doesn't really seem right/trivial. Greeks are not magic - they are derived from price, and it is not possible to understand the greek before you understand the price.

2. The fact that there is 10% drift on the stock yet it is valued currently at the spot reflects a higher risk aversion of the market, nothing else. That is the only thing that is counter-intuitive.

3. The derivative market is a market of replication and therefore cannot reflect anything counterintuitive by itself. Whatever the weird, counterintuitive dynamics of the market are, the derivative market just follows it. It has to be CONSISTENT with the stock market, even if the stock market is counter-intuitive. Consistency requires black-scholes.