Why Expected Return Does Not Enter Black–Scholes Option Pricing
Summary
The document explores whether an underlying asset’s expected return should affect a Black–Scholes option premium through the dealer’s expected losses from dynamic delta hedging. It contrasts a call priced under a zero drift assumption with a hypothetical higher expected return, arguing that the latter could place more probability near the strike, where delta and gamma are larger. This leads to the question of whether the dealer’s hedging costs and required premium should rise.
The replies emphasize that option pricing must remain consistent with replication and the underlying market, and that Greeks follow from the price rather than independently determine it. One response points to the Black–Scholes equation in reduced variables; another suggests testing hedging outcomes through simulations under different drift assumptions. The exchange does not present such simulation evidence or fully derive the pricing result. Its main limit is that the thought experiment’s intuition about gamma exposure does not itself establish a change in the replicating price.
Key ideas
- The question links expected return to the distribution of prices near the strike and the dealer’s gamma exposure.
- Black–Scholes pricing is described as consistent with replication and hedging, rather than set by expected return alone.
- Option Greeks derive from the option price and cannot independently settle the pricing question.
- The responses suggest examining hedge PnL across drift assumptions, but provide no simulation results.
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Full text
# No expected return in Black Scholes formula: But how about the gamma? # No expected return in Black Scholes formula: But how about the gamma? 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 by achirikhin (score 1) https://quant.stackexchange.com/a/79290 In the reduced variables, BS PDE becomes "theta equals gamma". ## Answer by Bennnn (score 0) https://quant.stackexchange.com/a/74068 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 by Arshdeep (score 0) https://quant.stackexchange.com/a/79288 - "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. - 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. - 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.
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