Why Option Pricing Uses Volatility but Not Expected Stock Return

# Expected return of the underlying not an explicit input in the binomial options pricing formula, but volatility is?


# Expected return of the underlying not an explicit input in the binomial options pricing formula, but volatility is?












I'm new to options (finance in general) and am trying to learn the theory.

I have read that the argument for why the underlying expected growth doesn't matter is that we're pricing the options relative to the underlying. That is, this information is already baked into the price of the underlying. Intuitively, I understand this as "no matter what the stock is trading at, we use that as our baseline for out option value calculation, and we will always be able to create the replication portfolio given our assumptions". Or, "if consensus is that the underlying company will do well in the near future with high certainty, then the stock will already become more expensive, i.e. reflect this consensus. We therefore need not price this in again in the option."

Fair enough, but then I started wondering why the same does not hold for volatility? E.g. in the binomial model, we have to explicitly input u and d.

Do stock prices not price in volatility? From my understanding, by the CAPM (which I know is perhaps not always the most realistic, but in theory at least) only systematic risk should be rewarded. So if we then consider the theoretical value of the stock as some discounted value of future cash flows, where the discount rate is obtained via CAPM, volatility in the stock has SOME effect on the price? Is the explicit volatility parameter in the option pricing formula then accounting for the idiosynchratic risk NOT accounted for in the stock price? I don't believe this is correct, but I also can't help but feel like we then double count volatility, as it's both somewhat incorporated into the stock price, and then input again. Although for the stock it should result in a cheaper valuation, whereas the option would be more valuable due to the limited downside - unlimited upside asymmetry.

Thanks!

## Answer by Barbab (score 2)

https://quant.stackexchange.com/a/82190

I think the key insight here is what option pricing models are actually trying to capture.

When we price options, what we're really doing is finding the fair value at which no arbitrage opportunities exist. The binomial model (and Black-Scholes) are built on the principle of constructing a replicating portfolio that exactly matches the option's payoff.

Expected return doesn't enter the formula because we're assuming risk-neutral pricing. In a risk-neutral world, all assets grow at the risk-free rate. This is a mathematical convenience that gives us the right price even though the real world isn't risk-neutral. The underlying stock price already incorporates market expectations about future returns, so we don't need to input those again.

Volatility, however, is different. It determines the range of possible outcomes, not the expected outcome. Even if two stocks have the same expected return, the one with higher volatility creates more value for options due to their asymmetric payoff structure.

The stock price doesn't fully "price in" volatility in the same way it prices in expected returns. Yes, CAPM suggests only systematic risk is rewarded, but that doesn't mean volatility (total risk) isn't relevant for option pricing. Option values increase with volatility regardless of whether that volatility is systematic or idiosyncratic, because options benefit from extreme movements in either direction.

So we're not double-counting volatility. The stock price might reflect a risk premium for systematic volatility, but the option price needs the total volatility parameter because options have convex payoffs that benefit from all types of price movement.