# Can the risk-neutral measure depend on the option type?

In an ideal Black-Scholes setting, the Risk-Neutral measure $$Q$$ is unique and so, obviously, does not depend on what derivative instrument we want to price.

Assume some deviation from perfect markets (e.g. trading costs), and also assume that the risk-neutral pricing formula is still valid: $$\text{Price}=E^Q[\text{future payoff}].$$

In this case, it seems that $$Q$$ may be option-specific, even though the underlying asset is the same. In particular, if replicating one option has different expected trading costs than another option, then it could be that to price the two options, we need to use two different measures $$Q_1$$ and $$Q_2$$ in the above formula.

Is there any formal analysis to support (or not) the above reasoning?

• Yes, good question. Risk-neutral measure can even depend on whether you are buying or selling an option. For instance for vanilla options you can have a bid RN measure and an ask RN measure so to speak. The difference being the different volatilities that take into account transaction costs for buying or selling. See Leland's paper "Option pricing and replication with transaction costs" for more details in a deterministic volatility setting. Oct 1, 2020 at 16:42
• Great reference, ilovevolatility, thanks! It's close to what I want but not exactly, I believe. The adjusted volatility (formula (13)) does not depend on the strike price, hence, it seems $Q_1$ and $Q_2$ would be the same for two options with different strikes. If you know of an example when this is not the case, this would answer my question Oct 1, 2020 at 18:40