@Mobius, thanks for the recap. It’s been years since I went through the math exercise but I remember about how you create the replicating portfolio and how it makes the option value independent of the expected value.
I also think that the rush to create the risk-neutral probability in order to make it easier to calculate makes it harder to get an intuitive feel for what’s going on. I remember staring at the risk neutral probability and trying to figure out how to interpret what it meant to get an intuition. I came up with something like “It’s underlying’s return distribution, transformed such that the risk-free asset is the baseline case” or somesuch. That doesn’t sound quite right now, but something in the ballpark to that.
So I think what you are saying is that the distribution of returns for the underlying is the same in the real world as in the risk neutral world (or at least it’s supposed to be, even though we often just assume it’s lognormally distributed, at least for BS), but that because of the risk-neutral transformation, the option price is not the same as the present value of the expected return of the underlying.
Is there an interpretation of risk-neutrality that makes good intuitive sense? Wilmott claims that that’s a big thing in his CQF program, and maybe I had it once, but don’t remember.
Finally, there’s the issue that option prices seem to depend primarily on the details of the contract, the current strike price, and the volatility. But of course, volatility is itself an expected value, since it can change in the future. Perhaps there is a risk premium in options that is as much about uncertainty in future volatility as it is about uncertainty in one’s ability and costs in keeping the replicating portfolio property hedged until expiration.
I know I’ve said a mouthful here, but your comments were really helpful, I thought.
Selling call spreads seems sensible here. A nicer way to get short and make your position behave well, since the deltas and vegas of each end of the spread tend to hedge each other (albeit not perfectly).
Bchad, I’ve been trying to find an intuitive explanation for the risk-neutral probability measure but haven’t really come across anything fully satisifying. I remember reading some threads on the topic on the Wilmott forums where Paul Wilmott was also a participant, and don’t think there was anything extraordinarily insightful to provide full clarity (although lots of good viewpoints from different angles). I almost think that one is best served by keeping their intuition about the no-arbitrage option price in the context of the replication strategy only, rather than trying to extend it and develop intuition about the risk-neutral measure. Some writers even object to refering to the risk-neutral measure as “probability” because of the confusion it creates when people try to make sense of it. It is certainly incorrect however to think about the risk-neutral probabilities as some sort of opinion that the option markets hold regarding the future of the underlying stock.
You can try to price options outside the risk-neutral world, by using the expected value of the option payoff under the real-world probability, discounted at a risk-adjusted rate (I’ve seen this referred to as the actuarial approach). You can still assume that the real-world probability distribution for the stock price is lognormal, just the risk-free rate drift is replaced by some expected return mu which you can estimate from CAPM for instance. The issue is that while you have an easy way to calculate the expected value of the option payoff, the risk-adjusted discount rate for the option cannot be mu because the option has different risk profile than the stock. The intuition behind this is that the option is in a way a levereged position in the underlying - it is equivalent to holding some stock and some bond, so the appropriate risk-adjusted discount rate for it is some sort of option “WACC” - a combination of the risk-adjusted discount rate for the stock mu (CAPM) and the risk-free cost of borrowing r. Moreover, the “option WACC” is not easily tractable analytically - it will vary with the moneyness of the option, the time to maturity, etc. So under the ‘actuarial approach’ you have some very intuitive and transparent expected option payoff, under the real-world measure, and a very unclear/untractable risk-adjusted discount rate that pairs with it.
Then some mathemagic happens that academics call the Girsanov theorem and the real-world probabilty is replaced by a risk-neutral measure. The expected option payoff now is under the risk-neutral measure so we’ve lost our intuition about real-world probabilities of future states, but we’ve gained computational tractability because the risk-adjusted discount rate is replaced by the risk-free rate. If someone understands the Girsanov theorem on an intuitive level, more power to them but I find next to impossible to make that step from real-world to risk-neutral expected values in a natural way. So I prefer to default to the replication strategy argument for my intuiton and keep in mind that it is equivalent to the risk-neutrality argument which allows me to actually calculate option prices easily.
Whew that’s about the longest post I’ve written on AF… lazy easter sunday