Replication, Hedging, and Risk Premiums in the Binomial Option Model
Summary
The document examines whether an option and its underlying stock require risk premiums in a binomial pricing model when their payoffs can be replicated using other instruments. The central issue is a confusion between replication and hedging: constructing a portfolio with the same payoff does not by itself remove the return associated with bearing the original exposure. Replication matches payoffs, while a hedge can reduce risk while retaining some exposure to expected return.
The response argues that an asset can have no risk premium when its exposure to expected return remains while its variance exposure falls disproportionately. Replicating the stock with the stock or a derivative removes the exposure to its mean along with the risk, so it does not demonstrate that the stock’s premium vanishes. The document offers a conceptual explanation rather than a full derivation, and its statements are tied to the simplified setting described; it does not work through a numerical binomial tree or address broader asset-pricing assumptions.
Key ideas
- Replication matches an instrument’s payoff, while hedging reduces risk while preserving some exposure.
- A replicating portfolio does not automatically eliminate the risk premium of the underlying exposure.
- Removing an asset’s risk by canceling its exposure also removes its expected-return exposure.
- A zero premium argument requires retaining mean exposure while reducing variance exposure disproportionately.
- The discussion is conceptual and does not provide a full binomial-tree derivation.
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Full text
# Do we need a risk premium for the assets in the binomial option pricing model? # Do we need a risk premium for the assets in the binomial option pricing model? Section 1.4 of Mark Joshi's book illustrates with a simple example, the idea that "the market will only compensate investors for [systemic risks]", which are not diversifiable (or) hedgeable. I'm trying to apply this idea to the assets in Binomial Option Pricing model. We have three of them there: An option, its underlying stock and a risk-free bond. My understanding is that the option is fully hedgeable, because we can replicate its payoff with a portfolio comprising of the stock and the bond. So, is it accurate to say that there won't be a risk premium for that? Similarly, we can also replicate the stock's payoff in the next time step with a portfolio comprising of the option and the bond. So, is the stock also fully hedgeable and hence no risk premium is required for it also? I'm pretty sure the answer is no, at least for the stock. But where did I go wrong in my argument? Edit 1: I realise from the responses that I was confused b/w hedging and replication. Replication has no effect on the risk premium. Because if you try to remove the risk completely with a replicating portfolio, your payoff also becomes zero. But hedging, on the other hand, can remove the risk, while maintaining some or all of the original payoff. So, my current understanding is that: In the coin toss example of Section 1.4 of the book, risk can be entirely removed while maintaining the same maximum payoff. But in the binomial option pricing model, only some of the payoff is maintained. So, the risk premium is reduced, but not eliminated entirely. ## Answer by Arshdeep (score 2) https://quant.stackexchange.com/a/79721 Stock is hedgeable by the stock itself, options don't even enter the picture. The point is that if you have anything ( a factor, stock, or anything) whose mean you can stay exposed to (1/n) while decreasing the exposure to it's variance very disproportionately(1/n^2), that anything will have a risk premium of 0. If you hedge the stock by the stock, you lose exposure to it's mean. Same with hedging with the derivative. This is why we call it "replication" - it's the same exact thing.
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