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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.

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  • $\begingroup$ (1) Unless you think all stocks earn the same expected return as risk free government bonds, then you need some kind of risk premium; a risk premium is what allows different assets to have different expected returns (2) the option has (varying) exposure to the same risk as the stock and the mathematics of option pricing allows you to price the option taking the stock as given. Part of the insight of option pricing is that you don't need some extra complete stochastic discount factor giving you the risk premium for everything; you get all you need to price option from the stock (&rf rate). $\endgroup$ Commented Jun 14 at 2:16

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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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  • $\begingroup$ Thanks. I think this makes sense. But should think about this a bit more. Do you have any sources that explain this specific idea? $\endgroup$ Commented Jun 14 at 3:09
  • $\begingroup$ Think of a new asset coming to the economy which is not related to any other asset and has a positive return. Every investor will want to buy this asset, and the return has to be 0 if demand has to match supply $\endgroup$
    – Arshdeep
    Commented Jun 14 at 13:58
  • $\begingroup$ Ok. So in the binomial option pricing model, the risk premium of the stock is reduced because of the introduction of the option, right? Can we quantify that somehow? What would be the risk premium for the option? $\endgroup$ Commented Jun 15 at 9:21

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