# Why Joshi defined option value to be discounted payoff using risk neutral expectation?

Currently I am reading Mark Joshi's The Concepts and Practice of Mathematical Finance.

At page $$59,$$ the author mentioned the following.

Instead of requiring that every portfolio should have expectation equal to today's value, we require that its expectation should be equal to the asset's value invested at the risk-free growth rate, or equivalently that its discounted expectation is equal to today's value. We thus want $$\mathbb{E}_{RN}\left( \frac{A_{\Delta T}}{B_{\Delta T}} \right) = \left( \frac{A_0}{B_0} \right)$$ for every asset where $$\mathbb{E}_{RN}$$ is an expectation with the risk-neutral probability $$p$$.

This equation is trivially satisfied for the bond and we have the chosen the risk-neutral probability so that it is satisfied by construction for the stock. This leaves us with the option we wish to price. We define Opt$$_0$$ to satisfy equation above: $$Opt_0 = \mathbb{E}_{RN}\left( \frac{A_{\Delta T}}{B_{\Delta T}} \right) = e^{-r\Delta t} \mathbb{E}_{RN}(f(S))$$ where $$f$$ is the option's payoff.

My question is the following:

Question: Why and how can Joshi define option value at time zero to be the discounted risk neutral expectation?

• The answer to this question is the entire rationale behind options pricing and a lot of chapters in a financial mathematics text build up to this. You should check out Shreve, vol II, chapter 5 to get a better picture. But the general idea is that you can make a hedging portfolio that matches the risk neutral price and that lets you make the amount of the payoff at maturity and so the risk neutral price should be the value in the real world as well. – Slade Sep 30 '19 at 4:29