Risk-Neutral Valuation

Risk neutral valuation is the pricing of financial instruments under a risk-neutral measure, or similarly an equivalent martingale measure (EMM) Q.



The claim’s price is then calculated as the expected discounted payoff under the risk-neutral probability measure Q. It is



Assuming a risk-neutral world has a clear advantage when it comes to pricing. In such a world, all cash-flows can be discounted by the risk-free interest rate, r, whereas, in a real-world, the discount rate should take into account the risk premium, which is more delicate.


Such valuation is straightforward in a complete market since completeness implies a unique risk-neutral probability measure (= EMM) Q and thus an unique option’s fair price, which can be expressed as an expectation with respect to the unique EMM Q. Finding Q therefore amounts to the equivalent of setting the constant market risk premium, Λ(S, v, t). 


In an incomplete model, the market risk premium is not constant. 

In the same way, in an incomplete market, there are derivatives that are not attainable. It follows that there exists many EMMs Q, and thus many arbitrage-free option prices. This results from the two following statements (Korn, Korn and Kroisandt (2010)):


• Having a non-empty set of EMMs guarantees no arbitrage opportunities.

• If Q is an equivalent martingale measure, then taking EQ(e−rTpayoff(T)) as an option price does not create any arbitrage opportunity.


Denoting the set of EMMs by Q, each of its element is thus a candidate pricing measure. Therefore, pricing an option amounts to the equivalent of selecting an adequate equivalent martingale measure.


Moving from the real-world probability measure P to an equivalent martingale measure is accomplished by using Girsanov’s theorem

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