Monday, February 11, 2019

option pricing - Does risk-neutral measure have anything to deal with risk-neutrality in utility theory?


Or simply: why do we call equivalent martingale measures as risk-neutral measures?


In the utility or game theory, when we consider a person's preferences to certain outcomes, we often deal with the utility functions. For example, if we consider an investor with a utility function U whose return on a portfolio Π is xΠ, we assume that he choose a portfolio that maximizes his expected utility Π such that EU(xΠ)=supΠEU(xΠ).

In particular, we say that an investor is risk-averse (risk-seeking) whenever U is concave (convex). We say that an investor is risk-neutral when U(x)=ax+b is an affine function.


The risk-neutral valuation - taking expectations w.r.t. martingale measures equivalent to the real-world ones - is used in quant finance a lot for the pricing purposes. I do understand the theory behind this method, and the relation with non-arbitrage arguments. I wonder though, whether there is any relation with the risk-neutrality as in the paragraph above.





I thought of the following idea: let us think of a fair price for a contract (when we write it) as the highest one at which the agent will buy it. The agent A with utility UA and expectation of prices EA has to make a choice between the zero utility (when he does not buy contract) and EAUA(erTCTC0)

where T is maturity of the contract, r is a rate used to compute present value of future cashflow, CT is the payoff of the contract, C0 is the price of the contract. Hence, we need to solve the equation EAUA(erTCTC0)=0
with unknown C0. Assuming that agent is risk-neutral, we obtain C0=erTEA(CT).
At the same time, pricing using the Δ-hedging in the Black \& Scholes framework gives us C0=erTEQ(CT)
where Q is a risk-neutral measure. Hence, if we assume that our agent is risk neutral, then his expectations (at least at any given time T) have to be given exactly by the measure Q.




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