Risk Neutral Pricing Thursday, May 12, 2011 2:03 PM We discussed last time how the Girsanov theorem allows us to reweight probability measures to change the drift in an SDE. This is used to construct a reweighted "equivalent risk-neutral" or "equivalent martingale" measure in financial market models in the following way: For a single asset, generalized GBM price model: The Girsanov theorem tells us that there exists an equivalent probability measure under which the price trajectories follow: where R(t) is the interest rate (also allowed to be a general random process adapted to the filtration generated by the Wiener process) that describes how cash appreciates in value: The Wiener process under the new risk-neutral probability measure is related to the original Wiener process by the transformation: (and so really we need the volatility to be bounded from below for this to work): Under the risk-neutral probability measure, every asset, including cash, is a martingale when discounted by the interest rate. 2011 Page 1
This in particular means that no matter how we manage a portfolio of assets (including cash), the portfolio will behave as a martingale. If we hold shares of the underlying asset then portfolio value: is always automatically a martingale no matter how we manage it. In particular, if T is the expiry time of the financial option, then: Now suppose that I have a perfect hedging strategy using such a portfolio so that I can perfectly replicate the payoff of some financial option that I'm considering: For a European option, this looks like: but one can contemplate more general financial options. Under this assumption: 2011 Page 2
And from this equation, it would seem that the fair price to charge for the financial option, at time t, is the value of the portfolio of assets and cash at that time that would perfectly replicate the option. (Otherwise there's arbitrage for the smarter person). The the value of the option at time t, using the fact that D(t) is measurable with respect to Risk-neutral pricing formula Recall that this risk-neutral pricing formula can only be expected to hold when: a. one can find an equivalent probability measure that makes the total market look risk neutral b. an active hedging strategy exists (a portfolio of cash and assets can replicate the financial option payoff) (market completeness) How does one know whether such a hedging strategy exists in general? The abstract way to approach this is through the Martingale Representation Theorem which says the following (Shreve Sec. 5.3) If M(t) is a martingale with respect to the filtration generated by Wiener process and M(t) is adapted to this filtration, then there exists a stochastic process which is also adapted to this same filtration such that This theorem is used in the following way: If we define V(t) as above, and if we specify that the filtration in our market model is just the filtration generated by the Wiener process under the riskneutral probability measure (note this excludes allowing other sources of uncertainty), then we can apply the Martingale Representation theorem to the discounted option price: Active hedge: 2011 Page 3
Active hedge: How does one actually use the risk-neutral pricing formula? To apply it for the European call option with geometric Brownian motion model with deterministic timedependent coefficients: One could in principle evaluate this risk-neutral expectation by inserting the Radon-Nikodym derivative (reweighting factor Z) and then taking averages with respect to the original probability measure. But an easier alternative is actually to express S(T) in terms of the Wiener process under the riskneutral probability measure, and complete the calculation in risk-neutral world. 2011 Page 4
Now the calculation is straightforward. One recovers the Black-Scholes-Merton formula but with: Note the Black-Scholes-Merton PDE is indeed the Kolmogorov backward equation for the risk-neutral probability measure. Even though the geometric Brownian motion model is used as the basis for option pricing, people don't really believe it. Volatility smile: The reason for the discrepancy is generally attributed to prices having larger probabilities for large fluctuations than the geometric Brownian motion model would give. Several modeling frameworks try to take this into account: stochastic volatility models: jump-diffusion models for price shocks 2011 Page 5
2011 Page 6 The simplest version is framed in terms of a compound Poisson process.
c.f. Paul and Bauschnagel, Stochastic Processes: From Physics to Finance, Ch. 5 These jump-diffusion models are still Markovian, so can still use the Markov process theory with infinitesmal generator How is the risk-neutral option-pricing formula affected? Girsanov theorem transformation to a risk-neutral probability measure still seems to be OK under the usual conditions Martingale representation theorem exists that might suggest market completeness, but be careful: if one wants to conclude this, then one would need the option price and the trading strategy to be adapted to the filtration generated jointly by the Brownian motion and the jump process But in practice, it would seem more reasonable to only impose that: For the case of jump-diffusions, Then one cannot perfectly hedge the risk of a financial option. The general method for finding prices then involves measuring the unhedged risk (say through a variance of the amount of money at risk) and charge a "risk premium" for that. 2011 Page 7