Models and Measures 1
A few years after Black, Scholes and Merton produced their path-breaking option pricing model, other researchers developed what has become known as risk-neutral valuation. The risk-neutral valuation principle is wonderfully simple. To value a derivative we do not need to know how the underlying market variables behave in the real world. All we need to know is how they behave in the (totally unreal) risk-neutral world where investors never require extra expected returns for bearing risk. In a risk-neutral world, the expected return from every investment asset is the risk free rate of interest. The risk-neutral valuation principle shows that a derivative can be correctly valued by calculating its expected payoff in a risk-neutral world and discounting that payoff at the risk-free rate.
It is important to emphasize that the value calculated using risk neutral valuation is the correct value in all worlds, not just in a world where investors are risk neutral. Consider a European call option on a stock. When we move from the risk-neutral world to the real world two things happen. The expected return on the stock increases so that the expected option payoff is higher. But the return provided by the payoff has positive systematic risk so that a discount rate higher than the risk-free rate is required. It happens that the higher discount rate exactly cancels out the effect of the higher expected payoff. Consider next a European put option, an option that pays off only when the stock price is low. When we move from the real world to the risk-neutral world, the higher growth rate in the stock price leads to a smaller expected payoff. However, the payoff has negative systematic risk. (When the market does badly, the return provided by the payoff is high; when the market does well, it is low.) As a result the discount rate is lower than the risk-free rate (and usually negative). Again we find that the impact of the lower discount rate exactly cancels out the impact of the smaller expected payoff.
Risk-neutral valuation is without doubt the most important single principle in derivative pricing. If we did not have risk-neutral valuation, we would have to make estimates of the return provided by underlying market variables in the real world and then quantify the riskiness of the payoff in some way to determine the correct discount rate.
In a famous theorem, Girsanov showed that moving from a world where investors have one set of risk preferences to a world where they have another set of risk preferences involves changing the expected drift or rate of return of a market variable, but the volatility remains the same.
This is a very useful result. It means that the volatility we estimate from real world data can be used in the risk-neutral world (or any other world we might be interested in). The expected return of a variable is $r + \lambda$ where $r$ is the risk-free rate, $\lambda$ market price of risk for the world being considered, and $\sigma$ the variable’s volatility (which as we have just explained is the same in all worlds). Consider a stock index. Suppose the risk-free rate of interest is 1%, the volatility of the index is 15% and the market price of risk is 0.4. The return provided by the index in the risk-neutral world is the risk-free rate or 1%. In the real world it is 1%+0.4×15% or 7%.
The risk-neutral valuation result can be generalized to say that if we rely on the expected return relationship, $r + \lambda\sigma$, we can calculate the value a derivative in any world (the real world, the risk-neutral world, or some other world). If we are careful in our calculations we always get the same answer no matter what the market price of risk. When interest rates are assumed to be constant (or deterministic), risk-neutral valuation, as we have described it above, usually works well. In this world the market price of risk, $\lambda$, is zero. When interest rates are stochastic, it can be useful to value a derivative in a world where the market price of risk has a particular non-zero value. (As just mentioned, we still get the right price.) What is known as the “equivalent martingale measure” result is sometimes useful. This states that, when we measure the prices of all securities in terms of the price of a numeraire security (i.e., we divide the dollar prices of all securities by the dollar price of the numeraire security), there is some value of the market price of risk for which the drift rates of all security prices are zero. We will refer this value of the market price of risk as the martingale market price of risk (MMPR).
The equivalent martingale measure result is particularly useful in the valuation of European-style derivatives. As a simple example consider the valuation of a two-year European option on a stock when interest rates are stochastic. We can choose as our numeraire security the price of a zero-coupon bond that pays off 1 in two years. Two years from now the numeraire security will be worth 1 so that the payoff from the option when it is measured in terms of the numeraire security and when it is measured in the usual way (i.e., in dollars) are the same. Furthermore, the equivalent martingale measure result tells us that the value of the option (or any other security) today divided by the price of a zero-coupon bond today equals the expected value in two years when the market price of risk equals the MMPR.
When the equivalent martingale measure result is used in conjunction with Monte Carlo simulation, the most difficult task is determining the MMPR. There is actually one MMPR for each source of uncertainty and so in practice several MMPRs may have to be calculated. Sometimes expressions for them can be calculated analytically and sometimes they cannot. This is where a result produced by Gibbs and Goyder (2013) is potentially useful. They recommend simulating each underlying market variables using the most convenient measure. (Often this will be the risk-neutral measure where the market price of risk is zero.) They then calculate the expected future value of a variable V that they know should be a martingale (i.e., have zero drift) under the MMPR. If the market price of risk had been chosen as the MMPR this would equal the current value of V. The Gibbs and Goyder approach forces equality by making a proportional adjustment to the values of V obtained in the simulation trials.
Consider our earlier example of the stock option. An analyst could simulate the stock price and the bond price in a convenient (e.g., $\lambda=0$) measure and calculate the stock price divided by the bond price at the two year point. Suppose that the stock price is initially 9 and the bond price is initially 0.9 so that the stock price divided by the bond price is initially 10. The final bond price is 1. Suppose that the final average stock price in the Monte Carlo simulation is 9.9. Every terminal stock price in the simulation would be multiplied by 10/9.9 so that the final stock price equals the initial value of the stock price divided by the initial value of the bond price. The expected payoff from option is then calculated. Suppose that this is 2.5. The equivalent martingale measure result tells us that the initial option price divided by the initial two-year zero-coupon bond price is also 2.5. The initial option price is therefore 2.5×0.9 = 2.25.
In this example we have used the stock price as the calibrating instrument. If we had the market price for another option providing a payoff at the two-year point we could calibrate to that market price in a similar way. In some circumstances the proportional adjustment to terminal values of V provides an adjustment that is exactly correct. In other situations, it provides a reasonable approximation. The attraction of the procedure is that the measure used to initially simulate variables can be independent of the derivative being valued and the procedure can be used for complex derivatives dependent on several market variables.
In our second blog, Models and Measures 2, we will discuss the growing importance of scenario analysis, which requires variables to be modeled in the real world rather than in the artificial worlds that are used for valuation. It might be thought that determining real world behavior is easier than determining behavior in a risk-neutral world. Often this is not the case. We will outline a number of different approaches to determining real-world stochastic processes.
Gibbs, Mark and Russell Goyder, “Automatic Numeraire Corrections for Generic Hybrid Simulations,” ssrn 2311740.
Hull, John, “Options, Futures, and Other Derivatives,” 9th edition, New York: Pearson.