The market price of risk is the return in excess of the risk-free rate that the market wants as compensation for taking risk.

Example

Historically a stock has grown by an average of 20% per annum when the risk-free rate of interest was 5%. The volatility over this period was 30%. Therefore, for each unit of risk this stock returns on average an extra 0.5 return above the risk-free rate. This is the market price of risk.

Long answer

In classical economic theory no rational person would invest in a risky asset unless they expect to beat the return from holding a risk-free asset. Typically risk is measured by standard deviation of returns, or volatility. The market price of risk for a stock is measured by the ratio of expected return in excess of the risk-free interest rate to the standard deviation of returns. Interestingly, this quantity is not affected by leverage. If you borrow at the risk-free rate to invest in a risky asset both the expected return and the risk increase, such that the market price of risk is unchanged. This ratio, when suitably annualized, is also the Sharpe ratio.

If a stock has a certain value for its market price of risk then an obvious question to ask is what is the market price of risk for an option on that stock? In the famous Black-Scholes world in which volatility is deterministic and you can hedge continuously and costlessly, then the market price of risk for the option is the same as that for the underlying equity. This is related to the concept of a complete market in which options are redundant because they can be replicated by stock and cash.

In derivatives theory we often try to model quantities as stochastic, that is, random. Randomness leads to risk, and risk makes us ask how to value risk, that is, how much return should we expect for taking risk. By far the most important determinant of the role of this market price of risk is the answer to the question, is the quantity you are modelling traded directly in the market?

If the quantity is traded directly, the obvious example being a stock, then the market price of risk does not appear in the Black-Scholes option-pricing model. This is because you can hedge away the risk in an option position by dynamically buying and selling the underlying asset. This is the basis of risk-neutral valuation. Hedging eliminates exposure to the direction that the asset is going and also to its market price of risk. You will see this if you look at the Black-Scholes equation. There the only parameter taken from the stock random walk is its volatility, there is no appearance of either its growth rate or its price of risk.

On the other hand, if the modelled quantity is not directly traded then there will be an explicit reference in the option-pricing model to the market price of risk. This is because you cannot hedge away associated risk. And because you cannot hedge the risk you must know how much extra return is needed to compensate for taking this unhedgeable risk. Indeed, the market price of risk will typically appear in classical option-pricing models any time you cannot hedge perfectly. So expect it to appear in the following situations:

• When you have a stochastic model for a quantity that is not traded. Examples: stochastic volatility; interest rates (this is a subtle one, the spot rate is not traded); risk of default.

• When you cannot hedge. Examples: jump models; default models; transaction costs.

When you model stochastically a quantity that is not traded, then the equation governing the pricing of derivatives is usually of diffusion form, with the market price of risk appearing in the 'drift term with respect to the non-traded quantity. To make this clear, here is a general example.

Suppose that the price of an option depends on the value of a quantity of a substance called phlogiston. Phlogiston is not traded but either the option s payoff depends on the value of phlogiston, or the value of phlogiston plays a role in the dynamics of the underlying asset. We model the value of phlogiston as

The market price of phlogiston risk is k®. In the classical option-pricing models we will end up with an equation for an option with the following term

The dots represent all the other terms that one usually gets in a Black-Scholes type of equation. Observe that the expected change in the value of phlogiston, /x<j>, has been adjusted to allow for the market price of phlogiston risk. We call this the risk-adjusted or risk-neutral drift. Conveniently, because the governing equation is still of diffusive type, we can continue to use Monte Carlo simulation methods for pricing. Just remember to simulate the risk-neutral random walk

and not the real one.

You can imagine estimating the real drift and volatility for any observable financial quantity simply by looking at a time series of the value of that quantity. But how can you estimate its market price of risk? Market price of risk is only observable through option prices. This is the point at which practice and elegant theory start to part company. Market price of risk sounds like a way of calmly assessing required extra value to allow for risk. Unfortunately there is nothing calm about the way that markets react to risk. For example, it is quite simple to relate the slope of the yield curve to the market price of interest rate risk. But evidence from this suggests that market price of risk is itself random, and should perhaps also be modelled stochastically.

Note that when you calibrate a model to market prices of options you are often effectively calibrating the market price of risk. But that will typically be just a snapshot at one point in time. If the market price of risk is random, reflecting people's shifting attitudes from fear to greed and back again, then you are assuming fixed something which is very mobile, and calibration will not work.

There are some models in which the market price of risk does not appear because they typically involve using some form of utility theory approach to find a person's own price for an instrument rather than the market's.

References and Further Reading

Ahn, H & Wilmott, P 2003 Stochastic volatility and mean-variance analysis. Wilmott magazine November 84-90

Markowitz, H 1959 Portfolio Selection: Efficient Diversification of Investment. John Wiley & Sons Ltd

Wilmott, P 2006 Paul Wilmott on Quantitative Finance, second edition. John Wiley & Sons Ltd

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