# Market price of risk

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

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 divided by standard deviation of returns. This quantity is not affected by leverage.

In derivatives theory we often try to model quantities as stochastic. 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}.

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.

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.

## References

- Apabhai, MZ, Choe, K, Khennach, F & Wilmott, P 1995 Spot-on modelling.
*Risk*magazine, December 8 (11) 59-63 - Wilmott, P 2006
*Paul Wilmott On Quantitative Finance, second edition*. John Wiley & Sons - Wilmott, P 2006
*Frequently Asked Questions in Quantitative Finance*. John Wiley & Sons