194 Frequently Asked Questions In Quantitative Finance
What is the Market Price of Risk?
Short Answer
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 divided by 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 theSharpe 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 acomplete marketin which options are redundant
because they can be replicated by stock and cash.
