Chapter 2: FAQs 195
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 quantityistraded 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 under-
lying asset. This is the basis ofrisk-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 isnot
traded); risk of default.