196 Frequently Asked Questions In Quantitative Finance
- 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 deriva-
tives 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 gen-
eral 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 phlo-
giston plays a role in the dynamics of the underlying
asset. We model the value of phlogiston as
d=μdt+σdX.
The market price of phlogiston risk isλ. In the classical
option-pricing models we will end up with an equation
for an option with the following term
...+(μ−λσ)
∂V
∂
+...= 0.
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,μ,has
been adjusted to allow for the market price of phlogis-
ton risk. We call this therisk-adjustedorrisk-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
d=(μ−λσ)dt+σdX.
and not the real one.
You can imagine estimating the real drift and volatility
for any observable financial quantity simply by looking
