204 Frequently Asked Questions In Quantitative Finance
tends to be very small in practice. The real question is
about variance, is it constant? If it is constant, and we
are hedging frequently, then we may as well work with
normal distributions and the Black–Scholes constant
volatility model. However, if it is not constant then
we may want to model this more accurately. Typical
approaches include the deterministic or local volatility
models, in which volatility is a function of asset and
time,σ(S,t), and stochastic volatility models, in which
we represent volatility by another stochastic process.
The latter models require a knowledge or specification
of risk preferences since volatility risk cannot be hedged
just with the underlying asset.
If the variance of returns is infinite, or there are jumps
in the asset, then normal distributions and Black–Scholes
are less relevant. Models capturing these effects also
require a knowledge or specification of risk preferences.
It is theoretically even harder to hedge options in these
worlds than in the stochastic volatility world.
To some extent the existence of other traded options
with which one can statically hedge a portfolio of deriva-
tives can reduce exposure to assumptions about distri-
butions or parameters. This is called hedgingmodel
risk. This is particularly important for market makers.
Indeed, it is instructive to consider the way market
makers reduce risk.
- The market maker hedges one derivative with
another one, one sufficiently similar as to have
similar model exposure.
- As long as the market maker has a positive
expectation for each trade, although with some model
risk, having a large number of positions he will
reduce exposure overall by diversification. This is
more like an actuarial approach to model risk.
