206 Frequently Asked Questions In Quantitative Finance
How Robust is the Black–Scholes
Model?
Short Answer
Very robust. You can drop quite a few of the assump-
tions underpinning Black–Scholes and it won’t fall over.
Example
Transaction costs? Simply adjust volatility. Time-
dependent volatility? Use root-mean-square-average
volatility instead. Interest rate derivatives? Black ’76
explains how to use the Black–Scholes formulæ in situa-
tions where it wasn’t originally intended.
Long Answer
Here are some assumptions that seems crucial to the
whole Black–Scholes model, and what happens when
you drop those assumptions.
Hedging is continuous: If you hedge discretely it turns out
that Black–Scholes is righton average.Inotherwords
sometimes you lose because of discrete hedging, some-
times you win, but on average you break even. And
Black–Scholes still applies.
There are no transaction costs: If there is a cost associated
with buying and selling the underlying for hedging this
can be modelled by a new term in the Black–Scholes
equation that depends on gamma. And that term is
usually quite small. If you rehedge at fixed time inter-
vals then the correction is proportional to the absolute
value of the gamma, and can be interpreted as simply
a correction to volatility in the standard Black–Scholes
formulæ. So instead of pricing with a volatility of 20%,
say, you might use 17% and 23% to represent the bid-
offer spread dues to transaction costs.
