108 Frequently Asked Questions In Quantitative Finance
(Probability) Measure: In layman’s terms, the measure
gives the probabilities of each of the outcomes in the
sample space.
Previsible: A previsible process is one that only depends
on the previous history.
Equivalent: Two measures are equivalent if they have
the same sample space and the same set of ‘possibil-
ities.’ Note the use of the word possibilities instead
of probabilities. The two measures can have different
probabilities for each outcome but must agree on what
is possible.
Another way of writing the above is in differential form
dW ̃t=dWt+γtdt.
One important point about Girsanov’s theorem is its
converse, that every equivalent measure is given by
a drift change. This implies that in the Black–Scholes
world there is only the one equivalent risk-neutral mea-
sure. If this were not the case then there would be
multiple arbitrage-free prices.
For many problems in finance Girsanov theorem is not
necessarily useful. This is often the case in the world
of equity derivatives. Straightforward Black–Scholes
does not require any understanding of Girsanov. Once
you go beyond basic Black–Scholes it becomes more
useful. For example, suppose you want to derive the
valuation partial differential equations for options under
stochastic volatility. The stock price follows the real-
world processes,P,
dS=μSdt+σSdX 1
and
dσ=a(S,σ,t)dt+b(S,σ,t)dWX 2 ,
