What is Girsanov's Theorem, and Why is it Important in Finance?
Girsanov s theorem is the formal concept underlying the change of measure from the real world to the risk-neutral world. We can change from a Brownian motion with one drift to a Brownian motion with another.
The classical example is to start with
with W being Brownian motion under one measure (the real-world measure) and converting it to
under a different, the risk-neutral, measure.
First a statement of the theorem. Let Wt be a Brownian motion with measure P and sample space Q. If yt is aprevisible process satisfying the constraint E-p ^exp (J fo vf") < 00 then there exists an equivalent measure Q on Q such that
is a Brownian motion.
It will be helpful if we explain some of the more technical terms in this theorem.
• Sample space: All possible future states or outcomes.
• (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 'possibilities.' 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
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 measure. 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,
where dX1 and 0X2 are correlated Brownian motions with correlation p(S, a, t).
Using Girsanov you can get the governing equation in three steps:
1. Under a pricing measure Q, Girsanov plus the fact that S is traded implies that
where X is the market price of volatility risk.
2. Apply Itô's formula to the discounted option price
V(S, a, t) = e-r(T-t)F(S, a, t), expanding under Q, using the formulae for dS and dV obtained from the Girsanov transformation
3. Since the option is traded, the coefficient of the dt term in its Itô expansion must also be zero; this yields the relevant equation
Girsanov and the idea of change of measure are particularly important in the fixed-income world where practitioners often have to deal with many different measures at the same time, corresponding to different maturities. This is the reason for the popularity of the BGM model and its ilk.
References and Further Reading
Joshi, M 2003 The Concepts and Practice of Mathematical Finance. Cambridge University Press
Lewis, A 2000 Option Valuation under Stochastic Volatility. Finance Press
Neftci, S 1996 An Introduction to the Mathematics of Financial Derivatives. Academic Press
What are the Greeks?
The 'greeks are the sensitivities of derivatives prices to underlyings, variables and parameters. They can be calculated by differentiating option values with respect to variables and/or parameters, either analytically, if you have a closed-form formula, or numerically.
Delta, A = |y, is the sensitivity of an option price to the stock price. Gamma, r = is the second derivative of the option price to the underlying stock, it is the sensitivity of the delta to the stock price. These two examples are called greek because they are members of the Greek alphabet. Some sensitivities, such as vega = are still called 'greek' even though they are n t in the Greek alphabet.