Measure change: Girsanov, Feynman–Kac and the fundamental theorems

SC · Chapter 513 min readAsked at Two Sigma, Citadel, DE Shaw, Optiver

Assumes SDEs: geometric Brownian motion, Ornstein–Uhlenbeck and the rest.

After this lesson you should be able to

  • State what Girsanov changes and what it leaves alone.
  • Explain the two fundamental theorems of asset pricing in plain terms.
  • Use Feynman–Kac to connect an expectation to a PDE.

Risk-neutral pricing rests on one manoeuvre: reweighting the probabilities so that every tradeable asset has the same expected return. Girsanov says exactly what that reweighting does to a diffusion, and the fundamental theorems say when it is possible and when it is unique.

Proposition 5.1

What Girsanov does

Changing to an equivalent measure shifts the drift of a Brownian motion and leaves its volatility untouched. Under the new measure W~t=Wt+∫0tλs ds\tilde{W}_t = W_t + \int_0^t \lambda_s\,ds is a Brownian motion, where λ\lambda is the market price of risk. The volatility is invariant because quadratic variation is a pathwise property — it is computed from the path itself, and reweighting which paths are likely cannot change any individual one.

Holds when

  • Equivalent means the two measures agree on what is possible, not on how likely.
  • For a stock, λ=(μ−r)/σ\lambda = (\mu - r)/\sigma — the Sharpe ratio — and the drift becomes rr.
  • Volatility is estimable from data without choosing a measure; drift is not, which is why one is modelled and the other assumed.
Drift, real world10Drift, risk-neutral4Volatility, either measure20
Figure 5.2 · What changing measure does, and what it cannot touch. Girsanov reweights the paths, which moves the drift to the risk-free rate. It cannot move volatility, because the quadratic variation of a path is a property of that path and not of how likely you think it is — and that invariance is the reason a hedging argument gives a price at all.

Why volatility cannot change. Quadratic variation is the limit of summed squared increments along a single path, so it is a number you could compute from a recording of what happened, with no reference to probability at all. A change of measure only alters how much weight you place on each path — it cannot alter the paths. That is why implied volatility is comparable to realised volatility despite living under different measures, and why the entire risk-neutral apparatus never asks you to adjust a volatility for risk preferences.

TheoremStatementConsequence
FirstNo arbitrage   ⟺  \iff an equivalent martingale measure existsPrices are discounted expectations under *some* measure
SecondThe market is complete   ⟺  \iff that measure is uniquePrices are unique, and everything can be hedged
Table 5.3 · The two fundamental theorems. Black–Scholes satisfies both: one source of randomness, one hedging instrument, one price. Add stochastic volatility and you keep the first but lose the second — prices are no longer unique, and a market price of risk must be estimated.

Proposition 5.4

The numeraire

A martingale measure is always relative to something. Under the usual risk-neutral measure it is the money-market account that prices are divided by; choose a zero-coupon bond instead and you get the forward measure, in which the forward price is a martingale and discounting comes out of the expectation.

Holds when

  • Changing numeraire is a computational convenience and never changes a price.
  • The forward measure is what makes Black-76 a clean statement about the forward.
  • For a quanto, the correct numeraire choice is what produces the drift adjustment.

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Measure change: Girsanov, Feynman–Kac and the fundamental theorems · QuantMax