Ziyuan Zhang
Ziyuan Zhang
Practice makes perfect

Stochastic Calculus Notes 06 | Risk-Neutral Pricing, Change of Measure and Girsanov

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Overview

Part 05 priced a call. This post prices any payoff with one formula,

\(V_t=E^{\mathbb Q}\big[e^{-r(T-t)}V_T\mid\mathcal F_t\big]\):

the price is a net present value — but under a different probability measure \(\mathbb Q\), one in which the stock earns the risk-free rate. Then we answer the obvious questions: what is a change of measure, why do \(\mathbb P\) and \(\mathbb Q\) agree on what can happen but not on how likely it is, and how does Girsanov’s theorem turn \(\mu\) into \(r\) by shifting the Brownian motion.

1. From the call to any contingent claim

Recap of the Black–Scholes setup. \(S\sim GBM(\mu,\sigma)\) under \(\mathbb P\); money market at constant \(r\); self-financing strategy \(\Delta\). Discounted wealth:

\[d(e^{-rt}X_t)=e^{-rt}\big((\mu-r)\Delta_tS_t\,dt+\sigma\Delta_tS_t\,dW_t\big).\]

For the call we found: with \(\widetilde S\sim GBM(r,\sigma)\) under \(\widetilde{\mathbb P}\),

\[c(t,s)=E^{\widetilde{\mathbb P}}\big[e^{-r(T-t)}(\widetilde S_T-K)^+\mid\widetilde{\mathcal F}_t,\ \widetilde S_t=s\big],\]

initial capital \(x=c(0,S_0)\), strategy \(\Delta_t=c_s(t,S_t)\), terminal wealth \(X_T=(S_T-K)^+\), so by no-arbitrage \(c(0,S_0)\) is the price. Now generalize.

Contingent claims. A claim pays \(V_T\) at \(T\), where \(V_T\) can be any \(\mathcal F_T\)-measurable random variable with enough integrability:

Claim Payoff \(V_T\)
Call \((S_T-K)^+\)
Put \((K-S_T)^+\)
Asian call \(\Big(\frac1T\int_0^TS_t\,dt-K\Big)^+\)
Lookback \(\max_{t\le T}S_t-S_T\)
Up-and-out call \((S_T-K)^+\,\mathbf 1_{\{\max_{t\le T}S_t\le L\}}\)

2. Two assumptions and the pricing formula

  1. Risk-neutral measure. There is a measure \(\mathbb Q\) on \((\Omega,\mathcal F)\) under which \(S\sim GBM(r,\sigma)\). Equivalently there is a \(\mathbb Q\)-Brownian motion \(W^{\mathbb Q}\) with

    \[\frac{dS_t}{S_t}=\underbrace{\mu\,dt+\sigma\,dW_t}_{\text{under }\mathbb P}=\underbrace{r\,dt+\sigma\,dW^{\mathbb Q}_t}_{\text{under }\mathbb Q}.\]

  2. Replication. There are \(x\) and \(\Delta\) with \(\mathbb P[X^{x,\Delta}_T=V_T]=\mathbb Q[X^{x,\Delta}_T=V_T]=1\).

Discounted wealth is a \(\mathbb Q\)-martingale. Under \(\mathbb Q\) the drift term vanishes:

\[d(e^{-rt}X_t)=e^{-rt}\sigma\Delta_tS_t\,dW^{\mathbb Q}_t .\]

That is why \(\mathbb Q\) is also called a martingale measure.

Replication over \([t,T]\). The no-arbitrage argument works from any starting time: if you start at \(t\) with \(X_t\) (known at \(t\)), trade self-financingly on \([t,T]\) and end with \(V_T\), then the price at \(t\) must be \(X_t\).

Determining \(V_t\). By replication, \(e^{-rT}X_T=e^{-rT}V_T\). By the martingale property,

\[e^{-rt}X_t=E^{\mathbb Q}\big[e^{-rT}X_T\mid\mathcal F_t\big]=E^{\mathbb Q}\big[e^{-rT}V_T\mid\mathcal F_t\big].\]

And since \(V_T=X_T=X_t+\int_t^TdX_u\), holding \(X_t\) at time \(t\) is exactly what it takes to deliver \(V_T\), so

\[\boxed{\ V_t=X_t=E^{\mathbb Q}\big[e^{-r(T-t)}V_T\mid\mathcal F_t\big]\ }\]

How to read it. The chain of reasoning is: (i) the replicating portfolio is the claim, so the claim’s price is the portfolio’s value; (ii) under \(\mathbb Q\) every discounted self-financing portfolio is a fair game, so today’s discounted value equals the expected discounted terminal value; (iii) the terminal value is the payoff. Prices are NPVs — not under the real-world measure, but under one where the risky and riskless assets have the same expected return.

The two assumptions raise five questions: how do we get \(\mathbb Q\); how are \(\mathbb P\) and \(\mathbb Q\) related; how do we switch between them; why is \(S\sim GBM(r,\sigma)\) under \(\mathbb Q\); and where does \(W^{\mathbb Q}\) come from. The rest of the post answers them.

3. What a change of measure changes

Not affected by the measure (they only involve sets and maps):

  1. \(\Omega\), the set of outcomes;
  2. \(\mathcal F\), the information (collection of subsets of \(\Omega\));
  3. \(\mathbb F\), the flow of information;
  4. a random variable \(X:\Omega\to\mathbb R\) — it is just a map;
  5. a process \(\{X_t\}\);
  6. adaptedness: \(\{\omega:X_t(\omega)\in E\}\in\mathcal F_t\).

Affected by the measure (they involve likelihood):

  1. probabilities \(P(A)=E^P[\mathbf 1_A]\);
  2. expectations \(E\big[\sum_i\alpha_i\mathbf 1_{A_i}\big]=\sum_i\alpha_iP(A_i)\);
  3. moment generating functions \(\alpha\mapsto E^P[e^{\alpha X}]\);
  4. distributions \(P(X\le t)\);
  5. the martingale property \(E[X_t\mid\mathcal F_s]=X_s\);
  6. independence \(P(A\cap B)=P(A)P(B)\).

So the stock price paths are the same; only the weights we put on them change — and with them, the drift.

4. Change of measure: the construction

Given \((\Omega,\mathcal F,\mathbb P)\), take any random variable \(Z\) with \(\mathbb P[Z>0]=1\) (write \(Z>0\)) and \(E^{\mathbb P}[Z]=1\). Define

\[\mathbb Q(A):=E^{\mathbb P}[Z\,\mathbf 1_A],\qquad A\in\mathcal F .\]

\(\mathbb Q\) is a probability measure: \(\mathbb Q(A)\ge0\); \(\mathbb Q(\Omega)=E^{\mathbb P}[Z]=1\); and countable additivity holds, e.g. for disjoint \(A,B\), \(\mathbb Q(A\cup B)=E^{\mathbb P}[(\mathbf 1_A+\mathbf 1_B)Z]=\mathbb Q(A)+\mathbb Q(B)\).

Example: fair coin to unfair coin. Two tosses, fair and independent under \(\mathbb P\). Set

\[Z(HH)=\tfrac49,\quad Z(HT)=Z(TH)=\tfrac89,\quad Z(TT)=\tfrac{16}9,\qquad E^{\mathbb P}Z=\tfrac14\Big(\tfrac49+\tfrac89+\tfrac89+\tfrac{16}9\Big)=1 .\]

Then \(\mathbb Q(HH)=\frac14\cdot\frac49=\frac19\), \(\mathbb Q(HT)=\mathbb Q(TH)=\frac29\), \(\mathbb Q(TT)=\frac49\). Under \(\mathbb Q\) the tosses are still independent (\(\frac19=\frac13\cdot\frac13\), etc.) but the coin is unfair: \(\mathbb Q(H)=\frac13\).

Equivalence. Because \(Z>0\): \(\mathbb P(A)>0\Rightarrow\mathbb Q(A)=E^{\mathbb P}[Z\mathbf 1_A]>0\), and conversely. So \(\mathbb P(A)>0\iff\mathbb Q(A)>0\) and we call \(\mathbb P,\mathbb Q\) equivalent. Equivalent is not equal: they agree on what can and cannot happen, not on how likely it is (\(\mathbb P(A)=0.01\), \(\mathbb Q(A)=0.99\) is fine).

Radon–Nikodym derivative. \(\dfrac{d\mathbb Q}{d\mathbb P}:=Z\), also called the likelihood ratio. On a finite space,

\[\frac{d\mathbb Q}{d\mathbb P}(\omega)=\frac{\mathbb Q(\{\omega\})}{\mathbb P(\{\omega\})}=Z(\omega).\]

5. Shifting the mean of a normal

Let \(X\sim N(0,\sigma^2)\) under \(\mathbb P\), \(\theta\in\mathbb R\), and \(Z=e^{-\theta X-\frac12\sigma^2\theta^2}\). Then \(Z>0\) and \(E[Z]=e^{-\frac12\sigma^2\theta^2}E[e^{-\theta X}]=1\). Under \(\mathbb Q(A)=E^{\mathbb P}[Z\mathbf 1_A]\):

\[E^{\mathbb Q}[e^{\alpha X}]=E^{\mathbb P}\big[e^{(\alpha-\theta)X}\big]e^{-\frac12\sigma^2\theta^2}=e^{\frac12\sigma^2(\alpha-\theta)^2-\frac12\sigma^2\theta^2}=e^{-\sigma^2\theta\alpha+\frac12\sigma^2\alpha^2},\]

so \(X\overset{\mathbb Q}{\sim}N(-\sigma^2\theta,\ \sigma^2)\), i.e. \(X+\sigma^2\theta\overset{\mathbb Q}{\sim}N(0,\sigma^2)\). Tilting by an exponential shifts the mean and leaves the variance alone.

General version. To move \(X\) from \(N(\mu_P,\sigma^2)\) under \(\mathbb P\) to \(N(\mu_Q,\sigma^2)\) under \(\mathbb Q\), use \(\frac{d\mathbb Q}{d\mathbb P}=\frac{e^{-\theta X}}{E^{\mathbb P}[e^{-\theta X}]}=e^{-\theta X-\frac12\sigma^2\theta^2+\theta\mu_P}\). Then

\[E^{\mathbb Q}[e^{\alpha X}]=\exp\Big\{\alpha(\mu_P-\theta\sigma^2)+\tfrac12\alpha^2\sigma^2\Big\},\qquad\text{so we need}\quad \theta=\frac{\mu_P-\mu_Q}{\sigma^2}.\]

6. Application to Brownian motion and the stock

Fix \(T\); \(W_T\sim N(0,T)\) under \(\mathbb P\). Take \(\theta=\frac{\mu-r}{\sigma}\) and

\[Z=e^{-\theta W_T-\frac12\theta^2T}.\]

By Section 5 (with \(\sigma^2=T\), \(\mu_P=0\)), \(W^{\mathbb Q}_T:=W_T+\theta T\sim N(0,T)\) under \(\mathbb Q\). Since \(\mu-\sigma\theta=r\),

\[S_T=S_0\exp\Big\{\big(\mu-\tfrac12\sigma^2\big)T+\sigma W_T\Big\}=S_0\exp\Big\{\big(r-\tfrac12\sigma^2\big)T+\sigma W^{\mathbb Q}_T\Big\}.\]

So at the fixed time \(T\), \(S_T\) looks like a \(GBM(r,\sigma)\) under \(\mathbb Q\). To have \(S\sim GBM(r,\sigma)\) under \(\mathbb Q\) for all \(t\le T\) we need more: is \(t\mapsto W^{\mathbb Q}_t=W_t+\theta t\) a \(\mathbb Q\)-Brownian motion on \([0,T]\)? Equivalently, is

\[\frac{dS_t}{S_t}=\mu\,dt+\sigma\,dW_t\overset{?}{=}r\,dt+\sigma\,dW^{\mathbb Q}_t,\qquad W^{\mathbb Q}_t=W_t+\frac{\mu-r}{\sigma}t ?\]

The core is to verify that \(W^{\mathbb Q}\) is a Brownian motion. (Note \(Z_t=E^{\mathbb P}[Z_T\mid\mathcal F_t]=e^{-\theta W_t-\frac12\theta^2t}\) is a \(\mathbb P\)-martingale — the exponential martingale of Part 04.)

7. Girsanov’s theorem

Simple version. If \(W\) is a \(\mathbb P\)-Brownian motion, \(Z_T=e^{-\theta W_T-\frac12\theta^2T}\) and \(\mathbb Q(A)=E^{\mathbb P}[\mathbf 1_AZ_T]\), then

\[t\mapsto W^{\mathbb Q}_t:=W_t+\theta t\ \text{ is a }\mathbb Q\text{-Brownian motion on }[0,T].\]

General version. Fix \((\widetilde\Omega,\widetilde{\mathcal F},\widetilde{\mathbb F},\widetilde{\mathbb P})\) with a \(\widetilde{\mathbb P}\)-Brownian motion \(\widetilde W\), and an adapted \(\widetilde\theta\). Let

\[\widetilde Z_t=\exp\Big(-\int_0^t\widetilde\theta_u\,d\widetilde W_u-\frac12\int_0^t\widetilde\theta_u^2\,du\Big),\quad t\le T,\qquad \frac{d\widetilde{\mathbb Q}}{d\widetilde{\mathbb P}}=\widetilde Z_T\ \text{on }\widetilde{\mathcal F}_T .\]

Then \(\widetilde W^{\widetilde{\mathbb Q}}_t=\widetilde W_t+\int_0^t\widetilde\theta_u\,du\), \(t\le T\), is a \(\widetilde{\mathbb Q}\)-Brownian motion.

Intuition: reweighting the paths by \(Z\) makes paths that drift down more likely and paths that drift up less likely, by exactly the amount that turns “Brownian motion plus drift \(\theta\)” back into a driftless Brownian motion.

Lévy’s characterization

Proving something is a Brownian motion straight from the definition is painful. Lévy’s characterization is the shortcut: if \(\widetilde M\) is a \(\widetilde{\mathbb Q}\)-martingale with continuous paths, \(\widetilde M_0=0\) and \([\widetilde M,\widetilde M]_t=t\), then \(\widetilde M\) is a \(\widetilde{\mathbb Q}\)-Brownian motion.

So to prove Girsanov we only need: (1) \([\widetilde W^{\widetilde{\mathbb Q}},\widetilde W^{\widetilde{\mathbb Q}}]_t=t\) and (2) \(\widetilde W^{\widetilde{\mathbb Q}}\) is a \(\widetilde{\mathbb Q}\)-martingale.

Proof of Girsanov (constant θ)

Step 1. Quadratic variation does not care about the measure, and the drift contributes nothing:

\[d[\widetilde W^{\widetilde{\mathbb Q}},\widetilde W^{\widetilde{\mathbb Q}}]_t=(d\widetilde W_t+\widetilde\theta\,dt)^2=(d\widetilde W_t)^2=dt .\]

Step 2. For \(s<t\) and \(A_s\in\widetilde{\mathcal F}_s\), move to \(\widetilde{\mathbb P}\), use Tower and the martingale \(\widetilde Z_t=E^{\widetilde{\mathbb P}}[\widetilde Z_T\mid\widetilde{\mathcal F}_t]\):

\[E^{\widetilde{\mathbb Q}}[\mathbf 1_{A_s}\widetilde W^{\widetilde{\mathbb Q}}_t]=E^{\widetilde{\mathbb P}}[\mathbf 1_{A_s}\widetilde W^{\widetilde{\mathbb Q}}_t\widetilde Z_T]=E^{\widetilde{\mathbb P}}[\mathbf 1_{A_s}\widetilde W^{\widetilde{\mathbb Q}}_t\widetilde Z_t]=E^{\widetilde{\mathbb P}}\big[\mathbf 1_{A_s}f(t,\widetilde W_t)\big],\]

with \(f(t,w)=(w+\widetilde\theta t)\,e^{-\widetilde\theta w-\frac12\widetilde\theta^2t}\). Direct differentiation gives

\[f_t=e^{-\widetilde\theta w-\frac12\widetilde\theta^2t}\,\widetilde\theta\big(1-\tfrac12\widetilde\theta w-\tfrac12\widetilde\theta^2t\big),\quad f_w=e^{-\widetilde\theta w-\frac12\widetilde\theta^2t}\big(1-\widetilde\theta w-\widetilde\theta^2t\big),\quad f_{ww}=e^{-\widetilde\theta w-\frac12\widetilde\theta^2t}\big(-2\widetilde\theta+\widetilde\theta^2w+\widetilde\theta^3t\big),\]

so \(f_t+\frac12f_{ww}=0\) and by Itô \(df(t,\widetilde W_t)=f_w\,d\widetilde W_t\): \(f(t,\widetilde W_t)\) is a \(\widetilde{\mathbb P}\)-martingale. Hence

\[E^{\widetilde{\mathbb Q}}[\mathbf 1_{A_s}\widetilde W^{\widetilde{\mathbb Q}}_t]=E^{\widetilde{\mathbb P}}[\mathbf 1_{A_s}f(s,\widetilde W_s)]=E^{\widetilde{\mathbb P}}\big[\mathbf 1_{A_s}(\widetilde W_s+\widetilde\theta s)e^{-\widetilde\theta\widetilde W_s-\frac12\widetilde\theta^2s}\big].\]

The same Tower/TOWK computation for \(\widetilde W^{\widetilde{\mathbb Q}}_s\) gives the identical right-hand side, so

\[E^{\widetilde{\mathbb Q}}[\mathbf 1_{A_s}\widetilde W^{\widetilde{\mathbb Q}}_t]=E^{\widetilde{\mathbb Q}}[\mathbf 1_{A_s}\widetilde W^{\widetilde{\mathbb Q}}_s]\quad\text{for all }A_s\in\widetilde{\mathcal F}_s .\]

Since \(\widetilde W^{\widetilde{\mathbb Q}}_s\) is \(\widetilde{\mathcal F}_s\)-measurable, partial averaging gives \(E^{\widetilde{\mathbb Q}}[\widetilde W^{\widetilde{\mathbb Q}}_t\mid\widetilde{\mathcal F}_s]=\widetilde W^{\widetilde{\mathbb Q}}_s\). A time-varying \(\widetilde\theta_t\) works the same way with more bookkeeping.

Price dynamics under \(\mathbb Q\)

By Girsanov, \(W^{\mathbb Q}_t=W_t+\theta t\) is a \(\mathbb Q\)-Brownian motion on \([0,T]\), and since \(\mu-\sigma\theta=r\),

\[dS_t=\mu S_t\,dt+\sigma S_t\big(dW^{\mathbb Q}_t-\theta\,dt\big)=rS_t\,dt+\sigma S_t\,dW^{\mathbb Q}_t\iff S\overset{\mathbb Q}{\sim}GBM(r,\sigma).\]

\(\theta=\dfrac{\mu-r}{\sigma}\) is the market price of risk: excess return per unit of volatility, i.e. the Sharpe ratio. This answers questions 1–5 of Section 2 for assumption 1.

8. Assumption 2: finding the hedge

Good news first: \(\frac{d\mathbb Q}{d\mathbb P}\big\vert_{\mathcal F_T}=Z_T>0\), so \(\mathbb P\) and \(\mathbb Q\) are equivalent and it suffices to show \(\mathbb Q[X_T=V_T]=1\).

Two facts:

  1. \(e^{-rt}X_t\) is a \(\mathbb Q\)-martingale: \(d(e^{-rt}X_t)=e^{-rt}\Delta_tS_t\sigma\,dW^{\mathbb Q}_t\).
  2. \(M^V_t:=E^{\mathbb Q}[e^{-rT}V_T\mid\mathcal F_t]\) is a \(\mathbb Q\)-martingale on \([0,T]\), by Tower: \(E^{\mathbb Q}[M^V_t\mid\mathcal F_s]=E^{\mathbb Q}[e^{-rT}V_T\mid\mathcal F_s]=M^V_s\).

We want to make these two martingales the same process. Suppose \(M^V\) can be written as a stochastic integral, \(dM^V_t=\Gamma_t\,dW^{\mathbb Q}_t\) for some adapted \(\Gamma\). Then choose

\[\Delta_t=\frac{\Gamma_t}{e^{-rt}\sigma S_t},\qquad t\le T,\]

so that \(d(e^{-rt}X_t)=e^{-rt}\Delta_tS_t\sigma\,dW^{\mathbb Q}_t=\Gamma_t\,dW^{\mathbb Q}_t=dM^V_t\). Starting both at \(x=M^V_0=E^{\mathbb Q}[e^{-rT}V_T]\), “same start + same dynamics” gives \(e^{-rt}X_t=M^V_t\) for all \(t\), and at \(T\), \(X_T=V_T\).

The remaining question — is every \(\mathbb Q\)-martingale of the form \(\int\Gamma\,dW^{\mathbb Q}\)? — is answered (yes, when the filtration is generated by the Brownian motion) by the martingale representation theorem. In words: in a Brownian world, every fair game is a trading strategy in the stock, so every claim can be hedged — the market is complete.

9. Quadratic covariation and integration by parts

For continuous \(X,Y\) and a partition \(\Pi\) of \([0,t]\),

\[[X,Y]^\Pi_t=\sum_{j}\big(X_{t_j}-X_{t_{j-1}}\big)\big(Y_{t_j}-Y_{t_{j-1}}\big),\qquad [X,Y]_t=\lim_{\Vert \Pi\Vert \to0}[X,Y]^\Pi_t .\]

For Itô processes driven by the same \(W\), \(dX_t=\Theta_tdt+\Delta_tdW_t\) and \(dY_t=\phi_tdt+\psi_tdW_t\), the heuristic \(d[X,Y]_t=dX_t\,dY_t\) gives

\[d[X,Y]_t=\Delta_t\psi_t\,dt,\qquad [X,Y]_t=\int_0^t\Delta_s\psi_s\,ds,\]

and \([X,Y]\equiv0\) if either \(\Delta\equiv0\) or \(\psi\equiv0\).

Proof via polarization

For any numbers, \(4(a-b)(c-d)=\big((a+c)-(b+d)\big)^2-\big((a-c)-(b-d)\big)^2\), so

\[[X,Y]=\tfrac14\big([X+Y,X+Y]-[X-Y,X-Y]\big).\]

\(X\pm Y\) are Itô processes with \(dW\)-coefficients \(\Delta_t\pm\psi_t\), so

\[d[X,Y]_t=\tfrac14\big((\Delta_t+\psi_t)^2-(\Delta_t-\psi_t)^2\big)dt=\Delta_t\psi_t\,dt .\]

Stochastic integration by parts.

\[d(X_tY_t)=X_t\,dY_t+Y_t\,dX_t+d[X,Y]_t=\big(X_t\phi_t+Y_t\Theta_t+\Delta_t\psi_t\big)dt+\big(X_t\psi_t+Y_t\Delta_t\big)dW_t,\]

again an Itô process. For smooth (no-\(dW\)) \(f,g\) the bracket vanishes and we recover the ordinary product rule \(d(f_tg_t)=f_t\,dg_t+g_t\,df_t\).

Series contents

  1. 01 · Probability Spaces, Filtrations, Conditional Expectation and Martingales
  2. 02 · Random Walks, Quadratic Variation and Brownian Motion
  3. 03 · Stochastic Integration
  4. 04 · Itô’s Formula, Itô Processes and Geometric Brownian Motion
  5. 05 · Black–Scholes: Replication, the PDE and the Greeks
  6. 06 · Risk-Neutral Pricing, Change of Measure and Girsanov (this post)
  7. 07 · SDEs, Feynman–Kac and Stopping Times

← Previous: Black–Scholes: Replication, the PDE and the Greeks　·　Next: SDEs, Feynman–Kac and Stopping Times →

This series collects my notes from QST MF 790 Stochastic Calculus at Boston University’s Questrom School of Business, in 7 parts. The content follows the order of the course; wording, intuition and a few corrections are my own. Throughout, \(N(\mu,\sigma^2)\) uses the variance as its second argument. Corrections are welcome. Chinese version of this post: 中文版.

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