From Stock Dynamics to Claim Dynamics
A stock price fluctuates, and a claim whose value depends on that price fluctuates with it. The claim’s instantaneous change has three sources: the direct change in calendar time, the stock’s drift, and a second-order contribution from the stock’s quadratic variation. The two-variable Itô formula collects these terms. Keeping them separate is useful: the drift of the stock is not the same thing as the drift of the claim.
Work on a filtered probability space \((\Omega,\mathcal{F},(\mathcal{F}_t)_{t\ge0},\mathbb{P})\) carrying a standard Brownian motion \(B_t\). On a finite horizon \([0,T]\), suppose the stock satisfies the geometric Brownian motion equation \(dS_t=\mu S_t\,dt+\sigma S_t\,dB_t\), with constants \(\mu\in\mathbb{R}\), \(\sigma>0\), and \(S_0>0\). Here \(\mu\) is the stock’s drift coefficient under \(\mathbb{P}\), and \(\sigma\) is its diffusion coefficient. The explicit solution is \(S_t=S_0\exp((\mu-\tfrac12\sigma^2)t+\sigma B_t)\), which is positive.
For each \(t\), an Itô integral \(\int_0^t H_s\,dB_s\) below is taken with a progressively measurable integrand \(H\) in \(\mathcal{L}^2\), meaning \(\mathbb{E}\int_0^T H_s^2\,ds<\infty\). Under these conditions the integral is a square-integrable martingale: for \(u\le t\), \(\mathbb{E}[\int_0^t H_s\,dB_s\mid\mathcal{F}_u]=\int_0^u H_s\,dB_s\). This conditional-expectation identity follows first for adapted step integrands from the independent, mean-zero Brownian increments, and then for all such \(H\) by their approximation in the Itô-isometry norm and continuity of conditional expectation in \(L^2\). If only local square integrability is available, the integral is in general only a local martingale, not necessarily a true martingale.
If \(C\) is a smooth European claim price, this calculation describes its dynamics on the interval where that smoothness holds. It does not impose a pricing equation or a terminal payoff condition. In particular, at a maturity where a payoff has a kink, the smooth two-variable formula applies before maturity on a smooth region, not automatically at the kink itself.
The Claim-Dynamics Formula
The two-variable Itô formula is the standard result from earlier material on Itô’s formula. Its second-order term is essential: Brownian quadratic variation satisfies \(dB_t\,dB_t=dt\), while \(dB_t\,dt=0\) and \(dt\,dt=0\). These rules are not ordinary differentiation rules. In the formula, the first-order stock contribution is \(C_s\,dS_t\), and the second-order contribution is \(\tfrac12 C_{ss}(dS_t)^2=\tfrac12\sigma^2S_t^2C_{ss}\,dt\).
Proof. For the function \(f(t,s)=C(t,s)\), its partial derivatives are \(f_t=C_t\), \(f_s=C_s\), and \(f_{ss}=C_{ss}\). The stock has \(dS_t=\mu S_t\,dt+\sigma S_t\,dB_t\), and its quadratic variation is \(d[S]_t=\sigma^2S_t^2\,dt\). Substitution into the two-variable Itô formula gives \(dC=f_t\,dt+f_s\,dS_t+\tfrac12 f_{ss}\,d[S]_t\). Replacing \(dS_t\) by its drift and diffusion terms yields the displayed formula, with drift \(C_t+\mu S_tC_s+\tfrac12\sigma^2S_t^2C_{ss}\) and diffusion \(\sigma S_tC_s\). \(\square\)
The drift can be read as three separate contributions. The partial time change is \(C_t\,dt\); the stock drift contributes \(\mu S_tC_s\,dt\); and curvature in the stock variable contributes \(\tfrac12\sigma^2S_t^2C_{ss}\,dt\). The diffusion coefficient of the claim is \(\sigma S_tC_s\). Thus the stock’s drift \(\mu S_t\) and diffusion coefficient \(\sigma S_t\) are not themselves the claim’s drift and diffusion coefficients.
Worked Example: The Square of a Geometric Brownian Stock
Take \(S_0=2\), \(\mu=\tfrac14\), and \(\sigma=\tfrac12\), and let \(C(t,s)=s^2\). Compute the dynamics and check its mean against the explicit stock solution.
The derivatives are \(C_t=0\), \(C_s=2s\), and \(C_{ss}=2\). The drift from the formula is \(\mu sC_s+\tfrac12\sigma^2s^2C_{ss}=\tfrac14s(2s)+\tfrac12(\tfrac14)s^2(2)=\tfrac12s^2+\tfrac14s^2=\tfrac34s^2\). The diffusion coefficient is \(\sigma sC_s=\tfrac12s(2s)=s^2\). Hence \(d(S_t^2)=\tfrac34S_t^2\,dt+S_t^2\,dB_t\).
For an independent check, the explicit stock solution is \(S_t=2\exp(t/8+B_t/2)\), since \(\mu-\tfrac12\sigma^2=\tfrac14-\tfrac18=\tfrac18\). Squaring gives \(S_t^2=4\exp(t/4+B_t)\). Since \(B_t\) is normal with mean \(0\) and variance \(t\), \(\mathbb{E}[e^{B_t}]=e^{t/2}\), so \(\mathbb{E}[S_t^2]=4e^{t/4}e^{t/2}=4e^{3t/4}\). The drift equation gives the same mean: its drift coefficient is \(\tfrac34S_t^2\), so the mean solves \(m'(t)=\tfrac34m(t)\), \(m(0)=4\), with solution \(m(t)=4e^{3t/4}\). The two calculations agree.
Discounting and the Choice of Measure
In the Black–Scholes market, the bank account with constant interest rate \(r\) is \(e^{rt}\). Discounting a claim means multiplying its value by \(e^{-rt}\). The claim’s drift depends on the probability measure used to specify the stock dynamics. Under \(\mathbb{P}\), the drift coefficient in the stock equation is \(\mu\). If the model is instead specified under a risk-neutral measure \(\mathbb{Q}\), the stock equation in the no-dividend model has drift \(r\). These are different measures and drift assumptions; switching between them requires a change-of-measure argument, not a relabelling of \(\mathbb{P}\) as \(\mathbb{Q}\).
Proof. The deterministic function \(g(t)=e^{-rt}\) has \(dg(t)=-re^{-rt}\,dt\). The product rule gives \(d(gC)=g\,dC+C\,dg\); there is no quadratic covariation term because \(g\) has finite variation. Substituting the claim dynamics gives the stated drift, with \(-rC\) contributed by \(C\,dg\), and the stated diffusion, which is multiplied by \(g\). If the stock equation is specified under \(\mathbb{Q}\), the same substitution uses drift \(rS_t\) in place of \(\mu S_t\), giving the second expression. \(\square\)
Worked Example: Discounted Stock as a Claim Process
Let \(C(t,s)=e^{-rt}s\), with \(r=\tfrac1{25}\), \(\mu=\tfrac1{10}\), \(\sigma=\tfrac15\), and \(S_0=100\). Compute its dynamics under \(\mathbb{P}\), then check the mean.
The derivatives are \(C_t=-\tfrac1{25}e^{-t/25}s\), \(C_s=e^{-t/25}\), and \(C_{ss}=0\). The time and stock-drift contributions to the drift sum to \(-\tfrac1{25}e^{-t/25}s+\tfrac1{10}s e^{-t/25}=(\tfrac1{10}-\tfrac1{25})e^{-t/25}s=\tfrac3{50}C(t,s)\). The quadratic-variation contribution is zero because \(C_{ss}=0\), and the diffusion coefficient is \(\tfrac15e^{-t/25}s=\tfrac15C(t,s)\). Thus \(dC_t=\tfrac3{50}C_t\,dt+\tfrac15C_t\,dB_t\), where \(C_t=C(t,S_t)\).
The explicit stock solution gives \(S_t=100\exp((\tfrac1{10}-\tfrac12\cdot\tfrac1{25})t+\tfrac15B_t)=100e^{(2/25)t+B_t/5}\). Therefore \(C_t=100e^{(2/25-1/25)t+B_t/5}=100e^{t/25+B_t/5}\), and \(\mathbb{E}[C_t]=100e^{t/25}e^{t/50}=100e^{3t/50}\), using \(\mathbb{E}[e^{B_t/5}]=e^{t/50}\). The drift equation predicts the same mean, since \(\mathbb{E}[C_t]\) solves \(m'(t)=\tfrac3{50}m(t)\), \(m(0)=100\).
The stochastic integral here is a true square-integrable martingale on every finite horizon: \(C_t^2=10000e^{2t/25+2B_t/5}\), so \(\mathbb{E}[C_t^2]=10000e^{2t/25+2t/25}=10000e^{4t/25}\). Consequently \(\mathbb{E}\int_0^T(\tfrac15C_s)^2\,ds=\tfrac{10000}{25}\int_0^T e^{4s/25}\,ds<\infty\). Under a model specified under \(\mathbb{Q}\) with stock drift \(r=\tfrac1{25}\), the discounted-stock drift coefficient is \((r-r)C_t=0\); that statement concerns \(\mathbb{Q}\), not \(\mathbb{P}\).
Separating the Three Contributions
Worked Example: A Time- and Stock-Dependent Smooth Function
Let \(C(t,s)=ts+s^2\), and evaluate its claim dynamics at \(t=1\), \(S_t=10\), with \(\mu=\tfrac1{10}\) and \(\sigma=\tfrac15\). This is a pointwise calculation of the coefficients; it does not assert that \(S_t=10\) with positive probability at that exact time.
The partial derivatives are \(C_t=s\), \(C_s=t+2s\), and \(C_{ss}=2\). At \((t,s)=(1,10)\), these are \(10\), \(21\), and \(2\), respectively, and \(C(1,10)=10+100=110\). The time contribution to the drift is \(10\). The stock-drift contribution is \(\mu sC_s=\tfrac1{10}\cdot10\cdot21=21\). The second-order contribution is \(\tfrac12\sigma^2s^2C_{ss}=\tfrac12\cdot\tfrac1{25}\cdot100\cdot2=4\). Thus the total drift coefficient is \(10+21+4=35\). The diffusion coefficient is \(\sigma sC_s=\tfrac15\cdot10\cdot21=42\).
As a direct algebraic check, write \(C(t,S_t)=tS_t+S_t^2\). The product rule gives \(d(tS_t)=S_t\,dt+t\,dS_t\). For the square, the first worked example’s calculation with general coefficients gives \(d(S_t^2)=(2\mu+\sigma^2)S_t^2\,dt+2\sigma S_t^2\,dB_t\). Adding the two expressions gives drift \(S_t+\mu tS_t+(2\mu+\sigma^2)S_t^2\) and diffusion \(\sigma tS_t+2\sigma S_t^2\). At \(t=1,S_t=10\), these are \(10+1+24=35\) and \(2+40=42\), matching the two-variable formula.
Common Pitfalls and Scope
A frequent mistake is to apply the ordinary chain rule and write only \(C_t\,dt+C_s\,dS_t\). That calculation omits \(\tfrac12 C_{ss}\,d[S]_t\). For the square example, it would give drift \(\tfrac12S_t^2\), since \(\mu=\tfrac14\), instead of the correct \(\tfrac34S_t^2\). The missing \(\tfrac14S_t^2\) is exactly \(\tfrac12\sigma^2S_t^2C_{ss}=\tfrac14S_t^2\). The correction is a consequence of Brownian quadratic variation, not an optional adjustment.
Another mistake is to treat every stochastic integral as a true martingale. The formula for \(C(t,S_t)\) is valid locally under the smoothness assumptions, but the martingale conclusion for its stochastic term requires the stated square-integrability condition. A local martingale need not have the conditional-expectation property of a true martingale. Nor does the formula alone say that the claim is priced correctly: it computes dynamics for a given smooth function \(C\), while pricing conditions and replication require additional arguments.
| Term in the claim dynamics | Expression | Source |
|---|---|---|
| Time contribution | \(C_t\,dt\) | Explicit time dependence |
| Stock-drift contribution | \(\mu S_tC_s\,dt\) | Drift of the stock under the specified measure |
| Quadratic-variation contribution | \(\tfrac12\sigma^2S_t^2C_{ss}\,dt\) | Curvature of the claim and stock quadratic variation |
| Claim diffusion | \(\sigma S_tC_s\,dB_t\) | Brownian fluctuation transmitted through stock sensitivity |
Check Your Understanding
For each question, state the assumptions needed for the formula or expectation you use.
- For \(C(t,s)=s^3\), compute \(C_t,C_s,C_{ss}\), then identify the drift and diffusion coefficients under \(dS_t=\mu S_t\,dt+\sigma S_t\,dB_t\).
- For \(C(t,s)=e^{-rt}s\), explain why the discounted drift is zero under a model specified under \(\mathbb{Q}\) with stock drift \(r\), but need not be zero under \(\mathbb{P}\).
- Which term is lost if one applies the ordinary chain rule to \(C(t,S_t)\), and what property of Brownian motion requires that term?
- State an integrability condition that makes the stochastic integral in the claim formula a square-integrable martingale.
- For \(C(t,s)=ts+s^2\), give the separate time, stock-drift and quadratic-variation contributions to its drift at a general \((t,s)\).