Portfolio Volatility from Itô’s Formula for an SDE

Summary

The document asks how to interpret the volatility term for a portfolio formed from two functions of a stochastic process. The process has drift terms and a Brownian component with coefficient B(t, X_t). Applying Itô’s formula to each portfolio component produces drift contributions, a term driven by the process b_t, and a Brownian increment whose coefficient combines the two functions’ sensitivities to the state, weighted by their portfolio shares and the diffusion coefficient.

The stated portfolio volatility is that Brownian coefficient. For an infinitesimal time interval, its square gives the instantaneous variance rate when the only random shock is the stated standard Brownian motion; over a finite interval, variance generally requires accounting for how the coefficient changes along the process. The document gives no worked calculation or clarification of whether b_t itself is random, so interpreting its contribution requires assumptions about that term. The answer depends on the model’s diffusion structure and should not be read as a constant volatility in general.

Key ideas

• Itô’s formula separates portfolio changes into drift and stochastic Brownian contributions.
• The Brownian coefficient combines each asset function’s state sensitivity with its portfolio weight and the process diffusion coefficient.
• That coefficient represents instantaneous volatility under the stated single-Brownian-shock model.
• Its square is the instantaneous variance rate, while finite-horizon variance may require integrating a time-varying coefficient.
• The interpretation of the b_t increment depends on assumptions not supplied in the document.

Tags

# Volatility of a stochastic Process given by an SDE


# Volatility of a stochastic Process given by an SDE












I am currently working on this thesis: http://arks.princeton.edu/ark:/88435/dsp01vd66w212h and i am stuck on page 199. There we have a portfolio $P=\alpha F+\beta G $ with $\alpha +\beta =1$ and underlying process $dX_t=db_t+A(t,X_t)dt+B(t,X_t)dW_t$, where $b_t$ is a function of time and $W_t$ a standard Brownian motion. With the It^{o} formula and the definition of $dX_t$ we get \begin{align*} dP_t=\bigg[\alpha \left(\frac{\partial F}{\partial t} (t,X_t) + \frac{\partial F}{\partial x} (t,X_t) A(t,X_t) + \frac{1}{2} \frac{\partial^2 F}{\partial^2 x^2} (t,X_t) B^2(t,X_t) \right) \ + \beta \left(\frac{\partial G}{\partial t} (t,X_t) + \frac{\partial G}{\partial x} (t,X_t) A(t,X_t) + \frac{1}{2} \frac{\partial^2 G}{\partial^2 x^2} (t,X_t) B^2(t,X_t) \right) \bigg]dt \ + \left[\alpha \frac{\partial F}{\partial x} (t,X_t) + \beta \frac{\partial G}{\partial x} (t,X_t) \right] dbt\ + \left[ \alpha \frac{\partial F}{\partial x} (t,X_t) B(t,X_t) + \beta \frac{\partial G}{\partial x} (t,X_t) B(t,X_t) \right] dW_t \end{align*} Now the author states that the volatility is given by \begin{align*} \alpha \frac{\partial F}{\partial x} (t,X_t) B(t,X_t) + \beta \frac{\partial G}{\partial x} (t,X_t) B(t,X_t) \end{align*} I understand that the volatility depends on this term but i dont unterstand that it is directly given by this term. Furthermore i am confused how to calculate the variance in this case. Many thanks for your help!

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