Deriving the Heston Option-Pricing PDE with Itô’s Lemma
Summary
The document derives the pricing equation for a portfolio value that depends on time, the underlying price, and stochastic variance in the Heston model. It applies Itô’s lemma to the two correlated processes, producing drift terms, variance terms, a mixed derivative term, and two Brownian risk terms. Substituting the asset dynamics into the expression makes those components explicit.
The derivation then assumes dynamic hedging can remove both sources of randomness, so the hedged portfolio earns the risk-free rate. Equating its drift to that rate yields a second-order PDE, including the correlation-driven cross derivative. This is a textbook no-arbitrage derivation, not an empirical test or a complete pricing recipe. Its hedge assumption matters: it presumes the relevant risks can be eliminated using available instruments. The final displayed equation also appears to contain derivative-notation errors, writing squared function symbols where second partial derivatives are intended.
Key ideas
- Itô’s lemma for a value function of price and variance creates a mixed derivative when the driving Brownian motions are correlated.
- Substituting the Heston SDEs expresses the portfolio change in drift and Brownian components.
- Dynamic hedging is assumed to eliminate both stochastic risk terms.
- Setting the hedged drift equal to the risk-free return gives the Heston pricing PDE.
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# Stochastic Vol Mathematical derivation
# Stochastic Vol Mathematical derivation
I want to understand the mathematical steps done. Can someone please simplify the derivation of d(pi) from Pi? Thanks in advance.
## Answer by Kevin (score 3)
https://quant.stackexchange.com/a/49203
I write down the solution for the Heston model. You can directly generalise the result.
Let $f=f(t,s,v)\in C^{1,2,2}(\mathbb{R}_+^3)$ be a real-valued function (portfolio value) and consider the two-dimensional stochastic process $(S_t,v_t)$ with \begin{align*} \mathrm{d}S_t&=(r-q) S_t \mathrm{d}t+\sqrt{v_t} S_t \mathrm{d}W_{1,t}, \\ \mathrm{d}v_t&=\kappa(\theta-v_t) \mathrm{d}t+\xi \sqrt{v_t} \mathrm{d}W_{2,t}, \end{align*} with $\mathbb{E}[\mathrm{d}W_{1,t}\mathrm{d}W_{2,t}]=\rho\mathrm{d}t$. Then, denoting partial derivatives by subscripts, we obtain from Ito's Lemma (which byouness mentioned in his comment) \begin{align*} \mathrm{d}f &= \left(f_t + \frac{1}{2}v_tS_t^2f_{ss} + \frac{1}{2}\xi^2v_tf_{vv}+\rho\xi S_tv_tf_{sv}\right) \mathrm{d}t + f_s \mathrm{d}S_t + f_v\mathrm{d}v_t. \end{align*} Following their definition as SDEs, the changes $\mathrm{d}S_t$ and $\mathrm{d}v_t$ can also be expressed in terms of $\mathrm{d}W_{1,t}$ and $\mathrm{d}W_{2,t}$ yielding \begin{align*} \mathrm{d}f &= \left(f_t + (r-q)S_tf_s + \kappa(\theta-v_t)f_v+ \frac{1}{2}v_tS_t^2f_{ss} + \frac{1}{2}\xi^2v_tf_{vv}+\rho\xi S_tv_tf_{sv}\right) \mathrm{d}t \\ & \;\;\;\;\; + f_s \sqrt{v_t}S_t \mathrm{d}W_{1,t} + f_v\xi\sqrt{v_t}\mathrm{d}W_{2,t}. \end{align*} The rest is identical to the derivation of the Black Scholes equation, we shall assume that both sources of risks can be eliminated by dynamic hedging forcing $\mathrm{d}f$ to be proportional to $\mathrm{d}t$. (choose the values you hold in the two assets such that the stochastic terms are zero). Thus, changes in $f$ are locally risk-free and hence, $\mathrm{d}f=rf\mathrm{d}t$.
We end up with the following linear, second-order, three-dimensional PDE \begin{align*} \frac{\partial f}{\partial t}+\frac{1}{2}v_tS_t^2\frac{\partial f^2}{\partial S_t^2}+\rho\xi v_tS_t\frac{\partial f^2}{\partial S_t\partial v_t}+\frac{1}{2}\xi^2v_t\frac{\partial f^2}{\partial v_t^2}+(r-q)S_t\frac{\partial f}{\partial S_t}+\kappa(\theta-v_t)\frac{\partial f}{\partial v_t}-rf=0. \end{align*}Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.