Dupire’s formula enables us to deduce the volatility function in a local volatility model from quoted put and call options on the market.
In a local volatility model the asset price model under a risk-neutral measure takes the form \[ dS_t=\mu(t)S_tdt+\tilde\sigma(t,S_t)S_tdW_t. \] The equivalent PDE statement is \[ \frac{{\partial C}}{\partial T}(T,K)=-rK\frac{{\partial C}}{\partial K}(T,K)+ \frac{1}{2}\sigma^2(T,K)K^2 \frac{{\partial^2C}}{\partial K^2}(T,K) \] with natural initial condition \[ C(0,K)=(S_{0}-K)_{+}, \] since when \(T=0\) option price is just the immediate payoff.
\begin{proof} Consider the European call option under the Local Volatility Model with time to maturity \(T\) and strike \(K\), denoting the option price \(C(T,K)\). Then clearly \[
C(T,K)=e^{-rT}\mathbb{E}_{\mathbb{Q}}(S_{T}-K)_{+}=e^{-rT}\int _{0}^\infty (x-K)_{+}\cdot p(T,x) \, dx ,
\] where \(p(T,\cdot)\) is the density of \(S_T\) under \(\mathbb{Q}\). Taking the derivative w.r.t. \(T\) we get \[
\frac{{\partial C}}{\partial T}=-re^{-rT}\int _{K}^\infty (x-K)\cdot p(T,x) \, d +e^{-rT}\int_{K}^\infty(x-K)\cdot\frac{{\partial p}}{\partial T}(T,x) \, dx,
\] where we notice that \(\frac{{\partial p}}{\partial T}(T,x)\) is the time derivative of the density of \(S_T\) under \(\mathbb{Q}\) and \(S_t\) has dynamics under \(\mathbb{Q}\) given by \[
dS_{t}=rS_{t}dt+\sigma(t,S_{t})S_{t}dW_{t}^\mathbb{Q}.
\] The time evolution of density is described by the Fokker-Planck Equation that \[
L_{t}^*p(t,x)= \frac{{\partial p}}{{\partial T}}(t,x),
\] where \(L_{t}^*\) is the adjoint of the infinitesimal generator \(L_t\) of the diffusion and \[
L_{t}=rx\partial_{x}+ \frac{1}{2}\sigma^2(t,x)x^2\partial_{{x x}}.
\] As a result \[
\begin{align}
e^{-rT}\int _{K}^\infty (x-K)\cdot \frac{{\partial P}}{\partial T}(T,x) \, dx & = \int _{0}^\infty e^{-rT}(x-K)_{+}\cdot L_{T}^*p(T,x) \, dx \\
& =\int _{0}^\infty L_{T}[e^{-rT}(x-K)_{+}]\cdot p(T,x) \, dx ,
\end{align}
\] from the definition of adjoint. Calculating the action of infinitesimal generator \[
L_{T}[e^{-rT}(x-K)_{+}]=e^{-rT}\left( rx \mathbb{1}_{x>K}+ \frac{1}{2}\sigma^2(T,x)x^2\delta_{\{K\}} \right),
\] so the integral has value \[
\int _{0}^\infty L_{T}[e^{-rT}(x-K)_{+}]\cdot p(T,x) \, dx = e^{-rT}\left( \int _{K}^\infty rxp(T,x) \, dx + \frac{1}{2}\sigma^2(T,K)K^2\cdot p(T,K)\right).
\] Combining these equations we have that \[
\begin{align}
\frac{{\partial C}}{\partial T} & = -r e^{-rT}\int _{K}^\infty (x-K)\cdot p(T,x) \, dx +e^{-rT}\left( \int _{K}^\infty r xp(T,x) \, dx+ \frac{1}{2}\sigma^2(T,K)K^2 \cdot p(T,K) \right) \\
& =r Ke^{-rT} \int _{K}^\infty p(T,x) \, dx + \frac{1}{2}\sigma^2(T,K)K^2e^{-rT}p(T,K),
\end{align}
\] hence by the use of the Black-Scholes PDE (BSPDE) we obtain the result \[
\frac{{\partial C}}{\partial T}(T,K)=-rK\frac{{\partial C}}{\partial K}(T,K)+ \frac{1}{2}\sigma^2(T,K)K^2 \frac{{\partial^2C}}{\partial K^2}(T,K).
\] \end{proof} ### Comments - Dupire’s ODE is a forward equation and it is much more efficient to use than BS-PDE since on knowing one point \((0,K)\) for some given strike \(K\) one can figure out the call price for all \((T,K)\) at all time to maturity.
- Furthermore, Dupire’s PDE also provides an explicit formula for implied volatility in the Local Volatility Model.
0.0.1 References
- https://www.csie.ntu.edu.tw/~d00922011/python/cases/LocalVol/DUPIRE_FORMULA.PDF