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There are numerous papers that describe the derivation of the Local-Vol equation using available market prices of options. For example:

  • Dupire's formula (see e.g. OpenGamma (2013)) gives us LV in terms of absolute strike & price:
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  • Gatheral (2003) gives us LV in terms of log-strike & total BS-implied variance:
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But what I am wondering is, given a pure Local Vol surface $\sigma_{LV}$, how does one recover Vanilla European market prices $C_{Mkt}$?

I can think of Monte-Carlo or PDE methods, but are there standard (semi)-analytical techniques which are faster than the first two? That is, are there ways to do:

  1. Convert local vols $\sigma_{LV}\rightarrow$ BS-implied vols $\sigma_{BS}\rightarrow$ market prices $C_{Mkt}$ (3 steps)
  2. Convert local vols $\sigma_{LV}\rightarrow$ market prices $C_{Mkt}$ (2 steps)
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  • $\begingroup$ I edited your question to invert the local vol formulas as the one giving LV = f(C) is the fundamental one. It is called Dupire's formula. $\endgroup$ – byouness Jan 29 at 17:16
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    $\begingroup$ For exact calculation the most accurate is to solve for the fokker planck equation (forward PDE) using finite differences. You get all the vanilla options in 1 pass so it is very efficient. For approximations of the implied vol surface in terms of the local vol surface see Jim Gatheral's most likely path approach faculty.baruch.cuny.edu/jgatheral/…. In essence square implied vol for strike $K$ at expiry $T$ is an "average" of square local vol along the "most likely path" that connects the initial spot $S_0$ to the final strike $K$. $\endgroup$ – Antoine Conze Jan 29 at 17:32
  • $\begingroup$ @AntoineConze do you have details on how to apply fokker planck? you are taking finite differences between what? $\endgroup$ – Phil-ZXX Jan 29 at 17:35
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    $\begingroup$ Use a finite differences scheme (e.g. Crank Nicolson) for solving the fokker plank en.wikipedia.org/wiki/Fokker%E2%80%93Planck_equation PDE to get the density of the stock price distribution at all expiries in 1 pass, then use the density to quickly compute all vanilla options prices. See e.g. papers.ssrn.com/sol3/papers.cfm?abstract_id=2008215 $\endgroup$ – Antoine Conze Jan 29 at 17:46
  • $\begingroup$ Hi Phil, I'll delete my answer bellow as I misunderstood your question. Looks like what you were looking for is an exact calculation using PDE resolution after all, and not a pricing using "neither PDE nor MC". $\endgroup$ – byouness Jan 29 at 20:36
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Let us denote $\mathcal{C}$ the European call prices and consider the map $K\mapsto\mathcal{C}(t,T,S_t,K)$ the market price of calls maturing at $T$. We can obtain a link between the risk-neutral probability distribution of the stock price and current call prices via the Breeden-Litzenberger formula: $$f^S_T(K)=f_{S_T|S_t}(S_T=K)=e^{r(T-t)}\frac{\partial^2\mathcal{C}_t}{\partial K^2}$$ Proof (sketch): Write the fair value of $\mathcal{C}_t$ as an integral over the risk-neutral probability density, differentiate twice wrt $K$.

This result tells us that we can infer all the risk-neutral distribution of the stock! Unlike the implied volatility that is a function of $K$ and $T$, the local volatility is a function of $S$ and $T$. We need to chose a Lipschitz function $\sigma(S,t)$ to guarantee existence and uniqueness of solution of the stock price SDE.

Let us denote $h(S_T)$ the payoff of the European option with a deterministic risk-free rate $r$ and dividend yield $q$. Then, the fair value is: $$V(S,t)=\mathbb{E}^{\mathbb{Q}}\left[e^{-\int_t^Tr_sds}h(S_T)|\mathcal{F}_t\right]$$ Note that this equation satisfies the Black-Scholes PDE: $$\frac{\partial V}{\partial t}(S,t)+(r-q)S\frac{\partial V}{\partial S}(S,t)+\frac{1}{2}\sigma(S,t)^2S^2\frac{\partial^2 V}{\partial S^2}(S,t)=rV(S,t)$$ $$V(S,T)=h(S)$$ Proof (sketch): Write the fair value of the payoff $h$ at time $T$ in integral form, apply Ito's lemma and to compute the infinitesimal change in $h$ and integrate on both sides, thus obtaining the forward Kolmogorov equation.

Assuming that we have an arbitrage-free and smooth implied volatility surface (so that we can make use of the Breeden-Litzenberger formula), we can find a non-parametric function $\sigma(S, t)$ to uniquely determine the local volatility function. We now introduce the Dupire formula: $$\sigma^2(K,T)=\frac{\frac{\partial\mathcal{C}}{\partial T}+q_T\mathcal{C}+(r_T-q_T)\frac{\partial\mathcal{C}}{\partial K}}{\frac{1}{2}K^2\frac{\partial^2\mathcal{C}}{\partial K^2}}$$

This gives us the relation between the local volatility function and call option function using market prices. Claim: The density $f_T^S$ of the SDE $dS_t=r_tS_tdt+\sigma(S_t,t)S_tdW_t$ is described by the Forward Kolmogorov equation: $$\frac{\partial f}{\partial T}(S,T)=-\frac{\partial}{\partial S}((r_T-q_T)S f(S,T))+\frac{1}{2}\frac{\partial^2}{\partial S^2}(\sigma^2(S,T)S^2f(S,T))$$ This tells you that the initial density at time $t=0$ is a Dirca delta function centered at $S=S_0$, and this equation drives the dynamics of the density up to time $T$.

Proof: Look at Dupire (1994) paper.

Finally, we can re-write the Dupire formula as a function of implied volatilities $$\sigma^2(K,T)=\frac{\sigma^2_{imp}+2\sigma_{imp}T\left(\frac{\partial\sigma_{imp}}{\partial T}+(r_t-q_T)K\frac{\partial\sigma_{imp}}{\partial K}\right)}{\left(1-\frac{K\ln\frac{k}{F_T}}{\sigma_{imp}}\frac{\partial\sigma_{imp}}{\partial K}\right)^2+K\sigma_{imp}T\left(\frac{\partial\sigma_{imp}}{\partial K}-\frac{1}{4}K\sigma_{imp}T(\frac{\partial\sigma_{imp}}{\partial K})^2+K\frac{\partial^2\sigma_{imp}}{\partial K^2}\right)}$$

Proof: Look at Gatheral(2006).

Finally, assuming $\sigma_{imp}$ constant in $K$ but time-varying, i.e. $\sigma_{imp}(K,T)=\sigma_{imp}(T)$, we can see by a Taylor expansion:

$$\sigma_{imp}(T)=\sigma^2_{imp}(T)+2T\sigma_{imp}(T)\sigma^\prime_{imp}(T)=\sigma^2_{imp}(T)+T(\sigma^2_{imp}(T))^\prime=(T\sigma^2_{imp}(T))^\prime(T)$$ integrating, we obtain: $$\sigma^2_{imp}(T)=\frac{1}{T}\int_0^T\sigma^2(t)dt$$ Therefore, assuming that the implied volatility does not depend on the strike, we don't need numerical techniques to compute the option prices, since we can use the Black-Scholes formula plugging in the average local volatility from time $0$ to $T$.

For $\sigma_{imp}(K,T)$ one has to compare different shapes of mappings $(K,T)\mapsto\sigma_{imp}(K,T)$ and $(S,T)\mapsto\sigma_{imp}(K,T)$. One needs to pay attention at the interpolation to have a smooth surface.

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