In this article we explore the Markovian functional approach to define a new class of SDEs. Although the real power of this approach is to simplify rich and sophisticated models, we will focus on single-underlying models: our starting point will be a local volatility model, which will then redefined through a simpler model that:

  • is a one-factor Markov model, with a Brownian driver and a simple structure; and
  • reproduces exactly the ame one-dimensional distributions of the local volatility model, that is all European options are priced exactly as in the local volatility model.

Let’s start from a local volatility model. We assume the risk-neutral dynamics

\[\frac{dS(t)}{S(t)} = \mu dt + \sigma(S(t), t) dW(t),\]

with $S(0) = S_0$; with standard notation, $W(t)$ is a Wiener process. It is well-known that, at a given maturity $T$, the probability distribution is fully determined by the European call prices,

\[f_{S}(s, T) = \left.\frac{\partial^2 C(K, T)}{\partial K^2}\right|_{K=s}.\]

We are interested in the behavior of our model only at a set of times

\[\mathcal{T} = \{ T_1, T_2, \ldots, T_n \}.\]

On each $T_i, i = 1, \ldots, n$, we will compute the cumulative density function $F_{S}(s, T_i)$ from $f_{S}(s, T_i)$. These $n$ cumulative distribution functions are the only components we need to keep from the local volatility model; everything else can be discarded.

To proceed, we need to define the (simpler) process that will drive our model. For simplicity here we use

\[dX(t) = \nu(t) dW(t),\]

with $X(0) = 0$ and a deterministic volatility $\nu(t) > 0$. The distribution of $X(t)$ is Gaussian,

\[X(t) \sim \mathcal{N}(0, v(t)^2),\]

with

\[v(t) = \int_0^t \nu(\tau)^2 d\tau.\]

The process $X(t)$ is Markov by construction; the idea is to define a functional mapping $\hat{S}(t) = f(X(t), t)$ such that $S(T)$ and $\hat{S}(T)$ have the same distributions for all $T \in \mathcal{T}$. For all times not included in $\mathcal{T}$, instead, the two processes will have different distributions; this means that the matching is only at discrete points, not for a continuous interval.

The mapping is defined as follows: for each maturity $T_i \in \mathcal{T}$, we impose the matching of cumulative density functions, that is each point $x = X(T_i)$ will correspond to the point $s = S(T_i)$ such that

\[F_S(s, T_i) = F_X(x, T_i).\]

Both $F_S$ and $F_X$ are monotonically increases and ranging from 0 to 1, so for every $x$ we can find one and only point $s$ such that the equation above is satisfied. That is, a function $g$ exists such that

\[s = g(x, T_i).\]

Because of our choice of $X(t)$, we have the analytic representation $F_X(x, T_i) = \Phi(x / v(T_i))$; also $F_S(s, T_i)$ is known, perhaps numerically. Therefore it is possible to compute

\[g(x, T_i) = \left( F_S \right)^{-1}\Phi \left( \frac{x}{v(T_i)} \right).\]