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2015 | OriginalPaper | Buchkapitel

Asymptotic Implied Volatility at the Second Order with Application to the SABR Model

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Abstract

We provide a general method to compute a Taylor expansion in time of implied volatility for stochastic volatility models, using a heat kernel expansion. Beyond the order 0 implied volatility which is already known, we compute the first order correction exactly at all strikes from the scalar coefficient of the heat kernel expansion. Furthermore, the first correction in the heat kernel expansion gives the second order correction for implied volatility, which we also give exactly at all strikes. As an application, we compute this asymptotic expansion at order 2 for the SABR model and compare it to the original formula.

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Fußnoten
1
This connection is similar to the connection which described the electromagnetic potential, except that the fibre of the gauge bundle is \(\mathbb {R}\) instead of U(1). This causes a difference of a factor i in equations.
 
2
There is a breaking of symmetry between time and spatial directions. The diffusion equation can be seen as a non-relativistic limit of a pure wave equation in imaginary time.
 
3
In finance we will usually consider noncompact manifolds, possibly with boundaries as in the SABR model for \(0<\beta <1\).
 
4
Using Feynman path integral, the solution to Eq. (7) can be written up to some normalization factor as
$$\begin{aligned} p(X,t) \propto \int [DX] e^{\displaystyle - \int _0^t \mathrm {d}t \left( \frac{1}{2} g_{ij} \dot{X}^i \dot{X}^j + A_i \dot{X}^i + Q \right) } \end{aligned}$$
where [DX] means integrating over all path X(s) going from \(X(0) = X_0\) to \(X(t) = X\). The normalization factor is the inverse of the same quantity with the integral computed over all paths with starting point \(X_0\), so that the total probability is 1. It is generally not possible to compute this integral exactly. However it gives some hints on the asymptotic solution at short time: the solution will be dominated by the path corresponding to the minimal value of the integrand inside the exponential, which will be close to the geodesic path.
 
5
We use the standard notation of \(\alpha \) for the initial value of the volatility variable in the SABR model instead of \(V_0\) as in the previous section.
 
6
In fact at very short maturities, the FDM scheme we use is less precise and less stable than this second order expansion, especially in the wings where the probability density is very small.
 
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Metadaten
Titel
Asymptotic Implied Volatility at the Second Order with Application to the SABR Model
verfasst von
Louis Paulot
Copyright-Jahr
2015
DOI
https://doi.org/10.1007/978-3-319-11605-1_2