Unifying the BGM and SABR Models: A Short Ride in Hyperbolic Geometry

Abstract

In this paper, using a geometric method introduced in (Henry-Labordère Large Deviations and Asymptotic Methods in Finance (2015) [12]) and initiated by (Avellaneda et al. Risk Mag. (2002) [4]), we derive an asymptotic swaption implied volatility at the first-order for a general stochastic volatility Libor Market Model. This formula is useful to quickly calibrate a model to a full swaption matrix. We apply this formula to a specific model where the forward rates are assumed to follow a multi-dimensional CEV process correlated to a SABR process. For a caplet, this model degenerates to the classical SABR model and our asymptotic swaption implied volatility reduces naturally to the Hagan-al formula (Hagan et al. Willmott Mag. 88–108 (2002) [11]). The geometry underlying this model is the hyperbolic manifold ${\mathbb H}^{n+1}$ with n the number of Libor forward rates.

Appendices

Appendix A: Heat Kernel Expansion

An short-time expansion of the conditional probability for a multi-dimensional Itô diffusion process can be achieved using the heat kernel expansion. In that purpose, the Kolmogorov equation is rewritten as the heat kernel equation on a (n)-dimensional Riemannian manifold ${\mathcal M}^{n}$ endowed with an Abelian connection as explained in [12, 13]. Let’s assume that our multi-dimensional stochastic equations (in ${\mathbb Q}^{\alpha \beta }$) are written as

$$\begin{aligned}dx_\mu =b_\mu (t,x)dt+\sigma _\mu (t,x)dW_\mu \end{aligned}$$

with $dW_\mu dW_\nu =\rho _{\mu \nu }(t)dt$. Then, the metric $g_{\mu \nu }$ depends only on the diffusion terms $\sigma _\mu$ and the connection ${\mathcal A}_\mu$ on the drift terms $b_\mu$ as well

$$\begin{aligned} g_{\mu \nu }(t,x)= & {} 2 {\rho ^{\mu \nu }(t) \over \sigma _\mu (t,x) \sigma _\nu (t,x)}, \; \mu ,\nu =1,\ldots , n \;, \; \rho ^{\mu \nu } \equiv [\rho ^{-1}]_{\mu \nu }\end{aligned}$$

(5.1a)

$$\begin{aligned} {\mathcal A}_\alpha (t,x)= & {} \sum _{\mu =1}^n g_{\alpha \mu } {1 \over 2}\left( b_\mu (t,x)-\sum _{\nu =1}^{n+1} g^{-{1 \over 2}}\partial _\nu \left( g^{1/2}g^{\mu \nu }(t,x)\right) \right) ,\; \alpha =1,\ldots , n\nonumber \\ \end{aligned}$$

(5.1b)

with $g(t,x) \equiv \det [g_{\mu \nu }(t,x)]$. In terms of these functions, the asymptotic solution to the Kolmogorov equation in the short-time limit is given by

$$\begin{aligned} p(x,t|x^0)={\sqrt{g(x)} \over (4\pi t)^{n \over 2}}\sqrt{\Delta (x,x^0)}\mathcal{P}(x,x^0)e^{-{\sigma (x,x^0) \over 2t}}\left( 1+\sum _{n=1}^\infty a_n(x,x^0)t^n \right) {,}\,t \rightarrow 0\nonumber \\ \end{aligned}$$

(5.2)

• Here, $\sigma (x,x^0)$ is the Synge function defined as

  $$\begin{aligned}\sigma (x,x^0)={d^2(x,x^0) \over 2} \end{aligned}$$

  The distance $d(x,x^0)$ is defined as the minimizer of

  $$\begin{aligned}d(x,x^0)^2=\min _{C} \int _0^T g_{\mu \nu }(t=0,x) {dx_\mu (t) \over dt} {dx_\nu (t) \over dt} dt \end{aligned}$$

  and t parameterizes the curve $\mathcal{C}(x,x^0)$ joining $x(t=0)\equiv x^0$ and $x(T)\equiv x$.

• $\Delta (x,x^0)$ is the so-called Van Vleck-Morette determinant

  $$\begin{aligned} \Delta (x,x^0)=g(0,x)^{-{1 \over 2}}\det (-{\partial ^2 \sigma (x,x^0) \over \partial x \partial x^0})g(0,x^0)^{-{1 \over 2}} \end{aligned}$$

  (5.3)

  with $g(x)=\det [g_{\mu \nu }(0,x)]$

• $\mathcal{P}(x,x^0)$ is the parallel transport of the Abelian connection along the geodesic $\mathcal{C}(x,x^0)$ from the point x to $x^0$.

  $$\begin{aligned} \mathcal{P}(x,x^0)=e^{-\int _{\mathcal{C}(x^0,x)} \mathcal{A}_\mu (t=0,x) dx^\mu } \end{aligned}$$

  (5.4)

• The $a_i(x,x^0)$ are smooth functions on ${\mathcal M}^{n}$ and depend on geometric invariants such as the scalar curvature R. More details can be found in [12].

Appendix B: Saddle-Point Method

The integration over ${\mathbb B}$ is obtained by using a saddle-point method which consists in approximating at the first order the integral $\int f(x) e^{ \epsilon \phi (x)} dx$ in the limit $\epsilon$ large by [8]

$$\begin{aligned}\int f(x) e^{ \epsilon \phi (x)} dx \sim _{\epsilon >>1}&f(x^*) e^{ \epsilon \phi (x^*)}\left( 1+ {1 \over \epsilon }\left( -{\partial _{\alpha \beta } f \over 2f} {A_{\alpha \beta } }\right. \right. \nonumber \\&+\left( {\partial _\alpha f \over 2f} \partial _{\beta \gamma \delta } \phi +{1 \over 8 } \partial _{\alpha \beta \gamma \delta } \phi \right) A_{\alpha \beta }A_{\gamma \delta } \\&\left. \left. -\,{\partial _{\alpha \beta \gamma } \phi \partial _{\delta \mu \nu } \phi \over 72} A_{\alpha \beta } A_{\gamma \delta } A_{\mu \nu }\right) \right) \end{aligned}$$

with $A^{\alpha \beta }=[\partial _{\alpha \beta } \phi ]^{-1}$, $dx \equiv \prod _{i=1}^n dx_i$ and $x^*$ the saddle-point (which minimizes $\phi (x)$). Here we have used Einstein summation convention. This expression can be obtained by developing $\phi (x)$ and f(x) in series around $x^*$. The quadratic part in $\phi (x)$ leads to a Gaussian integration over x which can be performed.