Accurate Pricing of Swaptions via Lower Bound

Abstract

We propose a new lower bound for pricing European-style swaptions for a wide class of interest rate models. This method is applicable whenever the joint characteristic function of the state variables is either known in closed form or can be obtained numerically via some efficient procedure. Our lower bound involves the computation of a one dimensional Fourier transform independently of the underlying swap length. Finally the bound can be used as a control variate to reduce the confidence interval in the Monte Carlo simulation. We test our bound on different affine models, also allowing for jumps. The lower bound is found to be accurate and computationally efficient.

A Appendix

1.1 A.1 Proof Proposition 1

We consider the lower bound to the swaption price as in formula (9.4) for affine models:

$$\displaystyle\begin{array}{rcl} & & LB(k;t,T,\{T_{h}\}_{h=1}^{n},R) = P(t,T) \cdot \mathbb{E}_{ t}^{T}\left [\left (\sum _{ h=1}^{n}w_{ h}\:e^{\mathbf{b}_{h}^{\top }\mathbf{X}(T)+a_{ h}} - 1\right )I(\mathcal{G})\right ]^{+} {}\\ \end{array}$$

where the set $\mathcal{G} =\{\omega \in \Omega: g(\mathbf{X}(T)) \geq k\} =\{\omega \in \Omega:\boldsymbol{\beta } ^{\top }\mathbf{X}(T)+\alpha \geq k\}$.

According to Carr and Madan (2000), we introduce the dampening factor e ^δk, then we apply the Fourier Transform with respect to the variable k to the T-forward expected value and we obtain:

$$\displaystyle\begin{array}{rcl} \psi _{\delta }(\gamma )& =& \int _{-\infty }^{+\infty }e^{i\gamma k+\delta k}\mathbb{E}_{ t}^{T}\left [\left (\sum _{ h=1}^{n}w_{ h}\:e^{\mathbf{b}_{h}^{\top }\mathbf{X}(T)+a_{ h}} - 1\right )I(g(\mathbf{X}(T)) \geq k)\right ]dk {}\\ & =& \mathbb{E}_{t}^{T}\left [\left (\sum _{ h=1}^{n}w_{ h}\:e^{\mathbf{b}_{h}^{\top }\mathbf{X}(T)+a_{ h}} - 1\right )\int _{-\infty }^{+\infty }e^{i\gamma k+\delta k}I(\boldsymbol{\beta }^{\top }\mathbf{X}(T)+\alpha \geq k)dk\right ] {}\\ & =& \mathbb{E}_{t}^{T}\left [\left (\sum _{ h=1}^{n}w_{ h}\:e^{\mathbf{b}_{h}^{\top }\mathbf{X}(T)+a_{ h}} - 1\right )\int _{-\infty }^{\boldsymbol{\beta }^{\top }\mathbf{X}(T)+\alpha }e^{i\gamma k+\delta k}dk\right ] {}\\ \end{array}$$

Since the dampening factor δ is positive, then the module of the integrand function decays exponentially for k → −∞ and the Fourier Transform is well defined, so:

$$\displaystyle\begin{array}{rcl} \psi _{\delta }(\gamma )& =& \mathbb{E}_{t}^{T}\left [\left (\sum _{ h=1}^{n}w_{ h}e^{\mathbf{b}_{h}^{\top }\mathbf{X}(T)+a_{ h}} - 1\right )e^{(i\gamma +\delta )(\boldsymbol{\beta }^{\top }\mathbf{X}(T)) }\right ]\frac{e^{(i\gamma +\delta )\alpha }} {i\gamma +\delta } {}\\ \end{array}$$

Using the characteristic function of X, calculated under the T-forward measure: $\Phi (\boldsymbol{\lambda }) = \mathbb{E}_{t}^{T}\left [e^{i\boldsymbol{\lambda }^{\top }\mathbf{X} }\right ]$, the function ψ _δ (γ) can be written as:

$$\displaystyle\begin{array}{rcl} \psi _{\delta }(\gamma )& =& \left (\sum _{h=1}^{n}w_{ h}e^{a_{h} }\Phi \left (-i\mathbf{b}_{h} + (\gamma -i\delta )\boldsymbol{\beta }\right ) - \Phi \left ((\gamma -i\delta )\boldsymbol{\beta }\right )\right )\frac{e^{(i\gamma +\delta )\alpha }} {i\gamma +\delta } {}\\ \end{array}$$

Finally the lower bound is the maximum with respect to k of the inverse transform of ψ _δ (γ):

$$\displaystyle\begin{array}{rcl} & & LB(k;t,T,\{T_{h}\}_{h=1}^{n},R) = P(t,T)\:\frac{e^{-\delta k}} {\pi } \int _{0}^{+\infty }e^{-i\gamma k}\psi _{ \delta }(\gamma )d\gamma {}\\ \end{array}$$

1.2 A.2 Proof of the Analytical Lower Bound for Gaussian Affine Models

$\mathbf{X} \sim N(\boldsymbol{\mu },V )$ in T-forward measure and $g(\mathbf{X}(T)) =\boldsymbol{\beta } ^{\top }\mathbf{X}+\alpha \sim N(\boldsymbol{\beta }^{\top }\boldsymbol{\mu }+\alpha,\:\boldsymbol{\beta }^{\top }V \boldsymbol{\beta })$

Then the approximate exercise region $\mathcal{G}$ becomes:

$$\displaystyle\begin{array}{rcl} \mathcal{G} =\{\omega \in \Omega: g(\mathbf{X}(T))> k\} =\{\omega \in \Omega: z> d\}& & {}\\ \end{array}$$

where z is a standard normal random variable and

$$\displaystyle{d = \frac{k -\boldsymbol{\beta }^{\top }\boldsymbol{\mu }-\alpha } {\sqrt{\boldsymbol{\beta }^{\top } V\boldsymbol{\beta }}}.}$$

The lower bound expression can be written using the law of iterative expectation:

$$\displaystyle\begin{array}{rcl} & & LB(k; t,T,\{T_{h}\}_{h=1}^{n},R) = P(t,T)\:\mathbb{E}_{ t}^{T}\left [\mathbb{E}_{ t}^{T}\left [\left (\sum _{ h=1}^{n}w_{ h}e^{\mathbf{b}_{h} ^{\top } \mathbf{X}(T)+a_{h} } -1\right )\vert z\right ]I(z> d)\right ] {}\\ \end{array}$$

Conditionally to the random variable z, the variable X is distributed as a multivariate normal with mean and variance:

$$\displaystyle\begin{array}{rcl} & & \mathbb{E}_{t}^{T}[\mathbf{X}\vert z] =\boldsymbol{\mu } +z \cdot \mathbf{v} {}\\ & & V ar(\mathbf{X}\vert z) = V -\mathbf{v}\mathbf{v}^{\top } {}\\ & &\mathbf{v} = \frac{V \boldsymbol{\beta }} {\sqrt{\boldsymbol{\beta }^{\top } V\boldsymbol{\beta }}} {}\\ \end{array}$$

We can now compute the inner expectation, using the normal distribution property:

$$\displaystyle\begin{array}{rcl} LB(k; t,T,\{T_{h}\}_{h=1}^{n},R)& =& P(t,T)\left (\sum _{ h=1}^{n}w_{ h}\:\mathbb{E}_{t}^{T}\left [e^{\mathbf{b}_{h}^{\top }\boldsymbol{\mu }+z\mathbf{b}_{h}^{\top }\mathbf{v}+\frac{1} {2} V _{h} }I(z> d)\right ] -\mathbb{E}_{t}^{T}\left [I(z> d)\right ]\right ){}\\ \end{array}$$

where V _h = b _h ^⊤(V − vv ^⊤)b _h .

Maximizing with respect to k, involved in the definition of d, we found the lower bound:

$$\displaystyle\begin{array}{rcl} LB(t,T,\{T_{h}\}_{h=1}^{n},R) =\max _{ k\in \mathbb{R}}\left (\sum _{h=1}^{n}w_{ h}e^{a_{h}+\mathbf{b}_{h}^{\top }\boldsymbol{\mu }+\frac{1} {2} V _{h}+\frac{1} {2} d_{h}^{2} }N(d_{h} - d) - N(-d)\right )& & {}\\ \end{array}$$

where d _h = b _h ^⊤ v and N(x) is the cumulative distribution function of standard normal variable.

To make faster the optimization of the lower bound with respect to the parameter k, we compute the first order approximation of maximum point as a starting point. Equation for stationary points:

$$\displaystyle\begin{array}{rcl} \frac{\partial LB(t,k)} {\partial k} & =& \frac{P(t,T)} {\sqrt{\boldsymbol{\beta }^{\top } V\boldsymbol{\beta }}} \left (\sum _{h=1}^{n}w_{ h}e^{a_{h}+\mathbf{b}_{h}^{\top }\boldsymbol{\mu }+\frac{1} {2} V _{h}+d_{h}^{2}/2 }N^{{\prime}}(-(d - d_{ h})) - N^{{\prime}}(-d)\right ) = 0{}\\ \end{array}$$

Taylor first order expansion:

$$\displaystyle{ N^{{\prime}}(-(d - d_{ h})) = N^{{\prime}}(-d) + d_{ h}N^{{\prime\prime}}(-d) + o(d_{ h}) = N^{{\prime}}(-d)(1 + dd_{ h}) + o(d_{h}) }$$

We substitute the first order expansion in the derivative expression and we obtain:

$$\displaystyle{ \sum _{h=1}^{n}w_{ h}e^{a_{h}+\mathbf{b}_{h}^{\top }\boldsymbol{\mu }+\frac{1} {2} V _{h}+d_{h}^{2}/2 }(1 + dd_{h}) - 1 = 0 }$$

So the first order guess of the maximum point is:

$$\displaystyle\begin{array}{rcl} d_{guess}& =& \frac{1 -\sum _{h=1}^{n}w_{h}e^{a_{h}+\mathbf{b}_{h}^{\top }\boldsymbol{\mu }+\frac{1} {2} V _{h}+d_{h}^{2}/2 }} {\sum _{h=1}^{n}w_{h}e^{a_{h}+\mathbf{b}_{h}^{\top }\boldsymbol{\mu }+\frac{1} {2} V _{h}+d_{h}^{2}/2 }d_{h}} {}\\ k_{guess}& =& \sqrt{\boldsymbol{\beta }^{\top } V\boldsymbol{\beta }}\:d_{guess} +\boldsymbol{\beta } ^{\top }\boldsymbol{\mu }+\alpha {}\\ \end{array}$$