The Collocating Local Volatility Model

In a recent paper [1] Lech Grzelak has introduced his Collocating Local Volatility Model (CLV). This model utilises the so called collocation method [2] to map the cumulative distribution function of an arbitrary kernel process onto the true cumulative distribution function (CDF) extracted from option market quotes.

Starting point for the collocating local volatility model is the market implied CDF of an underlying S_t at time t_i:

F_{S(t_i)}(x) = 1 + e^{r t_i}\frac{\partial V_{call}(t_0, t_i, K)}{\partial K}\mid_{K=x} = e^{r t_i}\frac{\partial V_{put}(t_0, t_i, K)}{\partial K}\mid_{K=x}

The prices can also be given by another calibrated pricing model, e.g. the Heston model or the SABR model. To increase numerical stability it is best to use OTM calls and puts.

The dynamics of the spot process S_t should be given by some stochastic process X_t and a deterministic mapping function g(t, x) such that

S_t=g(t, X_t)

The mapping function g(t, x) ensures that the terminal distribution of S_t given by the CLV model matches the market implied CDF. The model then reads

\begin{array}{rcl} S_t &=& g(t,X_t)  \nonumber \\ dX_t &=& \mu(X_t)dt + \sigma(X_t)dW_t\nonumber \end{array}

The choice of the stochastic process X_t does not influence the model prices of vanilla European options – they are given by the market implied terminal CDF – but influences the dynamics of the forward volatility skew generated by the model and therefore the prices of more exotic options. It is also preferable to choose an analytical trackable process X_t to reduce the computational efforts.

The collocation methods outlined in [2] defines an efficient algorithm to approximate the mapping function g(t, x) based on a set of collocation points x_{i,j}=x_j(T_i) for a given set of maturities T_i, i=1,...,m and j=1,...,n interpolation points per maturity T_i. Let F_{S_{T_i}}(s) be the market implied CDF for a given expiry T_i. Then we get

\begin{array}{rcl}  F_{X_{T_i}}\left(x_{i,j}\right) &=& F_{S_{T_i}}\left(g(T_i, x_{i,j})\right) = F_{S_{T_i}}  \left(s_{i,j}\right) \nonumber \\ \Rightarrow s_{i,j}&=&F^{-1}_{S_{T_i}}\left(F_{X_{T_i}}(x_{i,j})\right) \nonumber \end{array}

for the collocation points with s_{i,j}=g(T_i, x_{i,j}).

The optimal collocation points are given by the abscissas of the Gaussian quadrature for the terminal distribution of X_{T_i}. The simplest choice is a normally distribute kernel process X_t like the Ornstein-Uhlenbeck process

dX_t = \kappa(\theta-X_t)dt + \sigma dW_t.

The corresponding collocation points of the Normal-CLV model are then given by

\begin{array}{rcl} x_j(t) &=& \mathbb{E}\left[X_t\right] + \sqrt{\mathbb{V}ar\left[X_t\right]} x_j^{\mathcal{N}(0,1)}  \nonumber \\ &=& \theta + \left(X_0 - \theta)e^{-\kappa t}\right) + \sqrt{\frac{\sigma^2}{2\kappa}\left(1-e^{-2\kappa t}\right)} x_j^{\mathcal{N}(0,1)}, \ j=1,...,n\end{array}

in which the collocation points x_j^{\mathcal{N}(0,1)} of the standard normal distribution can be calculated by QuantLib’s Gauss-Hermite quadrature implementation

Array abscissas = std::sqrt(2)*GaussHermiteIntegration(n).x()

Lagrange polynomials [3] are an efficient interpolation scheme to interpolate the mapping function g(t, x) between the collocation points

g(t, X_t) = \sum_{j=1}^N s_j (t)\prod_{k=1, j\neq k}^N \frac{X(t)-x_j(t)}{x_k(t)-x_j(t)}

Strictly speaking Lagrange polynomials do not preserve monotonicity and one could also use monotonic interpolation schemes supported by QuantLib’s spline interpolation routines. As outlined in [2] this method can also be used to approximate the inverse CDF of an “expensive” distributions.

Calibration of the Normal-CLV model to market prices is therefore pretty fast and straight forward as it takes the calibration of g(t, x_t).

Monte-Carlo pricing means simulating the trackable process X_t and evaluate the  Lagrange polynomial if the value of the spot process S_t is needed. Pricing via partial differential equation involves the one dimensinal PDE

\frac{\partial V}{\partial t} + \mu(x)\frac{\partial V}{\partial x} + \frac{1}{2}\sigma^2(x)\frac{\partial^2 V}{\partial x^2}-rV = 0

with the terminal condition at maturity time T

V(T, x_T) = \text{Payoff}\left(S_T=g(T,x_T)\right)

For plain vanilla options the upper and lower boundary condition is

\frac{\partial^2 V}{\partial x^2} = 0 \ \ \forall x\in\left\{x_{min},x_{max}\right\}

Example 1: Pricing error for plain vanilla options

  • Market prices are given by the Black-Scholes-Merton model with

S_0=100, r=0.1, q=0.04, \sigma=0.25.

  • Normal-CLV process parameters are given by

\kappa=1.0, \theta=0.1,\sigma=0.5,x_0=0.1

Ten collocation points are used to define the mapping function g(t, x) and the time to maturity is one year. The diagram below shows the deviation of the implied volatility of the Normal-CLV price based on the PDE solution from the true value of 25%

clvpriceerror

Even ten collocation points are already enough to obtain a very small pricing error. The advice in [2] to stretch the collocation grid has turned out to be very efficient if the number of collocation points gets larger.

Example 2: Forward volatility skew

  • Market prices are given by the Heston model with

S_0=100, r=0.1, q=0.05, \nu_o=0.09, \kappa=1.0, \theta=0.06, \sigma=0.4, \rho=-0.75.

  • Normal-CLV process parameters are given by

\kappa=-0.075, \theta=0.05,\sigma=0.25,x_0=0.05

The diagram below shows the implied volatility of an forward starting European option with moneyness varying from 0.5 to 2 and maturity date six month after the reset date.

hestonforward

The shape of the forward volatility surface of the Normal-CLV model shares important similarities with the surfaces of the more complex Heston or Heston Stochastic Local Volatility (Heston-SLV) model with large mixing angles \eta. But by the very nature of the Normal-CLV model, the forward volatility does not depend on the values of \theta, \sigma or x_0, which limits the variety of different forward skew dynamics this model can create. CLV models with non-normal kernel processes will support a greater variety.

Example 3: Pricing of Double-no-Touch options

  • Market prices are given by the Heston model with

S_0=100, r=0.02, q=0.01, \nu_o=0.09, \kappa=1.0, \theta=0.06, \sigma=0.8\rho=-0.8.

  • Normal-CLV process parameters are given by different \kappa values and

\theta=100,\sigma=0.15,x_0=100.0

Unsurprisingly the prices of 1Y Double-no-Touch options exhibit similar patterns with the Normal-CLV model and the Heston-SLV model as shown below in the “Moustache” graph. But the computational efforts using the Normal-CLV model are much smaller than the efforts for calibrating and solving the Heston-SLV model.

moustache.png

The QuantLib implementation of the Normal-CLV model is available as a pull request #117, the Rcpp based package Rclv contains the R interface to the QuantLib implementation and the demo code for all three examples.

[1] A. Grzelak, 2016, The CLV Framework – A Fresh Look at Efficient Pricing with Smile

[2] L.A. Grzelak, J.A.S. Witteveen, M.Suárez-Taboada, C.W. Oosterlee,
The Stochastic Collocation Monte Carlo Sampler: Highly efficient sampling from “expensive” distributions

[3] J-P. Berrut, L.N. Trefethen,  Barycentric Lagrange interpolation, SIAM Review, 46(3):501–517, 2004.

Probability Distribution of the Heston Model, Part II

Starting point for a semi-analytical solution of the Fokker-Planck forward equation for the Heston model is the exact sampling algorithm of Broadie and Kaya [1] (for the notation please see [2])

\begin{array}{rcl} x_t &=& x_0 + m(t) + \sigma(t)Z \nonumber \\ \sigma^2(t) &=& (1-\rho^2)\int_0^t \nu_s ds \nonumber \\ m(t) &=& (r_t-q_t)t - \frac{1}{2}\int_0^t \nu_s ds + \rho\int_0^t \sqrt{\nu_s}dW_s^{(1)} \nonumber \end{array}

The probability distribution function can be described as

p(x_t, \nu_t, t) = p(\nu_t, t) p(x_t,t\mid \nu = \nu_t)

and p(\nu_t, t) is  given by a noncentral chi squared distribution. The distribution p(x_t, t \mid \nu = \nu_t) can be calculated using the exact simulation algorithm. In this algorithm the variable x_t is given as a function of two random variables \int_0^t \nu_s ds and Z.

The distribution of x_t can now be derived using the general transformation theorem for random variables: Let be a random variable with probability density function f. The transformed random variable Y=h(X) has the probability density function

p(y) = f(h^{-1}(y)) \left| \det \left( \frac{\partial h^{-1}_i(y)}{\partial y_j} \right)\right|

First step is now to rewrite the exact simulation method in terms of the two random variables

X_1 = \int_0^t \nu_s ds \ , X_2=Z .

The simulation scheme then becomes

\begin{array}{rcl} x_t &=& x_0+a(t) -\frac{1}{2}X_1 + \frac{\rho\kappa}{\sigma}X_1+\sigma(t) X_2 \nonumber \\\sigma^2(t)&=&(1-\rho^2)X_1 \nonumber \\a(t)&=&(r_t-q_t)t+\frac{\rho}{\sigma}\left( \nu_t-\nu_0-\kappa\theta t \right)\end{array}

or in terms of the transformed random variable

\begin{array}{rcl} Y_1 =h_1(X_1,X_2) &=& x_0+a(t)-\frac{1}{2}X_1 + \frac{\rho\kappa}{\sigma}X_1+\sqrt{\left(1-\rho^2\right)X_1}X_2 \nonumber \\ Y_2=h_2(X_1,X_2)&=& X_1 \nonumber\end{array}.

Let \phi(x_1) be the density function of X_1

\phi(x_1)=\frac{2}{\pi}\int_0^\infty \cos ux_1 \mathrm{Re}(\Phi(u))\mathrm{d}u

and X_2 follows by definition a normal distribution. The joint probability density function of (X_1, X_2) is then

f\left( \begin{matrix} x_1 \\ x_2 \end{matrix}\right)=\phi(x_1) \frac{1}{\sqrt{2\pi}}e^{-\frac{x_2^2}{2}}

with

\begin{array}{rcl}f\left(h^{-1}(y)\right)&=&f\left(\begin{matrix}y_2 \\ \frac{1}{\sqrt{\left(1-\rho^2\right)y_2}}\left(y_1-x_0-a(t)+\frac{1}{2}y_2-\frac{\rho\kappa}{\sigma}y_2\right)\end{matrix}\right)  \nonumber \\  \left|\det \left( \frac{\partial h^{-1}_i(y)}{\partial y_j} \right)\right| &=& \frac{1}{\sqrt{\left(1-\rho^2\right)y_2}}  \end{array}.

This yields to the semi-analytical formula for the solution of the Fokker-Planck equation because by definition p(x_t,t \mid \nu = \nu_t) is the distribution density function of Y_1, which is given by

p(x_t,t \mid \nu = \nu_t) = \int_0^\infty \mathrm{d}y_2 p(y_1, y_2)\mid_{y_1=x_t} = \int_0^\infty \mathrm{d}y_2 \left[f\left(h^{-1}(y)\right)\left|\det \left( \frac{\partial h^{-1}_i(y)}{\partial y_j} \right)\right| \right]_{y_1=x_t}

The integration over y_2 can be carried out using e.g. the Simpson integral rule together with the Cornish-Fisher expansion, which gives an upper bound for the truncation of the upper limit of the integration.

The contour plots below show the probability density function of the Heston model for some example parametrisations.

plot
\begin{array}{|c|c|c|c|c|c|c|c|c|} \hline {\rm Parameters} & x_0 & \nu_0 & r & q & \kappa & \theta & \sigma & \rho \\ \hline \hline  a & 4.6052 & 0.4 & 5\% & 2.5\% & 1.0 & 0.4 & 0.8 & -75\%  \\ \hline  b & 4.6052 & 0.4 & 5\% & 2.5\% & 1.0 & 0.4 & 0.8 & \ \ 75\%  \\ \hline  c & 4.6052 & 0.4 & 5\% & 2.5\% & 1.0 & 0.4 & 0.4 & -75\%  \\ \hline  d & 4.6052 & 0.4 & 5\% & 2.5\% & 1.0 & 0.4 & 0.4 & \ \ \ \ 0\%  \\ \hline  \end{array}

The example code is available here and depends on the upcoming QuantLib version 1.4.

[1] M. Broadie, Ö. Kaya, Exact Simulation of Stochastic Volatility and other Affine Jump Diffusion Processes

[2] K. Spanderen, Probability Distribution of the Heston Model, Part I

[3] R.U. Seydel, Tools for Computational Finance, pp 86