Andersen-Piterbarg Integration Limits for the Time Dependent Heston Model

The normalized characteristic function \phi_t(z) of a piecewise constant time dependent Heston model

\begin{array}{rcl} d \ln S_t&=& \left(r_t - q_t - \frac{\nu_t}{2}\right)dt + \sqrt\nu_t dW^{S}_t \nonumber \\ d\nu_t&=& \kappa_t\left(\theta_t-\nu_t \right ) dt + \sigma_t\sqrt\nu_t dW^{\nu}_t \nonumber \\ \rho_t dt &=& dW^{S}_tdW^{\nu}_t \end{array}

with n time intervals [t_0=0, t_1], ... ,[t_{n-1}, t_n] and constant parameters within these intervals

\kappa_t = \kappa_j \wedge \theta_t = \theta_j \wedge \sigma_t=\sigma_j \wedge \rho_t=\rho_j \forall j\in [1, n] \wedge t\in [t_{j-1}, t_j]

is given by the recurrence relation

\begin{array}{rcl}  k_j &=& \kappa_j - iz\rho_j\sigma_j \nonumber \\ d_j &=& \sqrt{k_j^2 +\sigma_j^2(z^2+iz)} \nonumber \\  g_j &=&\displaystyle \frac{k_j- d_j}{k_j + d_j} \nonumber \\ \tilde{g_j} &=& \displaystyle \frac{k_j- d_j - D_{j+1}\sigma_j^2}{k_j + d_j - D_{j+1}\sigma_j^2} \nonumber \\ D_j &=&  \displaystyle \frac{k_j + d_j}{\sigma_j^2}\frac{g_j-\tilde{g_j} e^{-d_j \tau_j}}{1-\tilde{g_j} e^{-d_j\tau_j}} \nonumber \\ C_j &=& \displaystyle \frac{\kappa_j\theta_j}{\sigma_j^2} \left( (k_j- d_j )\tau_j - 2\ln\left(\frac{1-\tilde{g_j}e^{-d_j\tau_j}}{1-\tilde{g_j}}\right)\right) + C_{j+1} \nonumber \\ \tau_j &=& t_j - t_{j-1} \nonumber \\ \phi_{t_n}(z) &=& \exp{\left(C_1(z)+D_1(z)\nu_0\right)}\end{array}

and the initial condition

C_{n+1} = D_{n+1} = 0

The should be noted that the complex logarithm in this formulation can be restricted to the principal branch without introducing any discontinuities [1]. It is important to know the asymptotic behaviour of C_1(z)+D_1(z)\nu_0 in order to calculate the truncation point for the integral over the characteristic function when using the Andersen-Piterbarg approach with control variate [2]. A Mathematica script gives

\begin{array}{rcl} \displaystyle \lim_{u\to\infty}\frac{D_1(u)}{u} &=& \displaystyle-\frac{i\rho_1 + \sqrt{1-\rho_1^2} }{\sigma_1} \nonumber \\ \displaystyle\lim_{u\to\infty}\frac{C_1(u)}{u} &=& \displaystyle -\sum_{j=1}^n \frac{\kappa_j\theta_j}{\sigma_j}\left(\sqrt{1-\rho_j^2} + i\rho_j\right)\tau_j\end{array}

The implementation of the Andersen-Piterbarg method for the piecewise constant time dependent Heston model is part of the pull request #251.

[1] Afshani, S. (2010) Complex logarithms and the piecewise constant extension
of the Heston model.

[2] Andersen, L. and Piterbarg, V. (2010) Interest Rate Modeling, Volume I: Foundations and Vanilla Models, (Atlantic Financial Press London).

New Semi-Analytic Heston Pricing Algorithms

The pricing engines for the Heston model in QuantLib have aged a bit. Meanwhile newer and better algorithms have been developed and discussed in the literature. For a comprehensive review please see [1][2]. Time to refurbish the existing engines.

The Heston model is defined by the following stochastic differential equation of the log spot

\begin{array}{rcl} d \ln S_t&=& \left(r_t - q_t - \frac{\nu_t}{2}\right)dt + \sqrt\nu_t dW^{S}_t \nonumber \\ d\nu_t&=& \kappa\left(\theta-\nu_t \right ) dt + \sigma\sqrt\nu_t dW^{\nu}_t \nonumber \\ \rho dt &=& dW^{S}_tdW^{\nu}_t \end{array}

The normalized characteristic function in the Gatheral formulation is given by

\phi_t(z) = \exp\left\{\frac{v_0}{\sigma^2}\frac{1-e^{-Dt}}{1-Ge^{-Dt}}\left(\kappa-i\rho\sigma z-D\right) + \frac{\kappa\theta}{\sigma^2}\left(\left(\kappa-i\rho\sigma z-D\right)t-2\ln\frac{1-Ge^{-Dt}}{1-G}\right) \right\}

\begin{array}{rcl} D&=&\sqrt{\left(\kappa - i\rho\sigma z\right)^2+\left(z^2+iz\right)\sigma^2} \nonumber \\ G&=&\displaystyle\frac{\kappa -i\rho\sigma z-D}{\kappa -i\rho\sigma z + D}\end{array}

Andersen and Piterbarg [3] introduced a Black-Scholes control variate to improve the numerical stability of Lewis’s formula (2001) for the price of a vanilla call option

\begin{array}{rcl} C(S_0, K, T)&=&BS\left(S_0, K, T, \sqrt{\nu_0}\right) \nonumber \\ &+& \frac{\sqrt{Se^{(r-q)t}K}e^{-rt}}{\pi}\displaystyle\int\limits_{0}^{\infty}{\Re\left( \frac{\phi^{BS}_T\left(u-\frac{i}{2}\right) - \phi_T\left(u-\frac{i}{2}\right)}{u^2+\frac{1}{4}} e^{i u \left(\ln\frac{S}{K}+(r-q)T\right) } \right)  \mathrm{d}u}\end{array}

with the Black-Scholes price BS\left(S_0, K, T, \sqrt(\nu_0)\right) of a vanilla call option with volatility \sqrt{\nu_0} and the characteristic function

\phi^{BS}_T(z)=e^{-\frac{1}{2}\nu_0 T (z^2 + iz)}

Different proposals for the volatility of the vanilla option have been brought up in order to achieve an optimal control variate:

  • \sigma_{BS}^2 = \nu_0
  • \sigma_{BS}^2 = \frac{1}{\kappa t}\left(1-e^{-\kappa t}\right)\left(\nu_0-\theta\right) + \theta
  • \sigma_{BS}^2 = c_2 with c_2 the second cumulant of \ln \frac{F}{K} [1].

The diagram below shows the resulting integrand for the different control variate volatilities, the Heston model parameters

r=7.5\%, q=5\%, S_0=100, \nu_0=0.08, \rho=-80\%,\sigma=0.5, \kappa=4, \theta=0.05

and for a vanilla call option with strike K=200 and maturity t=2.0.

control_variate

The best choice depends on the Heston- and option parameters but it seems that

\sigma_{BS}^2 = \frac{1}{\kappa t}\left(1-e^{-\kappa t}\right)\left(\nu_0-\theta\right) + \theta

is a good option for a large variety of parameters. A direct comparison of the integrand with and without control variate shows how effective the control variate is.

control_comparison

The central trick of Andersen-Piterbarg is now to truncate Lewis’s infinite integral at a finite u_{max} such that the remaining part is smaller than a given threshold based on the following inequation

\displaystyle\int_{u_{max}}^\infty \left\vert \frac{\phi_B(u-\frac{i}{2}) - \phi(u-\frac{i}{2})}{u^2+\frac{1}{4}} \right\vert du \le e^{-C_\infty u_{max}}\displaystyle\int_{u_{max}}^\infty \frac{1}{u^2} du

Please see the original paper or [2] for further details on how to get from here to an algorithm for u_{max}, especially for short dated options.

As Andersen and Piterbarg have pointed out, the simple trapezoidal rule works surprisingly well to carry out the resulting integral if a control variate is used. Alan Lewis’s test cases with high precision Heston reference prices should serve as a test bed here, in particular the call with strike 100. The Gauss-Laguerre method is very thankful for this particular test case but the point here is that the trapezoidal rule converges much faster than the higher order Simpson rule or the even more complex adaptive Gauss-Lobatto method.

convergence_speed

The diagram below compares the recently added COS engine with the Andersen-Piterbarg method using the trapezoid rule for this test-case. The Andersen-Piterbarg method is more accurate for a similar number of points.

convergence_comparison

The implementation is part of the pull request #251.

[1] M. Schmelzle, Option Pricing Formulae using Fourier Transform: Theory and Application.

[2] F. Le Floc’h, Fourier Integration and Stochastic Volatility Calibration.

[3] L. Andersen, and V. Piterbarg, 2010,  Interest Rate Modeling, Volume I: Foundations and Vanilla Models,  Atlantic Financial Press London.

The CLV Model with a Square Root Kernel Process

The kernel process of the Collocating Local Volatility (CLV) model [1] defines the forward skew dynamics of the model. Although the Ornstein-Uhlenbeck process has some nice analytical features it offers sometimes too little control over the forward skew dynamics. The square root process

d\nu_t=\kappa(\theta-\nu_t) dt + \sigma \sqrt{\nu_t}dW_t

is a promising kernel to get more control over the forward skew. First, exact sampling is pretty easy for the square root process. The probability density function of \nu_t given \nu_0 is

\nu_t = \frac{\sigma^2 (1- e^{-\kappa t})}{4\kappa} \chi_d^{'2}\left(\frac{4\kappa e ^{-\kappa t}}{\sigma^2(1-e^{-\kappa t})}\nu_0\right)

where \chi_d^{'2} denotes the noncentral chi-squared distribution with

d=\frac{4\theta\kappa}{\sigma^2}

degrees of freedom and the noncentrality parameter

\lambda = \frac{4\kappa e^{-\kappa t}}{\sigma^2(1-e^{-\kappa t})}\nu_0

The boost library provides an efficient and accurate implementation of the inverse of the cumulative noncentral chi-squared distribution function, which can be used for an exact Monte-Carlo sampling scheme.

The optimal collocation points are given by the Gaussian quadrature points, which are defined by the zeros of the corresponding orthogonal polynomials. The orthogonal polynomials are defined by a recurrence relation and the collocation points are given by the eigenvalues of a symmetric, tri-diagonal matrix with the diagonal \{\alpha_i\} and the minor diagonal \{\sqrt{\beta_i}\}. Again these vectors are defined by a recurrence relation [2]

\begin{array}{rcl} z_{k, i} &=& z_{k-1, i+1} + \alpha_k z_{k-1,i} - \beta_k z_{k-2,i} \\ \nonumber \alpha_{k+1} &=& \frac{z_{k-1,k}}{k-1,k-1}-\frac{z_{k,k+1}}{z_{k,k}} \\ \nonumber \beta_{k+1} &=& \frac{z_{k,k}}{z_{k-1,k-1}}\\ \nonumber z_{-1,i} &=& 0 \\ \nonumber z_{0,i} &=& \mu_i = \int_0^\infty x^i\chi_{d,\lambda}^{'2}(x) dx \\ \nonumber \alpha_1 &=& -\frac{\mu_1}{\mu_0} \\ \nonumber \beta_1 &=& \mu_0 = 1\end{array}

The first n moments \mu_i(d, \lambda), i=0,..,n of the noncentral chi-squared distribution can be calculated using Mathematica and exported as plain C code in order to be integrated into QuantLib.

m[n_] := CForm[ Expectation[X^n, X \[Distributed]
NoncentralChiSquareDistribution[d, lambda]] // Simplify]

Solving the recurrence relation is often ill-conditioned when using double precision. Therefore the resulting equations will be solved using the Boost.Multiprecision package with the cpp_dec_float back-end (header only and dependency free). The results have been tested accordingly to the proposal in [3]. For any reasonable number of collocation points a precision of up to 100 digits seems to be sufficient. On the other side the computation even with 100 digits is really fast. The eigenvalue calculation is carried out using standard double precision.

The implementation of the calibration routine follows the same way as for the Ornstein-Uhlenbeck kernel process. Again the calibration is pretty fast and accurate compared with other structured models even though it involves multiple precision arithmetic.

Example: Forward volatility skew

  • Market prices are given by a Heston model with

S_0=100, r=0.1, q=0.05, \nu_0=0.09, \kappa=1.0, \theta=0.06, \sigma=0.4, \rho=-0.75.

  • SquareRoot-CLV kernel process parameters are given by

\kappa=0.2, \theta=0.09, \sigma=0.1, x_0=0.09

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.

circlvforwardskew1

Same plot but with the following parameters of the square root kernel process

\kappa=0.1, \theta=0.09, \sigma=0.1, x_0=0.25

circlvforwardskew2

Source code is available here. The code is in an early state and needs some more testing/clean-up before a pull request can be made out of it. Next step will be to calibrate such a CIR-CLV model to the forward skew dynamics of an Heston Stochastic Local Volatility model.

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

[2] M. Morandi Cecchi and M. Redivo Zaglia, Computing the coefficients of a recurrence formula for numerical integration by moments and modified moments.

[3] Walter Gautschi, How and How not to check Gaussian Quadrature Formulae.

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.