Heston Stochastic Local Volatility Model
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1 Heston Stochastic Local Volatility Model Klaus Spanderen 1 R/Finance 2016 University of Illinois, Chicago May 20-21, Joint work with Johannes Göttker-Schnetmann Klaus Spanderen Heston Stochastic Local Volatility Model / 19
2 Motivation Combine two of the most popular option pricing models, the Local Volatility model with x = ln S t ( ) dx t = r t q t σ2 LV (x t, t) dt + σ LV (x t, t)dw t 2 and the Heston Stochastic Volatility model ( dx t = r t q t ν t 2 ) dt + ν t dw x t dν t = κ (θ ν t ) dt + σ ν t dw ν t ρdt = dwt ν dwt x to control the forward volatility dynamics and the calibration error. Klaus Spanderen Heston Stochastic Local Volatility Model / 19
3 Model Definition Add leverage function L(S t, t) and mixing factor η to the Heston model: ( ) dx t = r t q t L2 (x t, t) ν t dt + L(x t, t) ν t dwt x 2 dν t = κ (θ ν t ) dt + ησ ν t dwt ν ρdt = dwt ν dwt x Leverage L(x t, t) is given by probability density p(x t, ν, t) and L(x t, t) = σ LV (x t, t) E[νt x = x t ] = σ R p(x LV (x t, t) + t, ν, t)dν R νp(x + t, ν, t)dν Mixing factor η tunes between stochastic and local volatility Klaus Spanderen Heston Stochastic Local Volatility Model / 19
4 Package RHestonSLV: Calibration and Pricing Calibration: Calculate Heston parameters {κ, θ, σ, ρ, v t=0 } and σ LV (x t, t) Compute p(x t, ν, t) either by Monte-Carlo or PDE to get to the leverage function L t (x t, t) Infer the mixing factor η from prices of exotic options Package HestonSLV Monte-Carlo and PDE calibration Pricing of vanillas and exotic options like double-no-touch barriers Implementation is based on QuantLib, Klaus Spanderen Heston Stochastic Local Volatility Model / 19
5 Cheat Sheet: Link between SDE and PDE Starting point is a linear, multidimensional SDE of the form: dx t = µ(x t, t)dt + σ(x t, t)dw t Feynman-Kac: the price of a derivative u(x t, t) with boundary condition u(x T, T ) at maturity T is given by: t u + n µ i xk u k=1 n k,l=1 ( σσ T ) kl x k xl u ru = 0 Fokker-Planck: the time evolution of the probability density function p(x t, t) with the initial condition p(x, t = 0) = δ(x x 0 ) is given by: t p = n xk [µ i p] k=1 n k,l=1 xk xl [( σσ T ) kl p ] Klaus Spanderen Heston Stochastic Local Volatility Model / 19
6 Fokker-Planck Forward Equation The corresponding Fokker-Planck equation for the probability density p : R R 0 R 0 R 0, (x, ν, t) p(x, ν, t) is: t p = 1 [ ] 2 2 x L 2 νp η2 σ 2 ν 2 [νp] + ησρ x ν [Lνp] x [(r q 1 ) ] 2 L2 ν p ν [κ (θ ν) p] Numerical solution of the PDE is cumbersome due to difficult boundary conditions and the δ-distribution as the initial condition. PDE can be efficiently solved by using operator splitting schemes, preferable the modified Craig-Sneyd scheme. Klaus Spanderen Heston Stochastic Local Volatility Model / 19
7 Calibration: Fokker-Planck Forward Equation Zero-Flux boundary condition for ν = {ν min, ν max } 2κθ 1 Reformulate PDE in terms of q = ν σ 2 constraint is violated Prediction-Correction step for L(x t+ t, t + t) Non-uniform grids are a key factor for success or z = ln ν if the Feller Includes adaptive time step size and grid boundaries to allow for rapid changes of the shape of p(x t, ν, t) for small t Semi-analytical approximations of initial δ-distribution for small t Corresponding Feynman-Kac backward PDE is much easier to solve. Klaus Spanderen Heston Stochastic Local Volatility Model / 19
8 Calibration: Monte-Carlo Simulation The quadratic exponential discretization can be adapted to simulate the Heston SLV model efficiently. Reminder: L(x t, t) = σ LV (x t,t) E[νt x=x t ] 1 Simulate the next time step for all calibration paths 2 Define set of n bins b i = {xt i, x t i + x t i } and assign paths to bins 3 Calculate expectation value e i = E[ν t x b i ] over all paths in b i 4 Define L(x t b i, t) = σ LV (x t,t) e i 5 t t + t and goto 1 Advice: Use Quasi-Monte-Carlo simulations with Brownian bridges. Klaus Spanderen Heston Stochastic Local Volatility Model / 19
9 Calibration: Test Bed Motivation: Set-up extreme test case for the SLV calibration Local Volatility: σ LV (x, t) 30% Heston parameters: S 0 = 100, ν 0 = 0.09, κ = 1.0, θ = 0.06, σ = 0.4, ρ = 75% Feller condition is violated with 2κθ σ 2 = 0.75 Implied volatility surface of the Heston model and the Local Volatility model differ significantly. Klaus Spanderen Heston Stochastic Local Volatility Model / 19
10 Calibration: Fokker-Planck PDE vs Monte-Carlo Fokker-Planck Forward Equation, η=1.00 Monte-Carlo Simulation, η= Leverage Function L(x,t) Leverage Function L(x,t) Time ln(s) Time ln(s) Klaus Spanderen Heston Stochastic Local Volatility Model / 19
11 Calibration Sanity Check: Round-Trip Error for Vanillas Round-Trip Error for 1Y Maturity Implied Volatility (in %) Monte-Carlo Fokker-Planck Strike Klaus Spanderen Heston Stochastic Local Volatility Model / 19
12 Case Study: Delta of Vanilla Option Vanilla Put Option: 3y maturity, S 0 =100, strike=100 Delta of ATM Put Option Delta Heston Black-Scholes Heston Minimum-Variance Local Vol SLV η Klaus Spanderen Heston Stochastic Local Volatility Model / 19
13 Choose the Forward Volatility Skew Dynamics Interpolate between the Local and the Heston skew dynamics by tuning η between 0 and 1. Forward Starting Option: max(0, S 2y - α*s 1y ) Implied Forward Volatility (in %) η=1.00 η=0.50 η=0.25 η= Strike α Klaus Spanderen Heston Stochastic Local Volatility Model / 19
14 Case Study: Barrier Option Prices DOP Barrier Option: 3y maturity, S 0 =100, strike=100 Barrier Option Pricing Local Vol vs SLV NPV local - NPV SLV η = 1.0 η = 0.5 η = 0.2 η = Barrier Klaus Spanderen Heston Stochastic Local Volatility Model / 19
15 Case Study: Delta of Barrier Options DOP Barrier Option: 3y maturity, S 0 =100, strike=100 Barrier Option Δ local vs Δ SLV Δ local - Δ SLV η = 1.0 η = 0.5 η = 0.2 η = Barrier Klaus Spanderen Heston Stochastic Local Volatility Model / 19
16 Case Study: Double-No-Touch Options Knock-Out Double-No-Touch Option: 1y maturity, S 0 =100 Double No Touch Option Stochastic Local Volatility vs. Black-Scholes Prices NPV SLV - NPV BS η=1.00 η=0.75 η=0.50 η=0.25 η= NPV BS Klaus Spanderen Heston Stochastic Local Volatility Model / 19
17 Summary: Heston Stochastic Local Volatility RHestonSLV: A package for the Heston Stochastic Local Volatility Model Monte-Carlo Calibration Calibration via Fokker-Planck Forward Equation Supports pricing of vanillas and exotic options Implementation is based on QuantLib 1.8 and Rcpp Package source code including all examples shown is on github Klaus Spanderen Heston Stochastic Local Volatility Model / 19
18 Literature J. Göttker-Schnetmann and K. Spanderen. Calibration of the Heston Stochastic Local Volatility Model. K.J. in t Hout and S. Foulon. ADI Finite Difference Schemes for Option Pricing in the Heston Model with Correlation. International Journal of Numerical Analysis and Modeling, 7(2): , Y. Tian, Z. Zhu, G. Lee, F. Klebaner and K. Hamza. Calibrating and Pricing with a Stochastic-Local Volatility Model. A. Stoep, L. Grzelak and C. Oosterlee,. The Heston Stochastic-Local Volatility Model: Efficient Monte Carlo Simulation. Klaus Spanderen Heston Stochastic Local Volatility Model / 19
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