Empirical Approach to the Heston Model Parameters on the Exchange Rate USD / COP ICASQF 2016, Cartagena - Colombia C. Alexander Grajales 1 Santiago Medina 2 1 University of Antioquia, Colombia 2 Nacional University, Colombia June 17, 2016 Grajales - Medina (UdeA - UN) Heston - Empir.Calib. June 17, 2016 1 / 27
Outline 1 Introduction About this work Basic considerations of Heston Model 2 A very brief fundamentals of the Heston Model Heston under risk-neutral measure and PDE Heston call option formula Parameters calibration - some references 3 Proposal: Heston calibration to USD/COP Objective Function Stochastic optimisation problem inicial value of x : x 0 Results 4 Conclusions Grajales - Medina (UdeA - UN) Heston - Empir.Calib. June 17, 2016 2 / 27
Introduction Proposal: empirical calibration of Heston stochastic volatility model for USD / COP under physical measure of risk Parameter estimation: done by developing an algorithm that performs simulated trajectories for USDCOP Heston & matching pdf of simulated paths with pdf coming from the real exchange rate Calibration: two-sample KS test & Nelder Mead simplex direct search At the end: the results show that although achieving multiple optima parameter values -depending on an initial vector parameter - is posible, one of these could be chosen according to financial market information Grajales - Medina (UdeA - UN) Heston - Empir.Calib. June 17, 2016 3 / 27
Basic considerations of Heston Model Limitations of the Black-Scholes-Merton (BMS - 1973) in option valuation have been pointed out with rigour, e.g. volatility surfaces do not reflect the reality in markets. Facts given a financial asset, the volatility process is nonnegative and shows mean reversion the model provides leptokurtic and fat tails returns distribution explains the smile and skew effect appearing in the real implied volatility it is analytically tractable in aspects of vanilla option valuation BSM(1973), Heston(1993), Gathered(2006), Fabrice(2013), Fatone(2014), Hurn(2015) Grajales - Medina (UdeA - UN) Heston - Empir.Calib. June 17, 2016 4 / 27
Fundamentas of the Heston Model Seminal models in S.V: Hull & White (1987), Scott (1987), Stein & Stein (1991), and Heston (1993). Heston model: driven by the Itô differential equations: µ, θ, κ and σ are constants ds t = µ S t dt + v t S t dw (1) t (1) dv t = κ (θ v t ) dt + σ v t dw (2) t (2) S t is conditioned to S 0, v 0, and {v s, 0 s t} Corr(dW (1) t, dw (2) t ) = ρdt {v t } > 0 if 2κθ σ 2 Grajales - Medina (UdeA - UN) Heston - Empir.Calib. June 17, 2016 5 / 27
Heston under risk-neutral measure and PDE Risk neutral measure ds t = r S t dt + v t S t d W (1) t v t = κ (θ v t ) dt + σ v t d W (2) t where κ = κ + λ, θ = κθ/(κ + λ), and volatility risk premium λ(s t, v t, t) = λv t, λ constant Derivative price U is driven by the PDE 1 2 v ts 2 t 2 U S 2 t + ρσv t S t 2 U S t v t + 1 2 σ2 v t 2 U v 2 t + rs t U S t + (κ [θ v t ] λ(s t, v t, t)) U v t ru + U t = 0 Grajales - Medina (UdeA - UN) Heston - Empir.Calib. June 17, 2016 6 / 27
Heston call option formula where P j (x, v, T ; ln(k)) = 1 2 + 1 π C(S t, v t, t) = S t P 1 Ke rτ P 2 0 [ ] e iφ ln(k) f j (x, v, t; φ) Re dφ iφ f j (x, v, T : φ) = e C(τ; φ)+d(τ; φ)v+iφx, C(τ; φ) = rφiτ + a [ 1 ge dτ {(b σ 2 j ρσφi + d)τ 2 ln 1 g D(τ; φ) = b [ ] j ρσφi + d 1 e dτ, σ 2 1 ge dτ g = b j ρσφi + d b j ρσφi d, d = (ρσφi b j ) 2 σ 2 (2u j φi φ 2 ), ]}, u 1 = 1/2, u 2 = 1/2, a = κθ, b 1 = κ + λ ρσ, b 2 = κ + λ with λ = λ(s t, v t, t) = k v for a constant k. Grajales - Medina (UdeA - UN) Heston - Empir.Calib. June 17, 2016 7 / 27
Notes about the Heston call integral The parameters were taken as S = 100, r = q = λ = 0, κ = 10, θ = v 0 = 0.05, σ = 0.5, ρ = 0.5, T (0, 0.25] and K [50, 200] Figure 1: Case j = 2, Heston Integral for ATM call Grajales - Medina (UdeA - UN) Heston - Empir.Calib. June 17, 2016 8 / 27
Notes about the Heston call integral The parameters were taken as S = 100, r = q = λ = 0, κ = 10, θ = v 0 = 0.05, σ = 0.5, ρ = 0.5, T (0, 0.25] and K [50, 200] Figure 2: Case j = 2, Heston Integral for ITM call Grajales - Medina (UdeA - UN) Heston - Empir.Calib. June 17, 2016 9 / 27
Heston call surface on K T plane The parameters were taken as S = 100, r = q = λ = 0, κ = 10, θ = v 0 = 0.05, σ = 0.5, ρ = 0.5, T (0, 0.25] and K [50, 200] Figure 3: Case j = 2, Heston call C(K, T ) Grajales - Medina (UdeA - UN) Heston - Empir.Calib. June 17, 2016 10 / 27
Risk C-price sensitivities ρ: controls the skewness of the density of ln S T positive correlation implies an increase in variance for bullish markets and the contrary effect happens for negative correlation. Table 1: Call price relation in BSM and Heston model for different ρ ρ > 0 ρ < 0 OTM European Call Heston > BSM Heston < BSM ITM European Call Heston < BSM Heston > BSM Grajales - Medina (UdeA - UN) Heston - Empir.Calib. June 17, 2016 11 / 27
Risk C-price sensitivities σ: affects the kurtosis of the distribution of the ln S T and the curvature of the smile produced by the implied volatility of Heston model. kurtosis and curvature increase as sigma increases fat tails in the distribution of ln S T. So in ITM and OTM options: we have C Heston > C BSM in ATM options: we have C Heston < C BSM and such relations are kept in a deeply way as sigma increases. Grajales - Medina (UdeA - UN) Heston - Empir.Calib. June 17, 2016 12 / 27
Parameters calibration - some references Methods of Moments like SMM (Simulated Method of Moments), GMM (Generalized Method of Moments) and EMM (Efficient Method of Moments) in Duffie(1993), Andersen(1996) and (Gallant-1996) Other methods use Kalman filters in Harvey(1994), Harvey(1996) and Kim(1996) There is a method of moments conditioned to a diffusion equation in Bollerslev(2002) Bayesian methods like MCMC in Jacquier(1994), Eraker(2001) and Kim(1999) MLE (Maximum Likelihood Approach) in Aït-Sahalia(2007) Methods that use empirical characteristics functions in Singleton(2001) Another very wide range variety of methods use empirical calibration through implied volatility surfaces in the market or from the value of a set of tradable option prices, all of that even with heuristic, deterministic optimisation or stochastic optimisation innovations, e.g. in Gathered(2006), Fatone(2014), Hurn(2015). Grajales - Medina (UdeA - UN) Heston - Empir.Calib. June 17, 2016 13 / 27
Objective Function The Heston model - under physical measure - requires to calibrate the parameters µ, θ, σ, κ, ρ and v 0. This study will primarily focus on: the first five parameters values and consequently it is assumed that v 0 = θ. So if x = (µ, θ, σ, κ, ρ) R 5, empirical calibration to USD/COP consists of constructing an O.F.: Ψ(x) = (Ξ Λ Γ) (x) = mi=1 h i m, Ψ(x) aimed to obtain the probability of rejecting the null hypothesis H 0 : the two samples, real returns & each one of the simulated returns for the exchange rate are drawn from the same continuous distribution. Grajales - Medina (UdeA - UN) Heston - Empir.Calib. June 17, 2016 14 / 27
Objective Function - Implementation Γ : R 5 R m n, builds a matrix m n with m USDCOP returns simulated trajectories under Heston model, and for n successive days Λ : R m n R m performs a two-sample KS test: USDCOP returns & each of the m simulated trajectories The function Λ(Y ) returns a KS test p value for each trajectory, and a decision is taken: accept H 0 and do h i = 0 if p i 0.05, or reject H 0 and do h i = 1, with a significance level of 0.05. Finally, Ξ : R m [0, 1] returns the probability of rejecting the null hypothesis H 0 Grajales - Medina (UdeA - UN) Heston - Empir.Calib. June 17, 2016 15 / 27
Objective Function - Notes Under this line of reasoning, some natural questions arise: Is there an optimal value x opt that minimises the function Ψ(x)? If so, how to find it? If the value x opt exists, it could be a local minimum of the Ψ(x), near to zero but different from it, given the test level of significance of 0.05, and caused by the stochastic nature of the experiment. An auxiliary non-linear-optimisation algorithm can be used to find x opt, and such algorithm should use in its routine only values of Ψ(x) function without the use of any gradient, since there are no guaranteed conditions about differentiability of the function. Grajales - Medina (UdeA - UN) Heston - Empir.Calib. June 17, 2016 16 / 27
Objective Function, two-sample KS test and Nelder Mead simplex direct search This is therefore a stochastic optimisation problem that is solved by finding a parametric value x opt such that x opt = argmin x Ψ(x). (3) The existence of x opt will be determined if the proposed direct search algorithm, Nelder-Mead simplex direct search Nelder(1965), Lagarias(1998), Lewis(2000) can find it! This algorithm is implemented in Matlab Optimisation Toolbox under a function called fminsearch. Grajales - Medina (UdeA - UN) Heston - Empir.Calib. June 17, 2016 17 / 27
inicial value of x : x 0 First consideration: (µ, θ, σ, κ, ρ) [243 min φ(t), 243 max φ(t)] (0, max ( φ 2 (t) ) 243] (0, 1] (0, ) [ 1, 1] Second consideration: (µ, θ, σ, κ, ρ) = (243 φ(t), 243 var(φ(t)), 0.5, 10, 0.1) (ρ < 0: leverage effect, Aït-Sahalia et.al (2013)) Third consideration: (µ, θ, σ, κ, ρ) = (243 φ(t), 243 var(φ(t)), σ, κ, ρ), where: σ = 0.1 : 0.1 : 1.0, κ = 1 : 1 : 15, ρ = 0.20 : 0.1 : 0.20 Grajales - Medina (UdeA - UN) Heston - Empir.Calib. June 17, 2016 18 / 27
Results Figure 4: Ψ(x) 0.9 0.8 Average of Ψ(x) ρ=-.1, σ=.3,κ=1:1:15 ρ=.2, σ=.6,κ=1:1:15 ρ=.2, σ=.7,κ=1:1:15 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 5 10 15 κ Grajales - Medina (UdeA - UN) Heston - Empir.Calib. June 17, 2016 19 / 27
Results Figure 5: Ψ(x) 1 0.9 0.8 0.7 Average of Ψ(x) ρ=-.1,κ=15, σ=.1:.1:1 ρ=-.2,κ=14, σ=.1:.1:1 ρ=.2,κ=15, σ=.1:.1:1 ρ=-.2,κ=2, σ=.1:.1:1 ρ=-.1,κ=2, σ=.1:.1:1 ρ=0,κ=2, σ=.1:.1:1 ρ=.1,κ=1, σ=.1:.1:1 0.6 0.5 0.4 0.3 0.2 0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 σ Grajales - Medina (UdeA - UN) Heston - Empir.Calib. June 17, 2016 20 / 27
Results Figure 6: Ψ(x) 0.8 0.7 Average of Ψ(x) κ=1, σ=1, ρ=-.2:.1:.2 κ=14, σ=.7, ρ=-.2:.1:.2 κ=15, σ=.6, ρ=-.2:.1:.2 0.6 0.5 0.4 0.3 0.2 0.1 0-0.2-0.15-0.1-0.05 0 0.05 0.1 0.15 0.2 ρ Grajales - Medina (UdeA - UN) Heston - Empir.Calib. June 17, 2016 21 / 27
Results Table 2: Emp. Heston Calibration - Phy. Measure, (µ, θ, σ, κ, ρ) initial value x 0 x opt Ψ(x opt ) (.1268,.0126,.3, 3,.1) (.1291,.0127,.3129, 2.9489,.1010).22 = 22% (.1268,.0126,.7, 15,.1) (.1271,.0126,.7018, 15.1875,.1028).02 = 2% Grajales - Medina (UdeA - UN) Heston - Empir.Calib. June 17, 2016 22 / 27
Results Figure 7: Ψ(x) 0.45 Current Function Value: 0.22 0.4 Function value 0.35 0.3 0.25 Stop 0.2 0 20 40 60 80 100 120 140 160 180 Iteration Pause Grajales - Medina (UdeA - UN) Heston - Empir.Calib. June 17, 2016 23 / 27
Results Figure 8: Ψ(x) 0.07 Current Function Value: 0.02 0.065 0.06 0.055 Function value 0.05 0.045 0.04 0.035 0.03 0.025 Stop 0.02 0 20 40 60 80 100 120 140 160 180 Iteration Pause Grajales - Medina (UdeA - UN) Heston - Empir.Calib. June 17, 2016 24 / 27
Results Figure 9: Ψ(x) 1 0.9 RealData data Ret data fit 1 Real fit 2 Ret 0.8 0.7 Cumulative probability 0.6 0.5 0.4 0.3 0.2 0.1 0-0.06-0.04-0.02 0 0.02 0.04 0.06 Data Grajales - Medina (UdeA - UN) Heston - Empir.Calib. June 17, 2016 25 / 27
Results Figure 10: Ψ(x) 1 0.9 RealData data Ret data fit 1 Real fit 2 Ret 0.8 0.7 Cumulative probability 0.6 0.5 0.4 0.3 0.2 0.1 0-0.05-0.04-0.03-0.02-0.01 0 0.01 0.02 0.03 0.04 0.05 Data Grajales - Medina (UdeA - UN) Heston - Empir.Calib. June 17, 2016 26 / 27
Conclusions Calibration of Heston SVM (under physical measure of risk) seems to be adequate to describe USDCOP returns More than 95% of simulations achieved with FO & Nelder Mead pass two-ks test Questions and the future: Comparing with other Heston physical risk measure calibration Leverage effect on calibration of ρ Relationship σ κ: different combination give similar values of the O.F - Is Ψ flat close to the optimum? Bridge between physical and risk neutral measure - λ? Grajales - Medina (UdeA - UN) Heston - Empir.Calib. June 17, 2016 27 / 27