Option pricing a particle filtering approach

KWAME NKRUMAH UNIVERSITY OF SCIENCE AND TECHNOLOGY OPTION PRICING : A PARTICLE FILTERING APPROACH By Henry Nii Ayitey-Adjin (Bsc Mathematics) A THESIS SUBMITTED TO THE DEPARTMENT OF MATHEMATICS, KWAME NKRUMAH UNIVERSITY OF SCIENCE AND TECHNOLOGY IN PARTIAL FUFILLMENT OF THE REQUIREMENT FOR THE DEGREE OF MSc INDUSTRIAL MATHEMATICS October 15, 2015 Declaration I hereby declare that this submission is my own work towards the award of the MSc degree and that, to the best of my knowledge, it contains no material previously published by another person nor material which had been accepted for the award of any other degree of the university, except where due acknowledgement had been made in the text Henry Nii Ayitey-Adjin Signature Date Signature Date Prof S.K Amponsah Head of Department Signature Date Student Certified by: E Owusu-Ansah Supervisor Certified by: i Dedication With love and thankful heart to God, family I am becoming a better person each day ii Abstract Option pricing is a critical issue in the financial market An investigation into the use of Sampling Importance Resampling (SIR) filter for financial option pricing in the Black-Schole model is performed The impact of process noise, measurement noise, and the number of particles on the accuracy and performance of SIR filter is examined The Black-Schole model is solved by the finite difference scheme The SIR filter is implemented by the use of the GARCH model and the Black-Schole model with synthetic data The effect of different process noise, measurement noise, and number of particles on the SIR filter was examined It was found that the SIR filter performed well at lower process noise and high measurement noise when considering profitability of a call option Also, as the number of particle decrease the SIR filter performed very well iii Acknowlegdment Thanks be to God, who gives us the victory through our LORD JESUS CHRIST Firstly, I thank God for all the wisdom to come this far I am very grateful to Mr Owusu Ansah of the department of Mathematics, KNUST, my supervisor, for the faith in me, the guidance he provided and putting me to this challenge God richly bless you To all the lecturers at the department of mathematics, I am very grateful To my parent and family, thank you all for the support and encouragement God richly bless you all iv Contents Declaration v Dedication v Abstract v Acknowledgement v List of Tables viii List of Figures x Introduction 1.1 Background of the study 1.2 Statement of the Problem 1.3 Objectives of the Study 1.4 Methodology 1.5 Significance of the study 1.6 Organization of the study Literature Review 2.1 Introduction 2.2 Options 2.3 Valuation of Financial Options 2.4 Data Assimilation 2.5 Sequential Data Assimilation 10 2.6 Particle Filtering 12 v Methodology 3.1 3.2 13 INTRODUCTION 13 3.1.1 Black-Scholes 13 3.1.2 Black -Scholes Model 15 3.1.3 The Black-Scholes Equation 15 3.1.4 Portfolio 16 3.1.5 Transformation of Black-Scholes into the Diffusion equation 17 3.1.6 Pricing Call Option 19 3.1.7 Pricing Put Option 20 20 3.2.1 Finite Difference Methods 20 3.2.2 Numerical Scheme 21 DATA ASSIMILATION 23 3.3.1 Dynamical System 23 3.3.2 Deterministic System 24 3.3.3 Stochastic System 24 3.3.4 Stochastic Approach 25 3.3.5 Bayesian Framework 25 3.3.6 Framework 26 PARTICLE FILTERING 27 3.4.1 Sequential Importance Sampling 28 3.4.2 Degeneracy Problem 32 3.4.3 Resampling 32 3.4.4 Sampling Importance Resampling Filter 34 Application of Particle Filtering 36 Analysis 43 3.3 3.4 3.5 NUMERICAL SOLUTION 4.1 Introduction 43 4.2 Result and Discussion 44 4.2.1 44 Call Option vi 4.2.2 Put Option 46 Conclusion 49 5.1 Introduction 49 5.2 Conclusion 49 5.3 Recommendation 50 References 54 vii List of Tables 2.1 Determinants of Option value 3.1 Ito’s Multiplication Table 3.2 RMSE of Particle Filter for various process noise 38 3.3 RMSE of Particle Filter for various measurement noise 40 3.4 RMSE of Particle Filter for different number of particle 42 4.1 Estimated mean volatility and RMSE for a Call Option 46 4.2 Estimated mean volatility and RMSE for a Put Option 48 viii 14 List of Figures 3.1 The estimated volatility over time of 100 days with a process noise of P0 of at an underlying price S0 of $60 and a strike price K0 of $50 3.2 37 The estimated volatility over time of 100 days with a process noise of P0 of 0.5 at an underlying price S0 of $60 and a strike price K0 of $50 3.3 37 The estimated volatility over time of 100 days with a process noise of P0 of at an underlying price S0 of $60 and a strike price K0 of $50 3.4 38 The estimated volatility over time of 100 days with a measurement noise of M0 of at an underlying price S0 of $60 and a strike price K0 of $50 3.5 39 The estimated volatility over time of 100 days with a measurement noise of M0 of 0.5 at an underlying price S0 of $60 and a strike price K0 of $50 3.6 39 The estimated volatility over time of 100 days with a measurement noise of M0 of at an underlying price S0 of $60 and a strike price K0 of $50 40 3.7 The estimated volatility over time of 100 days with 1000 particles 41 3.8 The estimated volatility over time of 100 days with 10000 particles 41 3.9 The estimated volatility over time of 100 days with 100000 particles 41 4.1 The estimated volatility at an underlying price of $50 over time of 100 days ix 45 Figure 3.6: The estimated volatility over time of 100 days with a measurement noise of M0 of at an underlying price S0 of $60 and a strike price K0 of $50 Table 3.3: RMSE of Particle Filter for various measurement noise RMSE OF Particle Filter for M0 Underlying Price 0.5 40 0.2022 0.2131 0.2889 50 0.1907 0.1910 0.2540 60 0.4426 0.4328 0.4054 Table 3.3 reveals that the estimate best when the underlying asset price is equal to the strike price ie S0 = $50 = K0 The SIR performs poorly at S0 of $60 given the highest RMSE for all measurement noise tested In order to be profitable, the call option should have underlying asset S0 ¿ K0 the strike price With interest in the performance of SIR at S0 of $60, the SIR performance best at high measurement noise M0 as suggested by measurement noise M0 of with RMSE of 0.4054 and poorly at low measurement noise in the case of M0 of 0.5 with RMSE of 0.4426 Finally, an experiment to investigate the performance of SIR when the number of particle (nPF) in SIR is varied is performed The process noise and measurement noise is kept at respectively Figures 3.7,3.8,3.9 also shows volatility over time for 100 days A summary of result of the performance of SIR in terms of RMSE is displayed in Table 3.4 40 Figure 3.7: The estimated volatility over time of 100 days with 1000 particles Figure 3.8: The estimated volatility over time of 100 days with 10000 particles Figure 3.9: The estimated volatility over time of 100 days with 100000 particles 41 Table 3.4: RMSE of Particle Filter for different number of particle RMSE OF Particle Filter for nPF Underlying Price 1000 10000 100000 40 0.2158 0.2161 0.2211 50 0.1957 0.2139 0.2263 60 0.4352 0.4603 0.4862 The experimental result in Table 3.4 indicates that the RMSE of SIR increases as the number of particles nPF increases This indicate that the SIR performs best with a small number of particles 42 Chapter Analysis 4.1 Introduction In this chapter,a number of experiments are performed by the use of the particle filtering method The results of the predictions of volatility of financial options in the Black-Scholes models by the use of the particle filter specifically the bootstrap method is outlined The estimates were obtained for 100 samples and experiment was repeated 100 times given a mean value, with associated variances similar to zero Synthetic data is generated and used in evaluation of the pricing algorithm At the foundation of the option pricing is the famous Black-Scholes model The partial differential equation describes the Black-Scholes model best σ2S ∂ 2V ∂V ∂V + + rS − rV = ∂t ∂S ∂S (4.1) where V (S, t) is a European put or call option with an underlying asset price S and at time t, r(t) is the risk free interest rate with volatility of underlying asset σ The Black-Scholes model describes the European call option Vc (S, t) and put option Vp (S, t) with no-dividend payment on stocks as: Vc (S, t) = SΦ(d1 ) − Ke−r(T −t) Φ(d2 ) (4.2) Vp (S, t) = Ke−r(T −t) Φ(−d2 ) − SΦ(−d1 ) (4.3) 43 where K is the strike price, T − t is the time until expiration Φ(.) is the cumulative normal distribution function and d1 and d2 are: d1 = d2 = S In( K ) + (r + σ S ) In( K σ2 )(T (T − t) + ((r − σ − t) σ2 )(T − t) (T − t) The hidden states is the volatility of underlying states whereas the call and put options are considered as the output observation The input observations are the current value of underlying asset price and the time to maturity Thus the model setup represents a parameter estimation problem with the observation equation given by Equation 4.2 and Equation 4.3 which allows us to compute for the daily probability distributions for the volatility whiles keeping the risk free rate constant 4.2 4.2.1 Result and Discussion Call Option Figure 4.1 and figure 4.2 present the estimated volatility and the estimated price with respect to time at underlying price S0 = $50 and $60 respectively, with strike price K=$50, for a call option 44 Figure 4.1: The estimated volatility at an underlying price of $50 over time of 100 days Figure 4.2: The estimated volatility at an underlying price of $60 over time of 100 days Figure 4.3: The estimated price at an underlying price of $50 over time of 100 days 45 Figure 4.4: The estimated price at an underlying price of $60 over time of 100 days Table 4.1: Estimated mean volatility and RMSE for a Call Option Call Option Underlying Price Mean PF 50 0.0304 60 0.0063 RMSE PF 0.2157 0.3440 Table 4.1 provides the root mean square error RSME showing the performance of the particle filter 4.2.2 Put Option The estimated volatility and price over time for a put option is displayed in figure 4.3 and 4.4 respectively These estimates where obtained for an underlying stock price of $40 and $50 respectively at a strike price of $50 over a 100 days time period Figure 4.5: The estimated volatility at an underlying price of $40 over time of 100 days 46 Figure 4.6: The estimated volatility at an underlying price of $50 over time of 100 days Figure 4.7: The estimated price at an underlying price of $40 over time of 100 days Figure 4.8: The estimated price at an underlying price of $50 over time of 100 days 47 Table 4.2: Estimated mean volatility and RMSE for a Put Option Put Option Underlying Price Mean PF 40 0.0218 50 0.0339 RMSE PF 0.2823 0.2156 Table 4.2 provides the estimated mean volatility and the root mean square error RSME showing the performance of the particle filter Mean volatility decrease as the underlying price S decrease from the position where underlying price is equal to strike price K of $50 Whiles RMSE increase as underlying price decrease from $50 to $40 48 Chapter Conclusion 5.1 Introduction This study has investigated the performance of the Sampling Importance Resampling(SIR) also known as the bootstrap filter by taking into consideration the effect of different process noise, different measurement noise, different number of particles The bootstrap filter was implemented by use of synthetic data A comparison test for when the options are profitable (in the money) was performed for both put and call options This Chapter present conclusion and a number of recommendations 5.2 Conclusion Gorden et al (1993) proposed the SIR filter SIR filter was implemented under the following of scenarios: • effect of different underlying price • effect of different process noise on filter • effect of different measurement noise of filter • effect of different number of particles of SIR filter At different process noise, SIR filter was found to perform better when 49 P0 < In term being in-the-money for a call option, SIR filter still 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Data assimilation is often used to estimate the parameters There are two approaches to data assimilation These are variational and sequential data assimilation Variational data assimilation makes... uncertainty in the estimated states The two types of data assimilation schemes are the sequential and variational assimilation Variational data assimilation use all the observation available over a. .. j − r]∆t β = ac + bd = 3.3 DATA ASSIMILATION Data assimilation is primarily interested with the use of observational data into mathematical models The Bayesian view of data assimilation where