Three essays on asset pricing in financial market

THREE ESSAYS ON ASSET PRICING IN FINANCIAL MARKET SHAO DAN (M.Soc.Sci., SHUFE ) A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF ECONOMICS NATIONAL UNIVERSITY OF SINGAPORE Acknowledgements This thesis owes a great debt to Albert K.C. Tsui for his supervision and the National University of Singapore for the research scholarship. For Chapter 1, the author is grateful for the comments and suggestions of an anonymous referee who helps to improve this chapter greatly. Special thanks to Albert K.C. Tsui for his sharp comments, and Jingying Huang for her inspirations. Thanks also go to the participants of Numerical Methods in Finance, an Amamef conference by INRIA-Rocquencourt in France, 2006, for their helpful discussions. For Chapter 2, the author thanks Tim Bollerslev, George Tauchen, and two anonymous referees for their valuable comments which contribute substantially to this chapter. For Chapter 3, the author wants to extend the gratitude to Jerome Detemple, one associate editor, and one reviewer whose comments help to bring this chapter to a higher standard. Special thanks to Lou Jiann-Hua for his enlightenment. All substantive and typographical errors are solely the author’s responsibility. ii Contents Acknowledgements ii Summary v List of Tables vii List of Figures viii A Numerical Method for Pricing American-style Asian Options under GARCH Model 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 The GARCH Model and Dynamic Programming Formulation . . . . 1.3 Characterization of the Value Function . . . . . . . . . . . . . . . . . 1.3.1 The Value Function Vtn−1 . . . . . . . . . . . . . . . . . . . . 1.3.2 General Features of the Value Function . . . . . . . . . . . . . 1.4 Numerical Procedures for DP Equations . . . . . . . . . . . . . . . . 1.4.1 Trilinear Approximation . . . . . . . . . . . . . . . . . . . . . 1.4.2 Distribution Approximation . . . . . . . . . . . . . . . . . . . 1.4.3 Convergence Analysis . . . . . . . . . . . . . . . . . . . . . . . 1.4.4 Grid Choice . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 The Numerical Experiments . . . . . . . . . . . . . . . . . . . . . . . 1.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 8 11 11 15 17 20 22 30 Gaussian Estimation of Continuous Time Models of Interest Rate 2.1 Introduction . . . . . . . . . . . . . . . . 2.2 The QTSMs . . . . . . . . . . . . . . . . 2.3 The Gaussian Estimation Methods . . . 2.4 Implementation and Simulation . . . . . 2.5 Empirical Results . . . . . . . . . . . . . 2.6 Extension of Methodology Applicability . 2.7 Conclusion . . . . . . . . . . . . . . . . . 35 35 39 41 45 53 56 59 iii Quadratic Term Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Valuation of Mortgage-Backed Securities by a Copula Function Approach 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Cash Flow Functions of an MBS with Prepayment . . . . . . . . . . . 3.3 First Hitting Time Density . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Copula Function Based Dependence Modeling . . . . . . . . . . . . . 3.5 Numerical Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 60 66 69 78 85 97 Bibliography 99 iv Summary Asset pricing theory tries to understand the values of contingent claims with uncertain payments. More involved risks mean a higher rate of return expected to compensate the risk premium, which in turn leads to a lower present price. One can think of asset pricing theory as measuring the sources of aggregate risks that drive the price dynamics of asset in question. This thesis is aimed at studying three facets of asset pricing in financial markets. New numerical approach, semi-analytical method, and important extension of applicability of existing estimation method are proposed in this thesis. Chapter develops a new numerical method to price American-style Asian option in the context of the generalized autoregressive conditional heteroscedasticity (GARCH) asset return process. The development is based on dynamic programming coupled with the replacement of the normally distributed variable with a binomial one and the whole procedure is under the locally risk-neutral valuation relationship (LRNVR). We investigate the computational and implementation issues of this method and compare them with those of a candidate procedure which involves piecewise-polynomial approximation of the value function. Complexity analysis and computational results suggest that our method is superior to the candidate one and the generated GARCH option prices are capable of reflecting the changes in the conditional volatility of underlying asset. In Chapter we propose a Gaussian estimation method for the three-factor quadratic term structure models (QTSMs). Based on the recently developed Gaussian method we derive an exact discrete model of continuous time interest rate and the v exact Gaussian likelihood function of discrete observations and model parameters. Monte Carlo experiments show that the overall finite-sample performance of proposed method is satisfactory in terms of sample bias and mean square error (MSE). An empirical application to UK and US interest rates is also given. Moreover, to extract more information from entire term structure such as market price of risk premium we also discuss the extensibility of proposed method to deal with a panel of yields. Chapter studies the valuation of mortgage-backed securities (MBS) based on copula function approach which enables us to construct joint first hitting time distribution in a mathematically convenient way. While Nakamura (2001) solves the Volterra type integral equation by piecewise approximation, we provide an alternative semi-analytical copula based method which can construct joint distribution flexibly and can be implemented without computational difficulty. We also introduce the definition and some basic properties of copulas. Numerical experiments are made to demonstrate the applicability and efficiency of proposed method. We also discuss some possible model risks. vi List of Tables 1.1 1.2 1.3 2.1 2.2 2.3 Prices of American Call Option for Different Maturities, Exercise prices and Conditional Volatilities . . . . . . . . . . . . . . . . . . . . . . . Comparison with LSM when Pricing An American Call Option with Conditional Heteroscedasticity . . . . . . . . . . . . . . . . . . . . . . Prices of American Call Option As a Function of n . . . . . . . . . . 23 26 29 2.5 2.6 2.7 Parameter Setting for Hourly Observations . . . . . . . . . . . . . . . Parameter Setting and Sample Size . . . . . . . . . . . . . . . . . . . Properties of Gaussian Estimates after 1000 Replications for Monthly Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Properties of Gaussian Estimates after 1000 Replications for Weekly Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Properties of Gaussian Estimates after 1000 Replications for Daily Data Summary Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gaussian Estimates of the Three-factor Quadratic Interest Rate Model 51 52 54 55 3.1 MBS Prices Based on Different Copulas . . . . . . . . . . . . . . . . . 87 2.4 vii 47 49 50 List of Figures 1.1 Implied Volatility of the GARCH Option Price with a Low Initial Conditional Volatility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Implied Volatility of the GARCH Option Price with a High Initial Conditional Volatility . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 The GARCH Option Price As a Function of s . . . . . . . . . . . . . 1.4 The GARCH Option Price As a Function of θ . . . . . . . . . . . . . 1.5 The GARCH Option Price As a Function of λ . . . . . . . . . . . . . 28 30 31 32 2.1 2.2 2.3 48 57 57 Hump-shaped Condition Dynamics of Daily Data . . . . . . . . . . . Matching Hump-shaped Condition Dynamics of UK Data . . . . . . . Matching Hump-shaped Condition Dynamics of US Data . . . . . . . 3.1 Clayton Copula Based Joint Distribution Function . . . . . . . . . . . 3.2 Gaussian Copula Based Joint Distribution Function . . . . . . . . . . 3.3 t4 -Copula Based Joint Distribution Function with degrees of freedom 3.4 t8 -Copula Based Joint Distribution Function with degrees of freedom 3.5 t20 -Copula Based Joint Distribution Function with 20 degrees of freedom 3.6 Present Values of Cash Flows of Baseline Model . . . . . . . . . . . . 3.7 MBS Price Sensitivity to Initial Interest Rate with a Moving Threshold Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8 MBS Price Sensitivity to Initial Interest Rate with a Fixed Threshold Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.9 MBS Price Sensitivity to Copula Choice I . . . . . . . . . . . . . . . 3.10 MBS Price Sensitivity to Copula Choice II . . . . . . . . . . . . . . . 3.11 MBS Price Comparison with Different Dependence Parameter . . . . viii 28 82 83 83 84 84 89 91 92 93 95 96 To my parents, who offer me unconditional love and support, and to Jingying, who has been a great source of motivation and inspiration. ix Chapter A Numerical Method for Pricing American-style Asian Options under GARCH Model 1.1 Introduction Following the celebrated work of Black and Scholes (1973) and Merton (1973), researchers have developed the option valuation models to incorporate volatility which is an indisputable empirical fact. The time-varying volatility models can be generally classified into continuous-time ones and discrete-time Generalized Autoregressive Conditional Heteroscedasticity (GARCH) ones. The early attempts to model continuous-time stochastic volatility include Cox (1975), Merton (1976), and Geske (1979). Hull and White (1987) proposed an additional process to govern the evolution of volatility, known as bivariate diffusion model. However, all of these models face the difficulty of implementing and testing because of the nonobservability of variance. Since it was first proposed by Bollerslev (1986), GARCH process has increasingly gained prominence as a powerful econometric tool. Moreover, as pointed out by Heston and Nandi (2000), under a GARCH option model, one can calculate the volatilities directly from the historical data of asset returns, which makes it easier to value an option and estimate the model parameters from the discrete observations. 140 straight bond Clayton Gaussian t4 135 t8 130 t20 MBS price 125 ← straight bond 120 115 110 copulas ↓ 105 100 95 0.02 0.03 0.04 0.05 0.06 0.07 initial interest rate 0.08 0.09 0.1 Figure 3.7 MBS Price Sensitivity to Initial Interest Rate with a Moving Threshold Distribution 91 130 straight bond Clayton Gaussian t4 125 t8 ← straight bond t20 120 MBS price 115 110 copulas ↓ 105 100 95 90 0.04 0.05 0.06 0.07 initial interest rate 0.08 0.09 0.1 Figure 3.8 MBS Price Sensitivity to Initial Interest Rate with a Fixed Threshold Distribution 92 0.9999 price ratio 0.9998 0.9997 0.9996 Calyton/Gaussian t4/Gaussian 0.9995 t8/Gaussian t20/Gaussian 0.9994 0.1 0.2 0.3 0.4 0.5 innovation correlation ρ 0.6 Figure 3.9 MBS Price Sensitivity to Copula Choice I 93 0.7 Figure 3.9 illustrates that with the increase of innovation correlation ρ, the discrepancy between Gaussian copula price and other copula prices gets smaller and smaller. Since t-copula has a lower survival probability for τ1 and τ2 , it gives relatively low price for MBS. As ν → ∞, t-distribution approaches normal distribution the price, and difference between t-copula and Gaussian copula tends to shrink as expected. Consistent with Table 3.1, we find no firm evidence for a systematic price discrepancy between Gaussian copula and Clayton copula. As for the model choice, i.e. model misspecification, Figure 3.9 implies that if the true copula were a Gaussian copula but a Clayton copula were presumed instead, the price would not deviate from the true value. But if a t-copula were used, only with higher degrees of freedom and higher value of correlation ρ, the price difference would be sizable. Figure 3.10 shows a similar scenario where we assume the Clayton copula is the true copula. If a copula other than Clayton copula were used, we could draw the same conclusions as we did above based on Figure 3.9. For a comprehensive study of model risk of copula choice we have to examine the impact of copula choice on a large number of possible parameters and different MBS contracts. We have to scrutinize how the price will differ if we choose a different copula for a specific parameter set and contract. Another source of model misspecification is the choice of dependence parameter in our copula functions. Recall that we use the correlation coefficient of innovations of interest rate and house price as the dependence parameter of the two prepayment times τ1 and τ2 . However, this is not the correlation between τ1 and τ2 , and the true value should be much smaller than the innovation correlation. Li (2000) used asset correlation as a substitute of the correlation of survival times of default derivatives as in CreditMetrics™. We could also use the correlation between interest rate and house price themselves as the correlation of prepayment times in copula functions. To study this source of model risk, we compute the corresponding ρ0 from (3.3.6) for each ρ and calculate corresponding MBS price based on ρ0 . Then for each ρ we have a corresponding price ratio of the price calculated from ρ0 over the price calculated from ρ, which is plotted in Figure 3.11. Also note that for our baseline model parameter ρ0 < ρ and it is a decreasing function of time t. 94 1.0001 price ratio 0.9999 0.9998 0.9997 Gaussian/Calyton t4/Calyton 0.9996 t8/Calyton t20/Calyton 0.9995 0.1 0.2 0.3 0.4 0.5 innovation correlation ρ 0.6 Figure 3.10 MBS Price Sensitivity to Copula Choice II 95 0.7 1.025 1.02 price ratio 1.015 1.01 Clayton Gaussian t4 1.005 t8 t20 0.1 0.2 0.3 0.4 0.5 innovation correlation ρ 0.6 0.7 Figure 3.11 MBS Price Comparison with Different Dependence Parameter 96 As Figure 3.11 shows, if the true dependence parameter were the correlation coefficient between interest rate and house price and we used innovation correlation ρ (> ρ0 ) instead, all the MBS prices based on different copulas would have been underpriced. Furthermore, the discrepancy between true value and underpriced one is an increasing function of ρ. In other words, we must be very careful about the choice of copula dependence parameter. If we can’t obtain the parameter from past empirical information, the innovation correlation or underlying asset correlation are readily available substitutes. 3.6 Concluding Remarks In this chapter, we develop an alternative method to Nakamura (2001) and provide a semi-analytical valuation model of MBS by exploring the application of copula functions in constructing the joint distribution of first hitting times. Instead of piecewise approximation of the joint probability of prepayment times or Monte Carlo simulation of underlying assets, we propose to use appropriate copula functions to build the joint distribution of the first hitting times due to multiple prepayment triggers, because copula function approach has established itself as a practical and convenient way to deal with dependence relationship. The concept of copulas, their basic properties and some popularly used copula functions are introduced. In our case, we specify two possible prepayment triggers, interest rate and house price. With the underlying processes of interest rate and house price we derive the first hitting time distributions separately which are actually marginal distributions of overall prepayment time. Then we resort to copulas to construct joint probability distribution from each marginal distribution. Numerical experiments show that different copulas give similar results about MBS price to those given by simulation. They are capable of exhibiting the characteristic features of MBS such as monthly cash flows and duration. We also check some possible sources of model risk. It turns out that no systematic pattern of prices given by Clayton copula and Gaussian copula can be found while for t-copula with higher degree of freedom it can produce results convergent to those of Gaussian copula. Another source of model risk is wrong specification 97 of model parameter. 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Phillips (2001), ‘A gaussian approach for continuous time models of the short-term interest rate’, Econometrics Journal 4, 210–224. 106 [...]... n−1 where f is the conditional density function of τT and is continuous and bounded 2 over σT Then by Lebesgue’s dominated convergence theorem, the integral Vth is n−1 2 also continuous To show that Vth is nondecreasing in σT , one should note that n−1 2 equation (1.3.4) implies that it’s an increasing function of σT while equation (1.3.3) 2 is independent of σT 9 The value function 2 Vtn−1 (Stn−1... polynomial functions available, including a piecewise constant function, a piecewise linear function over cone, high-dimensional splines, and etc., the one we consider here is a linear function in all of its variables This is a trade-off in terms of the amount of calculation and a desirable precision The simple method such as piecewise constant requires much finer partitions to achieve good precision, whereas... chain approximation method for American option pricing They developed an explicit scheme for the GARCH model and proved its convergence But until recently applying GARCH process in the pricing of exotic options, such as Asian options, is not well studied Since Asian option’s payoff depends on the average price of a primitive asset over a certain time period, it is less sensitive to changes in underlying... )dτ, = ρ 0 2 where f is the conditional density function of τtm+1 and σtm+2 is a known continuous, 2 bounded, and increasing function of σtm+1 Since the integrand is continuous, strictly positive, and bounded, so is Vth Also note that Vth is a positively weighted average m m of Vth which is a nondecreasing function of its inputs, and that with the increase m+1 2 of σtm+1 the integral will allocate higher... function satisfies a dynamic programming (DP) recurrence We write the option value as a function of current time, current primitive asset price, current average price and asset return’s conditional variance, and solve the DP system recursively with backward induction We first formulate a numerical solution approach for our DP equation based on piecewise trilinear interpolation over finite grids, following... and Dynamic Programming Formulation Under the classical mathematical setting of Harrison and Pliska (1981), our discretetime market, consisting of one primitive asset and one default-free bond, is defined on the probability space (Ω, F, P) Let T be a positive real number (the terminal time), then we assume Ft |0≤t≤T is the P-completion of the filtration generated by a Brownian motion on (Ω, F, P) and Ft... version of the APARCH specification where p = q = 1 We use the following simplified volatility equation s s s σt = ω + ασt−1 (|ξt−1 − λ| − θ(ξt−1 − λ))s + βσt−1 (1.2.6) Now we introduce our second assumption Assumption 1.2.2 The value function of a contingent claim with one period to maturity can be calculated by Black-Scholes-Rubinstein formula This assumption can also be found in Duan (1995) and Heston... distribution of Stn−1 /S0 In our numerical analysis of next section, we run the algorithm repeatedly with increasingly finer partitions and we examine the changes in the option value The experiments stop when there is no significant change in option value Also note that the variance equation specification (i.e higher order terms) of APARCH doesn’t come into the convergence proof and only has effect on our... simulation results in Table 1.1, where we implemented the approximation method introduced in Section 1.4.2 with n = 40 in a MATLAB R14 23 programming environment Computations have been executed on a 1.70GHz Pentium M PC with 512 Mbytes of memory and running the Windows operating system The CPU times are given for every set of parameter values with different grid spaces and are reported in hours:minutes:seconds... is much longer than the time required by their finest grid The main reason behind this is that they considered a homogeneous volatility model with a 2-dimension value function, while we incorporate heteroscedastic conditional volatility and the additional variable leads directly to the huge calculation amount As analyzed in Section 1.4.1, if we applied the trilinear approximation, just following Ben-Ameur . potential polynomial functions available, including a piecewise constant function, a piecewise linear function over cone, high-dimensional splines, and etc., the one we consider here is a linear function in all. THREE ESSAYS ON ASSET PRICING IN FINANCIAL MARKET SHAO DAN (M.Soc.Sci., SHUFE ) A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF ECONOMICS NATIONAL UNIVERSITY OF SINGAPORE Acknowledgements This. 1))f(τ|F t m )dτ, where f is the conditional density function of τ t m+1 and σ 2 t m+2 is a known continuous, bounded, and increasing function of σ 2 t m+1 . Since the integrand is continuous, strictly positive,