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1 Title Pricing options and equity-indexed annuities in regimeswitching models by trinomial tree method Author(s) Yuen, Fei-lung; 袁飛龍 Citation Issue Date 2011 URL Rights unrestricted

2 PRICING OPTIONS AND EQUITY-INDEXED ANNUITIES IN REGIME-SWITCHING MODELS BY TRINOMIAL TREE METHOD by YUEN FEI LUNG A thesis submitted in partial fulfillment of the requirements for the Degree of Doctor of Philosophy at The University of Hong Kong. December 2010

3 Abstract of thesis entitled PRICING OPTIONS AND EQUITY-INDEXED ANNUITIES IN REGIME-SWITCHING MODELS BY TRINOMIAL TREE METHOD Submitted by YUEN FEI LUNG for the degree of Doctor of Philosophy at The University of Hong Kong in December 2010 Starting from the well-known paper in pricing vanilla European call and put options by Black and Scholes in 1973, there are many different research papers on option valuation. The basic assumption of the Black-Scholes model is that the price of the underlying asset, usually the stock, is a geometric Brownian motion. The Markov regime-switching model (MRSM) introduced by Hamilton (1989) improves the adaptability of the Black-Scholes model by allowing the parameters of the stock price process change according to the financial situation. However, due to the additional uncertainty brought by the Markov process, option pricing in the MRSM is complicated and is usually done by simulation, or by solving a system of partial differential equations.

4 Lattice model (or tree model) provides a simple way to price derivatives. In the existing literature, additional branches are required in the lattice for derivative pricing under the MRSM. In this case, the lattice is not recombining and its efficiency is reduced significantly. Based on the trinomial tree model introduced by Boyle (1988), a multi-state trinomial tree is introduced to price various options in the MRSM. The key idea is to use the same lattice to accommodate all different regimes by adjusting the probability measure. The method is simple and efficient. The numerical results of the option prices obtained by this method are analyzed. There are different MRSMs. The stock price in the MRSM of Elliott, Chan and Siu (2005) is a continuous process while the price of derivatives jumps when the regime switches. In order to study the jump risk of assets in details, the jump diffusion model of Naik (1993) is studied and extended. The multi-state trinomial tree is used to price options in this jump diffusion model and the nature of jump risk is discussed. Asian option is a strong path-dependent option. The payoff of Asian option depends on the whole path of the asset price process and pricing Asian options is not an easy task in the MRSM. Equity-indexed annuities (EIAs) are derivative products linked to the performance of an equity index and is now a popular product in the market. The payoff structure of EIAs can take different forms according to the needs of the investors. The multi-state trinomial tree model is

5 modified using the idea of Hull and White (1993) and applied to Asian options. The problem of quadratic approximation suggested by Hull and White (1993) is identified and solved. Using the result of Asian option, the price of Asian-optionrelated EIAs is obtained by an iterative equation.

6 Declaration I declare that this thesis represents my own work, except where due acknowledgements are made, and that it has not been previously included in a thesis, dissertation or report submitted to this University or to any other institution for a degree, diploma or other qualification. Signed YUEN FEI LUNG i

7 Acknowledgements I would like to express my most sincere gratitude to my supervisor, Prof. Yang, Department of Statistics and Actuarial Science, the University of Hong Kong, for his guidance. He always spends a large part of his valuable time to teach us and monitor our progress. I would also like to express my thanks to all of the department members for the help and support in these four years of study. ii

8 Table of Contents Declaration Acknowledgements Table of Contents List of Tables i ii iii v 1 Introduction 1 2 Multi-state Trinomial Tree Introduction Multi-state Trinomial Lattice Numerical Results and Analysis Alternative Models Hedging Risk of Regime Switching Conclusions 42 3 Pricing Regime-switching Risk Introduction Jump Diffusion Model Arrival Rates of Jumps under Risk Neutral Measure Trinomial Tree Pricing under Jump Diffusion Model Numerical Results and Analysis Jump Risks are Not Priced Jump Risks are Priced Conclusions 77 iii

9 4 Pricing Asian Option and Related EIAs Introduction A Modified Trinomial Lattice Pricing Equity-Indexed Annuities Numerical Results and Analysis Conclusions Concluding Remarks 107 References 110 iv

10 List of Tables 2.1 Comparison of different methods in pricing Euroean call option in MRSM Pricing European call option with trinomial tree Pricing European put option with trinomial tree Pricing American call option with trinomial tree Pricing American put option with trinomial tree Pricing down-and-out barrier call option with trinomial tree Pricing double barrier call option with trinomial tree Price of double barrier call options with different barrier levels Pricing European call option with trinomial tree: great derivation in volatilities Pricing European call option under model with three regimes Pricing European call option under model with three regimes using trinomial tree Pricing European call option under model with three regimes using combined trinomial tree Pricing European call option when jump risk is not priced Pricing European put option when jump risk is not priced Comparison of European call option prices in jump and non jump models Comparison of European put option prices in jump and non jump models Comparison of American call option prices in jump and non jump models 70 v

11 3.6 Comparison of American put option prices in jump and non jump models Comparison of down-and-out barrier call option prices in jump and non jump models Comparison of European call option prices with priced and nonpriced jump risk Comparison of European put option prices with priced and nonpriced jump risk Comparison of American call option prices with priced and nonpriced jump risk Comparison of American put option prices with priced and nonpriced jump risk Comparison of down-and-out call option prices with priced and non-priced jump risk Comparison of the prices of (Eurpean type) average price call options in simple BS model (linear approximation of representative value) Comparison of the prices of (American type) average price call options in simple BS model (linear approximation of representative value) Comparison of the prices of (Eurpean type) average price call options in simple BS model (quadratic and modified quadratic approximation, simple average asset price) Comparison of the prices of (American Type) average price call options in simple BS model (quadratic and modified quadratic approximation, simple average asset price) 97 vi

12 4.5 Comparison of the price of average price call options in MRSM found by different methods I Comparison of the price of average price call options in MRSM obtained by different methods II Price of average price call options with early exercise option in MRSM Price of one-year EIA in MRSM Price of unit annual reset EIA in MRSM 106 vii

13 Chapter 1 Introduction In the past decades, there are a lot of researches on option pricing and many models have been proposed for the underlying asset. Markov regime-switching model (MRSM), which allows the parameters of the market model controlled by a Markov process, have become one of the popular models recently. It is found to be consistent with the market data and has gained its popularity because it can reflect the information of the market environment which cannot be modelled by linear Gaussian process solely. Markov process can ensure the parameters changing according to the market environment and preserve the simplicity of the model. It is also consistent with the efficient market hypothesis, all the effects of the information about the stock price are reflected on the current stock price. Merton (1969) uses stochastic differential equation (SDE) to study the continuoustime portfolio theory; since then geometric Brownian motion (GBM) becomes one of the most commonly used stochastic process in financial mathematics because of its highly random nature and simplicity. Black and Scholes (1973), based on the work of Merton (1969), give the no-arbitrage price of an European option when the price of the underlying assets is a GBM. The development of the formulae of various derivatives is easy based on this model. However, when the parameters 1

14 of the SDE are not constant but controlled by a Markov process, the price of the options cannot be found easily. There are many papers about option pricing in multi-state models such as MRSM. Some of them use lattice model. Boyle (1988) uses a pentanomial tree lattice to find the price of derivatives with two states. Kamrad and Ritchken (1991) suggest a (2 k + 1)-branch model for k sources of uncertainty. Bollen (1998) constructs a pentanomial tree which is excellent in finding a fair price of European option and American option in two-regime situation. Aingworth, Das and Motwani (2006) use a lattice with 2k branches to study the k-state model. The increasing number of branches reduces the efficiency of the tree models and so some other methods are used by different researchers to price derivatives. Buffington and Elliott (2002) find the price of European option and American option using partial differential equations (PDE). Mamon and Rodrigo (2005) find an explicit solution to European options in regime-switching economy by considering the solution of the associated PDE. Elliott, Chan and Siu (2005) use Esscher transform to find the explicit price formula for European option. Boyle and Draviam (2007) find the price of exotic options under regime switching using PDE. PDE has become the focus of most researchers for option valuation in MRSM as it is flexible. Since the introduction of binomial tree model by Cox, Ross and Rubinstein 2

15 (1979), lattice model is one of the most popular methods to calculate the price of simple options like European option and American option. Various lattice models are suggested after that, see, for example, Jarrow and Rudd (1983) and Boyle (1986). Trinomial tree model of Boyle (1986) is highly flexible. The extra middle branch of it gives one degree of freedom to the lattice and that makes the lattice very useful in regime-switching model. Boyle and Tian (1998) use this property of the trinomial tree to price double barrier option and Bollen (1998) uses the similar idea to construct an efficiently recombining tree. In Chapter 2, a new method is introduced so that a trinomial tree can be used to find a fair price of options under MRSM efficiently. The MRSM used by Buffington and Elliott (2002) is a good and popular choice, and it is used to illustrate the idea of the multi-state trinomial tree model. Regime-switching market is not complete. There are many different ways to find a fair price of the options. Miyahara (2001) uses the minimal entropy martingale measure to find the price which maximizes the exponential utility. Elliott, Chan and Siu (2005) use Esscher transform to obtain a fair price. Guo (2001) introduces change-of-state (COS) contracts to complete the market. Naik (1993) shows that the price of options can also be found by the given market price of risks. In the MRSM of Buffington and Elliott (2002), stock price is a continuous process and jump risk seems not a systematic risk in a certain sense. 3

16 In Chapter 3, the two-regime model of Naik (1993) is extended to k states and the option prices under this market model are tested using the multi-state trinomial tree. Various modern insurance products are introduced into the market in these years, including equity-indexed annuities (EIAs) and variable annuities (VAs). Their payoffs are linked to the preformance of some assets and indices. Different special features such as minimum yearly return, ceiling rate and participation rate make the products more flexible and more complicated. Tiong (2000) gives a details study on EIAs and prices the products using Esscher transform. Lee (2002, 2003) proposes several new designs of EIAs and finds their explicit formulae under the Black-Scholes framework. Lin and Tan (2003) consider an Asian-optionrelated EIA and price it under a stochastic interest rate model. Asian option is a strong path-dependent option of which the value of payoff depends on the path of the asset price process. Its valuation is complex under MRSM. In Chapter 4, the multi-state trinomial tree model is used to price Asian options in MRSM using the idea of Hull and White (1993). The problem of quadratic approximation suggested by Hull and White (1993) is identified and solved. Using the Markovian property of the regime-switching process, the price of Asian-option-related EIA can be obtained by an iterative equation with the price of Asian option. The theoretical MRSM of Buffington and Elliott (2002) is presented here for 4

17 completeness. We consider the real world probability space (Ω, F, P ). We let T be the time interval [0, T ] that is being considered. {W (t)} t T is a standard Brownian motion on (Ω, F, P ). {X(t)} t T is a continuous-time Markov process with finite state space X := {x 1, x 2,..., x k }, which represents the economic condition and is observable. A set of unit vector {e 1, e 2,..., e k } where x i = e i = (0,..., 1,..., 0) R k is used to denote the current state of the Markov process. For simplicity, the state e i is called the state i. We denote the set of states to be K := {1, 2,..., k}. Let A(t) = [a ij (t)] i,j=1,...,k be the generator of the Markov process. By the semi-martingale representation theorem, X(t) = X(0) + t 0 X(s)A(s)ds + M(t), (1.1) where {M(t)} t T is a R k -valued martingale with respect to the P -augmentation of the natural filtration generated by {X(t)} t T. There are two basic investment tools in the model, one is bond and the other is stock. The market interest rate is denoted by {r(t, X(t))} t T which depends on the current state of economy only, r(t) := r(t, X(t)) = r, X(t), (1.2) where r := (r 1, r 2,..., r k ); r i > 0 for all i K and, denotes the inner product in R k. 5

18 The bond price process {B(t)} t T satisfies the equation db(t) = r(t)b(t)dt, B(0) = 1. (1.3) The rate of return and the volatility of the stock price process are denoted by {µ(t, X(t))} t T and {σ(t, X(t))} t T, respectively. Similar to the interest rate process, they are only affected by the state of economy, µ(t) := µ(t, X(t)) = µ, X(t), σ(t) := σ(t, X(t)) = σ, X(t), (1.4) where µ := (µ 1, µ 2,..., µ k ) and σ := (σ 1, σ 2,..., σ k ) with σ i > 0 for all i K. The stock price process {S(t)} t T is a Markov-modulated geometric Brownian motion. Z(t) is the cumulative rate of return of the stock over time interval [0, t], that is, Z(t) = ln(s(t)/s(0)). Then, we have S(t) = S(u) exp(z(t) Z(u)), (1.5) ds(t) = µ(t)s(t)dt + σ(t)s(t)dw (t), (1.6) t ( Z(t) = µ(s) 1 ) t 2 σ2 (s) ds + σ(s)dw (s). (1.7) 0 0 6

19 Chapter 2 Multi-state Trinomial Tree 2.1 Introduction Since the binomial tree model was introduced by Cox, Ross and Rubinstein (1979), the lattice model has become a popular way to calculate the price of simple options like the European option and the American option. It is mainly because the lattice method is simple and easy to implement. Various lattice models have been suggested after that, see, for example, Jarrow and Rudd (1983) and Boyle (1986). The trinomial lattice of Boyle (1986) is highly flexible, and has some important properties that the binomial lattice lacks. The extra branch of the trinomial tree gives one degree of freedom to the lattice and makes it very useful in the regime-switching model. Boyle and Tian (1998) use this property of the trinomial tree to price double barrier options, and propose an interesting method to eliminate the error in pricing barrier options. Bollen (1998) uses a similar idea to construct an efficiently recombining tree. There are many other researches using tree methods for derivative pricing in multi-state model. Boyle (1988) uses a pentanomial tree model to calculate the price of derivatives with two states. Kamrad and Ritchken (1991) suggest a 2 k + 1-branch model for k 7

20 sources of uncertainty. Aingworth, Das and Motwani (2006) uses a lattice with 2k branches to study the k-state regime-switching model. However, when the number of states is large, the tree models mentioned above are not efficient. We propose a multi-state trinomial tree to price the options in a regime-switching model. The trinomial tree that we propose is recombining. Instead of increasing the number of branches in the tree for different regimes, we use different sets of risk neutral probabilities for different regimes. Since it is a recombining tree, option valuation is fast, simple and efficient using this method. 2.2 Multi-state Trinomial Lattice In CRR binomial tree, when σ is the asset s volatility and is the size of time step, the ratios of change are given by e σ and e σ, the risk neutral probabilities of getting up and down are specified so that the expected rate of return of the stock matches the risk-free interest rate. In the trinomial model, with constant risk-free interest rate and volatility, the stock price is allowed to remain unchanged, go up or go down by a ratio. The upward ratio must be greater than e σ to ensure that a risk neutral probability measure exists. If π u, π m, π d are the risk neutral probabilities of the stock price increases, remains unchanged and decreases in the tree, r is the risk-free interest rate, then, for a constant λ, we 8

21 have π u e λσ + π m + π d e λσ = e r, (2.1) (π u + π d )λ 2 σ 2 = σ 2 (2.2) λ should be greater than 1 so that the risk neutral probability measure exists. In the literature, λ is usually taken to be 3 (Figlewski and Gao (1999), Baule and Wilkens (2004)) or 1.5 (Boyle (1988), Kamrad and Ritchken (1991)). After fixing the value of λ, the risk neutral probabilities can be found and the whole lattice can be constructed. However, in the Markov regime-switching model (MRSM), the risk-free interest rate and the volatility are not constant. They change according to the Markov process. More branches can be introduced into the lattice so that extra regimes and information can be incorporated in the tree, for example, Boyle and Tian (1988), Kamrad and Ritchken (1991) and Bollen (1998). The increasing number of branches makes the lattice model more complex. Bollen (1998) suggests an excellent recombining tree to solve the option prices in two-regime case, but the multi-regime problem still cannot be solved effectively. Here, we propose a different way to construct the tree. Instead of increasing the number of branches in the tree, we change the risk neutral probability measure under different regimes so that a recombining tree allows more regimes. The 9

22 method relies greatly on the flexibility of the trinomial tree model and the core idea of the multi-state trinomial tree model here is to change the probability measure to accommodate different regimes in the same recombining lattice. Assuming that there are k regimes in the MRSM, the corresponding risk-free interest rate and volatility of price of the underlying asset under these regimes be r 1, r 2,..., r k and σ 1, σ 2,..., σ k, respectively. The up-jump ratio of the lattice is taken to be e σ. For a lattice which can be used by all regimes, σ > max 1 i k σ i. (2.3) For the regime i, let π i u, π i m, π i d are the risk neutral probabilities of the stock price increases, remains unchanged and decreases in the branch of the tree. Then, similar to the simple trinomial tree model, the following set of equations can be obtained, for all i K, π i ue σ + π i m + π i de σ = e r i, (2.4) (π i u + π i d)σ 2 = σ 2 i. (2.5) If λ i is defined to be σ/σ i for each i, then, λ i > 1 and the values of π i u, π i m, π i d 10

23 can be found, in terms of λ i, πm i = 1 σ2 i σ = 1 1, 2 λ 2 i (2.6) πu i = eri e σ (1 1/λ 2 i )(1 e σ ), e σ e σ (2.7) πd i = eσ e ri (1 1/λ 2 i )(e σ 1). e σ e σ (2.8) Therefore, the set of risk neutral probabilities depends on the value of σ. In order to ensure that σ is greater than all σ i, we might take σ = max 1 i k σ i + ( 1.5 1) σ, (2.9) where σ is the arithmetic mean of σ i. Root mean square is another suitable choice of σ. The most efficient choice of σ is unknown. In this section, σ i are assumed to be not greatly different from each other; and the selection of σ is not important as long as it is comparable with the volatilities of different regimes. After the whole lattice is constructed, the main idea of the pricing method is presented here. We let T be the expiration time of the option, N be the number of time steps, then = T/N. At time step t, there are 2t+1 nodes in the lattice, the node is counted from the lowest stock price level, and S t,n denotes the stock price of the n th node (it starts from the 0 th node, for convenience) at time step t. As all the regimes are sharing the same lattice and the regime state cannot be reflected by the position of the nodes, each of the nodes has k possible derivative 11

24 prices corresponding to the regime state. Let V t,n,j be the value of the derivative at the n th node at time step t under the j th regime state. The transition probability of the Markov process can be found by the generator matrix. The generator matrix is assumed to be constant and taken to be A. We define p ij ( ) as the transition probability from regime state i to regime state j for the time interval with length ; and for simplicity, it is denoted by p ij. The transition probability matrix, denoted by P, can be found by the following equation, P ( ) = p 11 p 1k..... = e A = I + ( ) l A l /l!. (2.10) l=1 p k1 p kk With the transition probability matrix, the price of a derivative at each node can be found by iteration. We start from the expiration time, for example, for an European call option with strike price K, V N,n,i = (S N,n K) + for all states i, (2.11) where S N,n = S 0 exp[(n N)σ ]. We assume that the Markov process is independent of the Brownian motion under the real market measure and the transition probabilities are not affected by the use of risk neutral measure. With the derivative payoff at expiration, using 12

25 the following equation recursively, [ k ] V t,n,i = e r i p ij (πuv i t+1,n+2,j + πmv i t+1,n+1,j + πdv i t+1,n,j ), (2.12) j=1 the price of the option under all regimes can be obtained. Regime switching is another source of risk because we do not know the time of regime switching before it takes place. Moreover, due to regime switching, the market is incomplete and the derivatives do not have a unique no-arbitrage price. There are many ways to treat the additional risk from regime switching, for example, not pricing the regime-switching risk (Bollen (1998)), or introducing change-of-state (COS) contracts into the model (Guo (2001)). The first way is used in the previous calculation. Some derivatives benefit while some are suffered by the regime switching which depends on the initial regime, the transition probabilities and the structure of the derivatives. It is also hard to make a compromise in choosing appropriate transition probabilities if the market is not complete. It is a reasonable choice of not pricing the regime-switching risk as long as there is no arbitrage opportunity in the market and the Markov process is independent of the Brownian motion. New securities COS can be introduced into the model to complete the market. However, the regime is referring to the macroeconomic condition, hedging or pooling regime-switching risk is complicated in the incomplete market and insurance companies might not be willing to take the risk. We assume that there are not suitable COS securities in the market. The risk premium 13

26 comes from the risk of Brownian motion only. We focus on the multi-state trinomial tree model in this chapter and the regime-switching risk will be discussed in details in the other parts. If we have to price American option, the value of the option at each node under different regimes can be compared with the payoff of exercising the option immediately; and the larger value is used as the price for iteration. The calculation is similar to the valuation of American option in simple lattice model. For barrier option, the idea of Boyle and Tian (1998) can be applied. The whole lattice is constructed from the lower barrier. As the initial price of the underlying asset is not necessarily at the grid, a quadratic approximation is used to calculate the price of the down-and-out option. The price of a down-and-in option can be found using the idea that the sum of down-and-out option and down-and-in option is a vanilla option. For a double barrier option, we have used the flexibility of trinomial tree lattice but the value of σ is in fact not fixed. We can set both of the upper and lower barriers on the node level by a fine adjustment of the lattice parameter σ. The price of curved barrier option and discrete-time barrier option can also be found, using a similar method suggested by Boyle and Tian (1998). The regime is observable, the payoff of the derivatives can depend on the regime state, because the prices of the derivative under all regimes are found in 14

27 each node, the model is also applicable to price this kind of derivatives. 2.3 Numerical Results and Analysis Based on the model introduced in the last section, we calculate the prices of various options in different regimes. In this section we study the European option, the American option, the down-and-out barrier option, the double barrier option, and their prices are calculated by the multi-state trinomial tree. The results give us some insights into the price of derivatives in the MRSM and the effects of regime switching. First of all, the model is tested by comparing with the results given by Boyle and Draviam (2007). Table 2.1 shows that the option price obtained by using the trinomial lattice is very close to the value obtained by using the analytical solutions derived in Naik (1993), and also close to those obtained using partial differential equations in Boyle and Draviam (2007). This verifies that the trinomial tree method proposed in this chapter is applicable. We now study the values of different types of options in the MRSM. The underlying asset is assumed to be a stock with initial price of 100, following a geometric Brownian motion of a two-regime model with no dividend. In Regime 1, the risk-free interest rate is 4% and the volatility of stock is 0.25; in Regime 15

28 Table 2.1: Comparison of different methods in pricing Euroean call option in MRSM European Call Option I Regime 1 Regime 2 S 0 Naik B&D Lattice Naik B&D Lattice European Call Option II Regime 1 Regime 2 S 0 Naik B&D Lattice Naik B&D Lattice S 0 is the initial stock price and the strike price is set to be 100. The volatilities of the stock in Regime 1 and Regime 2 are 0.15 and 0.25 respectively. The option lasts for 1 year and the lattice is set to have 1000 time steps. The generators of the regime-switching process are and 1 1 for the above two sets of data respectively

29 2, the risk-free interest rate is 6% and the volatility of stock is All options expire in one year with strike price equal to 100. The generator for the regimeswitching process is taken to be The transition probabilities of the branch of state up, middle and down with 20 time steps are , and in Regime 1; , and in Regime 2, respectively. These values depend on the size of time step, but the values with other sizes of time step are not much different from these values because the time step is small in general. The values in 20-step case can already give the idea of the size of the risk neutral probabilities. We study the numerical results to see if there are any special characteristics of the prices. of these derivatives and the convergence properties of the model. Tables 2.2 and 2.3 show that the convergence rate of the European call and the European put options is fast. We know that the price of European call options and European put options found by the CRR model converges with order 1, that is, the error of the price is halved if the number of time steps is doubled (Baule and Wilkens (2004), Omberg (1987)). We can see from the tables that most of the ratios shown in the tables are close to 0.5. However, it is not the case for the European call option when the number of iterations is large for Regime 2. 17

30 Table 2.2: Pricing European call option with trinomial tree European Call Option Regime 1 Regime 2 N Price Diff Ratio Price Diff Ratio N is the number of time steps used in calculation. Diff is referring to the difference in price calculated using various numbers of time steps and ratio is the ratio of the difference. Table 2.3: Pricing European put option with trinomial tree European Put Option Regime 1 Regime 2 N Price Diff Ratio Price Diff Ratio

31 This is because the approximation errors for the two regimes are different and the round-off error. Boyle (1988) shows that using the trinomial tree model, the approximation error is smaller if the three risk neutral probabilities of the tree are almost equal with same number of time steps. In our case, we can see that the risk neutral probabilities of Regime 1 are not as close as those of Regime 2. Therefore, in Regime 2, the change in prices is smaller which implies a smaller approximation error as shown in the numerical results in the tables. The differences between the price changes for Regime 2 are less than one-tenth of that for Regime 1 most of the time. However, the prices of the asset in both regimes affect one another. The larger pricing error in Regime 1 affects the accuracy of the price in Regime 2. The result is that the value in Regime 2 converges in a faster, but more unstable way. On the other hand, the error in Regime 2 is small compared with that in Regime 1; thus the convergence patterns in Regime 1 are more stable. Moreover, the change of prices in Regime 2 is small when the number of time steps is large. The round-off error then becomes significant. When we apply the put-call parity to each of the regimes, the interest rate found in the two regimes are 4.37% and 5.63% respectively using the result of 5120 time steps. It is reasonable because both of them are between 4% and 6%, the interest rate found by Regime 1 data is close to Regime 1 rate while the same is true for Regime 2. Interestingly, the deviations between the current interest 19

32 Table 2.4: Pricing American call option with trinomial tree American Call Option Regime 1 Regime 2 N Price Diff Ratio Price Diff Ratio rate and the interest rate found by put-call parity in both regimes are equal to 0.37%. It is due to the symmetry of two regimes in terms of the transition probabilities. The mechanism behind and the meaning of it will be discussed in the next chapter. The result of the American option is similar to that of the CRR model. The prices of the American call option found by the trinomial tree is the same as the European call option. It is consistent with the fact that it is not optimal to exercise an American call option before expiration if the underlying asset pays no dividend. We know that this result is also true for the MRSM. The prices of the American put option in the table are larger than those of the European option, meaning that early exercise of the option is preferred and there are some 20

33 Table 2.5: Pricing American put option with trinomial tree American Put Option Regime 1 Regime 2 N Price Diff Ratio Price Diff Ratio N/A N/A N/A situations in which we have to exercise the American put option before expiration. The convergence pattern of the American put option is more complicated than the European one. The rate of convergence for Regime 2 is very fast, even faster than that of the European put option. The American put option is optimal to be exercised somewhere before the maturity, so the approximation error is smaller than that of the European option. The convergence pattern of Regime 2 is unstable, which is consistent with the results for the European option case; larger initial pricing error in Regime 1 and round-off error affect the convergence of the price in Regime 2. For the down-and-out barrier call option, the prices found in both regimes are smaller than those of the European call option due to the presence of the 21

34 Table 2.6: Pricing down-and-out barrier call option with trinomial tree Down-and-out Barrier Call Option Regime 1 Regime 2 N Price Diff Ratio Price Diff Ratio The barrier level is set to be 90. down-and-out barrier. The prices in the two regimes are closer to each other compared with the case of European option. Although the volatility in Regime 2 is greater and has a higher chance to achieve a higher value at expiration, the high volatility also increases the chance of hitting the down-and-out barrier and thus eliminates its advantage. The convergence pattern of barrier option is complex. It is difficult to get any conclusions from the data. However, we can see that apart from converging uniformly in one direction, the values of the option found in Regime 1 oscillate and the differences still have a decreasing trend in absolute value. It is likely due to the effect of quadratic approximation. The price of the double barrier option can also be obtained by the trinomial 22

35 Table 2.7: Pricing double barrier call option with trinomial tree Double Barrier Call Option Regime 1 Regime 2 N Price Diff Ratio Price Diff Ratio The barrier level is set to be 70 and 150. model. The method suggested by Boyle and Tian (1998) is adopted here. The lattice is built from the lower barrier and touches the upper barrier by controlling the value of σ used in the lattice. Table 2.7 shows the price of the double barrier option with different numbers of time steps being used. The lower barrier is 70 and the upper barrier is 150. The values decrease progressively and converge. Table 2.8 summarizes the values of the double barrier options with different barrier levels using 1000 time steps. When the difference between the upper and lower barriers is smaller, the price of the options is reduced as there is a higher chance of touching the barrier and becoming out of value. The effect of barriers is more significant for Regime 2 because the stock has a higher volatility in Regime 23

36 Table 2.8: Price of double barrier call options with different barrier levels Double Barrier Call Option in Regime Double Barrier Call Option in Regime The price of the double barrier options with lower barrier of 90, 80, 70, 60, 50 and upper barrier of 110, 120, 130, 140, 150, 200 in the two regimes are calculated using 1000 time steps. 24

37 Table 2.9: Pricing European call option with trinomial tree: great derivation in volatilities European Call Option Regime 1 Regime 2 N Price Diff Ratio Price Diff Ratio The volatilities of the two regimes are 0.10 and 0.50 respectively. 2, hence it has a greater chance of reaching the barriers. When the difference between the barriers increases, their effect on the barrier options is reduced and the options in Regime 2 with a larger volatility have a higher price than that of the same option in Regime 1. Their prices are lower than those of the vanilla call option, which has prices of and in the two regimes, respectively, found by trinomial tree with 1000 time steps. We now consider a few more examples. We predict that the convergence rate reduces if the volatilities of different regimes are largely different from each other. We want to find if the prediction is true. All the other conditions are assumed to be the same, but the volatilities of the asset under the two regimes become 25

38 0.10 and The prices of the European call option are found. The risk neutral probabilities of Regime 1 of 20 time steps case in the three branches are , , , respectively. Most of the probabilities are distributed on the middle branch. The price of European option is positively related to the volatility and so the value in Regime 1 decreases while the value in Regime 2 increases, when compared with the results of previous example. The pricing error in Regime 1 is larger compared with the results in the previous example as a large σ is being used to construct lattice. We can make use of the fact that the price of European option converges with order 1 so that a better result can be obtained even with a smaller number of time steps. Next we consider a three-regime example. This example is used to examine the efficiency of the trinomial tree under multi-state market. The interest rate and the volatility in the three regimes are 4%, 5%, 6% and 0.20, 0.30, 0.40, respectively. The initial price and strike price are both set as 100 and the generator matrix is taken as (2.13) The numerical results are shown in Table They show that the conver- 26

39 Table 2.10: Pricing European call option under model with three regimes European Call Option Regime 1 Regime 2 Regime 3 N Price Diff Price Diff Price Diff N is the number of time steps used in calculation. Diff is referring to the difference in price calculated using various numbers of time steps. gence pattern is similar to that of the two-regime case. That is, the convergence rate is still order 1 even for the three-regime case. The convergence property is very useful as it can help us approximate the price of vanilla options even with a small number of time steps. 2.4 Alternative Models Several amendments can be made to improve the rate of convergence or adaptability of the model under other situations. In the last section, it is assumed that the generator of the Markov process is a constant matrix and the volatilities of different regimes do not greatly deviate from the others. These two constraints 27

40 can be relaxed in some situations. The generator process can be a function of time. If it is continuous, an approximation can be used in the branches of each time point, for example, at the branches at time t to t+, the transition probability matrix can be approximated by the following equation, P (t, ) = p t,11 ( ) p t,1k ( )..... p t,k1 ( ) p t,kk ( ) e A(t). (2.14) The value of the options found by the lattice still converges to the value of the options under a continuous-time model. Apart from using I + l=1 ( )l A(t) l /l! to approximate the value of transition probability matrix, another expression can also be used, P (t, ) lim (I + A(t) n n )n = lim (I + A(t) ) 2n. (2.15) n 2 n This expression has also a good performance in approximating the value of P (t, ) using recursion in computer. It is important because the transition probability matrix has to be calculated for each time step. A good approximation method can greatly improve the efficiency of computation. When the number of regime states is large, the volatilities of the asset in different regimes might not be close to each other. The lattice in the last section is constructed by a value, σ, which is larger than the asset s volatilities in all 28

41 regimes, so that all regimes can be incorporated into this recombining lattice. This simplifies calculations. However, when the volatilities in different regimes largely deviate from one another, volatilities are relatively small in some regimes. But since the model still has to accommodate the largest σ i, the σ used in the model is large. For those regimes with small volatilities, due to the up and down ratios used in the tree are large, a high risk neutral probability has to be assigned to the middle branch. The initial error of these regimes is relatively larger. A recombining trinomial tree can be used to solve the problem. When we confront a number of regimes corresponding to quite different volatilities, we can divide the regimes into groups according to their size of volatility. The regimes with large volatility are grouped together, and so are the regimes with small volatility. The trinomial model can be applied to each group with regimes whose volatilities are close to each other. The trinomial lattices are then combined to form a multi-branch lattice, which is similar to the model suggested by Kamrad and Ritchken (1991) in the (2 k + 1)-branch model. More branches can be introduced to handle more complex situations in the market. All of them share the same middle branch. The problem is that the parameters σ in different trinomial lattices do not necessarily match. When the lattices are combined, the branches in each of the lattices need not meet each other, that is, the ratios used in one lattice are not multiples of the other lattices and the simplicity of 29

42 the model disappears because the branches do not recombine in the whole lattice efficiently and the number of nodes in the tree is very large. In order to preserve the simplicity of the model and improve the rate of convergence for the low-volatility regimes at the same time, a similar idea used in the lattices by Bollen (1998) can be adopted. All regimes are divided into two groups. In fact, they can be separated into more than two groups, but for purposes of illustration, we only use two groups here. Again, the σ used in trinomial lattice by the group with larger volatility is not necessarily a multiple of the σ used by the other group. The problem can be solved by adjusting the value of σ in either group or even both of the groups, depending on the situation. The volatility of the group with large volatility should be at least double that of the small volatility group; otherwise the multi-state trinomial tree in the previous section should be good enough for pricing. If the ratio between the two values is larger than 2, the values of lattice parameters σ in both groups are adjusted so that their ratio is set to 2. In practice, the ratio should not be very large. This model should be able to handle real data. Similar to the model that we propose in the last section, assume that there are k regimes and they are divided into two groups, k 1 of them in the low volatility group and k 2 of them in the high volatility group. The states of economy are 30

43 arranged in ascending order of volatility and so σ 1 σ 2... σ k1... σ k. We now construct the combined trinomial tree in which the stock can increase with factors e 2σ and e σ, remain unchanged, or decrease with factors e σ and e 2σ. At time step t, there are 4t + 1 nodes in the lattice, the node is counted from the lowest stock price level, and S t,n denotes the stock price of the n th node at time step t. Each of the nodes has k possible derivative prices corresponding to the regime states. Let V t,n,j be the value of the derivative at the n th node at time step t in the j th regime state. The regimes of group 1 use the middle three branches with ratios e σ, 1, and e σ. The regimes of group 2 use the branches with ratios e 2σ, 1, and e 2σ. We have to ensure that the combined trinomial tree can accommodate all regimes so that the risk neutral probabilities of all regimes exist. That is σ > max 1 i k 1 σ i and 2σ > max k 1 +1 i k σ i. (2.16) For the regime i, π i u, π i m, and π i d are the risk neutral probabilities for up, middle and bottom branches of the tree, respectively. Similar to the trinomial tree model, the following set of equations can be obtained. For 1 i k 1, π i ue σ + π i m + π i de σ = e r i, (2.17) (π i u + π i d)σ 2 = σ 2 i ; (2.18) 31

44 for k i k, π i ue 2σ + π i m + π i de 2σ = e r i, (2.19) (π i u + π i d)(2σ) 2 = σ 2 i. (2.20) Solving the equations above, we have, for 1 i k 1, πm i = 1 σ2 i σ, 2 (2.21) πu i = eri e σ πm(1 i e σ ), e σ e σ (2.22) πd i = eσ e ri πm(e i σ 1), e σ e σ (2.23) and for k i k, πm i = 1 σ2 i 4σ, 2 (2.24) πu i = eri e 2σ πm(1 i e 2σ ), e 2σ e 2σ (2.25) πd i = e2σ e ri πm(e i 2σ 1). e 2σ e 2σ (2.26) With the payoff of a derivative in different regimes at expiration, the price of the derivative at different regimes can be found using the following two equations recursively. For 1 i k 1, [ k ] V t,n,i = e r i p ij (πuv i t+1,n+3,j + πmv i t+1,n+2,j + πdv i t+1,n+1,j ), (2.27) j=1 32

45 For k i k, [ k ] V t,n,i = e r i p ij (πuv i t+1,n+4,j + πmv i t+1,n+2,j + πdv i t+1,n,j ). (2.28) j=1 A simple example is given here to illustrate the idea. We assume that there are three regimes in the market. The corresponding volatilities and risk-free interest rates in these regimes are 0.15, 0.40, 0.45 and 4%, 6%, 8%, respectively. The generator matrix of the regime-switching process is (2.29) Under the trinomial model in Section 2, the suggested value of σ is and the risk neutral probabilities of Regime 1 under the up, middle and down state with 20 time steps used are , , , respectively. The convergence rate of the price of derivatives in this regime is affected due to the volatility difference. If the three regimes are divided into two groups, Regime 1 forms the low volatility group and Regimes 2 and 3 form the high volatility group. By (2.9), the corresponding σ value in each of the trinomial trees is suggested to be σ(1) = ( 1.5 1)0.15 = , σ(2) = ( 1.5 1)( )/2 =