Markov functional model on a lattice

MARKOV-FUNCTIONAL MODEL ON A LATTICE PEE MENG HUAT (B.Science (Hons), NUS) A THESIS SUBMITTED FOR THE DEGREE OF MASTER OF SCIENCE DR OLIVER CHEN XIU-FEN DEPARTMENT OF MATHEMATICS NATIONAL UNIVERSITY OF SINGAPORE 2007/2008 Acknowledgements First and foremost, I would like to thank my family for providing such a conductive environment for me to studying in everyday after I returned home from school Most importantly to my wonderful girlfriend, Meiqi, who has never failed to give me support and encouragement whenever I need them You are really a gift from heaven and I truly appreciate everything that you have done for me deep down my heart Just can’t thank you enough with just words I would also like to extend my gratitude to my thesis supervisor, Dr Oliver Chen, who has shown me what it meant by doing research in Financial Mathematics You are the one who helps steer my career path into finance and I have really to thank you for that I am also forever in gratification of the knowledge that you have imparted to me over the last three years Once again, thanks so much for everything I would also like to thank the Department of Mathematics for giving me the chance to conduct tutorial classes even though I have no prior experience in teaching I really gain a great deal from the experience and it definitely made me a better person Last but not least, to all my friends that I have got to know while studying in NUS You made my time in NUS more enjoyable and fulfilling MENG HUAT NOVEMBER 2007 ii Author’s Contribution The author gave a brief review of the existing literature on interest rate modeling in Chapter The author then introduced Markov-Functional interest rate model in Chapter 2, explaining its advantages over the conventional models discussed in Chapter 1, including its perfect calibration to caps and floors as well as its efficient implementation The author then described the Continuous Time Lattice Method in detail in Chapter 3, including its derivation and proofs The application of the Continuous Time Lattice Method to the Markov-Functional interest rate framework is original This allows for efficient and exact numerical computations Algorithm for implementing the Continuous Time Lattice Method to obtain the probability kernel was proposed by the author Algorithms for implementing the MarkovFunctional model were also proposed, one in which a perfect calibration to digital caplets is obtained and another which focus more on the accuracy of the discount bond prices All the Matlab codes (Appendix C) in this thesis are the work of the author The numerical results, including tables and graphs, in Chapter were all produced by these codes iii Contents Acknowledgements ii Author’s Contributions iii Introduction vi List of Tables viii List of Figures ix Chapter 1 10 Review of Literature 1.1 1.2 1.3 Chapter Spot Rate Models HJM Model Market Rate Models 16 17 19 21 Markov-Functional Models 2.1 2.2 2.3 Chapter The Markov Process Caplets and Digital Caplets The LIBOR Markov-Functional Model 26 27 35 Continuous Time Lattice Method 3.1 3.2 Chapter Continuous Time Lattice Method Application to the Markov-Functional Model 38 38 45 Numerical Results 4.1 4.2 Implementation of the Algorithms An Alternative Approach Conclusion 57 Bibliography 59 Appendix A Theorems and Results 62 Appendix B Data 65 iv Appendix C 67 Matlab Codes v Introduction The traditional approach to interest rate modeling involves taking one mathematically convenient underlying interest rate and making certain distributional assumptions about it to reflect the future uncertainty of the rate Common choices of the underlying interest rate include the instantaneous spot and forward rates Interest rate models of this category are the focus of the first part of Chapter However, neither the instantaneous spot rate nor the instantaneous forward rate is directly observable in the market Moreover, there exists difficulty in calibrating either of the models to prices of actively traded instruments such as caps In the second part of Chapter 1, instead of using the spot or forward rates as the basis for modeling, the model uses rates which are actually traded in the market such as the LIBOR rates Hence, the model is commonly known as Market Models Even though the Market Model can fit the market prices of caps exactly if they are given by the Black’s formula, American-style instruments pose a problem and make efficient numerical implementation difficult Therefore, in Chapter 2, we introduce the Markov-Functional Interest Rate Model, which will be the main focus of this thesis The defining characteristic of Markov-Functional model is that prices of discount bonds are functions of some lowdimensional Markov process This allows for the efficient implementation of the model The model can also fit the market prices of liquid instruments such as caps perfectly In order to specify the functional form of the prices of discount bonds, it is necessary to make the assumption that the underlying Markov process can only take on a set of discrete values However, this would mean that there will be numerical integration vi over discrete set of lattice points, introducing discretisation error To overcome the problem, we introduce the Continuous Time Lattice Method in Chapter In this way, integrals turn into finite sums and numerical integration will be exact Algorithms are provided, both for the generation of the probability kernel under the Continuous Time Lattice Method and the implementation of the Markov-Functional model Chapter implements the algorithms described in Chapter and examines the numerical results The algorithm is able to produce an exact fit between the market and model prices of digital caplets while compromising on the accuracy of the prices of the discount bonds Since discount bonds are instruments traded more actively than digital caplets, it is more appropriate to ensure the accuracy of the prices of discount bonds at the expense of having a slight difference between the model and market prices of digital caplets Hence an alternative algorithm is proposed to achieve the above objective The algorithm involves specifying the functional form of the LIBOR rates first and obtaining the discount bonds prices from it The appendixes are provided to give additional information on the material covered in this thesis Appendix A lists all the theorems and results that will be used at one point or another in the thesis Appendix B provides the necessary data in the implementation of the algorithms in the form of graphs Finally, Appendix C provides the MATLAB codes for the various algorithms described in this thesis vii List of Tables 4.1 Accuracy of the Yield Curve 43 4.2 The Partitions 52 4.2 Objective Function Values 53 viii List of Figures 4.1 Probability Kernel 41 4.2 LIBOR Rates for various Markov process values 42 4.3 LIBOR Rates as a function of the Lattice Points 46 4.4 Worst fit at Ti = 54 4.5 Best fit at Ti = 55 B.1 Zero Coupon Yield Curve 65 B.2 Implied Volatilities of Caplets 66 ix Review of Literature In this chapter, I will give a brief review of the different types of interest rate models available, including the spot rate models, forward rate models and the market rate models I will also spend some time discussing the various advantages and disadvantages of each of the models before I introduce the Markov-Functional model, which will be the main focus of my thesis, in the following chapter 1.1 Spot Rate Models The first part of this chapter describes spot rate models These are models that utilize the instantaneous spot interest rates as the basis for modeling the term structure of interest rates The spot rate r, at time t is the rate of interest that one earns on a riskless investment in an infinitesimally short time period at time t That is, in a short time period between t and t + ∆t , investors earn a percentage r (t )∆t The spot rate is then modeled as a stochastic variable, governed by a stochastic differential equation of the form dr = µ (r , t )dt + σ (r , t )dW , where µ (r , t ) and σ (r , t ) are functions of spot rate r and time t Different choices for the functions µ and σ will give rise to different models Appendix A Theorems and Results Appendix A lists the set of theorems and results that will be used and quoted at some point in the thesis Unique Equivalent Martingale Measure Theorem Harrison and Kreps (1979) showed that, given a choice of numeraire M, we can find a unique probability QM measure such that the relative price process V ' (r , t ) = V (r , t ) / M (t ) is a Q M − martingale Hence, we have the fact that if a derivative has a payoff V (r , T ) at maturity T,  V (r , T )  V (r , t )  = E tM  M (t )  M (T )  The above result can be used to find the value of a derivative V (r , t ) at time t < T by  V (r , T )   V (r , t ) = M (t ) E tM  M T ( )   Change of Numeraire Theorem The above equation for the calculation of the value V (r , t ) of a derivative must be independent of the choice of the numeraire Geman (1995) showed that given two 62 numeraires N and M with equivalent martingale measure Q N and Q M respectively, we have  V (r , T )   V (r , T )    = M (t ) E tM  N (t ) E tN   M (T )   N (T )  Rearranging the above, we get  N (T ) / N (t )   E tN (G (T )) = E tM  G (T ) M (T ) / M (t )   where G (T ) = V (r , T ) The expectation of G under the measure Q N is thus equal to the N (T ) N (T ) / N (t ) under the measure Q M M (T ) / M (t ) expectation of G multiply by the random variable The random variable above is known as the Radon-Nikodym derivative and is denoted by dQ N dQ M Girsanov’s Theorem For any stochastic process κ (t ) such that t ∫ κ ( s) ds < ∞ dQ N = ρ (t ) is given by with probability one, then the Radon-Nikodym derivative dQ M t ρ (t ) = exp( ∫ κ ( s )dW M ( s ) − t κ ( s ) ds ) ∫ where W M is a Brownian motion under the measure Q M Under the measure Q N , the process t W N (t ) = W M (t ) − ∫ κ ( s )ds 63 is also a Brownian motion The above can also be written in differential form as dW M = dW N + κ (t )dt Ito’s Lemma Given a stochastic process x governed by the following stochastic differential equation dx = µ (t , x)dt + σ (t , x)dW If f (t , x) is a sufficiently differentiable function of t and x, then it follows the following stochastic differential equation  ∂f ∂f ∂f ∂ f  dt + σ (t , x) dW df =  + µ (t , x) + σ (t , x)  ∂x ∂x ∂x   ∂t 64 Appendix B Data Figure B.1 Zero Coupon Yield Curve 65 Figure B.2 Market Implied Volatilities of Caplets 66 Appendix C Matlab Codes Code C.1 Codes for generating the probability kernel and the terminal discount bond prices %Specifying the parameter values N = 10; M = 500; a = -3.5; b = 3.5; interval = (b-a)/M; sigma = 0.25; sigmax = 0.25; tau = 1; x0 = M/2; D = zeros(M+1,N+1); Lib = zeros(M+1,N); DC = zeros(M+1,N); p = zeros(N,M+1,M+1); D(:,N+1)=ones(M+1,1); %Probability Kernel Algorithm L = zeros(M+1,M+1); for i = 0:M A(i+1) = 0.5*(sigmax/interval)^2; C(i+1) = 0.5*(sigmax/interval)^2; B(i+1) = -(sigmax/interval)^2; end B(1) = -0.5*(sigmax/interval)^2; B(M+1) = -0.5*(sigmax/interval)^2; L1 = diag(A(1:M),1); L2 = diag(B); L3 = diag(C(2:M+1),-1); 67 L = L1 + L2 + L3; [V D1] = eig(L); inverse = inv(V); for i = 1:N D2 = diag(exp(i*tau*diag(D1))); p(i,:,:) = V*D2*inverse; end %Terminal Discount Bonds Prices Algorithm VerifyBond = zeros(N,1); for x = 0:M DC(x+1,N) = D0(N+1)*normcdf(-(a + x*interval)/(sigmax*sqrt(N*tau))); Lib(x+1,N) = L0(N)*exp(-sigma^2*N*tau/2 + sigma*sqrt(N*tau)*(a + x*interval)/(sigmax*sqrt(N*tau))); D(x+1,N) = (1 + tau*Lib(x+1,N))^(-1); VerifyBond(N) = VerifyBond(N) + p(N,x0+1,x+1)/D(x+1,N); end VerifyBond(N) = D0(N+1)*VerifyBond(N); for i = N-1:-1:1 for x = M:-1:0 sum = 0; for index = 0:M sum = sum + p(1,x+1,index+1)/D(index+1,i+1); end if x == M DC(x+1,i) = D0(N+1)*p(i,x0+1,x+1)*sum; else DC(x+1,i) = DC(x+2,i) + D0(N+1)*p(i,x0+1,x+1)*sum; end if DC(x+1,i)/D0(i+1) Rate(J) x = x - 1; end K = Lib0(x+1); sigmkt(J,i) = sigma(K); sigmdl(J,i) = fzero('volatility',sigma(K)); err(i) = err(i) + (sigmdl(J,i) - sigmkt(J,i))^2; end figure(i); plot(Rate,sigmkt(:,i),'bo-',Rate,sigmdl(:,i),'rx-'); title('Implied Volatility'); ylabel('Implied Volatility'); xlabel('Strikes'); end err Code C.6 Function for calculating the objective function value function y = objective(arg) global i; global x; global Lib0; global Rate; global K; M = 500; tau = 1; a1 = arg(1)^2; a2 = arg(2)^2; a3 = arg(3)^2; a4 = arg(4)^2; a5 = arg(5)^2; yintersect = intersect(arg); if yintersect < y = 1000000*(100*(a1 + a2 + a3) + 10*a4 + a5); else for x = 0:M Lib0(x+1) = LibAtOrigin(arg) + yintersect; end 72 if Lib0(M+1) < Rate(6) y = 1000000*(100*(0.055 - a1) + 10*(0.055 - a2) + (0.055 - a3) + (0.055 - a4) + (0.055 - a5)); elseif Lib0(1) > Rate(1) y = 1000000*a1; else y = 0; x = M; for J = 6:-1:1 while Lib0(x+1) > Rate(J) x = x - 1; end K = Lib0(x+1); sigmkt = sigma(K); sigmdl = fzero('volatility',sigma(K)); y = y + (sigmdl - sigmkt)^2; end end end Code C.7 Function for the functional form of the LIBOR rate function y = LibAtOrigin(arg) global x; global i; a1 = arg(1)^2; a2 = arg(2)^2; a3 = arg(3)^2; a4 = arg(4)^2; a5 = arg(5)^2; if i == 10 | i == if x [...]... by calibrating the model to prices of market instruments If we were to fit the model to digital caplets, we have the LIBOR Markov- Functional model Alternatively, we can also fit the model to digital swaptions, which will lead us to the Swap Markov- Functional model I will however only cover the LIBOR Markov- Functional model in this section A caplet with strike K, which sets at time Ti but pays out at... of caplets and digital caplets, which are the market instruments to which we are calibrating the Markov- Functional model The main 16 objective of the calibrating process is to obtain the functional forms of the terminal discount bonds prices This is the focus of Section 2.3 2.1 The Markov Process The main assumption made in the Markov- Functional model is that the state of the economy can be summarized... Market Model Models that use real market interest rates, in which traders are used to working with, as a basis for modeling interest rates, are known as market rate models One such example that I will elaborate in this section is the LIBOR market model, which as the name suggests, uses the LIBOR rate as a basis for modeling Market rate models are more complicated in their setup but the main advantage they... forward LIBOR rates are no longer martingales under the terminal measure Q N +1 They have a drift which depends on the forward LIBOR rates of longer maturities As the stochastic differential equation is fairly complicated, we are not able to solve it analytically but will have to resolve to using numerical methods such as Monte Carlo simulations 15 2 Markov- Functional Models In the previous chapter,... as the efficiency of calculating derivative prices as in the spot rate models This is the general class of Markov- Functional models, introduced by Hunt, Kennedy and Pelsser (2000) The defining characteristic of the Markov- Functional models is the assumption that the discount bonds prices at any time are a function of some low-dimensional Markovian process under some martingale measure This ensures its... disadvantages The first one being that both the spot rate and instantaneous forward rate are not directly observable in the market Another drawback is the difficulty in calibrating the model to prices of actively traded instruments such as caps To address these concerns, we consider a model proposed by Brace, Gatarek and Musiela (1997) which is known as the LIBOR market model 10 1.3.1 LIBOR Market Model. .. realized spot rate r (t ) The disadvantage of the HJM model is that Monte Carlo simulation such as the one mentioned above can be slow However, because the whole forward rate curve is calculated, the bonds prices are easily obtained 1.2.2 An Example: Ho-Lee Model In the previous section, we have seen that under the HJM framework, the stochastic differential equation governing the instantaneous forward... provides a perfect setting for the application of the Continuous Time Lattice Method The Continuous Time Lattice Method discretizes the state space of a stochastic process while maintaining continuous time Probability kernels become matrices so that evaluating expectations involve finite sums instead of integrals Application of the Continuous Time Lattice Method to the Markov- Functional interest rate framework... spot rate models discussed previously have HJM representation 1.3 Market Rate Models In the previous two sections of Chapter 1, we look at spot rate and forward rate models for interest rate, in which either the spot rate or instantaneous forward rate is used as a basis for modeling This is a mathematically convenient choice, leading to models which are highly tractable However, it does in fact have... now make the assumption that the one-dimensional Markov process follows dxt = σ (t )dWt N +1 17 where σ (t ) is a deterministic function of time and Wt N +1 is a Brownian motion under the measure Q N +1 For a time s > t , conditional on xt , we know that the random variable x s follows a normal distribution with mean xt and variance s ∫ σ (u) t 2 du The conditional probability density function of ... Implementation of the Algorithms An Alternative Approach Conclusion 57 Bibliography 59 Appendix A Theorems and Results 62 Appendix B Data 65 iv Appendix C 67 Matlab Codes v Introduction The traditional... Literature 1.1 1.2 1.3 Chapter Spot Rate Models HJM Model Market Rate Models 16 17 19 21 Markov- Functional Models 2.1 2.2 2.3 Chapter The Markov Process Caplets and Digital Caplets The LIBOR Markov- Functional. .. introduced Markov- Functional interest rate model in Chapter 2, explaining its advantages over the conventional models discussed in Chapter 1, including its perfect calibration to caps and floors as well