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1 Florida State University Libraries Electronic Theses, Treatises and Dissertations The Graduate School 2008 Stochastic Volatility Extensions of the Swap Market Model Milena G. (Milena Gueorguieva) Tzigantcheva Follow this and additional works at the FSU Digital Library. For more information, please contact

2 FLORIDA STATE UNIVERSITY COLLEGE OF ARTS AND SCIENCES STOCHASTIC VOLATILITY EXTENSIONS OF THE SWAP MARKET MODEL By MILENA G. TZIGANTCHEVA A Dissertation submitted to the Department of Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy Degree Awarded: Fall Semester, 2008

3 The members of the Committee approve the Dissertation of Milena G. Tzigantcheva defended on October 8, Craig Nolder Professor Directing Dissertation Fred Huffer Outside Committee Member Bettye Anne Case Committee Member Alec Kercheval Committee Member Jack Quine Committee Member De Witt Sumners Committee Member The Office of Graduate Studies has verified and approved the above named committee members. ii

4 To my daughter Adriana iii

5 ACKNOWLEDGEMENTS The author is grateful to all members of her ATE committee, namely Dr. Nolder, Dr. Case, Dr. Kercheval, Dr. Quine, Dr. Sumners and the outside committee member Dr. Huffer for their comments and suggestions on this dissertation. Also, Milena thanks both departmental chairs while she has been a graduate student: Dr. Sumners and Dr. Bowers. Most special and sincere thanks go to Dr. Nolder for his continuous and vital support in the process of preparing this dissertation. Milena thanks Dr. Case for her encouragement and inspiration. Regarding her teaching career and overall experience at FSU, the author also acknowledges Dr. Stiles and Esther Diaguila for their professionalism. Milena thanks Elaine Baxley for technical support with Matlab. Many thanks to the SBA librarian Sue McCauley for the research assistance with literature, and the following SBA co-workers for their support: Jim Francis, Kevin SigRist, John Benton, and Pam Noda. The author thanks her husband Dimitre and her daughter Adriana for their patience, encouragement and belief in her as most of the work was done at home. Last, but certainly not least, Milena thanks her parents Elena and George Stoyanov and her mother-in-law Adriana Tzigantcheva for their support from overseas. iv

6 TABLE OF CONTENTS List of Tables List of Figures vii viii Abstract ix 1. INTRODUCTION PRELIMINARY Interest Rates Interest-Rate Swaps Interest-Rate Swaptions Change of Numeraire Swap Market Model (SSM): Overview Fourier Transform and Characteristic Function Compound Poisson Process Stochastic Volatility Models: Overview STOCHASTIC VOLATILITY EXTENSIONS OF SMM Stochastic Volatility Extension Of SMM Without Jumps Stochastic Volatility Extension Of SMM With Log-Normal Jumps FAST FRACTIONAL FOURIER TRANSFORM Fast Fractional Fourier Transform (FRFT): Overview FRFT Algorithm Application Of The FRFT CALIBRATION Overview Data Calibration Step One Calibration Step Two RESULTS In-Sample Fit Out-Of-Sample Fit v

7 6.3 Forecasting CONCLUSION Future Work APPENDICES A. MATLAB CODE B. ABBREVIATIONS AND NOTATION REFERENCES BIOGRAPHICAL SKETCH vi

8 LIST OF TABLES 2.1 Black Implied Volatility of ATM Swaptions on 2/12/ Examples of Bivariate Diffusion Models for Option Pricing Swap Rates as of 2/12/ Discount Factors as of 2/12/ Fitted instantaneous volatility parameters on 2/12/ Scaling Factors on 2/12/ Fitted model parameters on 2/12/ Descriptive statistics for the forward swap rates Descriptive statistics for the fitted swap rate volatility parameters Fitted models parameters for the SV SMM Fitted models parameters for the LN-SV SMM Statistical Comparison of in-sample fit based on Diebold and Mariano criterion Statistical Comparison of in-sample fit based on BIC criterion Run time table Forecasting performance Of SV SMM Forecasting performance Of LN-SV SMM Forecasting performance Of BS SMM Statistical Comparisons of forecasting performance based on Diebold and Mariano criterion Statistical Comparison of forecasting performance based on BIC criterion.. 77 vii

9 LIST OF FIGURES 3.1 Time Series of Implied Volatilities for ATM Swaptions Implied Volatilities of ATM Swaptions Across Option Maturities for Different Swap Tenors ATM Implied Volatility Surface on 2/12/ Bloomberg SWPM Curve on 2/12/ Live Zero Coupon Rate and Discount Factor Calculator Descriptive statistics for the market ATM swaption volatility data SV SMM results for Swaptions with Total Maturity Less Than 10 years LN-SV SMM results for Swaptions with Total Maturity Less Than 10 years. 74 viii

10 ABSTRACT Two stochastic volatility extensions of the Swap Market Model, one with jumps and the other without, are derived. In both stochastic volatility extensions of the Swap Market Model the instantaneous volatility of the forward swap rates evolves according to a squareroot diffusion process. In the jump-diffusion stochastic volatility extension of the Swap Market Model, the proportional log-normal jumps are applied to the swap rates dynamics. The speed, the flexibility and the accuracy of the fast fractional Fourier transform made possible a fast calibration to European swaption market prices. A specific functional form of the instantaneous swap rate volatility structure was used to meet the observed evidence that volatility of the instantaneous swap rate decreases with longer swaption maturity and with larger swaption tenors. ix

11 CHAPTER 1 INTRODUCTION The two market models of interest rates dynamics, the LIBOR Market Model (LMM) and the Swap Market Model (SMM), have become gradually more accepted among practitioners and have attracted much of the academics interest. The construction of a mathematically consistent theory of a term structure with discrete LIBOR rates being lognormal was achieved by Miltersen, Sandmann and Sondermann (1997) and developed by Brace, Gatarek and Musiela (1997). In the LMM arbitrage-free dynamics are assigned to a set of non-overlapping forward LIBOR rates, leading to the well-known Black pricing formula for caps. Jamishidian (1997) developed a similar model for swap rates that prices swaptions with the well-known Black s swaption formula. However, the two market models are not compatible. If forward LIBOR rates are lognormal under its measure, as assumed by the LMM, forward swap rates cannot be lognormal at the same time under their measure, as assumed by the SMM. Most of the research has been focused on the LIBOR-based models: pricing, hedging, calibration and extensions. Very little has been published on its swap rate counterpart. Models based on dynamics assigned to forward swap rates are often considered less tractable than the LMM, both in theory and practice, although the two approaches are mathematically very similar. The LMM is used as a base to price both caps and swaptions. Academics have used approximations in order to price European swaptions within the LMM (see Hull and White (2000) and Jäckel and Rebonato (2003)). Although the SMM is the most natural model for swaptions, modeling the forward swap rates, very few researchers have studied its use for swaption pricing (see Galluccio and Hunter (2004) for the case of co-initial swap rates, Gallucicio et al. (2004, 2007) and De Jong et al. (2000) for that of co-terminal swap rates). The first stochastic volatility extension of the SMM was developed by Attaoui (2006). In this model, he makes a zero-correlation assumption between the changes in the swap rate 1

12 and the changes in the swap volatility. We explore two stochastic volatility extensions of the Swap Market Model: one with jumps (LN-SV SMM), and the other without (SV SMM), where the instantaneous volatility of the forward swap rates evolves according to a square-root diffusion process. First, we derive a stochastic volatility extension of the SMM à la Heston, similar to that of Attaoui (2006), where all instantaneous forward swap rate volatilities are equally affected by a multiplicative stochastic volatility factor. However, we assume that there is a non-zero correlation between the changes in the swap rate and the changes in the swap volatility. Stochastic Volatility models account for volatility clustering, dependence in increments, and long-term smiles and skews, but cannot generate jumps which might account for short-term instantaneous volatility patterns. These shortcomings are overcome by the introduction of jumps in either the underlying or in the volatility dynamics. The jump-diffusion SV model introduced by Bates (1996) adds proportional log-normal jumps to the Heston SV model (1993). Next, we derive the jump-diffusion stochastic volatility extension of the SMM à la Bates by adding jumps to the swap rate dynamics. We investigate the pricing power of each SV extension model to European swaptions. Using the fast fractional Fourier transform (FRFT), we calibrate both SMM extensions to market swaption data. It s been the practice to use the fast Fourier transform or FFT method to obtain option prices (see Carr and Madan (1999)). However, the method is not very flexible with respect to implementation. The integration grid and the log-strike grid cannot be chosen independently. This shortcoming is overcome if one uses the fast fractional Fourier transform method as shown by Chourdakis (2005). The FRFT algorithm has the advantage of using the characteristic function in a more efficient way than the standard FFT and saves computational time without loss of accuracy. Actually, the straight FFT can be seen as a special case of the FRFT. The FRFT implementation we follow is described in detail in Bailey and Swarztrauber (1991, 1994). We separate the calibration into a two-step process. The first step is the calibration for the instantaneous forward swap rate volatilities. We consider the parametric form suggested by Attaoui that takes into account the swap tenor as well as the option maturity. This feature allows the calibration to be performed to the whole swaption volatility matrix. The form also meets the observed evidence that volatility decreases with longer swaption maturity and larger swap tenors. The second step is to calibrate the SV SMM and LN-SV SMM 2

13 for the models parameters using the instantaneous volatility obtained in the first step. We can calibrate each model to the set of co-terminal swaptions with total maturity of 10 years for which Black formulas are available. For the best fit, we use Matlab build in function lsqnonlin(). Once we have the fitted parameters we can use the model to price swaptions with total maturity of less than 10 years. We then compare the swaption prices from the models to the observed on the market to investigate the pricing power of each model. Choosing the set of the model calibration instruments must be driven by practical considerations and should be the most informative in capturing the volatility. Calibrating both SMM extensions to the set of the co-terminal swaptions with total maturity of 10 years is enough to capture the volatility risk for all swaptions with total maturity less than or equal to 10 years. Our out-of-sample study shows that the set of the co-terminal swaptions is optimal for large portion of swaptions. Our empirical results show that to achieve a close fit to the market, one needs to incorporate both stochastic volatility and jumps into pricing models. The SV SMM does a better job overall, both in and out of sample pricing. The LN-SV SMM fits the short maturities better, but fails to do so for long maturities. Its overall fit is not as satisfactory as that of the SV SMM. There is strong evidence of negative correlation between forward swap rate changes and changes in forward swap rate volatility very similar to the observed for stock prices. Both SV SMM and LN-SV SMM have better prediction power than the BS SMM. However, LN-SV SMM produces better forecasts for swaptions with short option maturity and worse forecasts for swaptions with longer option maturity. 3

14 CHAPTER 2 PRELIMINARY [1] [2] [3] 2.1 Interest Rates Please refer to Appendix B for a list of Abbreviations and Notation. Definition Bank Account We define B(t) to be the value of a money-market account at time t 0. The bank account changes according to the following differential equation: where r t is a positive function of time. db(t) = r t B(t)dt, B(0) = 1, Investing one dollar at time zero yields B(t) = e t 0 rsds dollars at time t. The rate r t at which the bank account accrues is referred to as the instantaneous spot rate, or as the short rate. Definition Stochastic discount factor The (stochastic) discount factor D(t, T) is the amount at t that is equivalent to one dollar payable at time T, and is given by D(t, T) = B(t) B(T) = T e t r sds. In many pricing applications r is assumed to be deterministic function of time. However, when dealing with interest-rate products r is modeled through a stochastic process in which case the bank account and the discount factors will be stochastic processes as well. Definition Zero-coupon bond A zero-coupon bond (or discount bond) with maturity T is a contract that guaranties its holder the payment of one dollar at time T. We denote the contract value at time t < T by P(t, T). Obviously, P(T, T) = 1 for all T. If the rates r are deterministic, then D is deterministic as well and D(t, T) = P(t, T) for each pair (t, T). However, if rates are stochastic, D(t, T) is a random quantity at t 4

15 depending on the future evolution of rates r between t and T. Instead, the zero-coupon bond price P(t, T) has to be known at t. The relationship is that P(t, T) is the expectation of the random variable D(t, T) under a specific probability measure. Definition Continuously-compounded spot interest rate The continuouslycompounded spot interest rate current at time t for the maturity T is denoted by R(t, T) and is the constant rate at which an investment of P(t, T) dollars at t accrues continuously to yield one dollar at maturity time T: R(t, T) := ln P(t, T). T t The continuously-compounded interest rate is a constant rate that is consistent with the zero-coupon prices: e R(t,T)(T t) P(t, T) = 1, from which the bond price can be expressed in terms of continuously-compounded rate R: P(t, T) = e R(t,T)(T t). Definition Simply-compounded spot interest rate The simply-compounded spot interest rate current at time t for the maturity T is denoted by L(t, T) and is the constant rate at which an investment has to be made to produce one dollar at maturity, starting from P(t, T) dollars at time t. L(t, T) := 1 P(t, T) τ(t, T)P(t, T) (2.1) where τ(t, T) = T t is the year fraction of time from t to T. The market LIBOR rates are simply-compounded rates. LIBOR (London Interbank Offered Rate) is the most critical interbank rate (rate at which deposits are exchanged between banks) used as a reference for contracts. From (2.1) the bond price in terms of L is P(t, T) = L(t, T)τ(t, T). 5

16 Definition Annually-compounded spot interest rate The annually-compounded spot interest rate current at time t for the maturity T is denoted by Y (t, T) and is the constant rate at which an investment has to be made to produce one dollar at maturity, starting from P(t, T) dollars at time t, when reinvesting the obtained amounts once a year. Y (t, T) := 1 P(t, T) 1 τ(t,t) 1. The bond price in terms of annually-compounded rates is P(t, T) = 1 (1 + Y (t, T)) τ(t,t). All definitions of spot interest rates are equivalent in infinitesimal time periods: r(t) = lim R(t, T) = lim L(t, T) = lim Y (t, T). T t+ T t+ T t+ Fundamental Interest-Rate Curves A zero-coupon curve at a given time t is a fundamental curve that can be obtained from the market data of interest rates current at that time. Definition Yield curve The zero-coupon curve (yield curve) at time t is the graph of the function T { L(t, T), if t < T t + 1(years); Y (t, T), if T > t + 1(years). Such a zero-coupon curve is called the term structure of interest rates at time t. It is a plot at time t of simply-compounded interest rates for all maturities T up to one year and of annually compounded rates for maturities T larger than one year. Definition Zero-bond curve The zero-bond curve at time t is the graph of the function T P(t, T), T > t. The graph is a T-decreasing function starting from P(t, t) = 1 because interest rates are positive quantities. Such a curve is also referred as the term structure of discounted factors. 6

17 Forward Rates Forward rates are interest rates that can be locked today for an investment in the future, and are consistent with the current term structure of discount factors. Forward-rate agreement (FRA) is a contract that gives its holder a fixed payment at the maturity S based on a fixed rate K in exchange of a floating payment based on the spot rate L(T,S) resetting in time T and with maturity S. There are three time instants: current time t, the expiry time T and the maturity time S, with t T S. The value of the contract at time S is Nτ(T,S)(K L(T,S)), where N is the contract s nominal value. The contract value is positive when L < K at T and negative otherwise. Using (2.1) for L, the above contract s value at S is: N[τ(T,S)K 1 P(T,S) + 1]. The amount 1/P(T,S) at time S is equivalent to the amount P(t, T) at time t and the amount τ(t, S)K + 1 at time S is equivalent to P(t, S)τ(T, S)K + P(t, S) at time t. Therefore, the value of the contract at t is: FRA(t, T, S,N, K) = N[P(t, S)τ(T,S)K P(t, T) + P(t, S)]. At the time t when the contract is entered into, the delivery price K is chosen so that the FRA contract is fair, i.e. F RA(t, T, τ(t, S), N, K) = 0 at time t. Definition Simply-compounded forward interest ratethe simply-compounded forward interest rate current at time t for expiry T and maturity S is denoted by F(t; T,S) and is defined by 1 T) F(t; T,S) := (P(t, τ(t,s) P(t, S) 1). Then, the FRA can be written in terms of the forward rate: FRA(t, T, S,N, K) = NP(t, S)τ(T,S)(K F(t; T,S)). (2.2) Therefore, to value FRA, we just need to replace the LIBOR rate L(T,S) in the payoff with the corresponding forward rate F(t; T,S), and then take the present value. Hence, the forward rate may be viewed as an estimate of the future spot rate, which is random at t. 7

18 Definition Instantaneous forward rate The instantaneous forward rate current at time t for maturity T is denoted by f(t, T) and is defined as ln P(t, T) f(t, T) := lim F(t; T,S) =. S T+ T so that T P(t, T) = exp( f(t, u)du). t The instantaneous forward rate is a forward interest rate at t with maturity close to expiry T. 8

19 2.2 Interest-Rate Swaps [1] [2] [3] A generalization of FRA is the Interest-Rate Swap or IRS. An interest-rate swap is a contract to exchange payments between two differently indexed legs starting from a future time. At every instant T i in a pre-specified set of dates T α+1,...,t β the fixed leg pays out Nτ i K, where K is the fixed interest rate, N is the nominal value and τ i = T i T i 1 for every i = α + 1,...,β. The floating leg pays Nτ i L(T i 1, T i ) corresponding to L(T i 1, T i ) resetting at T i 1 for maturity T i. The floating leg resets at T α,...,t β 1 and pays at T α+1,...,t β. The dates T α,...,t β 1 are known as reset dates, and the dates T α+1,...,t β are known as settlement dates. We will refer to T α as the start date of a swap. The number β α, which is also the number of payments, is called the length of a swap. Let us define T := {T α,...,t β } and τ := {τ α+1,...,τ β } When the fixed leg is paid and the floating leg is received the IRS is called a Payer IRS (PFS) and in the other case the IRS is called a Receiver IRS (RFS). The PFS discounted payoff at time t < T α can be written as: β D(t, T i )Nτ i (L(T i 1, T i ) K), i=α+1 and the RFS discounted payoff at time t < T α can be written as: or alternatively as β i=α+1 D(t, T i )Nτ i (K L(T i 1, T i )), β D(t, T α ) P(T α, T i )Nτ i (K F(T α ; T i 1, T i )). (2.3) i=α+1 We can view the IRS contract as a basket of FRAs, value each FRA as in (2.2) and sum up the resulting values: 9

20 β RFS(t, T, τ, N,K) = FRA(t, T i 1, T i, τ i, N,K). i=α+1 And so β RFS(t) = NP(t, T i )τ i (K F(t; T i 1, T i )) i=α+1 β = NK τ i P(t, T i ) NP(t, T α ) + NP(t, T β ). (2.4) i=α+1 Analogously, β PFS(t, T, τ, N,K) = NK τ i P(t, T i ) + NP(t, T α ) NP(t, T β ). i=α+1 Alternatively, the IRS can be viewed as a contract for exchanging a coupon-bearing bond (fixed leg) for a floating note (floating leg). Definition Coupon-bearing bond A coupon-bearing bond is a contract that guarantees payments of deterministic amounts of cash-flows c = c α+1,...,c β at future times T α+1,...,t β. Typically, the cash-flows are defined as c i = Nτ i K for i < β and c β = Nτ β K+N, where K is a fixed interest rate and N is the nominal value of the bond. If K = 0, then the bond is the zero-coupon bond with maturity T β. Discounting each cash-flow to current time t we obtain the value of the bond: β CB(t) = c i P(t, T i ). i=α+1 We can write the IRS value in terms of the cashflows c i as: β RFS(t) = c i P(t, T i ) NP(t, T α ). (2.5) i=α+1 Analogously, β PFS(t) = NP(t, T α ) c i P(t, T i ). i=α+1 Definition Floating-rate note A floating-rate note is a contract that guarantees payments at future times T α+1,...,t β of the LIBOR rates that reset at the previous times T α,...,t β 1. In addition, the note pays a last cash-flow of the notional value at final time T β. 10

21 The floating-rate note is always equivalent to NP(t, T α ). It is simple to see this by changing the sign of the value of RFS in (2.4) assuming no fixed leg (K = 0) and then adding the final cash-flow N at T β. RFS(t, T, τ, N, 0) + NP(t, T β ) = NP(t, T α ). If t = T α, the value of the note is N, so that the value at the first reset time is always equal to the nominal value. This is true for all times T i when i = α + 1,...,β 1. The floating-rate note always trades at par. The fixed rate quoted on an IRS is the rate of interest that must be paid so that the present value of the fixed rate payments is equal to the expected future short-term LIBOR payments for the life of the swap. Definition Forward swap rate The forward swap rate S α,β (t) at time t for the sets of times T and year fractions τ is the rate of the fixed leg of the IRS at which the swap agreement has zero value, i.e. it is the fixed rate K such that RFS(t, T, τ, N,K) = 0. We obtain: S α,β (t) = P(t, T α) P(t, T β ) β i=α+1 τ ip(t, T i ). (2.6) Let us define the positive portfolio of zero coupon bonds C α,β (t) = β i=α+1 τ i P(t, T i ). (2.7) Often C α,β (t) is referred to as the present value of a basis point (PVBP). The swap rate is the forward swap rate at time T α and can be expressed in term of the PVBP as follows S α,β (T α ) = 1 P(T α, T β ) β i=α+1 τ ip(t α, T i ). (2.8) It will be useful to express the value at time t of a given forward swap in terms of the current value of the forward swap rate: RFS(t, T, τ, N) = NC α,β (t)[k S α,β (t)]. (2.9) Analogously, PFS(t, T, τ, N) = NC α,β (t)[s α,β (t) K]. 11

22 Let us introduce the forward zero-coupon-bond price at time t for maturity S as seen from expiry T: FP(t; T,S) = P(t,S) P(t,T). Then, where FP(t; T α, T i ) := P(t, T i) P(t, T α ) = FP j (t) = i j=α τ j F j (t), FP j (t), and FP denotes the forward discount factor. Then the formula (2.6) defining the forward swap rate can be written as: S α,β (t) = 1 FP(t; T α, T β ) β i=α+1 τ ifp(t; T α, T i ), or alternatively as S α,β (t) = 1 β β i=α+1 τ i j=α+1 i j=α τ j F j (t) 1 1+τ j F j (t). From the formula above we see that the simultaneous assumption of lognormal distributed forward rates and lognormal distributed swap rates is not consistent. An alternative expression is ω i (t) = S α,β (t) = β i=α+1 ω i (t)f i (t); τ i FP(t; T α, T i ) β k=α+1 τ kfp(t; T α, T k ) = τ i P(t, T i ) β k=α+1 τ kp(t, T k ). Since 0 ω i 1 for each i and β i=α+1 ω i = 1 it looks like that forward swap rates can be interpreted as weighted averages of spanning forward rates. However, the weights are functions of the F s and thus random at future times. Based on experimental studies one can approximate the ω s by their initial values and obtain 12

23 β S α,β (t) ω 0 (t)f i (t). i=α+1 This is useful for estimating the absolute volatility of swap rates from the absolute volatility of forward rates. Note that Definition of a forward swap rate implicitly refers to a swap contract of length T β T α which starts at time T α. A forward swap rate is a rather theoretical concept, as opposed to swap rates, which are quoted daily by financial institutions to their clients of interest rate swaps. Swap rates are important when it comes to pricing interest rate derivatives such as swaptions. The appropriate swap rate is commonly used as a strike level for swaptions. 13

24 2.3 Interest-Rate Swaptions [1] [2] [3] Swaptions are options on interest rates swaps. There are two main types of swaptions, a payer swaption and a receiver swaption. A European payer(receiver) swaption is an option giving the right to enter a payer(receiver) IRS at a given future time, the swaption maturity. The underlying IRS length T β -T α is called the tenor of the swaption. The set of reset and payment dates is called the tenor structure. Usually the swaption maturity is on the first reset date of the underlying IRS. The discounted payoff of a receiver swaption at first reset time (swaption maturity) T α can be written by considering the value of the underlying receiver IRS at T α. From (2.3) the IRS value is: β N τ i P(T α, T i )(K F(T α ; T i 1, T i )). i=α+1 The option will be exercised only if this is a positive value. Then, the receiver swaption payoff, discounted from the maturity T α to current time is β ND(t, T α )( τ i P(T α, T i )(K F(T α ; T i 1, T i ))) +. i=α+1 The positive part operator is not distributive with respect to sums, but is a piece-wise linear and convex function: ( β i=α+1 β i=α+1 τ i P(T α, T i )(K F(T α ; T i 1, T i ))) + τ i P(T α, T i )(K F(T α ; T i 1, T i )) +. Obviously, the additive decomposition is not possible for swaptions. The joint action of the rates involved in the contract payoff has to be considered when we value swaption contracts. This means that terminal correlation between different rates could be essential in dealing with swaptions. The market practice is to value swaptions with a Black-like formula. For a receiver swaption the market price formula at time zero is: RS Black (0, T, τ, N,K,σ Black α,β ) = NC α,β (0)BL(K, S α,β (0), σα,β Black Tα, 1), 14

25 where σ Black α,β is the Black market quoted volatility. A similar formula is used for the payer swaption: PS Black (0, T, τ, N,K,σ Black α,β ) = NC α,β (0)BL(K, S α,β (0), σα,β Black Tα, 1). Let us mention the put-call parity relationship for swaptions: P ayerswaption(t) ReceiverSwaption(t) = F orwardswap(t), provided that both swaptions expire at the same time T. Definition Let us consider a receiver (payer) swaption with strike K giving the holder the right to enter at time T α a receiver (payer) IRS with payment dates T α+1,...,t β and associated year fractions τ α+1,...,τ β. The swaption (receiver and payer) is said to be at-the-money (ATM) if and only if K = K ATM := S α,β (0) = P(0, T α) P(0, T β ). (2.10) C α,β (0) The receiver swaption is said to be in-the-money (ITM) if K > K ATM, and out-ofthe-money (OTM) if K < K ATM. The payer swaption is said to be in-the-money (ITM) if K < K ATM, and out-of-themoney (OTM) if K > K ATM. The discounted receiver swaption payoff at time t = 0 ND(0, T α )C α,β (T α )(K S α,β (T α )) + gives an intuitive meaning to the ITM and the OTM concept. If we substitute in the current forward swap rate S α,β (0) and evaluate (K S α,β (0)) + then the receiver swaption is ITM when S α,β (0) < K and it is OTM when S α,β (0) > K. 15

26 2.4 Change of Numeraire [1] [3] Definition A numeraire is any positive non-dividend paying asset. Theorem (Geman et al.) Let N be a numeraire and Q N a probability measure equivalent to Q 0 such that the price of any traded asset X relative to N is a Q N -martingale, i.e., X t N t = E N [ X T N T F t ], 0 t T. Let U be arbitrary numeraire. Then there exists a probability measure Q U, equivalent to Q 0, such that the price of any attainable claim Y normalized by U is a Q U -martingale, i.e., Y t U t = E U [ Y T U T F t ], 0 t T. In addition, the Radon-Nikodym derivative is given by dq U dq N = U TN 0 U 0 N T. Proof: It follows by the definition of the Q N that for any tradable asset price Z E N [ Z T N T ] = E U [ U 0Z T N 0 U T ] (2.11) (both being equal to Z 0 N 0 ). By definition of Radon-Nikodym derivative, we also know that for all Z E N [ Z T ] = E U [ Z T dq N N T N T dq ]. U By comparing the right-hand sides of the last two equalities we have that dqu dq N = U T N 0 U 0 N T. The general formula (2.11) follows from immediate use of the Bayes rule for conditional expectations. Corollary 1 The price of any asset divided by a reference asset is a martingale under the measure associated with the numeraire. For example, the forward LIBOR rate F 2 between expiry T 1 and maturity T 2 is given by F 2 (t) = P(t,T 1) P(t,T 2 ) (T 2 T 1 )P(t,T 2 ). If Q2 is the measure associated with P(., T 2 ), then F 2 will be a martingale under this measure. 16

27 A second example is the forward swap rate S α,β (t) defined in (2.6). If Q α,β is the measure associated with C α,β (t), then S α,β (t) is martingale under this measure. The forward swap rate is the ratio of a tradable asset (portfolio long one T α -zero coupon bond and short one T β -zero coupon bond) divided by the numeraire. Corollary 2 The risk-neutral price is invariant by change of numeraire, i.e Price t = E B [B(t) Payoff(T) B(T) ] = E S [S(t) Payoff(T) ]. S(T) Let Π(X T ) is the payoff on the underlying X at time T. There are two goals for changing numeraire: 1. Π(X T ) S(T) to be a simple quantity, 2. S(t)X t to be a tradable asset, so X t is martingale. The first condition ensures that the new numeraire does not complicate the computation E S [ Π(X T ) S(t)Xt ]. The second condition ensures that =X S(T) S(t) t is a martingale under Q S. Definition T-Forward measure The measure associated with the bond maturing at time T is called the T-forward measure and is denoted by P T. The expectation with respect to this measure is denoted by E T. The price of the derivative at t is π t = P(t, T)E T 1 [ P(T,T) Π(X T) F t ] = P(t, T)E T [Π(X T ) F t ], (2.12) Theorem For each 0 u t S < T, the forward rate is a martingale under the T-forward measure: E T [F(t; S,T) F u ] = F(u; S,T). In particular, the forward rate is the P T expectation of the future spot rate: Proof: E T [L(S,T) F t ] = F(t; S,T). From Definition follows that F(t; S,T)P(t, T) = 1 [P(t, T) P(t, S)]. This is the τ(s,t) price at t of a tradable asset, since it is a multiple of a difference of two bonds. Hence, by definition of T-forward measure F(t;S,T)P(t,T) P(t,T) measure. Note that F(S; S,T) = L(S,T). 17 = F(t; S, T) is a martingale under such a

28 2.5 Swap Market Model (SSM): Overview [1] [4] [5] [6] [7] [8] [9] [10] Recall that an IRS is a contract to exchange payments between two differently indexed legs. Let us assume a notional amount of one unit for simplicity. At every time T i from the set T α+1,...,t β the fixed leg pays out an amount τ i K corresponding to a fix rate K, where τ i is the year fraction of time from T i 1 to T i. The floating leg pays an amount corresponding to the forward rate F i (T i 1 ) set at time T i 1 for maturity T i. The floating leg resets at T α, T α+1,...,t β 1 and pays at T α+1,...,t β. The PFS payoff at time T α can be written as: β D(T α, T i )τ i (F i (T i 1 ) K). i=α+1 The PFS discounted payoff at time t < T α is β D(t, T i )τ i (F i (T i 1 ) K). i=α+1 The value of the contract at time t is computed as follows: β PFS(t, [T α,...,t β ], K) = E[ D(t, T i )τ i (F i (T i 1 ) K) F t ] i=α+1 β = P(t, T i )τ i E i [F i (T i 1 ) K F t ] i=α+1 β β = P(t, T i )τ i (F i (t) K) = [P(t, T i 1 ) (1 + τ i K)P(t, T i )]. i=α+1 i=α+1 Note that if the PFS T α -payoff β PFS(T α, [T α,...,t β ], K) = P(T α, T i )τ i (F i (T α ) K) i=α+1 is discounted to time t < T α β D(t, T α ) P(T α, T i )τ i (F i (T α ) K), i=α+1 18

29 then it leads to the same value under the risk-neutral expectation. The payer IRS discounted payoff at time zero for a K different than the swap rate can be written as: D(0, T α )(S α,β (T α ) K) β i=α+1 τ i P(T α, T i ). Let us consider a payer swaption with strike K giving the holder the right to enter at some future time T α a payer IRS with payment dates T α+1,...,t β and associated year fractions τ α+1,...,τ β. The payer swaption discounted payoff at time t = 0 is D(0, T α )(S α,β (T α ) K) + β i=α+1 τ i P(T α, T i ) = D(0, T α )(S α,β (T α ) K) + C α,β (T α ). (2.13) The swap rate S α,β is a martingale under the numeraire C α,β (t) as explained in the second example of Corollary 1 in Section 2.4. Indeed, the product C α,β (t)s α,β (t) = P(t, T α ) P(t, T β ) gives the price of a tradable asset and the forward rate S α,β (t) is a P α,β -martingale associated with the numeraire C α,β (t). The measure P α,β is called forward-swap measure or simply swap measure. A probability measure P α,β, equivalent to the historical probability measure P, is said to be the forward swap probability measure associated with dates T α and T β, if for i = α,..., β the relative bond price P(t,T i) C α,β (t), t [0, T i T α ] follows a local martingale process under P α,β. Since S α,β (t) is a P α,β martingale we postulate that under P α,β : ds α,β (t) = σ α,β (t)s α,β (t)dw α,β t, t [0, T α ] (2.14) where the instantaneous percentage volatility σ α,β (t) is deterministic and W α,β t motion under P α,β. is a Brownian Let us define the average percentage variance of the forward swap rate in [0, T] by υ 2 α,β(t) = T 0 (σ α,β (t)) 2 dt. 19

30 This model for the evolution of forward swap rates is known as the lognormal forwardswap model, since each swap rate S α,β has a lognormal distribution under its swap measure P α,β : ln S α,β (T α ) N(S α,β (t) 1 2 σ2 α,β(t α t), σ 2 α,β(t α t)). When pricing a swaption, the LSM is very convenient, since it yields the well-known Black formula for swaptions, as stated below. Theorem Equivalence between lognormal forward-swap model and Black s swaption prices The price of the payer/receiver swaption, as implied by the lognormal forward-swap model, coincides with that given by the Black-type formula for swaptions,i.e. PS/RS SMM (0, [T α,...,t β ], K) = PS/RS Black (0, T α, [T α,...,t β ], K) = C α,β (0)BL(K, S α,β (0), υ α,β (T α ), w) where BL(K, F, v, w) = FwΦ(wd 1 (K, F, v)) KwΦ(wd 2 (K, F, v)), d 1 (K, F, v) = ln(f/k) + v2 /2, v d 2 (K, F, v) = ln(f/k) v2 /2, v where w is either -1 (receiver) or 1 (payer) and is meant to be 1 when omitted. where For the payer swaption the Black s formula is: PS SMM (0, [T α,...,t β ], K) = C α,β (0)(S αβ (0)Φ(d 1 ) KΦ(d 2 )), (2.15) d 1,2 = ln(s αβ(0)/k) ± σ 2 αβ T α/2 σ αβ Tα. Proof: The swaption price is the risk-neutral expectation of the discounted payoff E[D(0, T α )(S α,β (T α ) K) + C α,β (T α )] = E B [ B(0) B(T α ) (S α,β(t α ) K) + C α,β (T α )] = E α,β [ C α,β(0) C α,β (T α ) (S α,β(t α ) K) + C α,β (T α )] = C α,β (0)E α,β [(S α,β (T α ) K) + ]. (2.16) 20

31 It follows from formula (2.11) for Z T = (S α,β (T α ) K) + C α,β (T α ), that U = B and N = C α,β. Now, given the lognormal distribution of S, computing (2.6) with (2.14) leads to the Black s formula for swaptions. Indeed, the above expectation is the classical Black and Scholes price for a Call option for the underlying asset S α,β, struck at K, with maturity T α, with zero constant risk-free rate and instantaneous percentage volatility σ α,β (t). In the special case where the swaption is ATM, K = S αβ (0), the formula reduces to: C α,β (0)(S αβ (0)Φ(d 1 ) S αβ (0)Φ( d 1 )) = C α,β (0)S αβ (0)(Φ(d 1 ) Φ( d 1 )) = C α,β (0)S αβ (0)(2Φ(d 1 ) 1) = (P(0, T α ) P(0, T β ))(2Φ(d 1 ) 1), (2.17) where d 1 = σαβ Black Tα /2. Since this payers swaption is ATM, the value of the corresponding receivers swaption is exactly the same. When an ATM swaption is quoted at an implied volatility, the actual price that is paid by the purchaser of the swaption is given by substituting the implied volatility into equation (2.17). Market Swaption Volatility The market convention is to quote swaptions in terms of their implied volatility which sets the Black-like model price to equal the market price. The reason is due to the fact that the implied volatilities tend to be more stable over time than the actual price of the swaption. The ATM swaptions volatilities are available through brokers daily in swaption tables where each row is indexed by the swaption maturity T α, where as each column is indexed in terms of the underlying swap length, T β T α. A x x y-swaption is the swaption in the table whose maturity is x years and whose underlying swap is y years long. For example, a 5 x 7 swaption is a swaption maturing in five years and giving the right to enter a seven-year swap. A typical example of swaption volatilities is shown in Table

32 Table 2.1: Black Implied Volatility of ATM Swaptions on 2/12/2008 Option/Swap 1yr 2yr 3yr 4yr 5yr 7yr 10yr 1yr yr yr yr yr yr yr The entries in the swaption table are not uniformly updated since the most liquid swaptions are updated more frequently. The Smile Effect [1] [11] [12] It is clear that the average volatility of the forward swap rate in [0, T], i.e. σ α,β (T α ), does not depend on the strike K of the swaption. Indeed, the volatility is a characteristic of the rate underlying the contract, and has nothing to do with the nature of the contract itself. Let us consider two different strikes K 1 and K 2 for the same swaption. Suppose the market prices are: PS market (0, T, τ, N,K 1, σ α,β ) and PS market (0, T, τ, N,K 2, σ α,β ). Does there exist a single volatility such that: PS market (0, T, τ, N,K 1, σ α,β ) = NC α,β (0)BL(K 1, S α,β (0), σ α,β Tα, 1) and PS market (0, T, τ, N,K 2, σ α,β ) = NC α,β (0)BL(K 2, S α,β (0), σ α,β Tα, 1) hold? In general, market prices do not behave like this. There are two different volatilities σ α,β (T α, K 1 ) and σ α,β (T α, K 2 ) at the market to match the observed market prices: PS market (0, T, τ, N,K 1, σ α,β ) = NC α,β (0)BL(K 1, S α,β (0), σ α,β (T α, K 1 ) T α, 1), PS market (0, T, τ, N,K 2, σ α,β ) = NC α,β (0)BL(K 2, S α,β (0), σ α,β (T α, K 2 ) T α, 1). Each swaption market price requires its own Black volatility σ α,β (K) depending on the strike rate K. The curve K σ α,β (T α, K) is the volatility smile of the swaption T α x T β T α. If Black formula were consistent across different strikes, this curve would be flat. Instead, 22

33 this curve is commonly smiley or skewed shaped. The term skew is used where for a fixed maturity, low-strikes implied volatilities are higher than high-strikes volatilities. A skew can be either monotonically decreasing or initially decreasing and then increasing, with a negative slop at the ATM level. The term smile is used where for a fixed maturity, the implied volatility has a minimum around the ATM level. Stochastic volatility models have been designed to capture the stochastic behavior of volatility and to accommodate market smiles and skews. The most well known model is Heston (1993) reviewed in Chapter 2, Section 2.8. Jump-diffusion models are employed with the purpose of calibrating smiles and skews of short maturities. We review the Bates (1996) model in Chapter 2, Section 2.8. Under a deterministic volatility structure, the forward swap rates are lognormally distributed, so that the corresponding swaption can be priced using a Black formula (Black 1976). However, there is no guarantee that dynamics (2.14) are well-defined in their natural martingale measure for a completely generic choice of the set of forward swap rates. The use of the Black formula was not supported by the existence of a reliable term structure model until Jamshidian established in 1997 the first structure model to do this. We describe this model next. 23

34 Co-terminal Swap Market Model [12] [13] [14] [7] Jamshidian (1997) developed the co-terminal swap market model described below. We follow closely Rutkowski s (1999) presentation of the lognormal model of forward swap rates. Let T = T α,...,t β be the tenor structure and τ j = T j T j 1 for j = α + 1,...,β. For any fixed α we consider a payer swap which starts at time T α and has β α payments. The fixed leg pays rate K at reset dates T α+1,...,t β. The value of such a swap is given in (2.5): β PFS(t) = P(t, T α ) c k P(t, T k ), k=α+1 where c k = τ k K for k < β and c β = 1 + τ β K for t [0, T α ]. Note that N is set to one. The associated swap rate S α,β (t) is given by (2.6). We consider a family of forward swap rates S i (t) := S(t, T i, T β ), t [0, T i ), which have the same maturity date T β for all i = α,..., β 1. Suppose we are given a family of zero-coupon bond prices P(t, T j ) 0 t Tj for j = α,..., β on a probability space (Ω,F, P) equipped with Brownian motion W. Let P β be the T β -forward measure and W β be the corresponding Brownian motion. For any m = α,..., β 1 we define the process C m (t) as follows: C m (t) = β j=β m+1 τ j P(t, T j ) = m 1 for t [0, T β m+1 ]. The forward swap rate is given by: We will make use of the following processes: υ i t := k=0 τ β k P(t, T β k ) S i (t) = P(t, T i) P(t, T β ). (2.18) C β i (t) β j=i+1 Here υ i t are the so-called discounted accrual factors. τ j P(t, T j ) P(t, T β ) = C β i(t) P(t, T β ). (2.19) 24

35 We will derive the relationship: For simplicity we denote Note that υ β t = S β (t) = 0. β 1 C β i (t) P(t, T β ) = ζ i t := j=i τ k+1 j k=i+1 (1 + τ k Sk (t)). i (1 + τ j+1 Sj+1 (t)). (2.20) j=α Then, using (2.18) we can rewrite (2.19) as follows: υt i = C β i(t) P(t, T β ) = τ i+1p(t, T i+1 ) + (τ i+2 P(t, T i+2 ) τ β P(t, T β )) P(t, T β ) = τ i+1 + τ i+1 ( P(t, T i+1) P(t, T β ) C β i 1 (t) C β i 1 (t) P(t, T β ) ) + C β i 1(t) P(t, T β ) = τ i+1 + τ i+1 Si+1 (t) C β i 1(t) P(t, T β ) + C β i 1(t) P(t, T β ) = τ i+1 + (1 + τ i+1 Si+1 (t)) C β i 1(t) P(t, T β ) = τ i+1 + (1 + τ i+1 Si+1 (t))υ i+1 t. (2.21) Multiplying both sides of equation (2.21) by ζ i 1 t leads to: ζt i 1 υt i = τ i+1 ζt i 1 + ζtυ i t i+1. By backward induction from β 1 we obtain the relationship: υ i t = β 1 j=i τ j+1ζ j 1 t ζt i 1 β 1 = j=i τ j+1 j k=i+1 (1 + τ k Sk (t)). (2.22) Also, the following relation holds: P(t, T i ) P(t, T β ) = 1 + υi t S i (t). (2.23) Definition For j = α,..., β and for every k = α,..., β, the relative bond price is defined as: Z β j+1 (t, T k ) = P(t, T k) C β j+1 (t) = P(t, T k ) τ j P(t, T j ) τ β P(t, T β ) 25

36 for t [0, T k T j ]. Definition For a fixed j = α,..., β a measure P j is said to be the forward-swap measure for the date T j if, for every k = α,..., β, the relative bond price Z β j+1 (t, T k ) follows a local martingale under P j. In other words: for any fixed m = α,..., β, the relative bond prices Z m (t, T β k ) = P(t, T β k), t [0, T β k T β m+1 ] C m (t) are local martingales under the forward swap measure P β m+1. Directly from the forward swap rate definition we have S β m (t) = P(t, T β m) P(t, T β ) C m (t) = Z m (t, T β m ) Z m (t, T β ). Thus S β m also is a local martingale under the forward swap measure P β m+1. Since obviously C 1 (t) = τ β P(t, T β ), then Z 1 (t, T β k ) = 1 τ β P(t,T β k ) P(t,T β ), and thus the measure P β can be chosen to coincide with the forward measure P β. We assume ν j for j = α,..., β 1 are the volatilities of the forward swap rates S j and will construct a family of forward swap rates in the following way: W j+1 t d S j (t) = S j (t)ν j (t)d W j+1 t (2.24) where is a standard Brownian motion under the corresponding forward swap measure P j+1. This model should also be consistent with the term structure at time zero: S j (0) = P(0, T j) P(0, T β ). (2.25) C β j (0) The construction of the model is based on backward induction. First, we suppose that S β 1 for the date T β 1 solves the SDE: d S β 1 (t) = S β 1 (t)ν β 1 (t)d W β t (2.26) for all t in [0, T β 1 ], where W β t = W β t = W, with the initial condition: S β 1 (0) = P(0, T β 1) P(0, T β ). τ β P(0, T β ) We have S β 1 (t) = P(0, T β 1) P(0, T β ) E t ( τ β P(0, T β ) t 0 ν β 1 (u)d W β u ). 26

37 Since P(0, T β 1 ) > P(0, T β ) it is clear that S β 1 is martingale under P β = P. The next step is to define S β 2. First we need to introduce a forward swap measure β 1 and the corresponding Brownian motion W t. By Definition P β 1 is equivalent to P such that Z 2 (t, T β k ) are local martingales. From Definition 2.5.2, for each k, we have: Z 2 (t, T β k ) = P(t, T β k ) τ β 1 P(t, T β 1 ) + τ β P(t, T β ) = Z 1 (t, T β k ) 1 + τ β 1 Z 1 (t, T β 1 ). Lemma Let G and H be real-valued adapted processes, such that dg t = a t dw t, dh t = b t dw t. Assume that H t > 1 for every t and denote Y t = (1 + H t ) 1. Then d(y t G t ) = Y t (a t Y t G t b t )(dw t Y t b t dt). We will now apply Lemma to G = Z 1 (t, T β k ) and H = τ β 1 Z 1 (t, T β 1 ). We have noticed above that Z 1 (t, T β k ) = P(t,T β k) τ β P(t,T β ) follows a strictly positive martingale under P β = P β : for some adapted process µ 1 (t, T β k ). Thus P β 1 dz 1 (t, T β k ) = Z 1 (t, T β k )µ 1 (t, T β k )d W β t (2.27) a t = Z 1 (t, T β k )µ 1 (t, T β k ) and b t = τ β 1 Z 1 (t, T β k )µ 1 (t, T β k ). Applying the Lemma gives us: Z 1 (t, T β k ) d( 1 + τ β 1 Z 1 (t, T β 1 ) ) = dz 2(t, T β k ) = 1 1+τ β 1 Z 1 (t,t β 1 ) (Z 1(t, T β k )µ 1 (t, T β k ) Z 1(t,T β k )τ β 1 Z 1 (t,t β 1 )µ 1 (t,t β k ) 1+τ β 1 Z 1 (t,t β 1 ) ) x(d W β t τ β 1Z 1 (t, T β 1 )µ 1 (t, T β k ) dt) 1 + τ β 1 Z 1 (t, T β 1 ) 27

38 So that dz 2 (t, T β k ) = η β 1 (t)(d W β t τ β 1Z 1 (t, T β 1 )µ 1 (t, T β k ) dt). 1 + τ β 1 Z 1 (t, T β 1 ) From definitions it is easy to see that Z 1 (t, T β 1 ) = P(t, T β 1) τ β P(t, T β ) = S β 1 (t) + Z 1 (t, T β ) = S β 1 (t) + 1 τ β. Differentiating both sides and using (2.26) and (2.27) we obtain: Therefore or It is enough to show that For t [0, T β 1 ]: And so Z 1 (t, T β 1 )µ 1 (t, T β 1 ) = S β 1 (t)ν β 1 (t). dz 2 (t, T β k ) = η β 1 (t)(d W β t dz 2 (t, T β k ) = η β 1 (t)(d W β t W β 1 t τ β 1 S β 1 (t)ν β 1 (t) 1 + τ β 1 ( S β 1 + τ 1 β )dt) τ β 1 S β 1 (t)ν β 1 (t) 1 + τ β 1 Sβ 1 + τ β 1 τ 1 β dt). W β 1 t defined below follows a Brownian motion under P β 1. t := W β t 0 W β 1 t 1 + τ β 1 τ 1 β = W β t τ β 1 Sβ 1 (u) + τ β 1 Sβ 1 (u) ν β 1(u)du. t where the definition of γ β 1 is clear from the context. 0 γ β 1 (u)dw u Now we can define the associated forward swap measure P β 1 by using Girsanov s theorem stated below. Theorem Let W t be a Wiener process on a probability space (Ω,F, P) and X t is a measurable process adapted to the filtration F W t. Define Z t = E(X) t, where E(X) is the stochastic exponential (Doleans exponential) of X with respect to W, i.e. E(X) t = exp(x t 1 2 [X t]). If Z t is a martingale, then probability measure Q can be defined such that Radon-Nikodyn derivative is dq dp F t = Z t = E(X) t. Then for each t: Q F W t P F W t. Furthermore, if Y is a local martingale under P, then Ỹt = Y t [W,X] t is a Q-local martingale. From the Theorem: P β 1 d dp = E t( t 0 28 γ β 1 (u)dw u ).

39 Next, we introduce the process S β 2 by assuming it solves the SDE: with the initial condition: We consider the process: Z 3 (t, T β k ) = d S β 2 (t) = S β 2 (t)ν β 2 (t)d S β 2 (0) = W β 1 t P(0, T β 2 ) P(0, T β ) τ β 1 P(0, T β 1 ) + τ β P(0, T β ). P(t, T β k ) τ β 2 P(t, T β 2 ) + τ β 1 P(t, T β 1 ) + τ β P(t, T β ) = Z 2 (t, T β k ) 1 + τ β 2 Z 2 (t, T β 2 ) so that W β 2 t It is useful to notice that: = W β 1 t t 0 τ β 2 Z 2 (u, T β 2 ) 1 + τ β 2 Z 2 (u, T β 2 ) µ 2(u, T β 2 )du. Z 2 (t, T β 2 ) = P(t, T β 2 ) τ β 1 P(t, T β 1 ) + τ β P(t, T β ) = S β 2 (t) + Z 2 (t, T β ), where in turn: Z 2 (t, T β ) = Z 2 (t, T β ) 1 + τ β 1 Z 1 (t, T β ) + τ β 1 Sβ 1 (t) and Z 1 (t, T β ) is already known from the previous step. Differentiating both sides we will find the volatility process of Z 2 (t, T β ), and then define P β 2. We consider the induction step with respect to m. Suppose that we have already defined the forward swap rates S β 1,..., S β m. This means, in particular, that the forward swap measure P β m+1 and the associated Brownian motion W β m+1 are already specified. We would like to determine the forward swap measure P β m, the Brownian motion W β m and the forward swap rate S β m 1. It is fairly easy to check that the processes Z m+1 (t, T β k ) = P(t, T β k) C m+1 (t) = Z m (t, T β k ) 1 + τ β m Z m (t, T β k ) follow local martingale under P β m. Applying Lemma to G = Z m (t, T β k ) and H = τ β m Z m (t, T β m ) we can set: W β m t = W β m+1 t t 0 τ β m Z m (u, T β m ) 1 + τ β m Z m (u, T β m ) µ m(u, T β m )du, 29

40 and thus P β m t can be easily found using Girsanov s theorem. From and P β m d d P = E t( β m+1 t Z m (t, T β m ) = P(t, T β m) C m (t) Z m (t, T β ) = 0 γ β m (u)d W β m+1 u ). = S β m (t) + Z m (t, T β ) Z m 1 (t, T β ) 1 + τ β m+1 Z m 1 (t, T β ) + τ β m+1 Sβ m+1 (t) it follows that Z m (t, T β ) are functions of S β 1,..., S β m+1. The process Z m 1 (t, T β ) is known from the preceding step and is a function of forward swap rates S β 1,..., S β m+1. Then, the process τ β m Z m (u, T β m ) 1 + τ β m Z m (u, T β m ) µ m(u, T β m ) can be express in terms of S β 1,..., S β m+1 and their volatilities ν β 1,...,ν β m+1. Having found W β m and P β m we introduce the forward swap rate S β m 1 through (2.24) and (2.25) and so forth. Swaption Pricing With Co-terminal SMM For a fixed date T j, j = α,..., β 1, we call jth-swaption the swaption with option maturity T j. The payer swaption payoff at time T j is Y = Recall that S j satisfies the SDE: β k=j+1 τ k P(T j, T k )[ S j (T j ) K] +. d S j (t) = S j (t)ν j (t)d W j+1 t, where W j+1 follows a standard Brownian motion under the forward swap measure P j+1. Recall that the definition of P j+1 implies that any process P(t,T k) C β j for k = α,..., β, is a local (t) martingale under this measure. The arbitrage price π t (X) of a claim X, which is a function of P(T j, T j+1 ),...,P(T j, T β ), equals π t (X) = C β j (t)ẽj+1 [C 1 β j (T j)x] 30

41 provided X settles at T j. We apply it to the swaption payoff: π t (Y ) = C β j (t)ẽj+1 [ S j (T j ) K] +. We assume that ν j : [0, T j ] R d is a bounded deterministic function. Theorem Black s swaption formula For any j = α,..., β 1, the arbitrage price at time t [0, T j ] of the jth-swaption is: where PS j t = β k=j+1 τ k P(t, T k )( S j (t)φ(d 1 ) KΦ(d 2 )) d 1,2 = ln ( S j (t)/k) ± v 2 j(t)/2 v j (t) with vj(t) 2 = T j ν t j (u) 2 du. In the fixed maturity forward swap market model swaptions with total maturity, defined as the swaption maturity plus the swap maturity, equal to the fixed maturity T β will have the Black-type formula and determination of the swap rate volatility is all that is necessary to price swaptions. For swaptions with underlying swap maturity dates other than a fixed maturity T β, closed form pricing formulas are not available. Fixed-length Swap Market Model [13] Rutkowski considers the family of forward swap rates with fixed-length as opposed to fixed maturity. For instance, for a forward swap rate which starts at time T j, j = α,..., β M, the first settlement date is T j+1, and its maturity date is T M+j. The value of such swap at time t [0, T j ] is PFS(t) = P(t, T j ) j+m k=j+1 c k P(t, T k ). The forward swap rate of length M, for the date T j, is that value of the fixed rate which makes the underlying swap starting at T j worthless at time t. From the last formula it follows: Sj M P(t, T j ) P(t, T j+m ) (t) = τ j+1 P(t, T j+1 ) τ j+m P(t, T j+m ). For any m = α,..., β M + 1, we define the process C m (t) = β m+1 i=β M+1 m+1 τ i P(t, T i ) = 31 m+m 2 k=m 1 τ β k P(t, T β k ).

42 A modified forward swap measure defined below corresponds to the choice of the process C m (t) as a numeraire. Definition For a fixed j = α,..., β M +1, a probability measure P j equivalent to P, is called the fixed-length forward swap measure for the date T j if, for every k = α,..., β, the relative bond price Z β M+1 j+1 (t, T k ) = follows a local martingale under P j. P(t, T k ) C β M+1 j+1 (t), t [T k T j ] It is clear that each forward swap rate S M j (t), j β M + 1, follows a local martingale under the forward swap measure for the date T j+1, since S M j (t) = Z β M+1 j+1 (t, T j ) Z β M+1 j+1 (t, T j+m ). We assume that ds M j (t) = S M j (t)λ j (t)d W j+1 t, where W j+1 t is the Brownian motion under P j+1 and the initial condition is S j M (0) = P(0, T j) P(0, T j+m ). C j+m (0) The construction of a fixed-length forward swap rates model also relies on backward induction. We start by postulating that P represents the forward swap measure for T β M+1, and then define recursively the forward swap measures corresponding to the preceding dates T β M, T β M 1,...,T α. Pricing Fixed-length Swaptions For a fixed date T j, j = α,..., β M, we consider the jth-swaption with option maturity at T j written on a swap with fixed length M. The payer swaption payoff at time T j is Y = j+m k=j+1 τ k P(T j, T k )[ S M j (T j ) K] +. Recall that S M j satisfies the SDE: d S M j (t) = S M j (t)λ j (t)d 32 W j+1 t,

43 where W j+1 follows a standard Brownian motion under the forward swap measure P j+1. Recall that the definition of P j+1 implies that any process P(t,T k ) C β M+1 j (t) for k = α,..., β, is a local martingale under this measure. The arbitrage price of the swaption payoff is π t (Y ) = C β M+1 j (t)ẽj+1 [ S M j (T j ) K] +. We assume that λ j : [0, T j ] R d is a bounded deterministic function. Theorem Black s swaption formula For any j = α,..., β M, the arbitrage price at time t [0, T j ] of the jth-swaption is: PS j t = j+m k=j+1 τ k P(t, T k )( S M j (t)φ(d 1 ) KΦ(d 2 )), where d 1,2 = ln ( S M j (t)/k) ± v 2 j(t)/2 v j (t) with v 2 j(t) = T j t λ j (u) 2 du. The backward induction is used systematically to produce various models of the forward swap rates. We have described the fixed-maturity forward swap market model developed by Jamshidian (1997) and the fixed-length forward swap market model developed by Rutkowski (1998). 33

44 [15] 2.6 Fourier Transform and Characteristic Function The Fourier transform of integrable function, f(x), is defined as: F f (u) = e iuy f(y)dy. Given the Fourier transform of f, the function f can be recovered by: f(x) = 1 2π e iux F f (u)du. A characteristic function is the Fourier transform of a probability distribution. If X has a probability density function q, then is its characteristic function. F q (u) = e iux q(x)dx = Ee iux Note than F q (0) = q(x)dx = 1. The critical property of characteristic functions is that the characteristic function of the sum of two independent random variables is the product of these variables characteristic functions. 34

45 [15] 2.7 Compound Poisson Process The Poisson process is an example of a stochastic process with discontinuous trajectories used to build complex jump processes. Definition Exponential random variables A positive random variable Y is said to follow an exponential distribution with parameter λ if it has a probability density function of the form λe λy 1 y 0. Definition Poisson Process Let (τ i ) i 1 be a sequence of independent exponential random variables with parameter λ and T n = n i=1 τ i. Process (N t, t 0) defined by: N t = n 1 1 t Tn is called a Poisson process with intensity λ. The Poisson process counts the number of random times (T n ) which occur in [0, t], where (T n T n 1 ) n 1 is an iid sequence of exponential variables. Definition Compound Poisson Process Compound Poisson process with intensity λ and jump size distribution f is a stochastic process Z t defined as: N t Z t = Y i, i=1 where jump sizes Y i are iid with distribution f and (N t ) is a Poisson process with intensity λ, independent from (Y i ) i 1. Characteristic Function of a Compound Poisson Process E[e iuzt ] = exp(tλ (e iuz 1)f(dz)), u R d, R d where λ denotes the jump intensity and f the jump size distribution. 35

46 2.8 Stochastic Volatility Models: Overview [15] [1] A widely studied class of stochastic volatility models is the class of bivariate diffusion models: ds t = µs t dt + σ t S t dw t, (2.28) where σ t is called the instantaneous volatility process. Important features for the instantaneous volatility process are positiveness and mean-reversion. Positiveness can be obtained by setting σ t = f(y t ), where f(t) is a positive function and y t is a random driving process. Mean-reversion is usually modeled by introducing a mean-reverting drift in the y t dynamics: dy t = κ(θ y t )dt +...dẑt, where Ẑt is a Wiener process correlated with W t. Let ρ denotes the correlation coefficient. In most practical cases it is taken to be a constant. κ is called the rate of mean-reversion and θ is the log-run average level of y t. The drift term pulls y t towards θ and thus σ t is pulled towards the long-run mean of f(y t ). Some common driving processes are: Geometric Brownian motion dy t = c 1 y t dt + c 2 y t dẑt, Gaussian Ornstein-Uhlenbeck dy t = κ(θ y t )dt + βdẑt, Cox-Ingersoll-Ross (CIR) dy t = κ(θ y t )dt + η y t dẑt. The log-normal model is not mean-reverting; the Gaussian OU process is mean-reverting but takes negative values. The most suitable choice from the tree listed is the CIR process. It is both mean-reverting and positive. Stochastic volatility models generically lead to implied volatility smiles and skews. The correlation coefficient ρ plays a fundamental role for generating the smiles and skews. A negative correlation is often interpreted in terms of the leverage effect : the empirical evidence that large downward moves in option prices are associated with upwards moves in volatility and more generally with periods of high volatility. Table 2.2 gives some examples of bivariate diffusion models used for option pricing. 36

47 Table 2.2: Examples of Bivariate Diffusion Models for Option Pricing Model Correlation f(y) y t Hull-White ρ = 0 f(y) = (y) Lognormal Scott ρ = 0 f(y) = e y Gaussian OU Stein-Stein ρ = 0 f(y) = y Mean-reverting OU Ball-Roma ρ = 0 f(y) = (y) CIR Heston ρ 0 f(y) = (y) CIR Among these models, the Heston model is quite commonly used in practice: ds t S t = µdt + V t dw S t, dv (t) = κ(θ V (t))dt + η V (t)dw V t. Applying Itô s formula for the log-normal process Y t = lns t we obtain: dy t = (r 1 2 V t)dt + V t dw S t. (2.29) The jump-diffusion stochastic volatility model introduced by Bates adds proportional lognormal jumps to the Heston stochastic volatility model. The model is: ds t S t = µdt + V t dw S t + dz t, where W S t and W V t dv (t) = κ(θ V (t))dt + η V (t)dw V t. are Brownian motions with correlation ρ, and Z t is a compound Poisson process with intensity λ and log-normal distribution of jump sizes such that if k is its jump size then ln(1+k) N(ln(1+ k) 1 2 δ2, δ 2 ). The no-arbitrage condition fixes the drift under the risk-neutral measure: µ = r λ k. Applying Itô s formula for the log-normal process Y t = lns t we obtain: dy t = (r λ k 1 2 V t)dt + V t dw S t + d Z t, (2.30) where Z t is a compound Poisson process with intensity λ and normal distribution of jump sizes. Note the difference between (2.29) and (2.30) in the shift λ k, the risk-neutral correction. We will use these two models as prototypes of the new extensions of the SMM. 37

48 CHAPTER 3 STOCHASTIC VOLATILITY EXTENSIONS OF SMM [15] [16] [17] [18] [19] [20] 3.1 Stochastic Volatility Extension Of SMM Without Jumps The Swap Market Model can be extended to a stochastic volatility model by introducing a common stochastic volatility factor for all forward-swap rates. We choose the multiplicative stochastic factor affecting the family of the swap rates to follow a mean reverting square-root diffusion process. We consider the at-the-money implied swaption volatility for the period March 31, 2003 to April 30, 2008 for swaptions 1x4, 2x2, 3x5, 4x10, 5x5, 2x10, 7x3, and 10x4. The time series of selected at-the-money implied volatility curves are shown in Figure 3.1. It is noticeable that all curves are akin in shape, i.e. volatilities react simultaneously to the same event and move in the same direction. However, their reactions have different amplitudes. The stochastic volatility extension has been justified in Attaoui (2006) [20] by constructing the correlation matrices with respect to each swap tenor of percentage changes in the implied volatilities and computing the eigenvalues and eigenvectors for each correlation sub-matrix. Most principal components display similar qualitative patterns across different option maturities and swap tenors. 38

49 Figure 3.1: Time Series of Implied Volatilities for ATM Swaptions Under the forward swap measure the SV extension of the SMM is: ds αβ (t) = S αβ (t) V (t)σ αβ (t)dw αβ (t), dv (t) = κ(θ V (t))dt + η V (t)dw V (t), (3.1) where W αβ (t) and W V (t) are the BM driving the swap rate and the swap volatility respectively. We denote by ρ the correlation between the two Brownian motions. Assuming that the characteristic function of the logarithm of the swap rate is known, the swaption price can be obtained by applying the fast fractional Fourier transform (FRFT) procedure. We will obtain an analytic expression for the characteristic function of the logarithm of the swap rate. In the next chapter we will introduce the FRFT method and will show how to use this procedure to compute swaption prices from the corresponding characteristic function. Let Y (t) = ln S αβ(t) S αβ, then the characteristic function Φ(y, v, T; u) is defined by: (0) Φ(y, v, t; u) = E[e iuy (T) Y (t) = y, V (t) = v]. 39

50 Consider Φ(y, v, t; u) = E[g(Y (T), V (T))/Y (t), V (t)] = E[g/F t ], where F t is the information set up to time t represented by the values of the stochastic process Y and V at time t. Considering a time s such that s > t and using the principle of iterated expectations, we obtain that Φ is a martingale as follows: E[Φ(y, v, s; u)/f t ] = E[E[g/F s ]/F t ] = E[g/F t ] = Φ(y, v, t; u). Applying Itô s formula to Φ(y, v, t; u) yields: dφ t = ( 1 2 σ2 αβv 2 Φ y + ρηv 2 Φ 2 y v η2 v 2 Φ v σ2 αβv Φ + κ(θ v) Φ y v + Φ )dt +... t We set the drift term to zero since Φ is martingale: 1 2 σ2 αβv 2 Φ y + ρηv 2 Φ 2 y v η2 v 2 Φ v σ2 αβv Φ + κ(θ v) Φ y v + Φ t = 0. (3.2) To determine the solution of equation (3.2), the final condition at time T should be satisfied: Φ(y, v, T; u) = e iuy. Together with the terminal condition the PDE allows computation of the characteristic function. Proposition The closed form of the characteristic function for Y (t) is: where = Proof: Φ t (u) = (κ iρηu) 2 + η 2 σαβ 2 u(u + i). exp( κθt(κ iρηu) (u2 +iu)σαβ 2 v η 2 coth( t 2 (cosh( t) + (κ iρηu) 2 )+κ iρηu + iuy) sinh( t 2 ))( 2κθ η 2 ) The computation of the characteristic function of the logarithm of the swap rate is original. First, we make the following guess: Φ(y, v, t; u) = e C(T t)+vd(t t)+iuy, (3.3) 40

51 where C and D are functions of one variable only. Then Substituting all into equation (3.2) leads to: Φ y = iuφ, Φ t = (C + vd )Φ, Φ v = DΦ, 2 Φ v = 2 D2 Φ, 2 Φ y = 2 u2 Φ, 2 Φ y v = iudφ. 1 2 σ2 αβu 2 v + iρηvd η2 D 2 v 1 2 σ2 αβviu + κ(θ v)d (C + vd ) = 0 or ( 1 2 η2 D 2 + (iρηu κ)d 1 2 σ2 αβu(i + u) D )v + (κθd C ) = 0. As v is stochastic, the expression above will be zero only if both the term multiplying v and the other one are zero independently. As a result of this we obtain two ODEs: D (s) = 1 2 η2 D 2 (s) + (iρηu κ)d(s) 1 2 σ2 αβu(i + u), C (s) = κθd with initial conditions D(0) = C(0) = 0. Lemma The explicit solutions to the equations for D and C are: C(s) = D(s) = κθs(κ iρηu) η 2 (u 2 + iu)σαβ 2 γ coth( s) + κ iρηu, 2 2κθ η ln(cosh( s 2 2 ) + κ iρηu sinh( s 2 )), 41

52 where = (κ iρηu) 2 + η 2 σαβ 2 u(u + i). Proof: Let R(s) = 1 2 η2, Q(s) = (iρηu κ), and P(s) = 1 2 σ2 αβu(i + u). For D we have to solve a Riccati equation of the form dd ds = R(s)D2 + Q(s)D + P(s). Let D 1 below be the solution of the quadratic equation RD 2 + QD + P = 0, where = (Q 2 4RP) = In terms of the new function Z defined by: we have: D 1 = Q + 2R, (3.4) (κ iρηu) 2 + η 2 σαβ 2 u(u + i) is the discriminant. Z = 1 D D 1 (3.5) dz ds = (Q + 2D 1R)Z R, dz = Z R. ds The solution of a first order linear ODE of the form y + p(s)y = q(s) is derived in the following manner: If µ(x) = e p(s)ds, then µy + µpy = µq, (µy) = µq, (µy) = µq, y = 1 µ In our case p(s) =, q(s) = R and hence µ(s) = e ds = e s. The solution is: Z = 1 e s ( R)ds e s 42 µq.

53 = 1 e s( R e s + c) = R + ce s. (3.6) At s = 0: Z(0) = R + c. And so Hence, 1 D(0) D 1 = R + c. c = 1 D(0) D 1 + R. Substituting the constant back into the solution (3.6) for Z we have: We can write it as: where Z = R + ( 1 D(0) D 1 + R )e s. Z = U + We s (3.7) U = R (3.8) and W = First, we simplify the expression for W: = W = 1 D(0) D 1 + R. 1 + R D(0) D 1 = 1 D(0) Q+ 2R + R 2R 2RD(0) + Q + R = R + 2RD(0) + Q (2 2RD(0) + Q ) = R (2RD(0) + + Q 2RD(0) + Q ) = g R, (3.9) 43

54 where Then, from (3.8) and (3.9) it follows that From (3.5) we have Substituting (3.4) in and simplifying further: Let s denote Hence, the solution for D is: g = 2RD(0) + + Q 2RD(0) + Q. (3.10) Z = U + We s = R (1 ge s ). D = D Z = D 1(U + We s ) + 1 U + We s. D = ( Q+ 2R )( R + R ge s ) + 1 R (1 ge s ) = ( Q+ )( R) + ( Q+ ) R 2R 2R ge s ) + 1 R(1 ge s ) = (Q 2 ) + ( Q+ 2 ge s ) + 1 R (1 ge s ) = (Q ) + ( Q+ 2 2 ge s ) R(1 ge s ) = Q + ( Q + ) ge s + 2 2R(1 ge s ) = Q+ ( Q + )( + Q+ ge s ). 2R(1 ge s ) D(s) = ( Q 2R where g and g are defined in (3.10) and (3.11) respectively. g = + Q Q. (3.11) + ge s )(g 1 ge s), (3.12) We can also write the expression for D in terms of P, Q and as follows: D(s) = ( Q 2R )(( +Q ) + ( +Q Q +Q )e s ) 1 ( +Q +Q )e s 44

55 = ( Q 2R )( Q + Qe s + e s Q Qe s e s ). The denominator simplifies as follows: Q Qe s e s = (e s + 1) Q(e s 1) Therefore, = (e s 1)( ( e s + 1 e s 1 ) + Q) = e s (1 e s )( ( e s + 1 e s 1 ) + Q) = e s (1 e s )( coth ( s 2 ) + Q). D(s) = ( Q 2R ) (e s 1)(Q + ) e s (1 e s )( coth ( s 2 ) + Q) = ( Q 2R )e s (1 e s )(Q + ) e s (1 e s )( coth ( s 2 ) + Q) = ( Q2 2 2R ) 1 coth ( s 2 ) + Q = ( 4PR 2R ) 1 coth ( s 2 ) + Q = 2P (3.13) coth ( s ) + Q. 2 Now we can find the solution for C (s) = κθd. We substitute in the solution (3.12) for D: C (s) = κθ( Q + ge s )(g 2R 1 ge s). Let M = κθ( Q ). Hence, 2R C(s) = M ( g + ge s 1 ge s)ds + c. We consider a change in variables: y = e s. Then dy = yds and C(s) = M ( g + gy 1 gy )dy y + c. Using g + gy (1 gy)y = g g gy + g gy + gy (1 gy)y = g(1 gy) + gy(g + 1) (1 gy)y = g y + (g + 1) g 1 gy 45

56 we have C(s) = M (g + 1) g ( 1 gy + g y = M ( (g + 1) g 1 gy dy + = M ( (g + 1) d(1 gy) + g 1 gy )dy + c g dy) + c y 1 dy) + c y = M ( (g + 1) ln(1 ge s ) + g ln(e s )) + c. (3.14) At s = 0: Therefore, C(0) = M (g + 1) ln(1 g) + c. c = C(0) M (g + 1) ln(1 g). Substituting the constant back into the solution (3.14) yields: C(s) = M ( (g + 1) ln(1 ge s ) + g( s)) + C(0) M (g + 1) ln(1 g) Substituting M and g in: C(s) = κθ( Q = M ((g + 1)(ln(1 ge s ) ln(1 g)) + g s) + C(0) = M 2R ) = M ge s ((g + 1) ln(1 )) + M g s + C(0) 1 g ( + Q Q ge s (g + 1) ln(1 ) + Mgs + C(0). 1 g ge s + 1) ln(1 ) κθ( Q 1 g 2R )( + Q )s + C(0) Q = κθ( Q) 2R = κθ R 2 ( Q ln(1 ge s 1 g ) ln(1 ge s 1 g ) κθ( + Q )s + C(0) 2R ) κθ( + Q )s + C(0) 2R = κθ ge s ( 2 ln(1 ) ( + Q)s) + C(0) 2R 1 g = κθ ge s (ln(1 ) 2 ( + Q)s) + C(0). (3.15) η2 1 g 46

57 Let s simplify further C by simplifying the expression under the logarithm: 1 ge s 1 g = 1 +Q +Q e s 1 +Q +Q = (Q ) ( + Q)e s Q Q = Q e s Qe s. 2 We multiply both the denominator and the numerator by e s : = Qe s + e s + + Q 2 e s Back to (3.15): = e s + Q(e s 1) 2 e s = (e s + 1) + Q(e s 1) 2 e s 2 e s = (e s 2 e s 2 + e s 2 e s 2 ) 2e s 2 e s 2 = 1 ( e s 2 + e s 2 e s Q(e s 2 e s 2 e s 2 e s 2 ) Q 2 e s 2 e s 2 e s 2 1 ) 2 = 1 (cosh( s e s 2 2 ) + κ iρηu sinh ( s 2 )). C(s) = 2κθ (ln( 1 η 2 = ( 2κθ )( s η 2 e s 2 2 ) + 2κθ η 2 (cosh( s 2 ) + κ iρηu (ln(cosh( s 2 ) + κ iρηu = ( 2κθ )(ln(cosh( s η 2 2 ) + κ iρηu sinh ( s 2 κθ ))) ( + Q)s η2 sinh ( s κθ s ))) κθqs 2 η 2 η 2 sinh ( s κθsq ))). (3.16) 2 η 2 By substituting the expression (3.16) for C and (3.13) for D, derived in Lemma 3.1.2, into the characteristic function (3.3) we obtain: Φ t (u) = exp( κθt(κ iρηu) (u2 +iu)σαβ 2 v η 2 coth( t 2 (cosh( t) + (κ iρηu) 2 + iuy) )+κ iρηu. (3.17) sinh( t 2 ))( 2κθ) η 2 47

58 3.2 Stochastic Volatility Extension Of SMM With Log-Normal Jumps [15] [21] [22] We will expand the model in Section 3.1 by adding jumps to the swap rate dynamics. Under the forward swap measure the jump-diffusion SV extension of the SMM (LN-SV SMM) is: ds αβ (t) = S αβ (t) V (t)σ αβ (t)dw αβ (t) + dz(t), dv (t) = κ(θ V (t))dt + η V (t)dw V (t) where W αβ (t) and W V (t) are Brownian motions with correlation ρ. Here Z(t) = N t i=1 J i is a compound Poisson process where J i are i.i.d. with distribution f J and N t is a Poisson process with intensity λ. We assume that the distribution of the jump sizes is log-normal with mean µ J and variance δj 2. If J is a jump size then ln(1 + J) N(ln(1 + µ) 1 2 δ2 J, δ 2 J), where µ is the average jump amplitude such that: µ J = ln(1 + µ) 1 2 δ2 J. Applying Itô s formula we obtain the equation for the log process Y t = ln S αβ(t) S αβ (0) : dy t = ( λ µ 1 2 σ2 αβv t )dt + σ αβ Vt dw αβ (t) + d Z t, where Z t is a compound Poisson process with intensity λ and Gaussian distribution of jump sizes. Proposition The characteristic function is exp( κθt(κ iρηu) Φ t (u) = exp(tλe 1 2 δ2 u 2 +i(ln(1+ µ) 1 2 δ2 J )u 1) where = Proof: (κ iρηu) 2 + η 2 σαβ 2 u(u + i). The characteristic function of the jumps is: (u2 +iu)σαβ 2 v η2 coth( t 2 (cosh( t) + (κ iρηu) 2 + iut( λ µ) + iuy) )+κ iρηu, sinh( t 2 ))( 2κθ) η 2 Φ J t (u) = exp(tλe 1 2 δ2 J u2 +i(ln(1+ µ) 1 2 δ2 J )u 1). (3.18) 48

59 Since jumps are homogeneous and independent from the continuous part the characteristics function of the swap rate process in the jump-diffusion SV extension of the SMM is obtained by the following multiplication: Φ t (u) = Φ J t (u)φ C t (u), where Φ C t (u) = exp( κθt(κ iρηu) (u2 +iu)σαβ 2 v η 2 γ coth( t 2 (cosh( t) + (κ iρηu) 2 + iut( λ µ) + iuy) )+κ iρηu. (3.19) sinh( t 2 ))( 2κθ) η 2 So that exp( κθt(κ iρηu) Φ t (u) = exp(tλe 1 2 δ2 u 2 +i(ln(1+ µ) 1 2 δ2 J )u 1) (u2 +iu)σαβ 2 v η2 coth( t 2 (cosh( t) + (κ iρηu) 2 )+κ iρηu + iut( λ µ) + iuy) sinh( t 2 ))( 2κθ η 2 ) Note that (3.17) and (3.19) are very similar. The difference lies in the shift λ µ, the risk-neutral correction. 49

60 CHAPTER 4 FAST FRACTIONAL FOURIER TRANSFORM [23] [24] [25] 4.1 Fast Fractional Fourier Transform (FRFT): Overview Using the characteristic function in order to price European options has become the practice after Heston (1993) [16] published his paper on stochastic volatility. An increasing number of recently developed models for asset prices use the characteristic function to price European options, including the stochastic volatility model of Heston (1993) [15] and the jump diffusion stochastic volatility model of Bates (1998) [15]. Carr and Madan (1999) [24] show how the power of the fast Fourier transform (FFT) method can be applied to a modification of the characteristic function in order to price options in an accurate and efficient manner. The FFT is an efficient method for computing the sums: D m (h) = N 1 j=0 e i2π N mj h j, m = 0,...,N 1, Dm 1 (h) = 1 N 1 e i2π N mj h j, m = 0,...,N 1. N j=0 where N is typically a power of 2. This algorithm reduces the number of multiplications in the required N sums from O(N 2 ) to O(N ln N), a very significant reduction. D m (h) is the discrete Fourier transform, or DFT, of a vector h and D 1 m (h) is its inverse. The FFT method is a way to approximate a continuous Fourier transform with a discrete counterpart, for a carefully chosen vector h = (h j ) N 1 j=0 : 0 e iku h(u)du N 1 j=0 50 e iku j h(u j )η, (4.1)

61 where u j = ηj, and η is the grid size of the vector u. Vector h corresponds to N function evaluations of h at u = (u j ) N 1 j=0. Let f be (f m) N 1 m=0, where each f m corresponds to an integral of the form (4.1) computed at k = k m, m = 0,...,N 1. Carr and Madan (1999) show that the vector f will contain the option prices that correspond to the log-strike prices at k = k m, m = 0,...,N 1 defined as k m = b + λm, m = 0,...,N 1, where λ is the grid size of k. This results in log-strike range [ b, b], where b = Nλ. 2 Then, where h j = e ibu j h j η. N 1 j=0 e ikmu j h j η = N 1 j=0 e i( b+λm)u j h j η = N 1 j=0 e iλmηj hj, To apply the fast Fourier transform, the following relationship must be satisfied: λη = 2π N. This inverse relationship between the grid sizes η and λ is well known (see Bailey and Swarztrauber (1991, 1994) [23], Carr and Madan (1999) [24]). It follows that to have a fine grid across k, one has to either increase η thus making the grid across u less refined, or increase the length of h, effectively filling it with zeros. Not only will the resulting series extend well beyond the range of k but also valuable computational time is lost. It is clear that out of the three parameters N, η, and λ only two can be chosen independently. Since η and N are chosen to make integral approximations more accurate, λ is restricted. Chouradakis (2005) [25] shows the use of a more efficient algorithm, the fast fractional Fourier transform, to option pricing. This algorithm uses the characteristic function in a more efficient way than the standard FFT approach. Therefore less function evaluations are typically needed and substantial savings in time can be made. The fast fractional Fourier transform (FRFT) [23] algorithm can be used to rapidly compute the sums: G m (h, α) = N 1 j=0 e i2πmjα h j, m = 0,...,N 1, (4.2) for any value of the parameter α. Although the FRFT is defined for all integer m it is usually used to compute the first N nonnegative values, i.e. for m = 0,...,N 1. The sums 51

62 are computed by invoking two normal and one inverse FFT algorithms. The advantage of using the FRFT method over the FFT is that the grid sizes η and λ can be chosen independently. The FFT method requires λη = 2π. The fractional FFT method is faster N due to the absence of this restriction allowing the use of sparser grids. It is very important in terms of computational time. We note that the ordinary DFT and its inverse are special cases of the fast fractional Fourier transform: D m (h) = G m (h, 1 ), m = 0,...,N 1, N Dm 1 (h) = 1 N G m(h, 1 ), m = 0,...,N 1. N The standard FFT can be seen as a special case of the FRFT for α = 1. We note that N the FRFT does not overcome the critical limitation of the FFT method because the grid points still need be equidistant. The number of operations under FRFT algorithm will be approximately 6N ln N since one N-point FRFT invokes three 2N-point FFT algorithms, approximately 2N ln N operations each. This implies that FFT and FRFT have a critical point at which both have similar operational requirements. According to Chourdakis the theoretical FRFT to FFT ratio is approximately equal to four. For instance, option pricing using a 256-point FRFT should demand a similar number of operations as a 1024-point FFT. 52

63 4.2 FRFT Algorithm As discussed in Bailey and Swarztrauber (1991, 1994), the fractional transform can be easily implemented by invoking three 2N-point FFT procedures. Suppose we want to compute an N-point FRFT on vector h = (h j ) N 1 j=0. Since 2jm = j 2 + m 2 (m j) 2 we have G m (h, α) = N 1 j=0 e i2πmjα h j = N 1 j=0 N 1 = e iπm2 α e iπj2α h j e iπ(m j)2 j=0 e iπ(j2 +m 2 (m j) 2 )α h j where y j = e πij2α h j and z j = e πij2α. N 1 = e iπm2 α y j z m j, (4.3) j=0 The sum above is a discrete convolution suggesting the use of the FFT procedure. However, the standard FFT method is applicable only for circular convolutions, wherein z m j = z m j+n. In our case we have z m j = z j m when m j < 0. It is possible however to convert the sum (4.3) into a circular convolution form by extending y and z to length 2N. In the original paper [23] the two vectors are extended to length 2p for an integer p N 1. We adopt Chourdakis s choice of p = N [25]. The following 2N-point vectors have to be defined: y = ((e iπj2α h j ) N 1 j=0, (0)N 1 j=0 ), z = ((eiπj2α ) N 1 j=0, (eiπ(n j)2α ) N 1 j=0 ). (4.4) Then, the FRFT is given by: for 0 m < N. G m (h, α) = (e iπm2α ) N 1 m=0d 1 m (D j (y)d j (z)) (4.5) The remaining N results of the final inverse FFT are discarded. Element-by-element vector multiplication is implied in the expression D j (y)d j (z). 53

64 4.3 Application Of The FRFT We adapt the methodology of Carr and Madan (1999) in order to use the fractional FFT in order to price European swaptions. The payers swaption price at time zero is: PS SMM (0, [T α,...,t β ], K) = C α,β (0)E α,β [(S α,β (T α ) K) + ], where E α,β stands for expectation under the forward swap rate. In terms of Y (t) = ln S αβ(t) S αβ (0) and k = ln K S αβ (0) we have: PS SMM (0, [T α,...,t β ], K) = C α,β (0)S α,β (0)E α,β [(e Y (Tα) e k ) + ]. (4.6) Let G(k) = E α,β [(e Y (Tα) e k ) + ], where Y (T α ) = ln S α,β(t α). S α,β (0) Then G(k) = k (es e k )q T (s)ds. We denote by q T (s) the risk-neutral density of Y (T) such that Φ T (u) = where Φ(.) is the characteristic function. e iuy q T (y)dy, (4.7) Since G(k) is not square-integrable due to the fact that G(k) k 1 we consider the following modification G(k) = e ak G(k). Now G(k) is square-integrable for a suitable a > 0 (called dampening parameter) satisfying E[S α,β (T α ) a+1 ] <. The accuracy of the prices with the FRFT method strongly depends on the parameter a. The Fourier transform of G(k) is defined by: Ψ(u) = e iuk G(k)dk. Carr and Madan (1998) link up the Fourier transform of the modified G(k) with the characteristic function Φ T of Y (T): Ψ(u) = Φ T(u (a + 1)i), a > 0. (4.8) (a + iu)(1 + a + iu) 54

65 Proof: Ψ(u) = = e iuk e ak G(k)dk = Using equation (4.7), we have: e iuk e ak (e s e k )q T (s)dsdk s q T (s)( (e s+(a+iu)k e (1+a+iu)k )dk)ds 1 = ( a + iu a + iu ) Ψ(u) = k q T (s)e (1+a+iu)s ds 1 (a + iu)(1 + a + iu) Φ T(u (a + 1)i). Using Fourier inversion, we express G(k) in terms of Ψ: G(k) = exp( ak) π 0 e iuk ψ(u)du, which we will evaluate numerically using the FRFT algorithm. We set equidistant points u j = ηj with grid size η > 0 and obtain the following approximation for G(k): G(k) exp( ak) π N 1 j=0 e iu jk ψ(u j )η. The value of η should be sufficiently small to approximate the integral well, while the value of Nη should be large enough so that the characteristic function is equal to zero for u > Nη. Application of the FRFT will result in a set of integral approximations computed for a set (k m ) N 1 m=0. These values will be equidistant around the ATM level of k = 0, with grid size λ. Then, the values (k m ) N 1 m=0 would assume the form: k m = Nλ 2 + λm, m = 0,...,N 1. This gives us strikes levels between exp(±k 0 ), where k 0 = Nλ. We also incorporate the 2 Simpson s rule weighting into our summation: G(k m ) exp( ak m) π N 1 j=0 exp( ak m) π Nλ iηj( e 2 +λm) ψ(u j ) η 3 (3 + ( 1)j δ j 1 ) N 1 55 j=0 e iηjλm h j,

66 where Here δ n is the Kronecker delta function: Nλ iηj h j = e 2 ψ(uj ) η 3 (3 + ( 1)j δ j 1 ). (4.9) δ n = { 1, n = 0; 0, n 0. One needs to make the appropriate choices for η and λ independently and then recover the value of α through the relation α = ηλ 2π. We set the FRFT parameters to the following values: a = 3, N = 64, and η = Then, we chose k 0 = 0.6, so that the range for the strikes is between exp{±0.60}. The grid size for strikes λ = 2(0.60) 64 = The choices above imply that the fractional parameter α is (0.25)( ) 2π = Let m and k denote the input and output grids, respectively. In order to implement FRFT, we will need to compute and store the following vectors: h = (h j ) 63 j=0, a = (exp{iαπj 2 }) 63 j=0, ā = (exp{iαπ(64 j) 2 }) 63 j=0, d = (exp{ ak m We now have to compute the fractional transform: π })63 j=0. y 1 = (h/a, (0) 63 j=0), ỹ 1 = fft(y 1 ), y 2 = (a, ā), ỹ 2 = fft(y 2 ), y 3 = ỹ 1 ỹ 2, ỹ 3 = ifft(y 3 ), y = ỹ 3 /a, and finally: c = yd = ifft(fft(y 1 )fft(y 2 ))d. Here y 1 and y 2 are y and z from (4.4) in Section 4.2 respectively. Let us recall the payer swaption price (expression (4.6)) at time zero: PS SMM (0, [T α,...,t β ], K) = C α,β (0)S α,β (0)G k (h, α), (4.10) where h is defined in (4.9) and G k is defined in (4.5). Here ψ is a modification of the characteristic function (see (4.8)) defined in (3.17). 56

67 CHAPTER 5 CALIBRATION [26] [6] [20] [2] [15] [9] [27] [10] [28] 5.1 Overview Once a stochastic volatility model is chosen, it must be calibrated against market data. Calibration is the process of identifying a set of model parameters that are determined by the observed data. In the stochastic volatility extension of the SMM without jumps (SV SMM), described in Chapter 3 Section 3.1, the set of model-specific parameters are Θ = {κ, θ,η, V 0, ρ}. In the stochastic volatility extension of the SMM with jumps (LN- SV SMM), described in Chapter 3 Section 3.2, the set of model-specific parameters are Θ = {κ, θ,η, V 0, ρ,µ J, δ J, λ}. The calibration of the instantaneous forward swap rate volatilities introduces additional forty-eight parameters, forty-five scaling factors and three vol parameters, common for the two extension models. We will separate the calibration into a two-step process. First step is the calibration for the instantaneous forward swap rate volatilities. We will look for shapes that replicate the initial term structure of Black implied swaption volatility. We would like the forward swap rate volatilities to be time stationary as much as possible since the term structure of the ATM volatility tends to be preserved in time. One extreme would be to assume time-constant volatility structure which clearly contradicts the historical market data. The other extreme is a totally time-homogeneous term structure of volatility which means that the volatility of the forward swap rates purely depends on the time to maturity. A better fit to the swaption implied Black volatility is obtained with a time-homogeneous function if the following condition is satisfied: σ α,β Tα must be monotone increasing function with option maturity T α of the swaption. For the 57

68 swaption implied Black volatility with total maturity of 10 years this is not always the case in the data. Hence, one imposes additional structure on the volatility of the forward swap rate: σ α,β (t) = f(t α t) = [a + b(t α t)]e c(tα t) + d. The function generates exact time-homogeneity and ensures non-negative values. It is flexible to fit monotone decreasing volatility structure as well. This specification of f(t α t) is suggested in the LMM framework by Brigo and Mercurio 2001 and Rebonato The above formulation can be extended into a richer parametric form: σ α,β (t) = g(t)φ α,β [(a + b(t α t))e c(tα t) + d]. (5.1) It reduces to the previous one when all Φ are set to one and g(t) is not present. The optional function g(t) is determined to reflect time-dependent movements in volatility levels. Usually it is modeled as a linear combination of a small number of sine waves multiplied by exponentially decaying factor. The presence of the scaling factors Φ ensures the exact recovery of the at-the-money prices. Modifications of the form (5.1) are used in Galluccio et al. [9] and in De Jong et al. [26] and for each swap period a separate calibration was done. Sami Attaoui proposed a new parametric form of the forward swap rates volatility that takes into account the swap tenor as well as the option maturity. This feature allows the calibration to be performed to the whole swaption volatility matrix, avoiding a separate calibration for each swap period. We consider a modification of the parametric form suggested by Attaoui in [20]: σ α,β (t) = Φ α,β {a(t β T α ) 1 2 e b(t α t) + c}, (5.2) where a, b and c are positive constants. This form meets the observed evidence that volatility decreases with longer swaption maturity and larger swap lengths as seen in Figure

69 Figure 5.1: Implied Volatilities of ATM Swaptions Across Option Maturities for Different Swap Tenors We note our data exhibits the excited pattern as described by Rebonato and Joshi in their swaption matrix pattern study in 2002 [7]. The parametric form used in their paper is a modification of the form (5.1) where the function g(t) is not considered. The forward-rate specific constant Φ α,β needed in order to ensure correct pricing of the ATM swaptions should be as close to unity as possible in the calibration process. First,we fit σ α,β (t) to the market implied swaptions volatility by adjusting the parameters a, b and c and keeping the scaling factors at one over swaptions with total maturity less or equal to 10 years: min a,b,c [(σblack α,β Tα ) 2 Tα σα,β(t)dt], 2 0 where σ α,β is described by the parametric form (5.2). 59

70 From the relationship: we have that: (σα,β Black Tα ) 2 Tα = σα,β(t)dt. 2 0 (σα,β Black Tα ) 2 Tα = Φ 2 α,β (a(t β T α ) 1 2 e b(t α t) + c) 2 dt = Φ 2 α,βi 2 (T α, a,b, c). 0 We set the scaling factors such that we have a good fit to the initial term structure of the swaption volatilities: Φ 2 α,β = (σblack α,β Tα ) 2 I 2 (T α, a,b, c). Next is the calibration for the model parameters Θ to minimize the sum of the square differences between the market and the model swaption prices: min [Π MODEL (T α, T β, σ αβ, K, Θ) Π MARKET (T α, T β, σαβ Black, K)] 2, Θ where Π MODEL (T α, T β, σ αβ, K, Θ) and Π MARKET (T α, T β, σαβ Black, K) are the model and the market swaption prices. This calibration is done over the set of swaptions with total maturity equal to 10 years. For all minimization procedures we use in Matlab the function lsqnonlin(). Matlab s least-squares, non-linear optimizer is the function lsqnonlin(fun,x0, lb,ub). It minimizes the vector-valued function, fun, using the vector of initial parameter values, x0, where the lower and upper bounds of the parameters are specified in vector lb and ub, respectively. lsqnonlin() uses an interior-reflective Newton method for large scale problems. Matlab defines a large scale problem as one containing bounded/ unbounded parameters, where the system is not under-determined, i.e., where the number of equations to solve is more than the required parameters. Matlab suggests Coleman and Li (1994, 1996) for further reference on the these methods. The result produced by lsqnonlin() is dependent on the choice x0, the initial estimate. This is, therefore a local optimizer, not a global one. There is no way to know whether the solution is a global or local minimum. 60

71 5.2 Data We use two types of data: swap data defining the term structure of interest rates and market implied volatilities for European swaptions. Together with the term structure data, these implied volatilities define the market prices of swaptions. The source of all data is Bloomberg. We have obtained 109 daily observations from 1/1/2008 to 5/31/2008 for the following data from Bloomberg: daily at-the-money (ATM) implied volatility of European swaptions with swaption maturity 1, 2, 3, 4, 5, 6, 7, 8 and 9 years and swaption tenor 1, 2, 3, 4, 5, 6, 7, 8 and 9 years, such that the total maturity is less or equal to 10 years, swap rates of maturities 1, 2, 3, 4, 5, 6, 7, 8 and 9 years to determine the forward swap rate curve. There is one-to-one relationship between the implied Black volatility of each swaption and its price, so that we are able to get prices for all swaptions. As described earlier the market convention is to quote swaption prices in term of their implied volatilities relative to the Black-type formula (Thm 2.5.6). The market prices are given by substituting the implied volatility into the Black formula. Hence, we have daily prices for 9 swaptions with total maturity 10 years (1x9, 2x8, 3x7, 4x6, 5x5, 6x4, 7x3, 8x2 and 9x1) and 36 swaptions with total maturity less than 10 years. The market for the ITM and OTM swaptions has not been liquid enough to obtain reliable historical values. Only ATM volatility is available for swaption data in Bloomberg. ATM swaption implied volatility surface for a particular date is shown in Figure

72 Figure 5.2: ATM Implied Volatility Surface on 2/12/2008 The swap rates are from the Bloomberg SWPM curve 23 (USD Swaps 30/360) based on piecewise linear interpolation method. Figure 5.3 shows the curve as of 2/12/2008. Figure 5.3: Bloomberg SWPM Curve on 2/12/

73 To download historical data from Bloomberg we use the Bloomberg Excel API s History Wizard. It generates BDH(Bloomberg Data History) functions to display the time series of the data. The general syntax is BDH(ticker, field, begin date, end date, options). Co-terminal swaptions tickers: USSV019 Index, USSV028 Index, USSV037 Index, USSV046 Index, USSV055 Index, USSV064 Index, USSV073 Index, USSV082 Index, and USSV091 Index. Swaption with total maturity less than 10 years tickers: option maturity in 1 year: USSV011 Index,USSV012 Index, USSV013 Index, USSV014 Index, USSV015 Index, USSV016 Index, USSV017 Index, USSV018 Index option maturity in 2 years: USSV021 Index, USSV022 Index, USSV023 Index, USSV024 Index, USSV025 Index, USSV026 Index, USSV027 Index option maturity in 3 years: USSV031 Index, USSV032 Index, USSV033 Index, USSV034 Index, USSV035 Index, USSV036 Index option maturity in 4 years: USSV041 Index, USSV042 Index, USSV043 Index, USSV044 Index, USSV045 Index option maturity in 5 years: USSV051 Index, USSV052 Index, USSV053 Index, USSV054 Index option maturity in 6 years: USSV061 Index, USSV062 Index, USSV063 Index option maturity in 7 years: USSV071 Index, USSV072 Index option maturity in 8 years: USSV081 Index USD Swap rates tickers: USSW1 Index, USSW2 Index, USSW3 Index, USSW4 Index, USSW5 Index, USSW6 Index, USSW7 Index, USSW8 Index, USSW9 Index and USSW10 Index. A zero curve is derived from the swap rates using a bootstrap method and then the discount factors are obtained. 63

74 Tables 5.1 and 5.2 present the swap rates for February 12, 2008 and the calculated discount factors respectively. Table 5.1: Swap Rates as of 2/12/2008 Maturity Rate Table 5.2: Discount Factors as of 2/12/2008 Maturity Rate

75 Figure 5.4 shows the live discount factor calculator developed in Excel. All links are live from Bloomberg. Figure 5.4: Live Zero Coupon Rate and Discount Factor Calculator The same mathematics is used to calculate the daily discount factors from the swap rate information for the period 1/1/08 to 5/31/08. 65

76 5.3 Calibration Step One The first step is the calibration of the swaption volatility matrix. parameters (a, b,c) to the form: We find the fitted σ α,β (t) = Φ α,β {a(t β T α ) 1 2 e b(t α t) + c} by first setting Φ to one and minimizing the square differences of the implied volatility to the model s. We use the Matlab function lsqnonlin(). The result for a particular date is shown in Table 5.3 below. Table 5.3: Fitted instantaneous volatility parameters on 2/12/2008 a b c As mentioned in the overview, a better calibration is achieved when scaling factors are introduced. Table 5.4 shows the scaling factors obtained from the whole swaption matrix calibration to market data on February 12, We note that the scaling factors are around one as expected. Table 5.4: Scaling Factors on 2/12/2008 Option/Swap The result is an indication that the future term structure of volatilities is not very different from the one observed today. It also implies that daily re-calibration of the parameters may not be necessary due to the time-homogeneity. 66

77 5.4 Calibration Step Two The second step is to calibrate the SV SMM and LN-SV SMM for the models parameters Θ using the obtained in step one instantaneous volatility. We can calibrate each model to the set of co-terminal swaptions with total maturity 10 years for which Black formulas are available. For the best fit we use Matlab build in function lsqnonlin(). Once we have the fitted parameters we can use the model to price swaptions with total maturity less than 10 years. We then compare the swaption prices from the models to the observed on the market to investigate the pricing power of each model. Table 5.5 presents the results from the calibration step two. Table 5.5: Fitted model parameters on 2/12/2008 Parameters SV SMM LN-SV SMM κ θ η V ρ µ J δ J λ RM SE RM SEvol RMSE is the Root Mean Square Error between the market and the model prices in bp. RMSEvol is the Root Mean Square Error between the market and the model implied volatilities in bp. The big advantage of having a two-step calibration procedure is that at each step fewer parameters are considered. 67

78 CHAPTER 6 RESULTS 6.1 In-Sample Fit We calibrate each of the extension models SV SMM and LN-SV SMM of the SMM to a set of co-terminal swaptions with total maturity of 10 years by applying the two step procedure described in the previous section for the period 1/1/2008 to 5/31/2008. Given the swaption prices we look for model parameters that can reproduce the data well. As a measure of the daily fit we used the Root Mean Square Error (RMSE) as defined below: RMSE = 1 n [FittedPrice i ActualPrice i ] n 2, (6.1) i=1 where n = 9 is the number of co-terminal swaptions used in the calibration. Averaging these daily time series of RMSEs over time produces the overall RMSE: RMSE = 1 N N RMSE i, (6.2) i=1 where N = 109 is the number of days and RMSE i for i = 1,...,N is calculated based on (6.1). Figure 6.1 shows the means and the standard deviations from 109 daily observations from 1/1/2008 to 5/31/2008 on implied Black swaption volatilities. Each row contains swaptions with fixed option maturity and different swap maturities. All maturities are in years. The Black volatilities are in annualized percent, volatility points. Table 6.1 contains the summary statistics for the forward swap rates from 109 daily observations from 1/1/2008 to 5/31/

79 Figure 6.1: Descriptive statistics for the market ATM swaption volatility data Table 6.1: Descriptive statistics for the forward swap rates Maturity Mean St Dev The fitted volatility parameters from calibration step one are presented in Table 6.2. The low standard deviations indicate that the volatility parameters are stable for the chosen period. It also indicates that daily re-calibration of these parameters may not be needed. Tables 6.3 and 6.4 present the fitted parameters for the SV SMM and LN-SV SMM from calibration step two respectively. The calibrated values seem to be stable over the period in consideration as indicated by their low standard deviations. We note that jumps take off some pressure from the vol of vol and mean-rev parameters. The results show a strong 69

80 Table 6.2: Descriptive statistics for the fitted swap rate volatility parameters Parameters Mean St Dev a b c Table 6.3: Fitted models parameters for the SV SMM Parameters Mean St Dev κ θ η V ρ RM SE Table 6.4: Fitted models parameters for the LN-SV SMM Parameters Mean St Dev κ θ η V ρ µ J δ J RM SE negative correlation between changes in swap rates and changes in swap rate volatility. It is similar to the observed relationship for stock prices. The fit of the models is measured by RMSE, the Root Mean Square Error, between the observed and the model prices in bp. We make a pairwise comparison between the models in-sample fit using the Diebold and Mariano (1995) method [29]. Suppose that two models generate time-series of RMSE pricing errors RMSE 1 (t) and RMSE 2 (t). We then compute the mean of the differences d(t) = RMSE 2 (t) RMSE 1 (t) and the associated statistic. A 70

81 significantly negative mean implies that model two has a significantly better fit than model one. The null of equal predictive accuracy is: H 0 : E[d(t)] = 0. The Diebold-Mariano test statistic is: S = d, LRV ˆ d/n where and LV R d = γ d = 1 N N d(t), t=1 γ j, γ j = cov(d t, d t j ), j=1 ˆ LRV d is a constant estimate of the asymptotic (long-run) variance of N d. Diebold and Mariano (1995) show that under the null of equal predictive accuracy: S A N(0, 1). So we will reject the null of equal predictive accuracy at the 5 percent level if S > Table 6.5 displays the pairwise comparison of in-sample fit using the Diebold and Mariano (1995) criterion. Table 6.5: Statistical Comparison of in-sample fit based on Diebold and Mariano criterion LN-SV SMM vs. SV SMM mean difference in RMSE DM statistic -9.7 We reject the null hypothesis that the difference between the two means is zero since 9.7 < The significantly negative mean difference implies that LN-SV SMM has a significantly better fit than SV-SMM. The number of parameters in a model plays an important role. Having more parameters usually means a better fit to data. In order to evaluate the in-sample fit of the two extension 71

82 models we apply the Bayesian Information Criterion (BIC). If one assumes for convenience that the model errors are normally distributed than BIC is an easy corrected version of the RSS, residual sum of squares of errors: BIC = RSS + k ln(n), where k is the number of parameters for estimation and n is the number of observations used. From (6.1) we have RSS = nrmse 2. Table 6.6 displays the pairwise comparison of in-sample fit based on the BIC criterion. Table 6.6: Statistical Comparison of in-sample fit based on BIC criterion SV SMM LN-SV SMM k RSS BIC The BIC criterion confirms that LN-SV SMM has a superior in-sample fit to our data compared to the SV SMM after taking in consideration the model parameters. Table 6.7 displays the run time of the calibration for SV SMM and LN-SV SMM. Table 6.7: Run time table SV SMM LN-SV SMM time in sec It took about two times more time to run calibration step two for LN-SV SMM. The calibration step one took 35 seconds. It is the same for both extension models. 72

83 6.2 Out-Of-Sample Fit In this section we observe how well the SV SMM and LN-SV SMM price the swaptions with total maturity of less than 10 years, i.e. swaptions not used for calibration, relative to the market practice. Black s formula is used in practice to price these swaptions. The SV SMM results for swaptions with total maturity less than 10 years are presented first in Figure 6.2. We notice than the model produces very similar prices to the ones derived through the Black-type swaption formula. Figure 6.2: SV SMM results for Swaptions with Total Maturity Less Than 10 years Differences between the model and the market prices are very small. The LN-SV SMM results for swaptions with total maturity less than 10 years are presented next in Figure

84 Figure 6.3: LN-SV SMM results for Swaptions with Total Maturity Less Than 10 years This model produces prices closer to the observed in the market. Differences between the model and the market prices are very small. Both results indicate that calibrating the models to the set of the co-terminal or diagonal swaptions only is enough to capture the volatility for swaptions to the left of the diagonal. 74