Stochastic Volatility Extensions of the Swap Market Model Milena G

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2008 Stochastic Volatility Extensions of the Swap Market Model Milena G. (Milena Gueorguieva) Tzigantcheva

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COLLEGE OF ARTS AND SCIENCES

STOCHASTIC VOLATILITY EXTENSIONS OF THE SWAP MARKET MODEL

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 The members of the Committee approve the Dissertation of Milena G. Tzigantcheva defended on October 8, 2008.

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 To my daughter Adriana

iii 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 acknowl- edges 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 TABLE OF CONTENTS

List of Tables ...... vii

List of Figures ...... viii

Abstract ...... ix

1. INTRODUCTION ...... 1

2. PRELIMINARY ...... 4 2.1 Interest Rates ...... 4 2.2 Interest-Rate Swaps ...... 9 2.3 Interest-Rate Swaptions ...... 14 2.4 Change of Numeraire ...... 16 2.5 Swap Market Model (SSM): Overview ...... 18 2.6 Fourier Transform and Characteristic Function ...... 34 2.7 Compound Poisson Process ...... 35 2.8 Stochastic Volatility Models: Overview ...... 36

3. STOCHASTIC VOLATILITY EXTENSIONS OF SMM ...... 38 3.1 Stochastic Volatility Extension Of SMM Without Jumps ...... 38 3.2 Stochastic Volatility Extension Of SMM With Log-Normal Jumps . . . . 48

4. FAST FRACTIONAL FOURIER TRANSFORM ...... 50 4.1 Fast Fractional Fourier Transform (FRFT): Overview ...... 50 4.2 FRFT Algorithm ...... 53 4.3 Application Of The FRFT ...... 54

5. CALIBRATION ...... 57 5.1 Overview ...... 57 5.2 Data ...... 61 5.3 Calibration Step One ...... 66 5.4 Calibration Step Two ...... 67

6. RESULTS ...... 68 6.1 In-Sample Fit ...... 68 6.2 Out-Of-Sample Fit ...... 73

v 6.3 Forecasting ...... 75

7. CONCLUSION ...... 78 7.1 Future Work ...... 79

APPENDICES ...... 80

A. MATLAB CODE ...... 80

B. ABBREVIATIONS AND NOTATION ...... 94

REFERENCES ...... 97

BIOGRAPHICAL SKETCH ...... 100

vi LIST OF TABLES

2.1 Black Implied Volatility of ATM Swaptions on 2/12/2008 ...... 22

2.2 Examples of Bivariate Diﬀusion Models for Option Pricing ...... 37 5.1 Swap Rates as of 2/12/2008 ...... 64 5.2 Discount Factors as of 2/12/2008 ...... 64

5.3 Fitted instantaneous volatility parameters on 2/12/2008 ...... 66 5.4 Scaling Factors on 2/12/2008 ...... 66 5.5 Fitted model parameters on 2/12/2008 ...... 67

6.1 Descriptive statistics for the forward swap rates ...... 69 6.2 Descriptive statistics for the ﬁtted swap rate volatility parameters ...... 70 6.3 Fitted models parameters for the SV SMM ...... 70

6.4 Fitted models parameters for the LN-SV SMM ...... 70 6.5 Statistical Comparison of in-sample ﬁt based on Diebold and Mariano criterion 71 6.6 Statistical Comparison of in-sample ﬁt based on BIC criterion ...... 72

6.7 Run time table ...... 72 6.8 Forecasting performance Of SV SMM ...... 75 6.9 Forecasting performance Of LN-SV SMM ...... 76

6.10 Forecasting performance Of BS SMM ...... 76 6.11 Statistical Comparisons of forecasting performance based on Diebold and Mariano criterion ...... 77 6.12 Statistical Comparison of forecasting performance based on BIC criterion . . 77

vii LIST OF FIGURES

3.1 Time Series of Implied Volatilities for ATM Swaptions ...... 39 5.1 Implied Volatilities of ATM Swaptions Across Option Maturities for Diﬀerent Swap Tenors ...... 59 5.2 ATM Implied Volatility Surface on 2/12/2008 ...... 62

5.3 Bloomberg SWPM Curve on 2/12/2008 ...... 62 5.4 Live Zero Coupon Rate and Discount Factor Calculator ...... 65 6.1 Descriptive statistics for the market ATM swaption volatility data ...... 69

6.2 SV SMM results for Swaptions with Total Maturity Less Than 10 years . . . 73 6.3 LN-SV SMM results for Swaptions with Total Maturity Less Than 10 years . 74

viii 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 square- root 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 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 J¨ackel 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 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 SMMa ` 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 SMMa ` 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 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 CHAPTER 2

PRELIMINARY

2.1 Interest Rates

[1][2][3] Please refer to Appendix B for a list of Abbreviations and Notation. Definition 2.1.1 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:

dB(t) = rtB(t)dt, B(0) = 1, where rt is a positive function of time. t r ds Investing one dollar at time zero yields B(t) = e 0 s dollars at time t. R The rate rt at which the bank account accrues is referred to as the instantaneous spot rate, or as the short rate. Deﬁnition 2.1.2 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

B(t) − T r ds D(t, T ) = = e t s . B(T ) R 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 2.1.3 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 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 2.1.4 Continuously-compounded spot interest rate The continuously- compounded 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 : ln P (t, T ) R(t, T ) := . − T t − The continuously-compounded interest rate is a constant rate that is consistent with the zero-coupon prices: eR(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 2.1.5 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.

1 P (t, T ) L(t, T ) := − (2.1) τ(t, T )P (t, T ) 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 1 P (t, T ) = . 1 + L(t, T )τ(t, T )

5 Deﬁnition 2.1.6 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. 1 Y (t, T ) := 1 1. P (t, T ) τ(t,T ) −

The bond price in terms of annually-compounded rates is 1 P (t, T ) = . (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 2.1.7 Yield curve The zero-coupon curve (yield curve) at time t is the graph of the function

L(t, T ), if t < T t + 1(years); T ≤ → 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 2.1.8 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 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: 1 N[τ(T,S)K + 1]. − P (T,S)

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. FRA(t, T, τ(T,S),N,K) = 0 at time t. Deﬁnition 2.1.9 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 P (t, T ) F (t; T,S) := ( 1). τ(T,S) P (t, S) − 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 Deﬁnition 2.1.10 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). − Zt

The instantaneous forward rate is a forward interest rate at t with maturity close to expiry T .

8 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 Ti in a pre-specified set of dates Tα+1, ..., Tβ the fixed leg pays out

NτiK, where K is the ﬁxed interest rate, N is the nominal value and τ = T T − for every i i − i 1 i = α + 1, ..., β. The ﬂoating leg pays

NτiL(Ti−1,Ti) corresponding to L(Ti−1,Ti) resetting at Ti−1 for maturity Ti. The ﬂoating leg resets at

Tα, ..., Tβ−1 and pays at Tα+1, ..., Tβ. The dates Tα, ..., Tβ−1 are known as reset dates, and the dates Tα+1, ..., Tβ are known as settlement dates. We will refer to Tα 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 , ..., T and τ := τ , ..., τ { α β} { α+1 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 < Tα can be written as:

D(t, T )Nτ (L(T − ,T ) K), i i i 1 i − i=Xα+1 and the RFS’ discounted payoﬀ at time t < Tα can be written as:

D(t, T )Nτ (K L(T − ,T )), i i − i 1 i i=Xα+1 or alternatively as

D(t, T ) P (T ,T )Nτ (K F (T ; T − ,T )). (2.3) α α i i − α i 1 i i=Xα+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 β

RFS(t, T, τ, N, K) = FRA(t, Ti−1,Ti, τi,N,K). i=Xα+1 And so β

RFS(t) = NP (t, T )τ (K F (t; T − ,T )) i i − i 1 i i=Xα+1

β = NK τ P (t, T ) NP (t, T ) + NP (t, T ). (2.4) i i − α β i=Xα+1 Analogously,

β PFS(t, T, τ, N, K) = NK τ P (t, T ) + NP (t, T ) NP (t, T ). − i i α − β i=Xα+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 2.2.1 Coupon-bearing bond A coupon-bearing bond is a contract that guarantees payments of deterministic amounts of cash-flows c = cα+1, ..., cβ at future times

Tα+1, ..., Tβ. Typically, the cash-flows are defined as ci = NτiK for i < β and cβ = NτβK+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 Tβ. Discounting each cash-ﬂow to current time t we obtain the value of the bond:

CB(t) = ciP (t, Ti). i=Xα+1

We can write the IRS value in terms of the cashﬂows ci as:

β RFS(t) = c P (t, T ) NP (t, T ). (2.5) i i − α i=Xα+1 Analogously, β PFS(t) = NP (t, T ) c P (t, T ). α − i i i=Xα+1 Deﬁnition 2.2.2 Floating-rate note A ﬂoating-rate note is a contract that guarantees payments at future times Tα+1, ..., Tβ of the LIBOR rates that reset at the previous times

Tα, ..., Tβ−1. In addition, the note pays a last cash-ﬂow of the notional value at ﬁnal time Tβ.

10 The floating-rate note is always equivalent to NP (t, Tα). 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 Tβ.

RFS(t, T, τ, N, 0) + NP (t, T ) = NP (t, T ). − β α

If t = Tα, 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 when i = α + 1, ..., β 1. The i − 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 2.2.3 Forward swap rate The forward swap rate Sα,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:

P (t, Tα) P (t, Tβ) Sα,β(t) = β − . (2.6) i=α+1 τiP (t, Ti) Let us deﬁne the positive portfolio ofP zero coupon bonds

Cα,β(t) = τiP (t, Ti). (2.7) i=Xα+1

Often Cα,β(t) is referred to as the present value of a basis point (PVBP).

The swap rate is the forward swap rate at time Tα and can be expressed in term of the PVBP as follows

1 P (Tα,Tβ) Sα,β(Tα) = β − . (2.8) i=α+1 τiP (Tα,Ti) It will be useful to express the valueP 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 (t)[K S (t)]. (2.9) α,β − α,β Analogously, PFS(t, T, τ, N) = NC (t)[S (t) K]. α,β α,β − 11 Let us introduce the forward zero-coupon-bond price at time t for maturity S as P (t,S) seen from expiry T : FP (t; T,S) = P (t,T ) . Then,

P (t, T ) i FP (t; T ,T ) := i = FP (t), α i P (t, T ) j α j=Yα+1 where 1 FPj(t) = , 1 + τjFj(t) and FP denotes the forward discount factor. Then the formula (2.6) deﬁning the forward swap rate can be written as:

1 FP (t; Tα,Tβ) Sα,β(t) = β − , i=α+1 τiFP (t; Tα,Ti) P or alternatively as 1 β 1 j=α+1 1+τj Fj (t) Sα,β(t) = − . β Qτ i 1 i=α+1 i j=α+1 1+τj Fj (t) P Q 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 β

Sα,β(t) = ωi(t)Fi(t); i=Xα+1

τiFP (t; Tα,Ti) τiP (t, Ti) ωi(t) = β = β . k=α+1 τkFP (t; Tα,Tk) k=α+1 τkP (t, Tk) P P Since 0 ω 1 for each i and β ω = 1 it looks like that forward swap rates can ≤ i ≤ i=α+1 i be interpreted as weighted averagesP 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 β S (t) ω (t)F (t). α,β ≈ 0 i i=Xα+1 This is useful for estimating the absolute volatility of swap rates from the absolute volatility of forward rates. Note that Deﬁnition 2.2.3 of a forward swap rate implicitly refers to a swap contract of length T T which starts at time T . A forward swap rate is a rather theoretical concept, β − α α 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 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 Tβ-Tα 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) Tα can be written by considering the value of the underlying receiver IRS at Tα. From (2.3) the IRS value is: β

N τ P (T ,T )(K F (T ; T − ,T )). i α i − α i 1 i i=Xα+1 The option will be exercised only if this is a positive value. Then, the receiver swaption payoﬀ, discounted from the maturity Tα to current time is

β + ND(t, T )( τ P (T ,T )(K F (T ; T − ,T ))) . α i α i − α i 1 i i=Xα+1

The positive part operator is not distributive with respect to sums, but is a piece-wise linear and convex function: β + ( τ P (T ,T )(K F (T ; T − ,T ))) i α i − α i 1 i i=Xα+1 β + τ P (T ,T )(K F (T ; T − ,T )) . ≤ i α i − α i 1 i i=Xα+1

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:

Black T Black Black RS (0, , τ, N, K, σα,β ) = NCα,β(0)BL(K,Sα,β(0), σα,β Tα, 1), p − 14 Black where σα,β is the Black market quoted volatility. A similar formula is used for the payer swaption:

Black T Black Black PS (0, , τ, N, K, σα,β ) = NCα,β(0)BL(K,Sα,β(0), σα,β Tα, 1). p 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 2.3.1 Let us consider a receiver (payer) swaption with strike K giving the holder the right to enter at time Tα a receiver (payer) IRS with payment dates Tα+1, ..., Tβ and associated year fractions τα+1, ..., τβ. The swaption (receiver and payer) is said to be at-the-money (ATM) if and only if

P (0,Tα) P (0,Tβ) K = KAT M := Sα,β(0) = − . (2.10) Cα,β(0)

The receiver swaption is said to be in-the-money (ITM) if K > KAT M , and out-of- the-money (OTM) if K < KAT M .

The payer swaption is said to be in-the-money (ITM) if K < KAT M , and out-of-the- money (OTM) if K > KAT M . The discounted receiver swaption payoﬀ at time t = 0

ND(0,T )C (T )(K S (T ))+ α α,β α − α,β α gives an intuitive meaning to the ITM and the OTM concept. If we substitute in the current forward swap rate S (0) and evaluate (K S (0))+ then the receiver swaption is ITM α,β − α,β when Sα,β(0) < K and it is OTM when Sα,β(0) > K.

15 2.4 Change of Numeraire

[1][3] Deﬁnition 2.4.1 A numeraire is any positive non-dividend paying asset. Theorem 2.4.2 (Geman et al.) Let N be a numeraire and QN a probability measure equivalent to Q0 such that the price of any traded asset X relative to N is a QN -martingale, i.e., Xt N XT = E [ Ft], 0 t T. Nt NT | ≤ ≤ Let U be arbitrary numeraire. Then there exists a probability measure QU , equivalent to Q0, such that the price of any attainable claim Y normalized by U is a QU -martingale, i.e.,

Yt U YT = E [ Ft], 0 t T. Ut UT | ≤ ≤ In addition, the Radon-Nikodym derivative is given by

U dQ UT N0 N = . dQ U0NT

Proof: It follows by the deﬁnition of the QN that for any tradable asset price Z Z U Z EN [ T ] = EU [ 0 T ] (2.11) NT N0UT

(both being equal to Z0 ). By deﬁnition of Radon-Nikodym derivative, we also know that for N0 all Z N N ZT U ZT dQ E [ ] = E [ U ]. NT NT dQ

U dQ UT N0 By comparing the right-hand sides of the last two equalities we have that N = . dQ U0NT 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 F2 between expiry T1 and maturity T2 is given by F (t) = P (t,T1)−P (t,T2) . If Q2 is the measure associated with P (., T ), then F will be a 2 (T2−T1)P (t,T2) 2 2 martingale under this measure.

16 α,β A second example is the forward swap rate Sα,β(t) deﬁned in (2.6). If Q is the measure associated with Cα,β(t), then Sα,β(t) is martingale under this measure. The forward swap rate is the ratio of a tradable asset (portfolio long one Tα-zero coupon bond and short one

Tβ-zero coupon bond) divided by the numeraire.

Corollary 2 The risk-neutral price is invariant by change of numeraire, i.e P ayoff(T ) P ayoff(T ) P rice = EB[B(t) ] = ES[S(t) ]. t B(T ) S(T )

Let Π(XT ) is the payoﬀ on the underlying X at time T . There are two goals for changing numeraire:

Π(XT ) 1. S(T ) to be a simple quantity,

2. S(t)Xt to be a tradable asset, so Xt is martingale. The ﬁrst condition ensures that the new numeraire does not complicate the computation

S Π(XT ) S(t)Xt S E [ S(T ) ]. The second condition ensures that S(t) =Xt is a martingale under Q .

Deﬁnition 2.4.3 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 ET . The price of the derivative at t is 1 π = P (t, T )ET [ Π(X ) F ] = P (t, T )ET [Π(X ) F ], (2.12) t P (T,T ) T | t T | t Theorem 2.4.4 For each 0 u t S < T , the forward rate is a martingale under ≤ ≤ ≤ the T-forward measure: ET [F (t; S,T ) F ] = F (u; S,T ). | u In particular, the forward rate is the P T expectation of the future spot rate:

ET [L(S,T ) F ] = F (t; S,T ). | t Proof: From Definition 2.1.9 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, F (t;S,T )P (t,T ) by definition of T -forward measure P (t,T ) = F (t; S,T ) is a martingale under such a measure. Note that F (S; S,T ) = L(S,T ).

17 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 Ti from the set Tα+1, ..., Tβ the fixed leg pays out an amount τiK corresponding to a fix rate K, where τi is the year fraction of time from Ti−1 to Ti. The floating leg pays an amount corresponding to the forward rate Fi(Ti−1) set at time Ti−1 for maturity Ti. The floating leg resets at

Tα,Tα+1, ..., Tβ−1 and pays at Tα+1, ..., Tβ.

The PFS’ payoﬀ at time Tα can be written as:

D(T ,T )τ (F (T − ) K). α i i i i 1 − i=Xα+1

The PFS’ discounted payoﬀ at time t < Tα is

D(t, T )τ (F (T − ) K). i i i i 1 − i=Xα+1 The value of the contract at time t is computed as follows:

PFS(t, [T , ..., T ],K) = E[ D(t, T )τ (F (T − ) K) F ] α β i i i i 1 − | t i=Xα+1

β i = P (t, T )τ E [F (T − ) K F ] i i i i 1 − | t i=Xα+1 β β

= P (t, T )τ (F (t) K) = [P (t, T − ) (1 + τ K)P (t, T )]. i i i − i 1 − i i i=Xα+1 i=Xα+1

Note that if the PFS’ Tα-payoﬀ

β PFS(T , [T , ..., T ],K) = P (T ,T )τ (F (T ) K) α α β α i i i α − i=Xα+1 is discounted to time t < Tα

β D(t, T ) P (T ,T )τ (F (T ) K), α α i i i α − i=Xα+1

18 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:

β D(0,T )(S (T ) K) τ P (T ,T ). α α,β α − i α i i=Xα+1 Let us consider a payer swaption with strike K giving the holder the right to enter at some future time Tα a payer IRS with payment dates Tα+1, ..., Tβ and associated year fractions

τα+1, ..., τβ. The payer swaption discounted payoﬀ at time t = 0 is

β D(0,T )(S (T ) K)+ τ P (T ,T ) = D(0,T )(S (T ) K)+C (T ). (2.13) α α,β α − i α i α α,β α − α,β α i=Xα+1

The swap rate Sα,β is a martingale under the numeraire Cα,β(t) as explained in the second example of Corollary 1 in Section 2.4. Indeed, the product C (t)S (t) = P (t, T ) P (t, T ) α,β α,β α − β α,β gives the price of a tradable asset and the forward rate Sα,β(t) is a P -martingale associated α,β with the numeraire Cα,β(t). The measure P 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 Tα and Tβ, if for i = α, ..., β

P (t,Ti) α,β the relative bond price , t [0,Ti Tα] follows a local martingale process under P . Cα,β (t) ∀ ∈ ∧ α,β α,β Since Sα,β(t) is a P martingale we postulate that under P :

dS (t) = σ (t)S (t)dW α,β, t [0,T ] (2.14) α,β α,β α,β t ∈ α

α,β where the instantaneous percentage volatility σα,β(t) is deterministic and Wt is a Brownian motion under P α,β . Let us deﬁne the average percentage variance of the forward swap rate in [0,T ] by

T 2 2 υα,β(T ) = (σα,β(t)) dt. Z0

19 This model for the evolution of forward swap rates is known as the lognormal forward- swap model, since each swap rate Sα,β has a lognormal distribution under its swap measure P α,β: 1 ln S (T ) N(S (t) σ2 (T t), σ2 (T t)). α,β α ∼ α,β − 2 α,β α − α,β α − When pricing a swaption, the LSM is very convenient, since it yields the well-known Black formula for swaptions, as stated below. Theorem 2.5.1 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.

SMM Black PS/RS (0, [Tα, ..., Tβ],K) = PS/RS (0,Tα, [Tα, ..., Tβ],K) = Cα,β(0)BL(K,Sα,β(0), υα,β(Tα), w) where BL(K, F, v, w) = F wΦ(wd (K, F, v)) KwΦ(wd (K, F, v)), 1 − 2 ln(F/K) + v2/2 d (K, F, v) = , 1 v ln(F/K) v2/2 d (K, F, v) = − , 2 v where w is either -1 (receiver) or 1 (payer) and is meant to be 1 when omitted. For the payer swaption the Black’s formula is:

PSSMM (0, [T , ..., T ],K) = C (0)(S (0)Φ(d ) KΦ(d )), (2.15) α β α,β αβ 1 − 2 where 2 ln(Sαβ(0)/K) σαβTα/2 d1,2 = ± . σαβ√Tα Proof: The swaption price is the risk-neutral expectation of the discounted payoﬀ

E[D(0,T )(S (T ) K)+C (T )] α α,β α − α,β α

B B(0) + = E [ (Sα,β(Tα) K) Cα,β(Tα)] B(Tα) −

α,β Cα,β(0) + = E [ (Sα,β(Tα) K) Cα,β(Tα)] Cα,β(Tα) −

= C (0)Eα,β[(S (T ) K)+]. (2.16) α,β α,β α − 20 It follows from formula (2.11) for Z = (S (T ) K)+C (T ), that U = B and N = C . T α,β α − α,β α 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” Sα,β, struck at K, with maturity Tα, with zero constant ”risk-free” rate and instantaneous percentage volatility σα,β(t).

In the special case where the swaption is ATM, K = Sαβ(0), the formula reduces to:

C (0)(S (0)Φ(d ) S (0)Φ( d )) α,β αβ 1 − αβ − 1 = C (0)S (0)(Φ(d ) Φ( d )) α,β αβ 1 − − 1 = C (0)S (0)(2Φ(d ) 1) α,β αβ 1 −

= (P (0,T ) P (0,T ))(2Φ(d ) 1), (2.17) α − β 1 −

Black where d1 = σαβ √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 Tα, where as each column is indexed in terms of the underlying swap length, T T .A x x y-swaption is the swaption β − α 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 2.1.

21 Table 2.1: Black Implied Volatility of ATM Swaptions on 2/12/2008

Option/Swap 1yr 2yr 3yr 4yr 5yr 7yr 10yr 1yr 28.08 26.75 25.20 24.01 23.22 21.54 19.77 2yr 25.36 24.04 22.95 22.13 21.46 20.26 18.85 3yr 22.96 22.12 21.36 20.71 20.16 19.18 18.01 4yr 21.22 20.65 20.05 19.55 19.05 18.22 17.20 5yr 20.06 19.53 19.03 18.57 18.13 17.41 16.45 7yr 18.04 17.60 17.20 16.83 16.46 15.86 15.10 10yr 15.74 15.38 15.04 14.75 14.44 14.02 13.40

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. σα,β(Tα), 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 K1 and K2 for the same swaption. Suppose the market market market prices are: PS (0, T, τ, N, K1, σα,β) and PS (0, T, τ, N, K2, σα,β). Does there exist a single volatility such that:

market PS (0, T, τ, N, K1, σα,β) = NCα,β(0)BL(K1,Sα,β(0), σα,β Tα, 1) p and market PS (0, T, τ, N, K2, σα,β) = NCα,β(0)BL(K2,Sα,β(0), σα,β Tα, 1) p hold? In general, market prices do not behave like this. There are two diﬀerent volatilities

σα,β(Tα,K1) and σα,β(Tα,K2) at the market to match the observed market prices:

market PS (0, T, τ, N, K1, σα,β) = NCα,β(0)BL(K1,Sα,β(0), σα,β(Tα,K1) Tα, 1), p market PS (0, T, τ, N, K2, σα,β) = NCα,β(0)BL(K2,Sα,β(0), σα,β(Tα,K2) Tα, 1). p Each swaption market price requires its own Black volatility σα,β(K) depending on the strike rate K. The curve K σ (T ,K) is the volatility smile of the swaption T x T T . 7→ α,β α α β − α If Black’ formula were consistent across diﬀerent strikes, this curve would be ﬂat. Instead,

22 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 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 , ..., T be the tenor structure and τ = T T − for j = α + 1, ..., β. α β j j − j 1 For any ﬁxed α we consider a payer swap which starts at time T and has β α payments. α − The ﬁxed leg pays rate K at reset dates Tα+1, ..., Tβ. The value of such a swap is given in (2.5): β PFS(t) = P (t, T ) c P (t, T ), α − k k k=Xα+1 where c = τ K for k < β and c = 1 + τ K for t [0,T ]. Note that N is set to one. The k k β β ∈ α associated swap rate Sα,β(t) is given by (2.6). We consider a family of forward swap rates

S˜ (t) := S(t, T ,T ), t [0,T ), i i β ∈ i which have the same maturity date T for all i = α, ..., β 1. Suppose we are given a family β − of zero-coupon bond prices P (t, Tj)0≤t≤Tj for j = α, ..., β on a probability space (Ω, F,P ) β β equipped with Brownian motion W . Let P be the Tβ-forward measure and W be the corresponding Brownian motion. For any m = α, ..., β 1 we deﬁne the process C (t) as follows: − m β m−1

Cm(t) = τjP (t, Tj) = τβ−kP (t, Tβ−k) j=βX−m+1 Xk=0 for t [0,T − ]. The forward swap rate is given by: ∈ β m+1

P (t, Ti) P (t, Tβ) S˜i(t) = − . (2.18) Cβ−i(t)

We will make use of the following processes:

β τ P (t, T ) C (t) υi := j j = β−i . (2.19) t P (t, T ) P (t, T ) jX=i+1 β β

i Here υt are the so-called discounted accrual factors.

24 We will derive the relationship:

β−1 j Cβ−i(t) = τk+1 (1 + τkS˜k(t)). P (t, Tβ) Xj=i k=Yi+1 For simplicity we denote

i i ˜ ζt := (1 + τj+1Sj+1(t)). (2.20) jY=α β ˜ Note that υt = Sβ(t) = 0. Then, using (2.18) we can rewrite (2.19) as follows:

i Cβ−i(t) τi+1P (t, Ti+1) + (τi+2P (t, Ti+2) + ... + τβP (t, Tβ)) υt = = P (t, Tβ) P (t, Tβ)

P (t, Ti+1) P (t, Tβ) Cβ−i−1(t) Cβ−i−1(t) = τi+1 + τi+1( − ) + Cβ−i−1(t) P (t, Tβ) P (t, Tβ)

Cβ−i−1(t) Cβ−i−1(t) = τi+1 + τi+1S˜i+1(t) + P (t, Tβ) P (t, Tβ)

Cβ−i−1(t) = τi+1 + (1 + τi+1S˜i+1(t)) P (t, Tβ)

˜ i+1 = τi+1 + (1 + τi+1Si+1(t))υt . (2.21)

i−1 Multiplying both sides of equation (2.21) by ζt leads to:

i−1 i i−1 i i+1 ζt υt = τi+1ζt + ζt υt .

By backward induction from β 1 we obtain the relationship: − β−1 j−1 β−1 j i j=i τj+1ζt υ = = τj+1 (1 + τkS˜k(t)). (2.22) t P ζi−1 t Xj=i k=Yi+1 Also, the following relation holds:

P (t, Ti) i ˜ = 1 + υtSi(t). (2.23) P (t, Tβ) Deﬁnition 2.5.2 For j = α, ..., β and for every k = α, ..., β, the relative bond price is deﬁned as: P (t, Tk) P (t, Tk) Zβ−j+1(t, Tk) = = Cβ−j+1(t) τjP (t, Tj) + ... + τβP (t, Tβ)

25 for t [0,T T ]. ∈ k ∧ j Definition 2.5.3 For a fixed j = α, ..., β a measure P˜j is said to be the forward-swap measure for the date Tj if, for every k = α, ..., β, the relative bond price Zβ−j+1(t, Tk) follows a local martingale under P˜j. In other words: for any fixed m = α, ..., β, the relative bond prices

P (t, Tβ−k) Zm(t, Tβ−k) = , t [0,Tβ−k Tβ−m+1] Cm(t) ∈ ∧ are local martingales under the forward swap measure P˜β−m+1. Directly from the forward swap rate deﬁnition we have

P (t, Tβ−m) P (t, Tβ) S˜β−m(t) = − = Zm(t, Tβ−m) Zm(t, Tβ). Cm(t) −

β−m+1 Thus S˜β−m also is a local martingale under the forward swap measure P˜ . Since 1 P (t,Tβ−k) β obviously C (t) = τ P (t, T ), then Z (t, T − ) = , and thus the measure P˜ can 1 β β 1 β k τβ P (t,Tβ ) be chosen to coincide with the forward measure P β. We assume ν for j = α, ..., β 1 are the volatilities of the forward swap rates S˜ and j − j will construct a family of forward swap rates in the following way:

˜ ˜ ˜ j+1 dSj(t) = Sj(t)νj(t)dWt (2.24)

˜ j+1 where Wt 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:

P (0,Tj) P (0,Tβ) S˜j(0) = − . (2.25) Cβ−j(0) The construction of the model is based on backward induction.

First, we suppose that S˜β−1 for the date Tβ−1 solves the SDE:

˜ ˜ ˜ β dSβ−1(t) = Sβ−1(t)νβ−1(t)dWt (2.26)

˜ β β for all t in [0,Tβ−1], where Wt = Wt = W , with the initial condition:

P (0,Tβ−1) P (0,Tβ) S˜β−1(0) = − . τβP (0,Tβ) We have t ˜ P (0,Tβ−1) P (0,Tβ) ˜ β Sβ−1(t) = − t( νβ−1(u)dWu ). τβP (0,Tβ) E Z0

26 β Since P (0,Tβ−1) > P (0,Tβ) it is clear that S˜β−1 is martingale under P = P . β−1 The next step is to define S˜β−2. First we need to introduce a forward swap measure P˜ ˜ β−1 ˜β−1 and the corresponding Brownian motion Wt . By Definition 2.5.3 P is equivalent to P such that Z2(t, Tβ−k) are local martingales. From Definition 2.5.2, for each k, we have:

P (t, Tβ−k) Z1(t, Tβ−k) Z2(t, Tβ−k) = = . τβ−1P (t, Tβ−1) + τβP (t, Tβ) 1 + τβ−1Z1(t, Tβ−1)

Lemma 2.5.4 Let G and H be real-valued adapted processes, such that

dGt = atdWt, dHt = btdWt.

Assume that H > 1 for every t and denote Y = (1 + H )−1. Then t − t t d(Y G ) = Y (a Y G b )(dW Y b dt). t t t t − t t t t − t t

We will now apply Lemma 2.5.4 to G = Z1(t, Tβ−k) and H = τβ−1Z1(t, Tβ−1).

P (t,Tβ−k) We have noticed above that Z (t, T − ) = follows a strictly positive martingale 1 β k τβ P (t,Tβ ) under P˜β = P β:

˜ β dZ1(t, Tβ−k) = Z1(t, Tβ−k)µ1(t, Tβ−k)dWt (2.27) for some adapted process µ1(t, Tβ−k). Thus

at = Z1(t, Tβ−k)µ1(t, Tβ−k) and

bt = τβ−1Z1(t, Tβ−k)µ1(t, Tβ−k).

Applying the Lemma gives us:

Z1(t, Tβ−k) d( ) = dZ2(t, Tβ−k) 1 + τβ−1Z1(t, Tβ−1)

1 Z1(t,Tβ−k)τβ−1Z1(t,Tβ−1)µ1(t,Tβ−k) = (Z1(t, Tβ−k)µ1(t, Tβ−k) ) 1+τβ−1Z1(t,Tβ−1) − 1+τβ−1Z1(t,Tβ−1)

˜ β τβ−1Z1(t, Tβ−1)µ1(t, Tβ−k) x(dWt dt) − 1 + τβ−1Z1(t, Tβ−1)

27 So that ˜ β τβ−1Z1(t, Tβ−1)µ1(t, Tβ−k) dZ2(t, Tβ−k) = ηβ−1(t)(dWt dt). − 1 + τβ−1Z1(t, Tβ−1) From deﬁnitions it is easy to see that

P (t, Tβ−1) 1 Z1(t, Tβ−1) = = S˜β−1(t) + Z1(t, Tβ) = S˜β−1(t) + . τβP (t, Tβ) τβ Diﬀerentiating both sides and using (2.26) and (2.27) we obtain:

Z1(t, Tβ−1)µ1(t, Tβ−1) = S˜β−1(t)νβ−1(t).

Therefore τ S˜ (t)ν (t) dZ (t, T ) = η (t)(dW˜ β β−1 β−1 β−1 dt) 2 β−k β−1 t ˜ −1 − 1 + τβ−1(Sβ−1 + τβ ) or τ S˜ (t)ν (t) dZ (t, T ) = η (t)(dW˜ β β−1 β−1 β−1 dt). 2 β−k β−1 t ˜ −1 − 1 + τβ−1Sβ−1 + τβ−1τβ ˜ β−1 ˜β−1 It is enough to show that Wt deﬁned below follows a Brownian motion under P .

For t [0,T − ]: ∈ β 1 t τ S˜ (u) W˜ β−1 := W˜ β β−1 β−1 ν (u)du. t t −1 ˜ β−1 − Z0 1 + τβ−1τβ + τβ−1Sβ−1(u) And so t ˜ β−1 ˜ β Wt = Wt γβ−1(u)dWu − Z0 where the deﬁnition of γβ−1 is clear from the context. Now we can deﬁne the associated forward swap measure P˜β−1 by using Girsanov’s theorem stated below.

Theorem 2.5.5 Let Wt be a Wiener process on a probability space (Ω, F,P ) and X is a measurable process adapted to the filtration FW . Define Z = (X) , where t t t E t (X) is the stochastic exponential (Doleans exponential) of X with respect to W , i.e. E (X) = exp(X 1 [X ]). If Z is a martingale, then probability measure Q can be defined E t t − 2 t t dQ such that Radon-Nikodyn derivative is F = Z = (X) . dP | t t E t Then for each t: Q FW P FW . | t ∼ | t Furthermore, if Y is a local martingale under P , then Y˜ = Y [W, X] is a Q-local martingale. t t− t From the Theorem: dP˜β−1 t = t( γβ−1(u)dWu). dP E Z0

28 Next, we introduce the process S˜β−2 by assuming it solves the SDE:

˜ ˜ ˜ β−1 dSβ−2(t) = Sβ−2(t)νβ−2(t)dWt with the initial condition:

P (0,Tβ−2) P (0,Tβ) S˜β−2(0) = − . τβ−1P (0,Tβ−1) + τβP (0,Tβ) We consider the process:

P (t, Tβ−k) Z2(t, Tβ−k) Z3(t, Tβ−k) = = τβ−2P (t, Tβ−2) + τβ−1P (t, Tβ−1) + τβP (t, Tβ) 1 + τβ−2Z2(t, Tβ−2) so that t ˜ β−2 ˜ β−1 τβ−2Z2(u, Tβ−2) Wt = Wt µ2(u, Tβ−2)du. − Z0 1 + τβ−2Z2(u, Tβ−2) It is useful to notice that:

P (t, Tβ−2) Z2(t, Tβ−2) = = S˜β−2(t) + Z2(t, Tβ), τβ−1P (t, Tβ−1) + τβP (t, Tβ) where in turn: Z2(t, Tβ) Z2(t, Tβ) = 1 + τβ−1Z1(t, Tβ) + τβ−1S˜β−1(t) and Z1(t, Tβ) is already known from the previous step. Differentiating both sides we will β−2 find the volatility process of Z2(t, Tβ), and then define P˜ . 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

P (t, Tβ−k) Zm(t, Tβ−k) Zm+1(t, Tβ−k) = = Cm+1(t) 1 + τβ−mZm(t, Tβ−k) follow local martingale under P˜β−m.

Applying Lemma 2.5.4 to G = Zm(t, Tβ−k) and H = τβ−mZm(t, Tβ−m) we can set:

t ˜ β−m ˜ β−m+1 τβ−mZm(u, Tβ−m) Wt = Wt µm(u, Tβ−m)du, − Z0 1 + τβ−mZm(u, Tβ−m)

29 ˜β−m and thus Pt can be easily found using Girsanov’s theorem.

˜β−m t dP β−m+1 = t( γβ−m(u)dW˜ ). β−m+1 u dP˜ E Z0 From P (t, Tβ−m) Zm(t, Tβ−m) = = S˜β−m(t) + Zm(t, Tβ) Cm(t) and Zm−1(t, Tβ) Zm(t, Tβ) = 1 + τβ−m+1Zm−1(t, Tβ) + τβ−m+1S˜β−m+1(t) it follows that Zm(t, Tβ) are functions of S˜β−1, ..., S˜β−m+1.

The process Zm−1(t, Tβ) is known from the preceding step and is a function of forward swap rates S˜β−1, ..., S˜β−m+1. Then, the process

τβ−mZm(u, Tβ−m) µm(u, Tβ−m) 1 + τβ−mZm(u, Tβ−m) can be express in terms of S˜β−1, ..., S˜β−m+1 and their volatilities νβ−1, ..., νβ−m+1. Having β−m β−m found W˜ and P˜ 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 = α, ..., β 1, we call jth-swaption the swaption with option j − maturity Tj. The payer swaption payoﬀ at time Tj is

β Y = τ P (T ,T )[S˜ (T ) K]+. k j k j j − kX=j+1

Recall that S˜j satisﬁes the SDE:

˜ ˜ ˜ j+1 dSj(t) = Sj(t)νj(t)dWt , 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,Tk) for k = α, ..., β, is a local Cβ−j (t) martingale under this measure. The arbitrage price πt(X) of a claim X, which is a function of P (Tj,Tj+1), ..., P (Tj,Tβ), equals

˜j+1 −1 πt(X) = Cβ−j(t)E [Cβ−j(Tj)X]

30 provided X settles at Tj. We apply it to the swaption payoﬀ:

j+1 + π (Y ) = C − (t)E˜ [S˜ (T ) K] . t β j j j − We assume that ν : [0,T ] Rd is a bounded deterministic function. j j → Theorem 2.5.6 Black’s swaption formula For any j = α, ..., β 1, the arbitrage − price at time t [0,T ] of the jth-swaption is: ∈ j β PSj = τ P (t, T )(S˜ (t)Φ(d ) KΦ(d )) t k k j 1 − 2 kX=j+1 where ˜ 2 ln (Sj(t)/K) vj (t)/2 d1,2 = ± vj(t) with v2(t) = Tj ν (u) 2du. j t | j | In the fixedR maturity forward swap market model swaptions with total maturity, defined as the swaption maturity plus the swap maturity, equal to the fixed maturity Tβ 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 Tβ, 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 = α, ..., β M, j − the first settlement date is Tj+1, and its maturity date is TM+j. The value of such swap at time t [0,Tj] is ∈ j+M PFS(t) = P (t, T ) c P (t, T ). j − k k kX=j+1

The forward swap rate of length M, for the date Tj, is that value of the ﬁxed rate which makes the underlying swap starting at Tj worthless at time t. From the last formula it follows: M P (t, Tj) P (t, Tj+M ) Sj (t) = − . τj+1P (t, Tj+1) + ... + τj+M P (t, Tj+M ) For any m = α, ..., β M + 1, we deﬁne the process − β−m+1 m+M−2

C˜m(t) = τiP (t, Ti) = τβ−kP (t, Tβ−k). i=β−MX+1−m+1 k=Xm−1

31 A modiﬁed forward swap measure deﬁned below corresponds to the choice of the process

C˜m(t) as a numeraire. Definition 2.5.7 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 Tj if, for every k = α, ..., β, the relative bond price

P (t, Tk) Zβ−M+1−j+1(t, Tk) = , t [Tk Tj] Cβ−M+1−j+1(t) ∈ ∧ follows a local martingale under P˜j. It is clear that each forward swap rate SM (t), j β M + 1, follows a local martingale j ≤ − under the forward swap measure for the date Tj+1, since

M S (t) = Z − − (t, T ) Z − − (t, T ). j β M+1 j+1 j − β M+1 j+1 j+M

We assume that M M ˜ j+1 dSj (t) = Sj (t)λj(t)dWt ,

˜ j+1 ˜j+1 where Wt is the Brownian motion under P and the initial condition is

˜M P (0,Tj) P (0,Tj+M ) Sj (0) = − . Cj+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 Tβ−M+1, and then deﬁne recursively the forward swap measures corresponding to the preceding dates

Tβ−M ,Tβ−M−1, ..., Tα. Pricing Fixed-length Swaptions For a fixed date T , j = α, ..., β M, we consider the jth-swaption with option maturity j − at Tj written on a swap with fixed length M. The payer swaption payoff at time Tj is

j+M Y = τ P (T ,T )[S˜M (T ) K]+. k j k j j − kX=j+1

˜M Recall that Sj satisﬁes the SDE:

˜M ˜M ˜ j+1 dSj (t) = Sj (t)λj(t)dWt ,

32 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,Tk) for k = α, ..., β, is a Cβ−M+1−j (t) local martingale under this measure. The arbitrage price of the swaption payoﬀ is

j+1 M + π (Y ) = C − − (t)E˜ [S˜ (T ) K] . t β M+1 j j j − We assume that λ : [0,T ] Rd is a bounded deterministic function. j j → Theorem 2.5.8 Black’s swaption formula For any j = α, ..., β M, the arbitrage − price at time t [0,T ] of the jth-swaption is: ∈ j j+M PSj = τ P (t, T )(S˜M (t)Φ(d ) KΦ(d )), t k k j 1 − 2 kX=j+1 where ˜M 2 ln (Sj (t)/K) vj (t)/2 d1,2 = ± vj(t) with v2(t) = Tj λ (u) 2du. j t | j | R 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 2.6 Fourier Transform and Characteristic Function

[15] The Fourier transform of integrable function, f(x), is deﬁned as:

∞ iuy Ff (u) = e f(y)dy. Z−∞ Given the Fourier transform of f, the function f can be recovered by:

∞ 1 −iux f(x) = e Ff (u)du. 2π Z−∞ A characteristic function is the Fourier transform of a probability distribution. If X has a probability density function q, then

∞ iux iuX Fq(u) = e q(x)dx = Ee Z−∞ is its characteristic function. ∞ Note than Fq(0) = −∞ q(x)dx = 1. The critical propertyR 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 2.7 Compound Poisson Process

[15] The Poisson process is an example of a stochastic process with discontinuous trajectories used to build complex jump processes. Deﬁnition 2.5.1 Exponential random variables A positive random variable Y is said to follow an exponential distribution with parameter λ if it has a probability density −λy function of the form λe 1y≥0.

Deﬁnition 2.5.2 Poisson Process Let (τi)i≥1 be a sequence of independent exponential random variables with parameter λ and T = n τ . Process (N , t 0) deﬁned by: n i=1 i t ≥ P

Nt = 1t≥Tn Xn≥1 is called a Poisson process with intensity λ.

The Poisson process counts the number of random times (Tn) which occur in [0, t], where

(T T − ) ≥ is an iid sequence of exponential variables. n − n 1 n 1 Deﬁnition 2.5.3 Compound Poisson Process Compound Poisson process with intensity λ and jump size distribution f is a stochastic process Zt deﬁned as:

Zt = Yi, Xi=1 where jump sizes Yi are iid with distribution f and (Nt) is a Poisson process with intensity

λ, independent from (Yi)i≥1. Characteristic Function of a Compound Poisson Process

E[eiuZt ] = exp(tλ (eiuz 1)f(dz)), u Rd, ZRd − ∀ ∈ where λ denotes the jump intensity and f the jump size distribution.

35 2.8 Stochastic Volatility Models: Overview

[15][1] A widely studied class of stochastic volatility models is the class of bivariate diﬀusion models:

dSt = µStdt + σtStdWt, (2.28) where σt is called the instantaneous volatility process. Important features for the instantaneous volatility process are positiveness and mean-reversion. Positiveness can be obtained by setting σt = f(yt), where f(t) is a positive function and yt is a random driving process.

Mean-reversion is usually modeled by introducing a mean-reverting drift in the yt dynamics:

dy = κ(θ y )dt + ...dZˆ , t − t t where Zˆt is a Wiener process correlated with Wt. 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 yt. The drift term pulls yt towards θ and thus σt is pulled towards the long-run mean of f(yt). Some common driving processes are:

Geometric Brownian motion dy = c y dt + c y dZˆ , • t 1 t 2 t t Gaussian Ornstein-Uhlenbeck dy = κ(θ y )dt + βdZˆ , • t − t t

Cox-Ingersoll-Ross (CIR) dyt = κ(θ yt)dt + η√ytdZˆt. • − 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 Table 2.2: Examples of Bivariate Diﬀusion Models for Option Pricing

Model Correlation f(y) yt Hull-White ρ = 0 f(y) = (y) Lognormal Scott ρ = 0 f(y) =pey 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) = p(y) CIR 6 p Among these models, the Heston model is quite commonly used in practice: dS t √ S = µdt + V tdWt , St

V dV (t) = κ(θ V (t))dt + η V (t)dWt . − p Applying Itˆo’s formula for the log-normal process Yt = ln St we obtain: 1 dY = (r V )dt + V dW S. (2.29) t 2 t t t − p 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 = µdt + V tdWt + dZt, St

V dV (t) = κ(θ V (t))dt + η V (t)dWt . − p S V where Wt and Wt are Brownian motions with correlation ρ, and Zt 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 δ2, δ2). The no-arbitrage condition fixes the drift under ∼ − 2 the risk-neutral measure: µ = r λk¯. Applying Itô’s formula for the log-normal process − Yt = ln St we obtain: 1 dY = (r λk¯ V )dt + V dW S + dZ˜ , (2.30) t 2 t t t t − − p 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 λk¯, the risk-neutral correction. We will use these two models as prototypes of the new extensions of the SMM.

37 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 con- structing 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 Figure 3.1: Time Series of Implied Volatilities for ATM Swaptions

Under the forward swap measure the SV extension of the SMM is:

αβ dSαβ(t) = Sαβ(t) V (t)σαβ(t)dW (t), p dV (t) = κ(θ V (t))dt + η V (t)dW V (t), (3.1) − p 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 Sαβ (t) , then the characteristic function Φ(y, v, T ; u) is deﬁned by: Sαβ (0) Φ(y, v, t; u) = E[eiuY (T ) Y (t) = y, V (t) = v]. | 39 Consider Φ(y, v, t; u) = E[g(Y (T ),V (T ))/Y (t),V (t)] = E[g/Ft], where Ft 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)/Ft] = E[E[g/Fs]/Ft] = E[g/Ft] = Φ(y, v, t; u).

Applying Itˆo’s formula to Φ(y, v, t; u) yields:

1 ∂2Φ ∂2Φ 1 ∂2Φ 1 ∂Φ ∂Φ ∂Φ dΦ = ( σ2 v + ρηv + η2v σ2 v + κ(θ v) + )dt + ... t 2 αβ ∂y2 ∂y∂v 2 ∂v2 − 2 αβ ∂y − ∂v ∂t

We set the drift term to zero since Φ is martingale: 1 ∂2Φ ∂2Φ 1 ∂2Φ 1 ∂Φ ∂Φ ∂Φ σ2 v + ρηv + η2v σ2 v + κ(θ v) + = 0. (3.2) 2 αβ ∂y2 ∂y∂v 2 ∂v2 − 2 αβ ∂y − ∂v ∂t To determine the solution of equation (3.2), the ﬁnal condition at time T should be satisﬁed:

Φ(y, v, T ; u) = eiuy.

Together with the terminal condition the PDE allows computation of the characteristic function. Proposition 3.1.1 The closed form of the characteristic function for Y (t) is:

(u2+iu)σ2 v exp( κθt(κ−iρηu) αβ + iuy) η2 ∆ coth( ∆t )+κ−iρηu − 2 Φt(u) = ∆t (κ−iρηu) ∆t ( 2κθ (cosh( 2 ) + ∆ sinh( 2 )) η2 )

2 2 2 where ∆ = (κ iρηu) + η σαβu(u + i). q − Proof: 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) = eC(T −t)+vD(T −t)+iuy, (3.3)

40 where C and D are functions of one variable only. Then ∂Φ = iuΦ, ∂y ∂Φ = (C′ + vD′)Φ, ∂t − ∂Φ = DΦ, ∂v ∂2Φ = D2Φ, ∂v2 ∂2Φ = u2Φ, ∂y2 − ∂2Φ = iuDΦ. ∂y∂v Substituting all into equation (3.2) leads to: 1 1 1 σ2 u2v + iρηvD + η2D2v σ2 viu + κ(θ v)D (C′ + vD′) = 0 −2 αβ 2 − 2 αβ − − or 1 1 ( η2D2 + (iρηu κ)D σ2 u(i + u) D′)v + (κθD C′) = 0. 2 − − 2 αβ − − 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: 1 1 D′(s) = η2D2(s) + (iρηu κ)D(s) σ2 u(i + u), 2 − − 2 αβ C′(s) = κθD with initial conditions D(0) = C(0) = 0. Lemma 3.1.2 The explicit solutions to the equations for D and C are:

(u2 + iu)σ2 D(s) = − αβ , γ coth( ∆s ) + κ iρηu 2 − κθs(κ iρηu) 2κθ ∆s κ iρηu ∆s C(s) = − ln(cosh( ) + − sinh( )), η2 − η2 2 ∆ 2

41 2 2 2 where ∆ = (κ iρηu) + η σαβu(u + i). q − Proof: Let R(s) = 1 η2, Q(s) = (iρηu κ), and P (s) = 1 σ2 u(i + u). 2 − − 2 αβ For D we have to solve a Riccati equation of the form dD = R(s)D2 + Q(s)D + P (s). ds

2 Let D1 below be the solution of the quadratic equation RD + QD + P = 0, Q + ∆ D = − , (3.4) 1 2R

2 2 2 2 where ∆ = (Q 4RP ) = (κ iρηu) + η σαβu(u + i) is the discriminant. − q − In terms ofp the new function Z deﬁned by: 1 Z = (3.5) D D − 1 we have: dZ = (Q + 2D R)Z R, ds − 1 − 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 R

µy + µpy = µq, ′ (µy) = µq, ′ (µy) = µq, Z 1 y = µq. µ Z In our case p(s) = ∆, q(s) = R and hence µ(s) = e ∆ds = e∆s. The solution is: − R 1 Z = e∆s( R)ds e∆s Z −

42 1 R = ( e∆s + c) e∆s −∆

R = + ce−∆s. (3.6) −∆ At s = 0: R Z(0) = + c. −∆ And so 1 R = + c. D(0) D −∆ − 1 Hence, 1 R c = + . D(0) D ∆ − 1 Substituting the constant back into the solution (3.6) for Z we have: R 1 R Z = + ( + )e−∆s. −∆ D(0) D ∆ − 1 We can write it as:

Z = U + W e−∆s (3.7) where R U = (3.8) −∆ and 1 R W = + . D(0) D ∆ − 1 First, we simplify the expression for W : 1 R 1 R W = + = −Q+∆ + D(0) D1 ∆ D(0) ∆ − − 2R 2R R R 2∆ + 2RD(0) ∆ + Q = + = ( − ) 2RD(0) ∆ + Q ∆ ∆ 2RD(0) ∆ + Q − − R 2RD(0) + ∆ + Q = ( ) ∆ 2RD(0) ∆ + Q − R =g ˜ , (3.9) ∆ 43 where 2RD(0) + ∆ + Q g˜ = . (3.10) 2RD(0) ∆ + Q − Then, from (3.8) and (3.9) it follows that R Z = U + W e−∆s = (1 ge˜ −∆s). −∆ − From (3.5) we have 1 D (U + W e−∆s) + 1 D = D + = 1 . 1 Z U + W e−∆s Substituting (3.4) in and simplifying further:

( −Q+∆ )( R + R ge˜ −∆s) + 1 D = 2R − ∆ ∆ R (1 ge˜ −∆s) − ∆ − ( −Q+∆ )( R ) + ( −Q+∆ ) R ge˜ −∆s) + 1 = 2R − ∆ 2R ∆ R (1 ge˜ −∆s) − ∆ − ( Q−∆ ) + ( −Q+∆ ge˜ −∆s) + 1 = 2∆ 2∆ R (1 ge˜ −∆s) − ∆ − ( Q−∆ ) + ( −Q+∆ ge˜ −∆s) + 2∆ = 2∆ 2∆ 2∆ R (1 ge˜ −∆s) − ∆ − Q ∆ + ( Q + ∆)˜ge−∆s + 2∆ = − − 2R(1 ge˜ −∆s) − − ( Q + ∆)( Q+∆ +ge ˜ −∆s) = − −Q+∆ . 2R(1 ge˜ −∆s) − − Let’s denote ∆ + Q g = . (3.11) ∆ Q − Hence, the solution for D is: Q ∆ g +ge ˜ −∆s D(s) = ( − )( ), (3.12) 2R 1 ge˜ −∆s − where g andg ˜ are deﬁned in (3.10) and (3.11) respectively. We can also write the expression for D in terms of P, Q and ∆ as follows:

∆+Q ∆+Q −∆s Q ∆ ( − ) + ( − )e D(s) = ( − )( ∆ Q ∆+Q ) 2R 1 ( ∆+Q )e−∆s − −∆+Q 44 Q ∆ Q ∆ + Qe−∆s + ∆e−∆s = ( − )(− − ). 2R Q ∆ Qe−∆s ∆e−∆s − − − The denominator simpliﬁes as follows:

Q ∆ Qe−∆s ∆e−∆s = ∆(e−∆s + 1) Q(e−∆s 1) − − − − − − e−∆s + 1 = (e−∆s 1)(∆( ) + Q) − − e−∆s 1 − e∆s + 1 = e∆s(1 e∆s)(∆( ) + Q) − − e∆s 1 − ∆s = e∆s(1 e∆s)(∆ coth ( ) + Q). − − 2 Therefore, Q ∆ (e−∆s 1)(Q + ∆) D(s) = ( − ) − 2R e∆s(1 e∆s)(∆ coth ( ∆s ) + Q) − − 2 Q ∆ (1 e∆s)(Q + ∆) = ( − )e∆s − 2R e∆s(1 e∆s)(∆ coth ( ∆s ) + Q) − − 2 Q2 ∆2 1 = ( − ) ∆s − 2R ∆ coth ( 2 ) + Q 4PR 1 = ( ) ∆s 2R ∆ coth ( 2 ) + Q

2P = ∆s . (3.13) ∆ coth ( 2 ) + Q Now we can ﬁnd the solution for C′(s) = κθD. We substitute in the solution (3.12) for D: ∆ Q g +ge ˜ −∆s C′(s) = κθ( − )( ). − 2R 1 ge˜ −∆s − Let M = κθ( ∆−Q ). Hence, − 2R g +ge ˜ −∆s C(s) = M ( )ds +c. ¯ Z 1 ge˜ −∆s − We consider a change in variables: y = e−∆s. Then dy = ∆yds and − M g +gy ˜ dy C(s) = ( ) +c. ¯ − ∆ Z 1 gy˜ y − Using g +gy ˜ g ggy˜ + ggy˜ +gy ˜ g(1 gy˜ ) +gy ˜ (g + 1) g (g + 1)˜g = − = − = + (1 gy˜ )y (1 gy˜ )y (1 gy˜ )y y 1 gy˜ − − − − 45 we have M (g + 1)˜g g C(s) = ( + )dy +c ¯ − ∆ Z 1 gy˜ y − M (g + 1)˜g g = ( dy + dy) +c ¯ − ∆ Z 1 gy˜ Z y − M (g + 1) 1 = ( d(1 gy˜ ) + g dy) +c ¯ − ∆ Z − 1 gy˜ − Z y − M = ( (g + 1) ln(1 ge˜ −∆s) + g ln(e−∆s)) +c. ¯ (3.14) − ∆ − − At s = 0: M C(0) = (g + 1) ln(1 g˜) +c. ¯ ∆ − Therefore, M c¯ = C(0) (g + 1) ln(1 g˜). − ∆ − Substituting the constant back into the solution (3.14) yields: M M C(s) = ( (g + 1) ln(1 ge˜ −∆s) + g( ∆s)) + C(0) (g + 1) ln(1 g˜) − ∆ − − − − ∆ − M = ((g + 1)(ln(1 ge˜ −∆s) ln(1 g˜)) + g∆s) + C(0) ∆ − − − M 1 ge˜ −∆s M = ((g + 1) ln( − )) + g∆s + C(0) ∆ 1 g˜ ∆ − M 1 ge˜ −∆s = (g + 1) ln( − ) + Mgs + C(0). ∆ 1 g˜ − Substituting M and g in:

κθ( ∆−Q ) ∆ + Q 1 ge˜ −∆s ∆ Q ∆ + Q C(s) = − 2R ( + 1) ln( − ) κθ( − )( )s + C(0) ∆ ∆ Q 1 g˜ − 2R ∆ Q − − − κθ(∆ Q) 2∆ 1 ge˜ −∆s ∆ + Q = − − ( ) ln( − ) κθ( )s + C(0) 2R∆ ∆ Q 1 g˜ − 2R − − κθ 1 ge˜ −∆s ∆ + Q = − ln( − ) κθ( )s + C(0) R 1 g˜ − 2R − κθ 1 ge˜ −∆s = ( 2 ln( − ) (∆ + Q)s) + C(0) 2R − 1 g˜ − − κθ 1 ge˜ −∆s = (ln( − )−2 (∆ + Q)s) + C(0). (3.15) η2 1 g˜ − − 46 Let’s simplify further C by simplifying the expression under the logarithm:

−∆s ∆+Q −∆s 1 ge˜ 1 − e − = − ∆+Q 1 g˜ 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 ∆e∆s + ∆ Q(e∆s 1) = − − 2∆e∆s ∆(e∆s + 1) Q(e∆s 1) = + − − 2∆e∆s 2∆e∆s ∆s ∆s ∆s − ∆s ∆s ∆s ∆s − ∆s (e 2 e 2 + e 2 e 2 ) Q(e 2 e 2 e 2 e 2 ) = ∆s ∆s + − ∆−s ∆s 2e 2 e 2 2∆e 2 e 2 ∆s −∆s ∆s 1 e 2 + e 2 Q e 2 1 = ∆s ( − ) e 2 2 − ∆ 2 1 ∆s κ iρηu ∆s = ∆s (cosh( ) + − sinh ( )). e 2 2 ∆ 2 Back to (3.15): 2κθ 1 ∆s κ iρηu ∆s κθ C(s) = − 2 (ln( ∆s (cosh( ) + − sinh ( ))) 2 (∆ + Q)s η e 2 2 ∆ 2 − η 2κθ ∆s 2κθ ∆s κ iρηu ∆s κθ∆s κθQs = (− )(− ) + − (ln(cosh( ) + − sinh ( ))) η2 2 η2 2 ∆ 2 − η2 − η2

2κθ ∆s κ iρηu ∆s κθsQ = (− )(ln(cosh( ) + − sinh ( ))) . (3.16) η2 2 ∆ 2 − η2 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:

(u2+iu)σ2 v exp( κθt(κ−iρηu) αβ + iuy) η2 ∆ coth( ∆t )+κ−iρηu − 2 Φt(u) = . (3.17) ∆t (κ−iρηu) ∆t ( 2κθ (cosh( 2 ) + ∆ sinh( 2 )) η2 )

47 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: αβ dSαβ(t) = Sαβ(t) V (t)σαβ(t)dW (t) + dZ(t), p dV (t) = κ(θ V (t))dt + η V (t)dW V (t) − p where W αβ(t) and W V (t) are Brownian motions with correlation ρ.

Nt Here Z(t) = i=1 Ji is a compound Poisson process where Ji are i.i.d. with distribution P fJ and Nt is a Poisson process with intensity λ. We assume that the distribution of the jump 2 sizes is log-normal with mean µJ and variance δJ . If J is a jump size then 1 ln(1 + J) N(ln(1 +µ ¯) δ2 , δ2 ), ∼ − 2 J J whereµ ¯ is the average jump amplitude such that: 1 µ = ln(1 +µ ¯) δ2 . J − 2 J Applying Itˆo’s formula we obtain the equation for the log process Y = ln Sαβ (t) : t Sαβ (0)

1 dY = ( λµ¯ σ2 V )dt + σ V dW (t) + dZ˜ , t 2 αβ t αβ t αβ t − − p where Z˜t is a compound Poisson process with intensity λ and Gaussian distribution of jump sizes. Proposition 3.1.3 The characteristic function is

2 2 κθt(κ−iρηu) (u +iu)σαβ v exp( 2 ∆t + iut( λµ¯) + iuy) 1 2 2 1 2 η ∆ coth( )+κ−iρηu − δ u +i(ln(1+¯µ)− δ )u − 2 − Φt(u) = exp(tλe 2 2 J 1) , − ∆t (κ−iρηu) ∆t ( 2κθ (cosh( 2 ) + ∆ sinh( 2 )) η2 )

2 2 2 where ∆ = (κ iρηu) + η σαβu(u + i). q − Proof: The characteristic function of the jumps is:

J − 1 δ2 u2+i(ln(1+¯µ)− 1 δ2 )u Φ (u) = exp(tλe 2 J 2 J 1). (3.18) t − 48 Since jumps are homogeneous and independent from the continuous part the character- istics function of the swap rate process in the jump-diﬀusion SV extension of the SMM is obtained by the following multiplication:

J C Φt(u) = Φt (u)Φt (u), where

(u2+iu)σ2 v exp( κθt(κ−iρηu) αβ + iut( λµ¯) + iuy) η2 γ coth( ∆t )+κ−iρηu ΦC (u) = − 2 − . (3.19) t ∆t (κ−iρηu) ∆t ( 2κθ (cosh( 2 ) + ∆ sinh( 2 )) η2 ) So that

2 2 κθt(κ−iρηu) (u +iu)σαβ v exp( 2 ∆t + iut( λµ¯) + iuy) 1 2 2 1 2 η ∆ coth( )+κ−iρηu − δ u +i(ln(1+¯µ)− δ )u − 2 − Φt(u) = exp(tλe 2 2 J 1) − ∆t (κ−iρηu) ∆t ( 2κθ (cosh( 2 ) + ∆ sinh( 2 )) η2 )

Note that (3.17) and (3.19) are very similar. The diﬀerence lies in the shift λµ¯, the risk-neutral correction.

49 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: N−1 −i 2π mj D (h) = e N h , m = 0, ..., N 1, m j − Xj=0 N−1 −1 1 i 2π mj D (h) = e N h , m = 0, ..., N 1. m N j − Xj=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. −1 Dm(h) is the discrete Fourier transform, or DFT, of a vector h and Dm (h) is its inverse. The FFT method is a way to approximate a continuous Fourier transform with a discrete N−1 counterpart, for a carefully chosen vector h = (hj)j=0 : ∞ N−1 −iku −ikuj e h(u)du e h(uj)η, (4.1) Z ≈ 0 Xj=0

50 where uj = ηj, and η is the grid size of the vector u. Vector h corresponds to N function N−1 N−1 evaluations of h at u = (uj)j=0 . Let f be (fm)m=0, where each fm corresponds to an integral of the form (4.1) computed at k = k , m = 0, ..., N 1. Carr and Madan (1999) show m − that the vector f will contain the option prices that correspond to the log-strike prices at k = k , m = 0, ..., N 1 deﬁned as m − k = b + λm, m = 0, ..., N 1, m − − where λ is the grid size of k. This results in log-strike range [ b, b], where b = Nλ . − 2 Then, N−1 N−1 N−1 −ikmuj −i(−b+λm)uj −iλmηj e hjη = e hjη = e h˜j, Xj=0 Xj=0 Xj=0

ibuj where h˜j = e hjη. To apply the fast Fourier transform, the following relationship must be satisfied: 2π λη = . 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:

N−1 G (h, α) = e−i2πmjαh , m = 0, ..., N 1, (4.2) m j − Xj=0 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 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 2π independently. The FFT method requires λη = N . The fractional FFT method is faster 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: 1 D (h) = G (h, ), m = 0, ..., N 1, m m N − 1 1 D−1(h) = G (h, ), m = 0, ..., N 1. m N m −N − 1 The standard FFT can be seen as a special case of the FRFT for α = N . We note that 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 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−1 N-point FRFT on vector h = (hj)j=0 . Since 2jm = j2 + m2 (m j)2 we have − − N−1 N−1 −i2πmjα −iπ(j2+m2−(m−j)2)α Gm(h, α) = e hj = e hj Xj=0 Xj=0

N−1 −iπm2α −iπj2α iπ(m−j)2 = e e hje Xj=0

N−1 −iπm2α = e yjzm−j, (4.3) Xj=0

−πij2α πij2α where yj = e hj and zj = e . 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 zm−j = z − . In our case we have z − = z − when m j < 0. It is possible however to m j+N m j j m − 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:

−iπj2α N−1 N−1 iπj2α N−1 iπ(N−j)2α N−1 y = ((e hj)j=0 , (0)j=0 ),z = ((e )j=0 , (e )j=0 ). (4.4)

Then, the FRFT is given by:

−iπm2α N−1 −1 Gm(h, α) = (e )m=0Dm (Dj(y)Dj(z)) (4.5) for 0 m < N. ≤ The remaining N results of the ﬁnal inverse FFT are discarded.

Element-by-element vector multiplication is implied in the expression Dj(y)Dj(z).

53 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:

PSSMM (0, [T , ..., T ],K) = C (0)Eα,β[(S (T ) K)+], α β α,β α,β α − where Eα,β stands for expectation under the forward swap rate. In terms of Y (t) = ln Sαβ (t) and k = ln K we have: Sαβ (0) Sαβ (0)

PSSMM (0, [T , ..., T ],K) = C (0)S (0)Eα,β[(eY (Tα) ek)+]. (4.6) α β α,β α,β −

S (T ) Let G(k) = Eα,β[(eY (Tα) ek)+], where Y (T ) = ln α,β α . α Sα,β (0) ∞ − Then G(k) = (es ek)q (s)ds. k − T R We denote by qT (s) the risk-neutral density of Y (T ) such that

∞ iuy ΦT (u) = e qT (y)dy, (4.7) Z−∞ where Φ(.) is the characteristic function.

Since G(k) is not square-integrable due to the fact that G(k) →−∞ 1 we consider the →k following modiﬁcation G˜(k) = eakG(k).

Now G˜(k) is square-integrable for a suitable a > 0 (called dampening parameter) satisfying E[S (T )a+1] < . The accuracy of the prices with the FRFT method strongly depends α,β α ∞ on the parameter a. The Fourier transform of G˜(k) is deﬁned by:

∞ Ψ(u) = eiukG˜(k)dk. Z−∞

Carr and Madan (1998) link up the Fourier transform of the modiﬁed G(k) with the characteristic function ΦT of Y (T ):

Φ (u (a + 1)i) Ψ(u) = T − , a > 0. (4.8) (a + iu)(1 + a + iu)

54 Proof:

∞ ∞ ∞ iuk ak iuk ak s k Ψ(u) = e e G(k)dk = e e (e e )qT (s)dsdk Z−∞ Z−∞ Zk −

∞ s s+(a+iu)k (1+a+iu)k = qT (s)( (e e )dk)ds Z−∞ Z−∞ − ∞ 1 1 (1+a+iu)s = ( ) qT (s)e ds a + iu − 1 + a + iu Z−∞ Using equation (4.7), we have: 1 Ψ(u) = Φ (u (a + 1)i). (a + iu)(1 + a + iu) T −

Using Fourier inversion, we express G(k) in terms of Ψ:

exp( ak) ∞ G(k) = − e−iukψ(u)du, π Z0 which we will evaluate numerically using the FRFT algorithm. We set equidistant points uj = ηj with grid size η > 0 and obtain the following approximation for G(k):

− exp( ak) N 1 G(k) − e−iuj kψ(u )η. ≈ π j Xj=0 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 N−1 (km)m=0. These values will be equidistant around the ATM level of k = 0, with grid size λ. N−1 Then, the values (km)m=0 would assume the form: Nλ k = + λm, m = 0, ..., N 1. m − 2 − This gives us strikes levels between exp( k ), where k = Nλ . We also incorporate the ± 0 0 2 Simpson’s rule weighting into our summation:

N−1 exp( akm) −iηj(− Nλ +λm) η j G(k ) − e 2 ψ(u ) (3 + ( 1) δ − ) m ≈ π j 3 − − j 1 Xj=0

− exp( ak ) N 1 − m e−iηjλmh , ≈ π j Xj=0

55 where

iηj Nλ η j h = e 2 ψ(u ) (3 + ( 1) δ − ). (4.9) j j 3 − − j 1

Here δn is the Kronecker delta function: 1, n = 0; δ = n { 0, n = 0. 6 One needs to make the appropriate choices for η and λ independently and then recover the ηλ value of α through the relation α = 2π . We set the FRFT parameters to the following values: a = 3, N = 64, and η = 0.25. Then, we chose k = 0.6, so that the range for the strikes is between exp 0.60 . The grid 0 {± } 2(0.60) size for strikes λ = 64 = .01875. The choices above imply that the fractional parameter (0.25)(0.01875) −4 α is 2π = 7.46.10 . 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:

63 h = (hj)j=0, a = (exp iαπj2 )63 , { } j=0 a¯ = (exp iαπ(64 j)2 )63 , { − } j=0 ak d = (exp − m )63 . { π } j=0 We now have to compute the fractional transform:

63 y1 = (h/a, (0)j=0), y˜1 = fft(y1),

y2 = (a, a¯), y˜2 = fft(y2),

y3 =y ˜1y˜2, y˜3 = ifft(y3),

y =y ˜3/a, and ﬁnally:

c = yd = ifft(fft(y1)fft(y2))d.

Here y1 and y2 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:

SMM PS (0, [Tα, ..., Tβ],K) = Cα,β(0)Sα,β(0)Gk(h, α), (4.10) where h is defined in (4.9) and Gk is defined in (4.5). Here ψ is a modification of the characteristic function (see (4.8)) defined in (3.17).

56 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 , ρ . In the stochastic volatility extension of the SMM with jumps (LN- { 0 } SV SMM), described in Chapter 3 Section 3.2, the set of model-specific parameters are Θ = κ, θ, η, V , ρ, µ , δ , λ . The calibration of the instantaneous forward swap rate { 0 J J } 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 con- tradicts 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: σα,β√Tα must be monotone increasing function with option maturity Tα of the swaption. For the

57 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 (t) = f(T t) = [a + b(T t)]e−c(Tα−t) + d. α,β α − α − 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 t) is α − suggested in the LMM framework by Brigo and Mercurio 2001 and Rebonato 1998. The above formulation can be extended into a richer parametric form:

σ (t) = g(t)Φ [(a + b(T t))e−c(Tα−t) + d]. (5.1) α,β α,β α − 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]:

− 1 −b(T −t) σ (t) = Φ a(T T ) 2 e α + c , (5.2) α,β α,β{ β − α } 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 5.1.

58 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 Φα,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 σα,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: Tα Black 2 2 min[(σα,β Tα) σα,β(t)dt], a,b,c p − Z0 where σα,β is described by the parametric form (5.2).

59 From the relationship: Tα Black 2 2 (σα,β Tα) = σα,β(t)dt. p Z0 we have that:

Tα 2 − 1 − − Black 2 2 b(Tα t) 2 2 2 (σα,β Tα) = Φα,β (a(Tβ Tα) e + c) dt = Φα,βI (Tα, a, b, c). p Z0 − We set the scaling factors such that we have a good ﬁt to the initial term structure of the swaption volatilities: Black√ 2 2 (σα,β Tα) Φα,β = 2 . I (Tα, 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:

MODEL MARKET Black 2 min [Π (Tα,Tβ, σαβ,K, Θ) Π (Tα,Tβ, σαβ ,K)] , Θ X − MODEL MARKET Black where Π (Tα,Tβ, σαβ,K, Θ) and Π (Tα,Tβ, σαβ ,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 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 5.2.

61 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 2/12/2008

62 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 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 1 0.027325 2 0.027430 3 0.030190 4 0.033000 5 0.035520 6 0.037720 7 0.039580 8 0.041120 9 0.042450 10 0.04356

Table 5.2: Discount Factors as of 2/12/2008

Maturity Rate 1 0.972755 2 0.946847 3 0.913807 4 0.876542 5 0.837027 6 0.796501 7 0.755891 8 0.716604 9 0.678397 10 0.641637

64 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 5.3 Calibration Step One

The first step is the calibration of the swaption volatility matrix. We find the fitted parameters (a, b, c) to the form:

− 1 −b(T −t) σ (t) = Φ a(T T ) 2 e α + c α,β α,β{ β − α } 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 0.302145 b 0.390982 c 0.160012

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, 2008. We note that the scaling factors are around one as expected.

Table 5.4: Scaling Factors on 2/12/2008

Option/Swap 1 2 3 4 5 6 7 8 9 1 0.9775 0.9974 1.0420 1.0320 1.0764 1.0662 1.0453 1.0310 1.0298 2 0.9814 0.9908 1.0335 1.0541 1.0630 1.0499 1.0479 1.0601 3 0.9810 1.0127 1.0457 1.0662 1.0664 1.0667 1.0564 − 4 0.9641 1.0376 1.0400 1.0613 1.0550 1.0565 − − 5 0.9733 1.0125 1.0413 1.0491 1.0455 − − − 6 0.9780 1.0016 1.0200 1.0216 − − − − 7 0.9663 1.0021 0.9993 − − − − − 8 0.9388 0.9513 − − − − − − 9 0.9179 − − − − − − − − − − − − − − −

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 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 2.4308 1.4000 θ 0.0061 0.0201 η 2.5921 0.1624 V0 1.5032 0.0040 ρ -0.5011 -0.8833 µJ - 0.0081 δJ - 0.0045 λ - 0.36 RMSE 2.9201 1.0526 RMSEvol 0.0035 0.0001

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 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:

1 n RMSE = v [F ittedP rice ActualP rice ]2, (6.1) un i − i u Xi=1 t 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:

1 N RMSE = RMSE , (6.2) N i Xi=1 where N = 109 is the number of days and RMSEi 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 5/31/2008.

68 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 1 0.0280 0.0044 2 0.0286 0.0033 3 0.0311 0.0028 4 0.0337 0.0024 5 0.0360 0.0022 6 0.0380 0.0020 7 0.0396 0.0019 8 0.0410 0.0018 9 0.0422 0.0017 10 0.0432 0.0017

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 Table 6.2: Descriptive statistics for the ﬁtted swap rate volatility parameters

Parameters Mean St Dev a 0.301032 0.001 b 0.391649 0.023 c 0.158908 0.006

Table 6.3: Fitted models parameters for the SV SMM

Parameters Mean St Dev κ 2.068545 0.262158 θ 0.006780 0.000769 η 2.591967 0.001637 V0 1.502181 0.002469 ρ -0.500598 0.001289 RMSE 3.23 0.02

Table 6.4: Fitted models parameters for the LN-SV SMM

Parameters Mean St Dev κ 1.402530 0.001719 θ 0.020226 0.00264 η 0.159780 0.00636 V0 0.001016 0.000018 ρ -0.915870 0.004135 µJ 0.009530 0.002665 δJ 0.040802 0.002638 RMSE 1.05 0.023

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 RMSE1(t) and RMSE2(t). We then compute the mean of the differences d(t) = RMSE (t) RMSE (t) and the associated statistic. A 2 − 1

70 significantly negative mean implies that model two has a significantly better fit than model one. The null of equal predictive accuracy is:

H0 : E[d(t)] = 0.

The Diebold-Mariano test statistic is: d¯ S = , ˆ LRV d¯/N q where 1 N d¯= d(t), N Xt=1 ∞

LV Rd¯ = γ0 + 2 γj, γj = cov(dt, dt−j), Xj=1 ˆ ¯ and LRV d¯ is a constant estimate of the asymptotic (long-run) variance of √Nd. 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 > 1.96. | | 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 -2.18 DM statistic -9.7

We reject the null hypothesis that the difference between the two means is zero since 9.7 < 1.96. 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 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 = nRMSE2. 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 53 56 RSS 1137 120 BIC 1385 382

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 126 286

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 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 6.3.

73 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 6.3 Forecasting

[30][29] We now evaluate the forecasting performance of SV SMM and LN-SV SMM models. We consider a forecasting horizon of 4 weeks. Using the ﬁtted parameters for SV SMM and LN- SV SMM we calculate the swaption prices for swaptions with total maturity less or equal to 10 years for the month of June 2008 and then compare them to the ones observed on the market. Forecast errors are the daily diﬀerences between the model (predicted) and the market (actual) prices. For the term structure forecast we use the random walk model, that is, the best forecast of future term structures is today’s term structure. This is a standard in the forecasting literature. Tables 6.8 and 6.9 below report the forecasting RMSE in bp across expiries and swap periods for the SV SMM and LN-SV SMM respectively.

Table 6.8: Forecasting performance Of SV SMM

Option/Swap 1 2 3 4 5 6 7 8 9 1 1.97 2.62 4.38 1.14 4.34 1.16 3.04 1.24 2.84 2 2.35 2.16 6.47 2.89 8.29 9.42 3.47 0.78 3 2.49 2.15 4.08 9.24 3.06 8.79 3.07 − 4 1.16 1.97 3.22 5.26 2.91 4.43 − − 5 2.87 4.08 2.12 3.10 2.40 − − − 6 0.71 1.03 1.05 2.33 − − − − 7 0.55 4.17 1.29 − − − − − 8 1.84 2.32 − − − − − − 9 0.79 − − − − − − − − − − − − − − −

The overall RMSE for the whole swaption matrix over the 21 days is 7.8 bp.

75 Table 6.9: Forecasting performance Of LN-SV SMM

Option/Swap 1 2 3 4 5 6 7 8 9 1 1.69 2.56 4.38 6.36 10.80 9.34 11.80 8.65 14.90 2 1.57 1.47 6.00 5.16 11.10 12.70 7.52 5.30 3 1.42 1.32 3.68 8.08 1.91 6.76 1.25 − 4 1.30 1.51 3.04 5.56 4.39 6.19 − − 5 1.20 2.07 2.48 4.65 4.03 − − − 6 0.42 0.61 1.14 5.80 − − − − 7 0.16 1.13 0.28 − − − − − 8 1.29 0.54 − − − − − − 9 0.21 − − − − − − − − − − − − − − −

The overall RMSE for the whole swaption matrix over the 21 days is 8.7 bp. The BS SMM forecasting RMSE in bp across expiries and swap periods are shown in Table 6.10.

Table 6.10: Forecasting performance Of BS SMM

Option/Swap 1 2 3 4 5 6 7 8 9 1 2.84 3.62 5.57 3.46 5.82 3.56 5.97 2.61 8.22 2 3.37 3.42 6.89 2.99 8.00 9.71 4.07 0.59 3 4.00 2.93 5.14 9.21 3.32 8.43 3.90 − 4 2.16 2.54 4.02 5.51 3.53 5.09 − − 5 3.76 5.04 2.60 4.42 3.99 − − − 6 1.25 1.58 1.77 6.56 − − − − 7 0.87 3.97 1.11 − − − − − 8 2.85 2.06 − − − − − − 9 0.22 − − − − − − − − − − − − − − −

The accuracy of the results is very good, with all errors being well within the ask-bid spread of the swaption prices. We make pairwise comparisons between the model’s forecasting performance using the Diebold and Mariano (1995) method. First we calculate 21 daily RMSE for each of the three models: SV SMM, LN-SV SMM and BS SMM. Then, we calculate the Diebold-Mariano statistics. Table 6.11 displays the pairwise comparisons of forecasting performance using the Diebold and Mariano (1995).

76 Table 6.11: Statistical Comparisons of forecasting performance based on Diebold and Mariano criterion

SV vs. BS LN-SV vs. BS LN-SV vs. SV mean diﬀerence in RMSE -3.1 -2.3 0.8 DM statistic -12.26 -10.40 5.17

The critical value of 1.96 means that if the standardized difference between the two means is larger than 1.96 it can be concluded that there is a significant difference between the means. Table 6.12 displays the pairwise comparison of forecasting performance based on the BIC criterion.

Table 6.12: Statistical Comparison of forecasting performance based on BIC criterion

SV LN-SV BS k 53 56 49 RSS 1277 1589 2495 BIC 1439 1759 2644

Based on the two criterions we conclude that both SV SMM and LN-SV SMM have better prediction power than the BS SMM. We note that the LN-SV SMM produces better forecasts for swaptions with short option maturity and worse forecasts for swaptions with longer option maturity.

77 CHAPTER 7

CONCLUSION

We have derived the characteristic function of the stochastic volatility extension of the SMM where all instantaneous forward swap rate volatilities are equally affected by a multiplicative stochastic factor. We assume that there is a non-zero correlation between the changes in the forward swap rate and the changes in the forward swap rate volatility in contrast to the assumption of zero correlation made by Attaoui in his SV SMM extension. Then, we derive the jump-diffusion stochastic volatility extension characteristic function of the SMM a la Bates by adding log-normal jumps to the forward swap rate dynamics. Using the fast fractional Fourier transform (FRFT) both models SV SMM and LN-SV SMM are calibrated to market swaption data. The FRFT algorithm has the advantage of using the characteristic function in a more efficient way than the standard FFT method widely used in option pricing and saves computational time without loss of accuracy. A specific functional form for the instantaneous forward swap rate volatilities is chosen to meet the observed evidence that volatility decreases with longer swaption maturity and larger swap tenors. 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. The choice of set of the model calibration instruments must be driven by practical considerations and should be the most informative in capturing the volatility. 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 over all 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

78 the SV SMM. There is a 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. We recommend using SV SMM or LN-SV SMM over any LMM version for pricing swaptions due to easier calibration with fewer parameters. LIBOR rates are not directly quoted by the market, where as swap rates are. As mention above a critical property of the SV SMM and LN-SV SMM is that the set of the co-terminal swaptions is optimal for large portion of IR derivatives. In comparison LMM must be calibrated to a larger set even though some calibration instruments are redundant as far as volatility risk is concerned [9]. The price of a swaption within LMM depends on a correlation matrix among LIBOR rates and so the calibrating the LMM to a set of swaptions is more complex process. We recommend using SV SMM or LN-SV SMM over the Black model since once these models are calibrated, they can be used to price any swaption with a different maturity or swap tenor. In both SMM versions the same calibration is used for pricing all 45 swaptions (9 with total maturity 10 years and 35 with total maturity less than 10 years). In contrast, separate specifications of the Black model are needed to price the 45 swaptions. Black’s model cannot be extended to other swaptions with different maturities or swap tenor. In contrast, once calibrated, the SV SMM and the LN-SV SMM can be used to price any swaption with a different maturity or exercise date.

7.1 Future Work

The immediate extension of this study would be to include in-the-money and out-of-the- money swaptions in the analysis. It would be interesting to see which of the two extension models generates the most realistic strike volatility function for swaptions. Future work may involve finding a calibration procedure for the LN-SV SMM so that the model fits longer maturity swaption data well. The jump parameters could be found by calibrating to short swaption maturities and then kept fix for a second calibration procedure to find the rest of the parameters by fitting data to longer swaption maturities.

79 APPENDIX A

MATLAB CODE

The FRFT procedure is tested for correctness when applied to the Black-Scholes characteristic function and the swaption prices from the model are compared to the prices calculated from the BS closed-form solution. The code for the new characteristic functions is tested for correctness by setting the model speciﬁc parameters to simpliﬁed values and stepping through the code and comparing to similar Excel implemented formulas. xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx SV SMM CALIBRATION xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx close all; clear all; global tau; global K; % swaption strike global sigma; % quoted ATM volatility global sigmaM; % fwd swap rate inst vol global FSR; % forward swap rate global C; global swaption_price; % from implied Black swap volatility global model_price; global M; % swaption maturity global TN; % swap tenor global eta; % FrFT grid parameter global a; % FrFT dampening parameter global N; % FrFT parameter

80 global SwN; % number co-terminal swaptions global IVm; % implied vol from model %------parameters tau=1; SIG=importdata(’c:/sig.xls’)/100; % swaption volatility 1/1/08-5/31/08 DF=importdata(’c:/df.xls’); % discount factors 1/1/08-5/31/08 M=[1 2 3 4 5 6 7 8 9]; SwN=9; t=0; % today TN=10; %co-terminal swaptions with total maturity 10 yrs % FrFT parameters a=3; N=64; eta=.25; %------alpha = M; beta = TN; for dind=1:109 % workdays %discount curve data P= DF(dind,1:10); %set to appropriate discount curve sigma=SIG(dind,1:9); %CALIBRATION STEP 1 parPhi=[1 1 1 1]; parPhiL=[-Inf -Inf -Inf -Inf]; parPhiU=[Inf Inf Inf Inf]; y=lsqnonlin(@sumsqvol,parPhi,parPhiL,parPhiU); Ap=y(1); Bp=y(2); Cp=y(3); %volatility parameters for indx1=1:9 for indx2=1:9 IVpar(indx1,indx2)=Ap*((indx2)^(-.5))*exp(-Bp*(indx1))+Cp; end end

81 Phi=sigma./IVpar; %scaling factors sigmaM=Phi(M,10-M).*(Ap.*((10-M).^(-0.5)).*exp(-Bp*(M-t))+Cp); alpha = M; beta = TN; F=zeros(9,1); for i=1:size(F), F(i)=(P(i)/P(i+1)-1)/tau; end for z=1:SwN tmpsum=0; for i=alpha(z):beta-1, tmpsum=tmpsum+P(i+1); end C(z)=tmpsum; FSR(z)=GetSwapRate(F,alpha(z),beta); K(z) = FSR(z); % ATM swaption swaption_price(z) =10000*C(z)*Black (K(z),FSR(z),sigma(z),M(z)); end %CALIBRATION STEP 2 par=[2.28 .006 2.59 -0.5 1.5]; % k theta nu rho v0 parL=[0.5 0.0001 0.01 -0.99 0.001]; parU=[inf inf inf 0.99 inf]; [x,resnorm,residual,exitflag] = lsqnonlin(@sumsq10,par,parL,parU, optimset(’Display’,’iter’,’TolFun’,1e-16,’TolX’,1e-12,’MaxIter’,1000000)); kappa(dind)=x(1); theta(dind)=x(2); nu(dind)=x(3); rho(dind)=x(4); v0(dind)=x(5); end disp(sprintf(’\n average kappa=%f’,mean(kappa))); disp(sprintf(’\n st dev kappa=%f’,std(kappa)));

82 disp(sprintf(’\n average theta=%f’,mean(theta))); disp(sprintf(’\n st dev theta=%f’,std(theta))); disp(sprintf(’\n average nu=%f’,mean(nu))); disp(sprintf(’\n st dev nu=%f’,std(nu))); disp(sprintf(’\n average rho=%f’,mean(rho))); disp(sprintf(’\n st dev rho=%f’,std(rho))); disp(sprintf(’\n average v0=%f’,mean(v0))); disp(sprintf(’\n st dev v0=%f’,std(v0))); xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx function ret = sumsq10(par) global FSR; global C; global model_price; global M; global K; global sigmaM; global sigma; global swaption_price; global SwN; global IVm; for q=1:SwN model_price(q) = 100*C(q)*FSR(q)*SVMprices (par,K(q),FSR(q),M(q),sigmaM(q)); IVm(q)=fzero(@dbs,0.1,optimset(’fzero’),K(q),M(q),model_price(q),FSR(q),C(q)); end ret=sigma-IVm; disp(ret); end xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx FAST FRACTIONAL FOURIER TRANSFORM IMPLEMENTATION xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

83 function ret=SVMprices(par,K1,FS,t,sig) global a; global N; global eta; k0=0.6; lambda=(2*k0)/N; fft_alpha=eta*lambda/(2*pi); jvec=0:N-1; v=eta*jvec; tmp=SVMzhi(par,t,v,sig).*exp(1i*v*k0)*eta; tmp=(tmp/3).*(3+(-1).^jvec+ ((jvec-1)==0) ); %applying simpson’s rule ku=-k0+lambda*jvec; ku=ku’; avec=exp(1i*fft_alpha*pi*(jvec).*jvec); abarvec=exp(1i*fft_alpha*pi*(N-jvec).*(N-jvec)); dvec=exp(-a*ku/pi); y1=[tmp./avec,zeros(1,N)]; y2=[avec,abarvec]; y1w=fft(y1); y2w=fft(y2); y3=y1w.*y2w; y3w=ifft(y3); y3n=y3w(1:N); y=y3n./avec; % Gk(h,alpha) cpvec=y.*dvec’; %expectation based on ku x=exp(ku); y=(cpvec); xi=(K1/FS); %moneyness ret=real(interp1(x,y,xi)); end xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

84 MODIFIED CHAR FUN EQ. (4.7) xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx function ret = SVMzhi(par,t,v,sig) global a; ret= SVMcharecteristicFn2(par,t,(v-(a+1)*1i)/(a+1i*v).*(1+a+1i*v),sig); end xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx SV SMM CHARACTERISTIC FUNCTION- EQ. (3.17) xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx function ret=SVMcharecteristicFn2(par,t,u,sig) kappa=par(1); theta=par(2); nu=par(3); rho=par(4); v0=par(5); gamma= sqrt((nu^2)*(sig^2)*(u.*u+i*u)+(kappa-(i*rho*nu)*u).*(kappa-(i*rho*nu)*u)); num = kappa*theta*t*(kappa-i*rho*nu*u)/(nu^2); power=2*kappa*theta/nu^2; o1=cosh(0.5*gamma*t)+((kappa-i*rho*nu*u)./gamma).*sinh(0.5*gamma*t); denom = (o1).^(power); frac=-((u.*u+i*u)*sig)./(gamma.*coth(0.5*gamma*t)+kappa-i*rho*nu*u); ret=exp(num).*exp(v0*frac)./denom; end xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx LN-SV SMM CHARACTERISTIC FUNCTION EQ. (3.19) xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx function ret=SVMcharecteristicFnJumps(par,t,u,sig) lambdaJ=.36; % intensity of jumps

85 kappa=par(1); theta=par(2); nu=par(3); rho=par(4); v0=par(5); jumpmu=par(6); jumpsig=par(7); gamma= sqrt((nu^2)*(sig^2)*(u.*u+i*u)+(kappa-(i*rho*nu)*u).*(kappa-(i*rho*nu)*u)); num = kappa*theta*t*(kappa-i*rho*nu*u)/(nu^2); power=2*kappa*theta/nu^2; o1=cosh(0.5*gamma*t)+((kappa-i*rho*nu*u)./gamma).*sinh(0.5*gamma*t); denom = (o1).^(power); frac=-((u.*u+i*u)*sig)./(gamma.*coth(0.5*gamma*t)+kappa-i*rho*nu*u); tret=exp(num).*exp(v0*frac).*exp(i*t*(-lambdaJ*jumpmu)*u)./denom; charJ=exp(t*lambdaJ*exp(-0.5*jumpsig^2*(u.*u)+i*(log(1+jumpmu)-0.5*jumpsig^2)*u)-1); ret=charJ.*tret; end xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx BLACK FORMULA FOR CO-TERMINAL SWAPTIONS xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx function ret= Black(K,S,sigma,t) d1 =(log(S / K) + 0.5 * t * sigma^2) / (sigma * sqrt(t)); d2 =(log(S / K) - 0.5 * t * sigma^2) / (sigma * sqrt(t)); Nd1=normcdf(d1); Nd2=normcdf(d2); ret =(S * Nd1 - K * Nd2); end xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx FORWARD SWAP RATE FORMULA xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx function ret = GetSwapRate(F,alpha,beta)

86 global tau;

tmp_sum=0; tmp=1; for j=alpha:beta-1, tmp=tmp*(1/(1+tau*F(j))); end SR=1-tmp; %numerator for i=alpha:beta-1, tmp=1; for j=alpha:i, tmp=tmp*(1/(1+tau*F(j))); end tmp_sum=tmp_sum + (tau*tmp); end SR=SR/tmp_sum; ret=SR; end xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx FORECASTING WITH SV SMM xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx close all; clear all; global tau; global K; % swaption strike global sigma; % quoted ATM volatility global sigmaM; global FSR; % forward swap rate global C; global swaption_price; % from implied Black swap volatility global model_price; global M; % swaption maturity

87 global TN; % swap tenor global eta; % FrFT parameter global a; % FrFT parameter global N; % FrFT parameter global SwN; % number co-terminal swaptions global IVm; %------parameters tau=1; sigma=importdata(’c:/data060108063008.xls’); % implied vols for 6/1/08-6/30/08 Phi=importdata(’c:/PhiMatrix053108.xls’); SwN=9; M=[1 2 3 4 5 6 7 8 9]; TN=[1 2 3 4 5 6 7 8 9]; % volatility parameters Ap=0.303159; Bp=0.397343; Cp=0.159281; % FrFT parameters a=3; N=64; eta=.25; %------%discount curve data PCurve= [1 0.9720 0.9380 0.9005 0.8617 0.8228 0.7841 0.7464 0.7096 0.6742 0.6371

88 0.6195 ]; P=PCurve; % set discount curve alpha = M; beta = [ 2 3 4 5 6 7 8 9 10; 3 4 5 6 7 8 9 10 0; 4 5 6 7 8 9 10 0 0; 5 6 7 8 9 10 0 0 0; 6 7 8 9 10 0 0 0 0; 7 8 9 10 0 0 0 0 0; 8 9 10 0 0 0 0 0 0; 9 10 0 0 0 0 0 0 0; 10 0 0 0 0 0 0 0 0; ]; kappa=2.432106; theta=0.005720; nu=2.596049; rho=-0.5; v0=1.502109; par=[kappa theta nu rho v0]; % k theta nu rho v0 F=zeros(size(P,1)-1,1); for i=1:size(F), F(i)=(P(i)/P(i+1)-1)/tau; end for indt=1:21 %21 workdays for z=1:9 for r=1:9 tmpsum=0; for i=alpha(z):beta(z,r)-1, tmpsum=tmpsum+P(i+1); end C(z,r)=tmpsum;

89 SR(z,r) = GetSwapRate(F,alpha(z),beta(z,r)); K(z,r) = SR(z,r); % ATM swaption day=1/365; s=day:day:30*day; sigmaM(z,r)=Phi(z,r).*(Ap.*((10-M(z)).^(-0.5)) .*exp(-Bp*(M(z)-s(indt)))+Cp); quoted_price(z,r) =10000 * C(z,r)* Black (K(z,r), SR(z,r), sigma(9*(indt-1)+z,r),M(z)); swaption_price(z,r) =10000 * C(z,r)* Black (K(z,r), SR(z,r),sigmaM(z,r),M(z)); model_price (z,r)= 100 * C(z,r)* SR(z,r)* SVMprices (par,K(z,r),SR(z,r),M(z),sigmaM(z,r)); end end bMSE=0; mMSE=0; for m=1:9 for rr=1:10-m bMSE=bMSE+(quoted_price(m,rr)-swaption_price(m,rr))^2; mMSE=mMSE+(quoted_price(m,rr)-model_price(m,rr))^2; end end bRMSE=sqrt(bMSE/21); mRMSE=sqrt(mMSE/21); disp(sprintf(’\nday %f’,indt)); disp(sprintf(’\n BS Model to Quoted RMSE=%f’,bRMSE)); disp(sprintf(’\n SVM Model to Quoted RMSE=%f’,mRMSE)); end xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx PRICING SWAPTIONS WITH TOTAL MATURITY LESS THAN 10 YEARS USING FITTED PARAMETERS FROM CALIBRATION TO CO-TERMINAL SWAPTIONS xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

90 close all; clear all; global tau; global K; % swaption strike global sigma; %forward swap rate quoted ATM volatility global SR; % swap rate global C; global swaption_price; % from implied Black swap volatility global model_price; global M; % swaption maturity global TN; % swaption tenor global eta; global a; global N; %------parameters tau=1; SIG=importdata(’c:/sigMatrix.xls’)/100; % swaption volatility 1/1/08-5/31/08 DF=importdata(’c:/df.xls’); % discount factors 1/1/08-5/31/08 M=[1 2 3 4 5 6 7 8 9]; TN=[1 2 3 4 5 6 7 8 9]; % FFT parameters a=3; N=64; eta=.25; %------parPhi=[1 1 1 1]; parPhiL=[-Inf -Inf -Inf -Inf]; parPhiU=[Inf Inf Inf Inf]; y=lsqnonlin(@sumsqvol,parPhi,parPhiL,parPhiU); Ap=y(1); Bp=y(2); Cp=y(3); for indx1=1:9 for indx2=1:9 IVpar(indx1,indx2)=Ap*((indx2)^(-.5))*exp(-Bp*(indx1))+Cp;

91 end end Phi=sigma./IVpar; %scaling factors P= DF(dind,1:10); %set to appropriate discount curve F=zeros(size(P,1)-1,1); for i=1:size(F), F(i)=(P(i)/P(i+1)-1); end alpha = M; beta = [ 2 3 4 5 6 7 8 9; 3 4 5 6 7 8 9 0; 4 5 6 7 8 9 0 0; 5 6 7 8 9 0 0 0; 6 7 8 9 0 0 0 0; 7 8 9 0 0 0 0 0; 8 9 0 0 0 0 0 0; 9 0 0 0 0 0 0 0; ]; kappa=2.068545; theta=0.006780; sigmav=2.591967; rho=-0.500598; v0=1.502181; par=[kappa theta sigmav rho v0]; Ap=0.301032; Bp=0.391649; Cp=0.158908; for z=1:8 for r=1:8 tmpsum=0; for i=alpha(z):beta(z,r)-1, tmpsum=tmpsum+P(i+1); end C(z,r)=tmpsum;

92 SR(z,r) = GetSwapRate(F,alpha(z),beta(z,r)); K(z,r) = SR(z,r); % ATM swaption swaption_price(z,r) =10000 * C(z,r)* Black (K(z,r), SR(z,r), sigma(z,r),M(z)); t=0; sigmaM(z,r)=Phi(z,r).*(Ap.*((beta(z,r)-M(z)).^(-0.5)) .*exp(-Bp*(M(z)-t))+Cp); model_price(z,r) = 100 * C(z,r)* SR(z,r)* SVMprices (par,K(z,r),SR(z,r),M(z),sigmaM(z,r)); end end for m=1:8 for rr=1:9-m IV(m,rr)=fzero(@dbs,0.1,optimset(’fzero’),K(m,rr),M(m), model_price(m,rr),SR(m,rr),C(m,rr)) end end plot(K,IV); % plot(TN(1:8),model_price(1,:));hold; % plot(TN(1:8),swaption_price(1,1:8)); differences=swaption_price-model_price; disp(differences); disp(sprintf(’\nmodel price (bp)=%f’,model_price)); disp(sprintf(’\nmarket price(bp)=%f’,swaption_price)); xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

93 APPENDIX B

ABBREVIATIONS AND NOTATION

ATM = At the money; bps = Basis Point (1bps=0.0001) ; BM = Brownian motion; CIR = Cox-Ingersoll-Ross model; HW = Hull-White model; IRS = Interest Rate Swap (either payer or receiver); PFS = A Payer IRS; RFS = A Receiver IRS; ITM = In the money; LMM = LIBOR market Model; SMM = Swap Market Model; SV SMM = Stochastic volatility SMM; LN-SV SMM = SV SMM with log-normal jumps added to the swap rate dynamics; OTM = Out of the money; PDE = Partial diﬀerential equation; PVBP = Present Value per Basis Point; SDE = Stochastic diﬀerential equation; FRA = Forward-rate agreement; FFT = Fast Fourier Transformation; FRFT = Fast Fractional Fourier Transformation; RMSE = Root Mean Square Error; B(t): Bank account at time t; r(t): Instantaneous spot interest rate at time t;

94 D(t,T): Stochastic discount factor at time t for the maturity T; P(t,T): Bond price at time t for the maturity T; R(t,T): Continuously-compounded spot rate at t for the maturity T; L(t,T): Simply-compounded (LIBOR) spot interest rate at t for the maturity T; f(t,T): Instantaneous forward rate at time t for the maturity T; Y(t,T): Annually-compounded spot interest rate at t for the maturity T; FP(t;T,S): 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). F(t;T,S): Simply-compounded forward (LIBOR) rate at t for expiry T and maturity S; f(t,T,S): Continuously-compounded forward rate at t for expiry T and maturity S;

T1,T2, ..., Ti−1,Ti, ..: An increasing set of maturities;

τi: The year fraction between Ti−1 and Ti;

Fi(t): F (t, Ti−1,Ti);

S(t; Ti,Tj),Si,j(t): Forward swap rate at time t for a swap with ﬁrst reset date Ti and payment dates Ti+1, ..., Tj;

Ci,j(t): Present value of basis point associated to the forward swap rate Si,j(t). Q: Risk-neutral measure; QU : Measure associated with the numeraire U; QT : T-forward adjusted measure associated with the numeraire P (., T ); i Q : Ti-forward adjusted measure; i,j Q : Swap measure between Ti, Tj associated with the numeraire Ci,j; E: Expectation under risk-neutral measure; EQ: Expectation under the probability measure Q; ET : Expectation under the T-forward adjusted measure; i E : Expectation under the Ti-forward adjusted measure; BL(K, F, v, w): Black’s formula, BL(K, F, v, w) = F wΦ(wd (K, F, v)) KwΦ(wd (K, F, v)) 1 − 2 ln(F/K)+v2/2 ln(F/K)−v2/2 where d1(K, F, v) = v , d2(K, F, v) = v , where w is either -1 (receiver) or 1 (payer) and is meant to be 1 when omitted. Φ: Cumulative distribution function of the standard Gaussian distribution; FRA(t, T, S, τ, N, K): Price at time t of a forward-rate agreements with reset date T and payment date S at the ﬁxed rate K; As usual τ is the year fraction between T and S and can be omitted, and N is the nominal amount and can be omitted;

95 PFS(t, T, τ, N, K): Price at time t of a payer interest rate swap with first reset date T1 and payment dates T2, .., Tn at the fixed rate K; As usual τi is the year fraction between Ti−1 and Ti and can be omitted, and N is the nominal amount and can be omitted; RFS(t, T, τ, N, K): Same as above but for a receiver swap; PS(t, T, τ, N, K): Price of a payer swaption maturing at time T, which gives its holder the right to enter at time T an interest rate swap with first reset date T1 and payment dates

T2, .., Tn at the ﬁxed rate K; As usual τi is the year fraction between Ti−1 and Ti and can be omitted, and N is the nominal amount and can be omitted; RS(t, T, τ, N, K): Same as above but for a receiver swaption;

96 REFERENCES

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99 BIOGRAPHICAL SKETCH

Milena G. Tzigantcheva

Milena Gueorguieva Tzigantcheva was born in 1971 in Haskovo, Bulgaria. Under the advisement of Dr. Ivanova-Karatopraklieva, she obtained her Master’s degree in 1994 from the Department of Mathematics and Informatics at Soﬁa University St. Kliment Ohridski, Soﬁa, Bulgaria. In 2002 Milena enrolled in the doctoral program in Financial Mathematics at Florida State University where Dr. Craig A. Nolder became her Ph.D. advisor. She has been working with him in the area of Interest Rate Models. Milena currently lives in Tallahassee, Florida with her daughter Adriana and her husband Dimitre Tzigantchev and is a Director of Risk Management at the Florida State Board of Administration.

100