Pricing and hedging swaptions in the LIBOR market model
by
Atsushi Kawai
Submitted for the degree of Doctor of Philosophy at the University of New South Wales
December 2001 ACKNOWLEDGEMENTS
First and foremost I would like to thank my supervisors Dr. Beniamin Goldys and Dr. Rob Womersley. This thesis could not have been completed without their constant help and guidance. I would like to thank Dr. Marek Musiela. I learnt the Black Scholes option pricing formula from his lectures for the first time. His interesting lectures and my Honors project with him motivated me to do my PhD study. I would like to thank Dr. Marek Rutkowski for useful comments on my work and interesting discussions on credit derivatives (not part of this thesis). My PhD study has been financially supported by an Overseas Postgraduate Research Scholarship ( OPRS) and a School of Mathematics Scholarship. Some material in Section 2.3, Section 2.6 and Section 2. 7 of this thesis will appear in
Journal of Computational Finance; "Analytical and Monte Carlo swaption pricing un der the forward swap measure". Some material in Section 2.3, Section 2.4, Section 2.6,
Section 2.7 and Section 3.3 of this thesis was submitted for publication to Applied Mathe matical Finance; "A new approximate swaption formula in the LIBOR market model: an asymptotic expansion approach". ABSTRACT
This thesis develops new pricing and hedging methods for European payer swaptions in the LIBOR market model. The LIBOR market model has been recently considered as one of the most important multi-factor interest rate term structure models. It models the dynamics of the forward LIBOR rate directly so that it is arbitrage-free among bonds and between bonds and cash. A log-normal LIBOR market model prices caplets in accordance with the Black market caplet formula. However, a closed form pricing formula for European payer swaptions is not available in the LIBOR market model. We develop both analytical and numerical methods to price European payer swaptions in the LIBOR market model. First we develop a new approximate pricing formula for European payer swaptions. The main tool for the development is an asymptotic expansion method. Alternatively, we use Jarrow and Rudd's method based on the Edgeworth seriesi expansion for the development. Second we develop a new Monte Carlo simulation method to find European payer swaption prices. The Monte Carlo method is carried out under the forward swap measure and we incorporate a control variate which significantly speeds up the computation. Both developments are considered in the log-normal LIBOR market model first. Then we consider the developments in more general LIBOR market models. We expand our discussion to hedging methods for European payer swaptions. We propose a delta hedging method. To use the method, we require the values of Greeks for European payer swaptions. We outline how to estimate the Greeks using the pricing methods developed. CONTENTS
1. Introduction .... 6 1.1 Introduction . 6 1.2 Arbitrage pricing theory 7
1.3 Interest rate models . . . 11 1.3.1 Short rate models . 12 1.3.2 The HJM model 15 1.3.3 The LIBOR market model 17 1.3.4 The swap market model 26
2. Swaption pricing in a log-normal LIBOR market model . 30 2.1 Introduction ...... 30 2.2 Review of approximate swaption formulae 32 2.3 A valuation formula under the forward swap measure 34 2.4 An asymptotic expansion method 40 2.4.1 Introduction ...... 41 2.4.2 An asymptotic formula . 44 2.4.3 Approximate formulae 55 2.4.4 An alternative approach 59 2.5 J arrow and Rudd method 63 2.5.1 Introduction . . . . 63 2.5.2 An approximate formula 66 Contents 5
2.6 Monte Carlo simulation ...... 71
2.6.1 Monte Carlo under the terminal measure 72
2.6.2 Monte Carlo under the forward swap measure 74
2.7 Numerical results ...... 78
2.7.1 Details of numerical examples 79
2.7.2 Comparison of Monte Carlo methods 80
2.7.3 Comparison of approximate formulae 86
2.7.4 Other numerical results ...... 90
3. Swaption pricing in extended LIBOR market models 95
3.1 Introduction ...... 95
3.2 An n5-period log-normal LIBOR market model. 95
3.3 LIBOR market models with more general volatility functions 102
3.3.1 An asymptotic expansion method 103
3.3.2 Monte Carlo simulation 111
3.3.3 Numerical results .... 116
4. Swaption hedging in the LIBOR market model . 124
4.1 Introduction ...... 124
4.2 Continuous time hedging 124
4.3 Discrete-time hedging . 125
4.3.1 Delta hedging 125
4.4 Estimation of Greeks 126
4.5 Numerical results 128
4.6 Conclusion . 137
Bibliography 138 1. INTRODUCTION
1.1 Introduction
Since the advent of the celebrated option pricing formula by Black and Scholes (1973) and Merton (1973), not only derivative business but also research on mathematical modeling of derivative securities has expanded very rapidly. As various derivative securities such as equity, interest rate, currency and commodity derivatives have been created, various mathematical models have been developed to model such products. Nowadays, derivative securities such as credit, weather and energy derivatives are getting popular and many people are working on modeling these products. This thesis develops new pricing and hedging methods for European payer swaptions in the LIBOR market model. The LIBOR market model has been developed by Brace, Gatarek and Musiela (1997), Musiela and Rutkowski (1997a) and Jamshidian (1997), and has been recently considered as one of the most important interest rate models for LIBOR and swap derivatives. The thesis is organized as follows. Chapter 1 introduces the theoretical framework for the arbitrage pricing theory and reviews various mathematical models for interest rate derivatives. Chapter 2 develops analytical and numerical pricing methods for European payer swaptions in a log-normal LIBOR market model. Chapter 3 considers some exten sions of the log-normal LIBOR market model and develops analytical and numerical pricing methods for European payer swaptions in the extended LIBOR market models. Chapter 4 develops a hedging method for European payer swaptions in the LIBOR market model. 1. Introduction 7
1.2 Arbitrage pricing theory
This section explains the arbitrage pricing theory, which is a fundamental theory for pric ing derivatives securities. The discussion of this section relies on Harrison and Kreps (1979), Harrison and Pliska (1981), Geman, El Karoui and Rochet (1995) and Musiela and Rutkowski ( 1997b). We consider a continuous trading economy with a trading interval [O, T*] for a fixed T* > 0. The uncertainty in the economy is characterized by the fil tered probability space (D, lF, IP) where n is the state space, lF is the O'-algebra representing measurable events, and IP E P, where P is a collection of mutually equivalent probability measures on (D, Fr•). Each individual in the economy is characterized by a subjective probability measure IP from P. Information evolves over the trading interval according to the right continuous, complete filtration Ft : t E [O, T*]. We also assume a frictionless and competitive economy. By frictionless we mean that there are no transaction costs in buy ing and selling financial securities, there are no bid/ ask spreads, there are no restrictions on trade such as margin requirements or short sale restrictions, and there are no taxes. Competitiveness means that each trader believes that he can buy/sell as many shares of a traded security as he desires without influencing its price. This implies, of course, that the market for any financial security is perfectly liquid. There are n primary traded secu rities whose price processes are given by stochastic processes Z 1 , ... , zn. We assume that
Z = (Z1 , ... , zn) follows a continuous, Rn-valued semimartingale on (D, lF, IP). Follow ing Harrison and Pliska (1981), for T :S T*, consider an Rn-valued predictable stochastic process Definition 1.2.1. is called a self-financing trading strategy over time interval [O, T] if the wealth process V ( ), which is n ½()= Gt( We write r to denote the class of all self-financing trading strategies over the time interval [O, T]. Similarly, = Ur::;r• Definition 1.2.2. A strategy E r is called an arbitrage opportunity if the wealth process V () satisfies, for some (hence for all) IP E P, the following set of conditions: Vo( The existence of an arbitrage opportunity implies that an investor can create something out of nothing. It is, roughly speaking, the chance for a free lunch. One example of an arbitrage opportunity would be a portfolio that requires zero investment, but has a strictly positive payoff in some states of the world. The basic idea underlying the arbitrage (risk neutral) pricing methodology is that given the initial prices and the evolution of a set of securities, we construct a portfolio of these securities, i.e., ½(), perhaps rebalancing it across time ( trading strategy), such that the portfolio's cash flows and value match the cash flows and value of a secondary traded asset, say an interest rate option. Then, to avoid riskless profit opportunities, i.e., arbitrage opportunities, the cost of constructing this portfolio must equal the market price of the traded interest rate option. This argument only works, however, under complete markets, where we are able to find a portfolio that replicates the cash flows and value of the interest rate option. Such a replicating portfolio, if it exists, is called a synthetic interest rate option. In the case of a continuous-time model, the class of all self-financing strategies is usually too large; that is, arbitrage opportunities are not excluded a priori from . To prevent this drawback, for any T ~ T*, it is necessary to restrict attention to a certain subclass of , leading to the introduction of admissible trading strategies. Let us introduce a benchmark security, or a numeraire first. 1. Introduction 9 Definition 1.2.3. A numeraire is a price process Xt almost surely strictly positive for each t E [O, T]. Theorem 1.2.4 (Geman et al., 1995, Proposition 1). Self-financing portfolios remain self-financing after discounting primary securities by a numeraire. Let Z x stand for the relative price process of primary securities when the process X is chosen as a numeraire, i.e., Zf = Zt/ Xt = (Zf / Xt, ... , z;:- / Xt). For any trading strategy ¢, we write V x (¢) to denote its relative wealth v/ (¢) = ½(¢)/ Xt = Definition 1.2.5. A probability measure px on (D, Fr•), equivalent to Ill>, is called a martingale measure if Z x follows a local martingale under pX. Definition 1.2.6. A trading strategy ¢ E (1.2.4) follows a martingale under px_ The class of all px_admissible trading strategies from It can be shown that the market model Mx is arbitrage-free, then the question of pricing and hedging a contingent claim reduces to the existence of replicating self-financing trading strategies. Definition 1.2. 7. A European contingent claim H which settles at time T (a random cash-fl.ow H paid at time T) is an arbitrary non-negative Fr-measurable random variable. Definition 1.2.8. A strategy¢ E j replicates a European contingent claim H if Vr( ¢) = H. If a claim H admits at least one replicating strategy¢ from j, it is said to be attainable in Mx and the wealth process½(¢) is referred to as the replicating process of H. 1. Introduction 10 Definition 1.2.9. The market model Mx is said to be complete if any contingent claim is attainable. Then the following fundamental theorem in the arbitrage pricing theory can be proved. Theorem 1.2.10 (Musiela and Rutkowski, 1997b, Proposition 10.1.1). For any mar tingale measure JP>X E P(Mx), the market model Mx is arbitrage-free. Any contingent claim H attainable in Mx admits a unique replicating process in Thus, if a contingent claim H is attainable, H can be replicated by a portfolio ¢ E }. This means that holding the portfolio and holding the contingent claim are equivalent from a financial point of view. Consequently, we obtain an arbitrage-free price of a contingent claim H. Definition 1.2.11. An arbitrage-free price of a contingent claim H in MX at time t, 7rt(HIJP>x), is the wealth process V(¢) of any JP>X-admissible trading strategy which repli cates H. Theorem 1.2.12 {Musiela and Rutkowski, 1997b, Proposition 10.1.2). For any Eu ropean contingent claim H which is attainable in Mx, we have (1.2.5) where Ex denotes the expectation corresponding to JP>X. In some cases, another numeraire, say Yt, may be better than Xt to simplify pricing and hedging problems, leading to the introduction of the change of numeraire formula. Theorem 1.2.13 {Geman et al., 1995, Corollary 1). If a contingent claim is attain able in a given numeraire, it is also attainable in any other numeraire and the replicating strategy is the same. 1. Introduction 11 Theorem 1.2.14 (Geman et al., 1995, Corollary 2). Let H be a contingent claim which can be priced by arbitrage in standard market models Mx and MY. More specially, we as sume that H is JP>X-attainable and jp>Y -attainable, where IP'x E P(Mx) and jp>Y E P(MY). Then for every t E [O, T], (1.2.6) 1.3 Interest rate models This section reviews various models for interest rate derivatives. We briefly look at short rate models and the HJM model first. Then we give an in-depth review for the LIBOR market model and a quick review for the swap market model. A zero-coupon bond ( a discount bond) of maturity T :::; T* means a financial security paying to its holder one unit of cash at a prescribed date T in the future. This means that, by convention, the bond's principal (known also as the face value or nominal value) is one dollar. We assume that bonds are default-free; that is, the possibility of default by the bond's issuer is excluded. The price of a zero-coupon bond of maturity T at any instant t :::; T is denoted by B(t, T); it is thus obvious that B(T, T) = 1 for any maturity date T :::; T*. We assume throughout that for any fixed maturity T :::; T*, the price process B(t, T), t E [O, Tl, follows a strictly positive and adapted process on (D, lF, JP>). The purposes of term structure modeling are firstly to price all zero-coupon bonds of varying maturities from a finite number of economic fundamentals, called state variables. The second is to price all interest rate derivatives, taking as given the prices of the zero coupon bonds. 1. Introduction 12 1.3.1 Short rate models An instantaneous interest rate rt (also referred to as a short-term interest rate, or a spot interest rate) is an interest rate for risk-free borrowing or lending prevailing at time t over the infinitesimal time interval [t, t + dt]. We assume the process rt to be an adapted stochastic process on the probability space (n, lF, IP). We also assume that rt is integrable for almost all sample paths on [O, T*] with respect to the Lebesgue measure. An accumulation factor or a savings account Bt, is defined by Bt = exp (1t rudu), Vt E [O, T*]. (1.3.7) Bt is an adapted process of finite variation and with continuous sample paths. Equivalently, Bt solves the differential equation dBt = rtBtdt, with the initial condition B 0 = 1. It can be interpreted as the amount of cash accumulated up to time t by starting with one unit of cash at time 0. As discussed in Section 1.2, the absence of arbitrage opportunities between all bonds with different maturities and a saving account is equivalent to the existence of the following probability measure. Definition 1.3.1. A probability measure IP* on (n, Fr•) equivalent to IP is called the spot martingale measure, if, for any maturity T E [O, T*], the relative bond price B(t, T)/ Bt, Vt E [O, T], (1.3.8) follows a martingale under IP*. Thus, we choose the saving account Bt as a numeraire. Then, Theorem 1.2.12 gives the arbitrage-free price 7rt(H) under the probability measure IP*. Corollary 1.3.2. Let H be a IP* -attainable European contingent claim which settles at time T. Then the arbitrage-free price 1ft(H) at time t E [O, T] is given by the risk-neutral valuation formula (1.3.9) 1. Introduction 13 where lE* denotes the expectation corresponding to IP*. In particular, the price of H at time 0 equals 1ro(H) = lE*(Bi1 H). Using this formula, the bond price is given by B(t, T) Bt lE*(Bi11Ft) lE* (e-ftr,,dulFt), Vt E [O,T]. (1.3.10) A classic approach to interest rate modeling is to take the short rate rt to be a state variable. We assume that rt follows a diffusion process, that is, under the spot measure IP* the dynamics of rt follow drt = v(rt, t)dt + a(rt, t)dW/, (1.3.11) where v, a : ~ x [O, T*] -. ~ and Wt is a one-dimensional Brownian motion. Among some popular models Vasicek (1977) specified v(rt, t) = a - brt and a(rt, t) = a, where a, b and a are strictly positive constants, thus assuming that r follows an Ornstein-Uhlenbeck process with constant coefficients. Cox, Ingersoll, and Ross (1985) used v(rt, t) = a - brt and a(rt, t) = a .jri, where a, b and a as before. In each model, closed-form solutions for prices of a zero-coupon bond price, a European option written on a zero-coupon bond and on a coupon-bearing bond are available. For the Vasicek model, equation (1.3.11) is linear and can be solved explicitly, and the solution is a Markov process with continuous sample paths and Gaussian increments. The model is mean reverting to a/b, that is, the short rate is pulled to a level a/b. However, as any Gaussian model, rates can assume negative values with positive probability. Cox, Ingersoll, and Ross (CIR) model implies non-negative interest rates, and the short rates is characterized by a non-central chi-squared distribution. Both models are time-homogeneous, meaning that the assumed short rate dynamics depend only on constant coefficients. These models produce an endogenous term structure of interest rates, meaning that the current term-structure of rates is an output rather than 1. Introduction 14 an input of the model, thus the initial (today's) term structure does not necessarily match that observed in the market, no matter how the model parameters are chosen. By choosing the parameters judiciously, they can be made to provide an approximate fit to many of the term structures that are encountered in practice. But the fit is not usually an exact one and in some cases there are significant errors. A 1% error in the price of the underlying bond can lead to a 25% error in a bond option price. An exact fit to the currently observed yield curve can be achieved by time-inhomogeneous models, that is, the models allowing the coefficients of (1.3.11) to depend explicitly on time t. Hull and White (1990) assumed that dr(t) = [a(t) - b(t)r(t)] dt + O"(t)rf dWt, (1.3.12) for some constant (3 ~ 0, where Wt is a one-dimensional Brownian motion, and a, b, O": R+ -. R are locally bounded functions. When (3 = 0, we obtain the generalized Vasicek model, on the other hand, when (3 = 1/2, we obtain the generalized CIR model. Both models can be fitted exactly to the current term structure of interest rates. The generalized Vasicek model is also possible to fit exactly to the term structure of forward rate volatilities, although the future volatility structures implied by the model are likely to be unrealistic in that they do not conform to typical market shapes. Closed-form solutions for bond and bond option prices are easily available for the generalized Vasicek model, but not easily for the generalized CIR model. The single-factor Markovian models discussed so far are often criticized since the long rate is a deterministic function of the spot rate, and that the prices of bonds of different maturities are perfectly correlated. A more general approach to term structure modelling is, by introducing more state variables and sources of uncertainty, to construct multi-factor models. For the details of these models, see, for example, Duffie and Kan (1996). Suppose that X = (X1, ... , xn) is a multidimensional diffusion process, defined as a unique strong 1. Introduction 15 solution of the stochastic differential equation (1.3.13) where W is a n-dimensional Brownian motion, and the coefficients µ and CJ take values in [O, T*] x ?Rn and [O, T*] x ?Rn@?Rn, respectively. Then, the short rate is defined by rt = g(Xt) for some g : ?Rn _, ?R. As an example, assume that Xt follows dXt = (a(t) + b(t)Xt) dt + CT(t) · dWt, (1.3.14) where a : [O, T*] _, ?Rn and b, CT : [O, T*] -, ?Rn ® ?Rn are bounded functions. Then, define the short rate so that rt = ~~=l Xi. All the short rate models presented so far belong to an important class of term structure models, called affine term-structure models. Definition 1.3.3. An adapted process Y(t, T) defined by the formula 1 Y(t, T) = --T ln B(t, T), Vt E [O, T), (1.3.15) -t is called the yield-to-maturity on a zero-coupon bond maturing at time T. Definition 1.3.4. Affine term-structure models are interest rate models where the bond's yield Y(t, T) is an affine function in the short rate rt, i.e., Y(t, T) = a(t, T) + (3(t, T)rt, (1.3.16) where a and (3 are deterministic functions of time t. 1.3.2 The HJM model Heath, Jarrow and Morton (1990, 1992) built up a framework that encompass many of the interest rate models. Instead of modeling a short-term interest rate, they start with an exogenous specification of the dynamics of instantaneous, continuously compounded forward rates f(t, T). Assume that the bond prices B(t, T) is sufficiently smooth with respect to maturity T. 1. Introduction 16 Definition 1.3.5. The instantaneous, continuously compounded forward rate, or for short, instantaneous forward rate f(t, T) at time t for date T > t is f( T) = _ oln B(t, T) t, 8T . (1.3.17) It corresponds to the rate that one can contract for at time t, on a riskless loan that begins at date T and is returned an instant later. Solving the differential equation of expression (1.3.17) yields B(t, T) = exp (-lT f(t, u)du), Vt E [O, T]. (1.3.18) The spot rate at time t, rt, is the instantaneous forward rate at time t for date t, i.e., rt= f(t, t), Vt E [O, T*]. (1.3.19) Consequently, the saving account equals Bt = exp (1t f(u, u)du), Vt E [O, T*]. (1.3.20) We can find the spot martingale measure of Definition 1.3.1, hence ensures no-arbitrage across all bonds, and between bonds and the saving account. Then we obtain arbitrage-free dynamics of instantaneous forward rates. Theorem 1.3.6 (Heath, Jarrow and Morton, 1992). For any fixed maturityT ~ T*, the dynamics of the forward rate f(t, T) under the spot martingale measure JP* are df(t, T) = a (t, T, f(t, T)) · lT a (t, u, f(t, T)) dudt + a (t, T, f(t, T)) · dWt, (1.3.21) for a Borel-measurable function f(O, ·) : [O, T*] --t ~, and a : C x ~ --t ~d, where C = { (t, T) IO ~ t ~ T ~ T*}. For any maturity T, a(·, T) follow adapted processes such that 1T la (u, T, f(u, T))l2 du< oo. Furthermore, W*(t) is ad-dimensional Brownian motion under JP*. 1. Introduction 17 Then, from equation (1.3.18), the dynamics of bond prices B(t, T) follow dB(t, T) = B(t, T) (rtdt - lT a (t, u, f(t, T)) du· dW*(t)) , (1.3.22) and the short rate rt = f (t, t) is given by the expression rt= f(0, t) + 1t a (u, t, f(t, T)) · 1T a (u, s, f(u, T)) dsdu + 1t a (u, t, f(u, T)) · dW*(u), (1.3.23) If a (t, T, f(t, T)) = a(t, T), that is, the the volatility function does not depend on J(t, T), the forward rate f(t, T) and the spot rate rt have Gaussian probability laws in IP'*. We refer to this case as the Gaussian HJM model. We give a few examples of the Gaussian HJM models and show that the HJM model is a general framework for interest rate models. Taking d = land a(t, T) = a for a strictly positive constant a > 0, we obtain the continuous-time Ho-Lee model (1986), d [) f (0, T) I 2 ) d d * Tt = ( [)T T=t + a t t + a wt . (1.3.24) As another example, taking a(t, T) = ae--y(T-t) for strictly positive real numbers a, 'Y > 0, we get an extended version of Vasicek model (c.f. (1.3.12)), (1.3.25) One of the problems with the Gaussian HJM models is that the rates can become negative with a positive probability. To overcome this, Heath, Jarrow and Morton (1992) proposed a volatility function a(t, T) = TJ(t, T) min(f(t, T), M), where TJ(t, T) is a deterministic function and M is a large positive constant. With this volatility, the rates are positive and do not explode. However, no known distribution for f (t, T) is available. 1.3.3 The LIBOR market model As presented in the previous section, the HJM methodology for term structure modeling is based on the instantaneous, continuously compounded forward rates. However, in the ac tual market, the rates applicable to interest-rate derivatives, foremost among them LIBOR 1. Introduction 18 and swap derivatives, are quoted for accrual periods of at least a month, commonly three or six months, and their calculation is simpler than continuously compounded. Moreover, the market quotes liquid caps and swaptions in terms of implied Black-Scholes volatilities, implicitly assuming that forward LIBOR and swap rates follow log-normal processes with the quoted volatilities. Yet, the resulting formulas for caps and swaptions in the HJM model have no resemblance to the Black-Scholes formula. This leads to the calibration of the model being highly computationally intensive, and the result is often unsatisfactory. Brace, Gatarek and Musiela (1997) and Musiela and Rutkowski (1997a) introduced a new arbitrage-free model that models the forward LIBOR rate directly as the primary process rather than a secondary process derived from instantaneous forward rates. The model is called the LIBOR market model or the BGM model (referring to the authors). The model is arbitrage-free among bonds and between bonds and cash, and it prices caplets in accordance with the Black market caplet formula. In this section, we introduce the discrete tenor LIBOR market model developed by Musiela and Rutkowski (1997a) and Jamshidian ( 1997), and the thesis will be developed based on this model. For the construction of the continuous-tenor LIBOR market model, see Brace, Gatarek and Musiela (1997a). We set up the tenor structure as the finite set of dates 0 =To< T1 <··· T;,, i = 0, ... , N, are all equal to a fixed <5 (e.g., <5 = 0.5 years). We define a right-continuous function 'l/J: (0, TN+il---. {1, ... , N + 1} so that 'lj;(t) is the unique integer satisfying T,t,(t)-1 :S t < T,t,(t). Let S denote a set of continuous semimartingales on [0, TN+il and let s+ ={XE S: X(t) > 0, Vt}. 1. Introduction 19 For i = 1, ... , N + l, let Bi E s+ and Bi(t) be the price of a time t E [0, I';] zero-coupon bond maturing at Ti ~ TN+l with Bi(T;) = 1, and Bi(t) = 0 for any t E (Ti, TN+1l and any maturity Ti, i = 0, ... , N + l. Note that for simplicity, we use a new notation Bi(t) for a zero-coupon bond instead of B(t, T;). Definition 1.3. 7. The forward LIB OR rate at time t ~ Ti for the accrual period [Ti, Ti+il is 1 ( Bi(t) ) . Li (t) = ~ ( ) - 1 , i = 1, . . . , N. (1.3.26) u Bi+1 t At a tenor date n the price of any bond Bi(Tk), k > i, that has not yet matured is given by i-1 1 B/Tk) = II 1 + 6L-(r. r (1.3.27) j=k J k More generally, at an arbitrary time t < Tk, we have i-1 1 Bi(t) = B,j,(t)(t) II l + 6L-(t)' (1.3.28) j=,/J(t) J Following Section 1.2, let us choose zero-coupon bond prices B = (B1, ... , BN+i) as primary traded securities. A bond trading strategy is an RN+ 1-valued predictable stochastic process (t) = ( 1(t), ... , N+l V(t) = (t) · B(t) = L V(t) = V(0) + 1t Let us give a definition of a special function which characterizes contingent claims in the LIBOR market model. 1. Introduction 20 Definition 1.3.8. A function f(x1 , ... , xn) is homogeneous of degree m if (1.3.31) In the construction of the LIBOR market model by Musiela and Rutkowski (1997a) and Jamshidian (1997), the most important point is as follows. Any European LIBOR contingent claim H which matures (random payoff H is decided) at time Tn and settles (payoff H occurs) at time Tk is homogeneous of degree 1 in bond prices, which implies that the contingent claim is attainable with a self-financing portfolio of a finite number of bonds, that is, at time Tn the value of the portfolio is the same as the value of the contingent claim. Hence, we do not require a money market account or continuously compounded instantaneous interest rates. Therefore, if there exists a numeraire X E s+ and a probability measure jp>X,....., IP such that a family of bond prices Bi(t), i = 1, ... , N + l relative to X(t), i.e., Bi(t)/X(t), i = 1, ... , N + l, are all martingales with respect to IPx, then Theorem 1.2.12 gives the arbitrage-free price 7ft of a European LIBOR contingent claim H. Corollary 1.3.9. Let H be a jp>X -attainable European contingent claim which matures at time Tn and settles at time Tk. Then the arbitrage-free price 1ft at time t E [O, Tn] is given by (1.3.32) As the first kind of numeraire X, we introduce the spot LIBOR measure, which was de fined by Jamshidian (1997) and shares many characteristics with the risk neutral measure. The spot LIBOR measure takes a numeraire given by ,t,(t)-1 1 B*(t) B,w)(t) g Bi+1(T;) ,t,(t)-1 B,/;(t)(t) IJ (1 + 6Li(T;)). (1.3.33) i=O 1. Introduction 21 Definition 1.3.10. The spot LIBOR measure JP>L rv lP' on (fl, FrN+i) is a martingale mea sure such that a family of bond prices Bi(t), i = 1, ... , N + l relative to B*(t), i.e., Bi(t)/ B*(t), i = 1, ... , N + l are all martingales with respect to JP>L_ For the explicit representation of the spot LIBOR measure, let us assume that dBi(t) = Bi(t) (ai(t)dt + bi(t) · dWt), (1.3.34) with Rd-valued adapted processes ai(t) and bi(t), and a standard d-dimensional lP' Brownian motion. Then, the spot LIBOR measure is defined by the Radon-Nikodym derivative, (1.3.35) where the process h satisfies the relation ai(t) - a,w)(t) = (b,;,(t)(t) - ht) · (bi(t) - b,;,(t)(t)). (1.3.36) Theorem 1.3.11 (Jamshidian, 1997). Under the spot LIBOR measure, the arbitrage free process of the forward LIBOR rate is dL-(t) = ~ J(i(t)' (j(t) dt + r.(t)' dWL . l N • ~ 1 + JL-(t) -,, t ' i = , ... , , (1.3.37) j=,/J(t) J in which Wf = Wt - J; hudu is a standard d-dimensional Brownian motion under JP>L, Li(0) is deterministic and (i(t)' denotes the transpose of (i(t). It is assumed that (i(t) = (i (t, Li(t), ... , LN(t)) satisfies some conditions so that both coefficients of equation (1.3.37) satisfy Lipschitz and linear growth conditions. Then there exists a unique non-exploding global solution to equation ( 1. 3. 37). This expression is given by a general form, and the log-normal model corresponds to the case when (i(t) = >..i(t)Li(t) where >..i is a bounded Rd-valued function. In this case, equation (1.3.37) simplifies to dLi(t) _ ~ J>..i(t)' >..j(t)Lj(t) d ·( )' L L(t) - ~ l+JL-(t) t+>..,t dWt, i = 1, ... ,N. (1.3.38) i j=,/J(t) J 1. Introduction 22 The time-t price of a European contingent claim paying H at time Tk is given by (1.3.39) where JEL denotes the expectation corresponding to pL. At time 0, this reduces to no - IE' ( H Il I +o~,(T,)) . (1.3.40) On the other hand, Musiela and Rutkowski (1997a) introduced the terminal measure pN+l in which a numeraire is given by the bond price that matures last in our setting, Definition 1.3.12. The terminal measure pN+l ,....., IP on (0, FrN+i) is a martingale mea sure such that a family of bond prices Bi(t), i = 1, ... , N + l relative to BN+i(t), i.e., Bi(t)/ BN+i(t), i = 1, ... , N + l, are all martingales with respect to pN+l_ As in the spot LIBOR measure, the terminal measure is defined by the Radon-Nikodym derivative, dJP'N+l ( {TN+l l 1TN+l ) exp - hu · dWu - lhul 2du , (1.3.41) ~ = Jo 2 0 where the process h satisfies the relation (1.3.42) Theorem 1.3.13 (Musiela and Rutkowski, 1997a). Under the terminal measure, the arbitrage-free process of the forward LIBOR rate is dL-(t) = - ~ b(i(t)' (j(t) dt + (-(t)' dWN+l . 1 N (1.3.43) ' ~ 1 + 5L-(t) ' t ' i = ' ... ' ' j=i+l J in which wt+1 = Wt - J; hudu is a standard d-dimensional Brownian motion under pN+l and Li(O) is deterministic. It is assumed that (i satisfies some conditions so that both coefficients of equation (1.3.43) satisfy Lipschitz and linear growth conditions. Then there exists a unique non-exploding global solution to equation (1.3.43). 1. Introduction 23 As before, the log-normal model corresponds to the case when (i(t) = >-.i(t)Li(t) where >-.i(t) is a bounded ~d-valued function. In that case, equation (1.3.43) simplifies to dLi(t) = _ ~ 5>-.i(t)' >-.j(t)Lj(t) d >-.-( )' dWN+l · = l N (1.3.44) L-(t) ~ l+5L-(t) t+ it t ' i , ... , . i j=i+l J The time-t price of a European contingent claim paying H at time Tk is given by 1rt = BN+1(t)IE N+l ( BN+1(Tk)H IFt ) , (1.3.45) where JEN+l denotes the expectation corresponding to ]!DN+l. At time 0, this reduces to "' = BN +1 (O)JEN+l ( H fi (! HL,(T,))) . (1.3.46) Obviously, equations (1.3.39) and (1.3.45) lead to the following change of numeraire formula (c.f. (1.2.14)) between the spot LIBOR measure and the terminal measure, B*(t)IEL ( B*7Tk) I Ft) = BN+1(t)JEN+l ( BN+~(Tk) I Ft). (1.3.47) The measure relationship between the spot LIBOR measure and the terminal measure is given by the following Radon-Nikodym derivative dPL I (1.3.48) dPN+l Ft We now describe how the log-normal LIBOR market model prices caplets in accordance with the Black market caplet formula. An interest rate cap is a contractual arrangement where the grantor (seller) has an obligation to pay cash to the holder (buyer) if a particular interest rate exceeds a mutually agreed level at some future date or dates. Specifically, a forward start cap is a strip of caplets, each of which is a call option on a forward LIBOR rate. Let K, to denote the cap strike rate. In a forward cap settled in arrears at dates ~+1, i = n, ... , M, the cash flow at time ~+1 is (Li(~)-K,)+J. Thus, from the formula (1.3.45), the arbitrage-free price at time t E [O, ~] of a caplet with strike K, and maturity~ is Cpl(t) = B (t)JEN+1 ( (Li(Ti) - K,t 51 ,r:) (1.3.49) i N+l B ('T' ) .rt ' N+l 1 i+l 1. Introduction 24 where Li(T;) is given by the solution of equation (1.3.44). It seems not to be simple to evaluate the expectation (1.3.49). However, by changing the terminal measure jpN+l to a different forward measure, the evaluation becomes very simple. Definition 1.3.14. For i = 0, ... , N -1, the forward measure JPi+1 "'JPN+l on (0, Fr;+i), with the Radon-Nikodym derivative given by BN+1 (0)Bi+l (t) (1.3.50) Bi+1(0)BN+1(t)' is a martingale measure such that a family of bond prices Bi(t), j = 1, ... , N + 1 relative to Bi+l (t), i.e., Bi(t)/ Bi+1(t), j = 1, ... , N + 1, are all martingales with respect to JPi+ 1 . Then the change of numeraire formula (Theorem 1.2.14) gives Cpli(t) (1.3.51) and the dynamics of the forward LIBOR rate Li(t) under the forward measure JPi+1 follow dLi(t) = Li(t)>.i(t)' dWi+ 1(t). (1.3.52) Hence, in view of equation (1.3.51) and (1.3.52), the next result follows immediately. Theorem 1.3.15 (Musiela and Rutkowski, 1997b, Lemma 16.3.1). The price at time t E [0, Ti] of a caplet with strike "' and maturing T; equals Cpli(t) = 0Bi+1(t) ( Li(t)N(hi) - "'N ( hi - ~)), (1.3.53) where and T Ai= lt 'j>.i(u)l2du. 1. Introduction 25 Here, N stands for the standard Gaussian cumulative distribution function 2 N(x) = r,c_1 lx e-z 12 dz, 't:/x E ~- (1.3.54) v 27r -oo By taking l>-i(u)I = >.i, Vu E [t, Ti], for some constants >.i, i = 1, ... , n, we can easily see that formula (1.3.53) reduces to the Black market caplet formula, (1.3.55) where ln(Li(t)/,..:) + ½Ai hi= -/A: ' and Eventually, we obtain the following pricing formula for a forward cap. Theorem 1.3.16 (Musiela and Rutkowski, 1997b, Proposition 16.3.1). The arbitrage free price at time t::; Tn of a forward cap, denoted by FC(t), is i=n M LCpli(t) i=n M L 6Bi+1(t) ( Li(t)N(hi) - KN ( hi - A)), (1.3.56) i=n where and 1. Introduction 26 1.3.4 The swap market model To calibrate swaptions more easily and accurately, Jamshidian (1997) proposed a new model, in which he took the forward swap rates directly as the primary processes and the model prices swaptions in accordance with the Black market swaption formula. The model is often referred as the swap market model. An interest rate swap is an agreement between two parties to exchange a fixed interest rate with a floating interest rate at some future dates. Specifically, a forward start payer swap settled in arrears is a swap agreement entered at the traded date t :S Tn such that a fixed interest rate,.,, is paid and a floating interest rate Li(T;,), i = n, ... , M, is received at the consecutive dates Tn+l, ... , TM+l· Theorem 1.3.17. The value at time t :S Tn of a forward start payer swap, FS(t), is M FS(t) = L Bi+1(t) (Li(t) - K,) 5. (1.3.57) i=n Proof. From Theorem (1.3.45), the arbitrage-free price of a forward start payer swap is (1.3.58) Then, by the change of numeraire formula (Theorem 1.2.14), M F S(t) = ~L..,. Bi+1 (t)JE i+l ( (Li(T;,) - ,.,,) 51 .Ft) . (1.3.59) i=n Since Li(t) is a JPi+l martingale, it turns out that M FS(t) = L Bi+1(t) (Li(t) - K,) 5. D (1.3.60) i=n The forward swap rate is the fixed rate ,.,, which makes the value of a forward swap zero, i.e., the value of,.,, for which FS(t) = 0. Definition 1.3.18. The forward swap rate Sn,M(t) at time t for the date Tn is S = I:~n Bi+1(t)Li(t) [ ] n,M (t) B (t) , t E 0, Tn , (1.3.61) n,M 1. Introduction 27 where M Bn,M(t) = LBi+1(t). i=n A payer swaption with strike rate "', maturing at time Tn, gives the right to receive cash flows corresponding to a time Tn forward payer swap settled in arrears. Hence, using equation (1.3.45), the time t ~ Tn price of a European payer swaption PS(t) with strike "', maturity Tn and accrual period M - n + 1 is (1.3.62) or using (1.3.61) (1.3.63) Jamshidian (1997) focused on a certain family of forward swap rates, that is, by fixing M = N, he looked at a family of forward swap rates whose underlying swaps differ in length, but have a common expiry date TN+l· Then, he constructed a family of arbitrage free forward swap rate processes. Theorem 1.3.19 (Jamshidian, 1997). The arbitrage-free process of the forward swap rate under the terminal measure is, for n = 1, ... , N, dS (t) = _ I:[:n I:~=n+l o~k(t)' ~n(t)rr;=n+l,#k (1 + osj,N(t)) dt + C (t)' dWN+l. n,N -.;;:-'N . ( ( )) ',,n t ~i=n rrj=n+l 1 + oSj,N t (1.3.64) in which wf +1 is a standard d-dimensional Brownian motion under JP>N+l, Sn,N(O) are deterministic. It is assumed that ~n satisfies some conditions so that both coefficients of equation ( 1. 3. 64) satisfy Lipschitz and linear growth conditions. Then there exists a unique non-exploding global solution to equation (1.3.64). 1. Introduction 28 This expression is given by a general form, and the log-normal model corresponds to ~n(t) = vn(t)Sn,N(t), where vn(t) is a bounded Rd-valued function. In this case, equa tion (1.3.64) simplifies to _ ~~n ~~=n+l ovk(t)' sk,N(t)vn(t)Sn,N(t)(t)II;=n+l,j;=k (1 + oSj,N(t)) dt dSn,N(t) = ~~n rr;=n+l (1 + oSj,N(t)) +vn(t)' Sn,N(t)dwt+1. (1.3.65) Remark: Rutkowski (1999) constructed a family of forward swap rates with fixed length, but different expiry dates. J amshidian also defined the forward swap measure pn,N. Definition 1.3.20. For n = l, ... , N, the forward swap measure JP>n,N,....., JP>N+l on (D, FrJ, with the Radon-Nikodym derivative given by BN+l (O)Bn,N(t) (1.3.66) Bn,N(O)BN+1(t)' is a martingale measure such that a family of bond prices Bj(t), j = 1, ... , N + l relative to Bn,N(t), i.e., Bj(t)/ Bn,N(t), j = 1, ... , N + l, are all martingales with respect to JP>n,N_ Then the change of numeraire formula (Theorem 1.2.14) gives PS(t) OBN+1(t)JEN+l ( :::~~)) (Sn,N(Tn) - ~)+I Ft) oBn,N(t)JEn,N ( (Sn,N(Tn) - ~ti Ft), (1.3.67) where JEn,N denotes the expectation corresponding to JP>n,N. The dynamics of the forward swap rates Sn,N under the forward swap measure JP>n,N follow dSn,N(t) - ( )'dwn,N Sn,N(t) - Vn,N t t , (1.3.68) where vn,N(t) is deterministic and wt,N is a standard d-dimensional Brownian motion under the forward swap measure. Thus, the next result follows immediately from equa tions (1.3.67) and (1.3.68). 1. Introduction 29 Theorem 1.3.21. The price at time t E [O, Tn] of a European payer swaption with strike K, and maturity Tn is PS(t) = 8Bn,N(t) ( Sn,N(t)N(h) - fi,N ( h - Jr)) , (1.3.69) where ln ( Sn,N (t) / fi,) + ½Y h= vY ' and We then see that taking lvn,M(u)I = Vn,M for some constant Vn,M > 0 reduces for mula (1.3.69) to the Black market formula for a European payer swaption, PS(t) = 8Bn,N(t) ( Sn,N(t)N(h) - fi,N ( h - Jr)) (1.3.70) where h = ln(Sn,N(t)/fi,) + ½Y vY and 2. SWAPTION PRICING IN A LOG-NORMAL LIBOR MARKET MODEL 2.1 Introduction One discrepancy between the two market models, the LIBOR market model and the swap market model, is that, as Jamshidian (1997) pointed out, the LIBOR and swap market models are inconsistent with each other, that is, forward LIBOR and swap rates can not simultaneously have deterministic volatilities. Hence the resulting Black formulae are inconsistent with each other. Rebonato (1999a) argued that this discrepancy is almost negligible. Thus it might be satisfactory to use the LIBOR market model for the LIBOR derivatives and the swap market model for swap derivatives separately. However if one wants to capture the dynamics of both LIBOR and swap markets simultaneously, for ex ample if one wants to capture the dynamics of a portfolio of caplets and swaptions, then the LIBOR market model may be chosen to price and hedge both LIBOR and swap derivatives. Unfortunately in the LIBOR market model we cannot price swaptions in accordance with the Black formula, and a closed form formula for swaptions is not available. Thus to be able to deal with swaptions in the LIBOR market model, we need a good approximation method. A good and preferably simple analytical approximate formula for swaptions can be used not only for pricing but also for efficient calibration and estimation of Greeks. An efficient numerical method is also useful to obtain accurate swaption prices. We are going to discuss these two issues one by one. We especially focus on the pricing meth ods in the log-normal LIBOR market model in this chapter, and the next chapter deals with the pricing methods in extended LIBOR market models. An approximate formula 2. Swaption pricing in a log-normal LIBOR market model 31 for swaptions in the log-normal LIBOR market model was derived by Brace, Gatarek and Musiela (BGM) (1997). One disadvantage of their formula is that it needs a solution of a nonlinear equation first to find the approximate swaption price. This introduces a compu tationally intensive numerical root search when calibrating the parameters to the market quoted prices. To overcome this problem, Brace, Dun and Barton (1998) introduced a sim pler formula, which does not require a solution of any equation first and is a form of the Black market swaption formula. They discussed the accuracy of their formula in terms of prices and hedge parameters. Rebonato (1999a), Hull and White (1999), Sidenius (2000), Zi.ihlsdorff (2000) and Andersen and Andreasen (2000a) also developed approximate pricing formulae for European payer swaptions in the log-normal LIBOR market model. Using a new approach, this chapter derives an approximate formula for European payer swaptions in the log-normal LIBOR market model. To obtain the formula, we apply an asymptotic expansion method recently introduced by Kunitomo and Takahashi (2001). Numerical results show that our approximate formula is more accurate than other approximate for mulae by Brace, Gatarek and Musiela (1997), Brace, Dun and Barton (1998) and Andersen and Andreasen (2000a). This chapter also develops an approximate formula for European payer swaptions using the method by Jarrow and Rudd (1982). The second issue considered in this chapter is finding accurate swaption prices using an efficient numerical method. Because of the high dimensionality of the LIBOR market model, finite difference and lattice methods are not good choices. The only computa tionally feasible method will be a Monte Carlo simulation. However, recently Glasserman and Zhao (2000) claimed that the standard discretization of the forward LIBOR process is not arbitrage-free since the discrete-time deflated bond price processes are not martin gales. Then, focusing on the process of the difference of deflated bonds rather than the LIBOR rate process itself, they introduced an arbitrage-free discretization scheme which makes all discrete-time deflated bond prices be positive martingales. Brace, Musiela and 2. Swaption pricing in a log-normal LIBOR market model 32 Schlogl (1999) introduced a different method. We introduce an arbitrage-free discretiza tion method under the forward swap measure. Like Glasserman and Zhao's method, our method also discretizes a process which relates a difference of deflated bond prices. However a significant advantage of our discretization scheme is that we can use our ap proximate formula straightforwardly as a control variate. Combining our discretization method with the control variate we achieve a significant reduction in simulation variance, hence we are able to find accurate swaption prices quite rapidly. Using this method, it will be possible to find accurate hedge parameters for European swaptions and also accurate prices of Bermudan swaptions with feasible computational efforts. For pricing Bermudan swaptions in the LIBOR market model, we refer to Andersen (2000), Andersen and Andreasen (2000b), Noubir (1999) and Tang and Lange (2001). More generally, for pricing American style options using a Monte Carlo simulation, we refer to Barraquand and Martineau (1995), Broadie and Glasserman (1997a, 1997b), Broadie, Glasserman and Jain (1997) and Longstaff and Schwartz (1998). 2.2 Review of approximate swaption formulae In this section, before presenting our new approximate formula, we review two approximate swaption pricing formulae in the log-normal LIBOR market model. The first is by Brace, Gatarek and Musiela (1997), and the second is by Brace, Dun and Barton (1998). Recall the valuation formula (1.3.63) for a European payer swaption price under the terminal measure, (2.2.1) where S (T.) = L~n Bi+1(Tn)Li(Tn) n,M n B n,M (T.)n , 2. Swaption pricing in a log-normal LIBOR market model 33 and M Bn,M(Tn) = LBi+1(Tn), i==n Brace, Gatarek and Musiela (1997) transformed (2.2.1) into M PS(t) = oL.J~ Bi+1(t)lE i+l ( (Li(Tn) - K,) JAi Ft), (2.2.2) i=n where Since each Li(Tn) is a log-normal random variable under the corresponding forward measure pi+l, by finding an approximate region that the event A occurs, we can evaluate each expectation in (2.2.2) using normal cumulative distribution functions. Hence, with our notation, we obtain the following approximate formula for a European payer swaption. Lemma 2.2.1 (Brace, Gatarek and Musiela, 1997). An approximate price at time t E [O, Tn] of a European payer swaption with strike K,, maturity Tn and accrual period M-n+l is M PS(t) ~ oL Bi+1(t) (Li(t)N(-s0 - i + ri) - K,N(-s0 - i)), (2.2.3) i=n where s0 is given by Ci+l = K,O, i = n, ... , M -1, CM+l = 1 + K,O, and ri = ~ Vi+l-n, i = n, ... , M, where µmax is the largest eigenvalue and v is the corresponding eigenvector for the matrix Tn A1+1-n,m+l-n = lt A1(s) 1 Am(s)ds, l, m=n, ... ,M, and for i = n, ... , M, di is given by 2. Swaption pricing in a log-normal LIBOR market model 34 Brace, Dun and Barton (1998) tried to express a volatility of the forward swap rate approximately in terms of a function of volatilities of the forward LIBOR rates, Ai(t), and gave the following approximate price for a European payer swaption. The detailed derivation can be found in Brace and Womersley (2000). Lemma 2.2.2 (Brace, Dun and Barton, 1998). An approximate price at time t E [O, Tn] of a European payer swaption with strike K, maturity Tn and accrual period M -n+ 1 ZS PS(t) ~ 5Bn,M(t) ( Sn,M(t)N(h) - KN ( h - JA)), (2.2.4) where ln (Sn,M(t)/K) + ½A h= v'A , and 2.3 A valuation formula under the forward swap measure As a preliminary for deriving our approximate pricing formula for a European payer swap tion, this section constructs a family of forward swap rate processes in the log-normal LIBOR market model. Since the basic model is the LIBOR market model, we do not fix M, but only fix the horizon date TN+l so that we are able to consider a family of forward swap rates expiring within TN+l with any swap length. To make further developments eas ier, we use two kinds of tricks. Firstly, like Jamshidian's method in the previous chapter, we introduce the forward swap measure and consider each forward swap rate process under the forward swap measure. Secondly, we introduce a new state variable v of a function of the forward LIBOR rates, and derive the forward swap processes in terms of v. 2. Swaption pricing in a log-normal LIBOR market model 35 Let F;,(t) denote the bond prices discounted by the bond price BN+i(t), that is, Z::(t) Bi(t) r, = ( ) , i = 1, ... , N + l, (2.3.5) BN+l t which are martingales under the terminal measure jp>N+l_ Musiela and Rutkowski (1997a) showed that the processes Fi(t) and the forward LIBOR rate processes Li(t) under the terminal measure jp>N+l are given by, for i = 1, ... , N, (2.3.6) and (2.3.7) where (2.3.8) Let ai+i(t) denote the bond prices discounted by Bn,M(t), that is, Bi+1(t) Bn,M(t) FH1(t) i=n, ... ,M. (2.3.9) From Ito's lemma, 2. Swaption pricing in a log-normal LIBOR market model 36 and applying Ito's lemma again, dai+l (t) ( 'Yi+1,N(t) - a;+1,N(t)'Y;+1,N(t))' dW/v+l ai+1(t) t. -t. U;+ 1 ( th;+1,N(t) ( 'Yi+i,N( t) - t. ll;+l ( t)'Y;+1,N( t) )' dt ( 'Yi+l,N(t) - t. a;+1(th;+1,N(t))' ( dW,N+l - t. a;+1(th;+1,N(t)dt) , Also using ~~n ai+1 = 1, some algebra shows that M M 'Yi+l,N(t) - L ai+1(thi+1,N(t) = 'Yi+l,M(t) - L ai+1(thi+1,M(t). j=n j=n Hence, ( dW,N+' - t. ai+l (t)'Y;+1,N(t)dt) . (2.3.10) Let us define a new martingale measure, the forward swap measure pn,M. Definition 2.3.1. We call a martingale measure pn,M rv JIDN+l on (D, :FTJ, with the Radon-Nikodym derivative given by 2 ::: IF, = exp (l t. a;+1(uh;+1,N(u)' dW.;'+' - ~ l t. a;+1(uh;+i,N(u) du) BN+l (O)Bn,M(t) ( ) ( ) , (2.3.11) Bn,M O BN+l t the forward swap measure for the maturity Tn and accrual period M - n + 1. Then, the process tM wt,M = wt+1 -1 ~aj+1(uhj+1,N(u)du, Vt E [O,Tnl, (2.3.12) 2. Swaption pricing in a log-normal LIBOR market model 37 follows a standard d-dimensional Brownian motion under the forward swap measure ]pm,M. Thus, under the forward swap measure, the dynamics of the discounted bond price pro cesses ai+ 1 ( t) are i=n, ... ,M, (2.3.13) where M T/i(t) = 'Yi+I,M(t) - Lai+1(thj+I,M(t), (2.3.14) j=n and hence the processes ai+1(t) are all ]pm,M martingales. Note that the processes T/i(t) are stochastic and depend on both n and M, and so does ai+1(t), but, for simplicity, the indexes are omitted. From equation (2.2.1), the change of numeraire formula (Theorem 1.2.14) gives the price of a payer swaption under the forward swap measure Jpm,M as PS(t) (2.3.15) The forward LIBOR processes Li(t) in ]pm,M satisfy the equations (2.3.16) Note that Li(t) are now dependent on both n and M. From equation (2.2.1), the forward swap rate can be expressed as 'E~n Bi+1(t)Li(t) Bn,M(t) M L ai+I (t)Li(t) i=n M L vi(t), (2.3.17) i=n 2. Swaption pricing in a log-normal LIBOR market model 38 where we introduced new state variables vi(t), i = n, ... , M. An application of Ito's lemma shows that the dynamics of the processes vi(t) under JP>n,M are (2.3.18) Notice that each vi(t) depends on both n and M. The next Lemma shows that f/i(t) can be expressed in terms oft and Vn+i(t), ... , vM(t). Lemma 2.3.2. f/i (t) = f/i (t, V ( t)) , (2.3.19) where v(t) = (vn+1(t), ... , VM(t)). Proof. From the definition of vi(t), (2.3.20) Also, from the definition of the forward LIBOR rates (1.3.26), we obtain 1 ( ai(t) ) Li (t) = ~ ( ) - 1 , (2.3.21) u ai+l t or 5vi(t) = ai(t) - ai+1(t), i = n + 1, ... , M. (2.3.22) It is obvious that (2.3.23) Then, the relations (2.3.22) and (2.3.23) give the system of equations 2. Swaption pricing in a log-normal LIBOR market model 39 1 -1 0 0 0 0 an+l (t) Vn+1(t) 0 1 -1 0 0 0 an+2(t) Vn+2(t) =15 (2.3.24) 0 0 0 0 1 -1 aM(t) VM(t) 1 1 1 1 1 1 1 aM+l (t) J Solving this system gives r-l r-2 r-3 2 1 1 Vn+1(t) an+l (t) -1 r-2 r-3 2 1 1 Vn+2(t) an+2(t) -1 -2 r-3 2 1 1 Vn+3(t) 15 r aM(t) -1 -2 -3 -r+2 1 1 VM(t) aM+1(t) 1 -1 -2 -3 -r+2 -r+ 1 1 J (2.3.25) where r = M - n + l. This gives the expression for each ai(t) in terms of v(t). Each T/i(t) in (2.3.14) can be written as M T/i(t) = 'Yi+1,M(t) - L aj+1(th1+1,M(t) j=n (2.3.26) It follows that T/i(t) = T/i (t, v(t)). Hence the result follows. D As a result, the next lemma formulates our problem. Lemma 2.3.3. A price at time t E [O, Tn] of a European payer swaption with strike ,-.,, maturity Tn and accrual period M - n + l is (2.3.27) 2. Swaption pricing in a log-normal LIBOR market model 40 where M Sn,M(Tn) = I: Vi(Tn)- (2.3.28) i==n The processes Vi (u) for t :S u :S Tn solve (2.3.29) where Fort :S u 1 :Su and i = n, ... , M, and 2.4 An asymptotic expansion method From the expression of the processes vi(u), i = n, ... , M, in Lemma 2.3.3, the process of the forward swap rate is very complicated. To obtain an approximate European payer swaption formula, we apply an asymptotic expansion approach recently introduced by Kunitomo and Takahashi (2001). The method is applicable to a wide range of option valuation problems, and when an analytic solution to an option price is not available it is a good tool to obtain an approximate formula for the option price. 2. Swaption pricing in a log-normal LIBOR market model 41 2.4.1 Introduction Let us consider the following stochastic differential equation which depends on a certain parameter E E R+, (2.4.30) where X(O) E R, F : [O, T] x Rd x R ---t R, G : [O, T] x Rd x R ---t Rd and Wt is a d-dimensional Brownian motion. The following result is very standard. Theorem 2.4.1. Suppose that the coefficients F and G satisfy Lipschitz and linear growth conditions IF(t, x, E) - F(t, y, E)I + IG(t, x, E) - G(t, Y, E)I ~ Clx - YI, for O ~ t ~ T, where C is a positive constant. Then for each E > 0 there exists a unique non-exploding global solution to equation (2.4-30} and moreover x(e:l(t) E £ 2 (!1 x [O, Tl). Proof. The proof is omitted as it is standard. D Then under some conditions we can prove the differentiability of x(e:l(t) with respect to E. Theorem 2.4.2. If the functions F and G have continuous derivatives up to order k with respect to x, then x(e:) ( t) has continuous derivatives up to order k with respect to E. Proof. We set x(e:l(t) = X(t). Let us consider a mapping (E, X) - T(E, X) given by T(E, X)(t) = X(O) + 1t F(u, X(u), E)du + 1t G(u, X(u), E) 1 dWu. (2.4.31) 2. Swaption pricing in a log-normal LIBOR market model 42 Note that X(t) is the unique solution of the equation X(t) = T(E, X)(t). Let X(t) and Y(t) be arbitrary processes, then //T(E, X) -T(E, Y)l! 2 IE [1T (F(u,X(u),£) - F(u, Y(u),£))du+ 1T (G(u,X(u),£) - G(u, Y(u),£))' dW.]' < 21E [1T (F(u,X(u),E)-F(u, Y(u),E))dur +21E [1T (G(u,X(u),£) - G(u, Y(u),£))' dW.]' < 2TIE [1T (F(u, X(u), E) - F(u, Y(u), c))2 du] +21E [1T (G(u, X(u), c) - G(u, Y(u), c))2 du] < 2(T+ l)C21E [1T (X(u)-Y(u)/du] 2(T + 1)c2 11x - Yl1 2 . (2.4.32) Consider subintervals [O, h], [h, 2h], ... for [O, T] so that h satisfies 2(1 + h)C2 < 1. Then the result follows from Theorem 9.4 in Da Prato and Zabczyk (1992). D To briefly introduce the asymptotic expansion method, let us consider, for simplicity, the following one-dimensional stochastic differential equation with no drift. (2.4.33) where x and f7 E R, Wt is a one-dimensional Brownian motion and E E (0, 1]. f7 satisfies the conditions in Theorem 2.4.1 and Theorem 2.4.2 so that X(0 )(t) is twice continuously differentiable with respect to E. Our problem is to find an approximation to the expectation given by (2.4.34) where his an R - R function. Using a Taylor's series expansion, we obtain a second-order asymptotic expansion for the process X(0 )(t). 2. Swaption pricing in a log-normal LIBOR market model 43 Theorem 2.4.3 (Kunitomo, Takahashi, 2001). A second-oder asymptotic expansion for X(0 l(t) is, for c - 0, (2.4.35) Here, O(c3) is meant in the sense of the L2 (n X [t, Tn])-norm 11x(e)(u) 11 2 = Jtn lE 1x(e)(u) 12 du. Let us write an asymptotic expansion for x(c:)(t) at time t = T as, for c - 0, (2.4.36) where - - E)X(e) (T) I - ~ 32 x(e) (T) I 9o - x' 91 - 8 ' and 92 - 2 8 2 . c t:=0 c e=O It can be found that 91 is a stochastic integral with a deterministic integrand and 92 is a stochastic double integral with a deterministic integrand. Thus, 91 is normally distributed, say, 91 r.., N (0, E), provided that E > 0. Now, define y(c:)(T) by (2.4.37) Then, we are able to obtain an asymptotic expansion for the density function of y(c:) (T). Theorem 2.4.4 (Kunitomo, Takahashi, 2001). An asymptotic expansion for the den sity function ofY(c:)(T) is, fore - 0, f~\x) = (2.4.39) 2. Swaption pricing in a log-normal LIBOR market model 44 Consequently, using the asymptotic density function J~\x), we are able to obtain an asymptotic expansion for the expectation (2.4.34), that is, +00 lE (h (x(c:)(T))) = 1_ h(g0 + cx)J~\x)dx. (2.4.40) 00 2.4.2 An asymptotic formula To apply an asymptotic method to our problem in Lemma 2.3.3, we assume the arbitrage free dynamics of the forward LIBOR rates associated with a small disturbance c « 1, L?\u), t::; u::; Tn, follow dL~c:\u) = -c2 t ,fo-j(u)' ~f (u) ai(u)L~c:\u)du + Eai(u)' L~c:\u)dWt'+1, i = 1, ... , N, j=i+l l+oLj (u) (2.4.41) where (2.4.42) Note that when we obtain the original dynamics of the forward LIBOR rate processes (1.3.44). If we assume that there exists a constant K < 1 such that sup (l,\n(u1)j 2 , ... , j,\M(u1)12) < K 2 , then c < K < 1. In the limit as c - 0 each process L~0\u) = Li(t) fort ::; u::; Tn, Using L~c:) (u), for t ::; u ::; Tn the processes i=n, ... ,M, (2.4.43) i=n, ... ,M, (2.4.44) 2. Swaption pricing in a log-normal LIBOR market model 45 where Fort:::; u1 :::; u and i = n, ... , M, where and An asymptotic price at time t E [O, Tn] of a European payer swaption with strike ,-,,, maturity Tn and accrual period M - n + 1 is (2.4.45) where M si:~(Tn) = L v}';\Tn)- i=n It is easily seen that a unique solution for each Lf0\u) exists, since each Li(u) has a unique solution. Lemma 2.4.5. For each u, t :::; u :::; T;, the processes L?\u), i 1, ... , N, have all continuous derivatives with respect to E. Proof. It can be easily shown that the condition in Theorem 2.4.2 is satisfied with any order. Alternatively, we can prove it in the following way. For i = N, the process Lt\u) is (2.4.46) 2. Swaption pricing in a log-normal LIBOR market model 46 The solution to this equation is LW(u) = LN(t) exp (-ic2 iu IO'N(u1)12du1 + Ciu O'N(u1) 1 dwti'+l) . (2.4.47) Thus, it is obvious that L W(u) has all continuous derivatives with respect to c. Then for i = N - l, the process Lt~ 1 (u) is 1 dLt~1(u) __ 2 6CTN-1(u) CTN(u)Lt\u)d ( )'dwN+I () - c ( ) U + cO'N-1 U u . (2.4.48) L!r_1(u) 1 + JL!r (u) The solution to this equation can be written as (e)_ (u) = L () ( (60"N-1(ui)'CTN(u1)Lt\u1) ll ( )l 2) d LN 1 N-1 t exp -E 21u ( ) + -2 O'N-1 U1 U1 t 1 + JL!r (u 1) +c 1" D"N-1( u,)' dW,'.'.+~ . (2.4.49) Since Lt\u) has all continuous derivatives with respect to c, equation (2.4.49) implies that Lt~ 1(u) has all continuous derivatives with respect to c. Continuing similarly for i = N - 2, ... , 1, it can be shown that Lie\u), i = 1, ... , N - 2, also have all continuous derivatives with respect to c. D Relations between v(e) and L(e) are (e) Lie)(u)IJ~i+1 (l+JL?\u)) i=n, ... ,M. (2.4.50) vi (u) = M M ( (e) ) ' ~j=n+l ITk=j 1 + JL1 (u) + 1 Since each Lie) (u) has all continuous derivatives with respect to c, it follows that each vie) (u) also has all continuous derivatives with respect to c. For notational simplicity let us define the following partial derivatives. For i, j, k = n, ... , M, 8 .. ( (e)( )) - ocpi (u,v(el(u)) d 8 i 'k (t v(e)(u)) = a2cpi (u,v(el(u)) 'Pi,J u, V U - n (e) ( ) , an 'P ,J, , n (e) ( ) n (e) ( ) . uv1 u uv1 u uvk u Theorem 2.4.6. The third order asymptotic expansion for vt)(u), u E [t, Tnl, is, for c - 0, + O(c4), (2.4.51) 2. Swaption pricing in a log-normal LIBOR market model 47 where = iu 'Pi (u1, v(t)) 1 dW,:';M, €=0 and 83 (e1( ) I M M 1 1 1 ~€3u :/~~}tr (lt r' (lt r2 cpk(u3,v(t)) dW,:;M ) cp1 (u2,v(t)) dW,:';M ) 8cpi,J,k(u1,v(t)) dW,:';M 0 +3 t. t 1u (1u, cp1 (u2, v(t)) 1 cpk(u2, v(t)) du2) 8cpi,j,k(u1, v(t))' dW,:';M +6 t. t 1u (1u, (1u 2 cpk(u3, v(t)) 1 dW,:';M) Ocpj,du2, v(t)) 1 dW,:';M) acpi,j (u1, v(t)) 1 dW,:';M. Here, O(c4) is meant in the sense of the L 2 (0 x [t, Tn])-norm llv(")(u)ll 2 = Jtn E lv(0 )(u)j2 du. Proof. By a direct differentiation of v;"\ u) with respect to E, 8c €=0 Note that v( 0)(u1) = v(t) fort::; u 1 ::; u. 2. Swaption pricing in a log-normal LIBOR market model 48 Since the third derivative follows as above. 0 Using the third order asymptotic expansion for v;c:\u), let us write si:~(u) - K, ~;'!n v;c:\u) - K, at time u = Tn as, for c-+ 0, (2.4.52) where 9o = Sn,M(t) - K,, 2. Swaption pricing in a log-normal LIBOR market model 49 Assuming a non-degeneracy condition that, (2.4.53) g1 is distributed as N (0, I;), where M M T, 1 I;= LL1 n 'Pi (u1, v(t)) <{}j (u1, v(t)) du1. i=n j=n t Let us define X~c:~(Tn), by (2.4.54) To find an asymptotic expansion for the density function of X~c:~(Tn),, we will need the following result of Nualart, Ustunel and Zakai (1988) on the conditional expectation of a multiple Weiner-Ito integral. Lemma 2.4.7. Let T = [O, oo), f(u1, ... , up) E L2(TP), and let Ip(!) denote the multiple Winer-Ito integral of the kernel f. Leth E L2(T) and consider h®P; the pth tensor product of h (h®P (u1, ... , up)= h(u1) · h(u2) · · · · h(up)). Then we have (2.4.55) where K = < f, h®P >L2(TP) (llhlli2(T)r - 2. Swaption pricing in a log-normal LIBOR market model 50 Then Lemma 2.4. 7 gives the following formulae. Lemma 2.4.8. Let Wu bead-dimensional Brownian motion, x ER and q(u) be a R-+ Rdx 1 deterministic function such that E = JT q( u1) 1 q( u1)du1 > 0. (i) Let q1(u), q2(u) and q3(u) be R-+ Rdxl deterministic functions. Then fort~ u3 ~ (2.4.56) where where = a3 (x 3 - 3Ex), (2.4.58) where 2 1 1 1 a3 = ; 3 Ju (Jui (Ju q(u3) q3(u3)du3) q(u2) q2(u2)du2) q(u1) q1(u1)du1. (ii) Let q1(u) and q2(u) be R -+ Rdxn deterministic functions and 1 be ad-dimensional vector of ones. Then fort~ u2 ~ u1 ~ u ~ T, 2 IE [ ([ ([' q,(u,)' dw.,) q1(u1)'dw.,) 1T q(u1)'dW., ~ x] (2.4.59) 2. Swaption pricing in a log-normal LIBOR market model 51 where 2 a4 ; 21u (1ui (1u q2(u3) 1 q2(u3)du3) q1(u2) 1 q(u2)du2)' q1(u1) 1 q(u1)du1 + ;, 1" (1"' (1"' q,(u,)' q( u,)du,)' q1 ( u,)' q,( u,)du2) q1( u1)' q( u1)du1 1 2 + ; 21u l 1 q1(u1) (1u (1u q2(u3) 1 q(u3)dt3) q(u2) 1 q2(u2)du2) q1(u1) 1 ldu1, and Proof. A direct application of Lemma 2.4.7 gives the results in part (i). For the result in part (ii), using Ito's lemma we express in an semimartingale form, then Lemma 2.4. 7 gives the result. D For notational simplicity let us denote the following: M q(u) = L 'Pi (u, v(t)), i=n and M 8qi(u) = L O<{)k,i(u, v(t)). k=n Using Lemma 2.4.8, we obtain the following conditional expectations. Lemma 2.4.9. There exist constants c1, Ji, h, h, d1 and d2 such that (2.4.60) IE (g3lg1 = x) = Ji (x 3 - 3Ex) + hEx + h (x 3 - 3Ex), (2.4.61) 2. Swaption pricing in a log-normal LIBOR market model 52 and (2.4.62) where Proof. A direct application of Lemma 2.4.8 to the stochastic integrals 91, 92 , and 93 gives the results immediately. D Then, using the conditional expectations in Lemma 2.4.9, we obtain an asymptotic expansion for the density function of xi:ivf(Tn)- 2. Swaption pricing in a log-normal LIBOR market model 53 Theorem 2.4.10. An asymptotic expansion for the density function of x~:h(Tn) is, for € - 0, f1;\x) = 2 4 2 2 2 +c /i; h (x - 6Ex + 3E2) ¢E(x) + c h (x - E) Proof. Let 'l/Jx(t) be a characteristic function of x~:h(Tn)- Then, 'l/Jx(t) ]E [eitX;,:~(Tn)] 9 2 2 3 lE [eit i ( 1 + lE [ it (c92 + c 93) + ~ (it)2 E 9~ + 0 (c ) I 91])] lE [ it9 i ( 1 + itcc1 (gi - E) + itc2 (!1 (gr - 3Eg1) + /2Eg1 + h (gr - 3Eg1)) +~ (it)2 c2 (ci (gt - 6giE + 3E2) + d1E (gi - E) + d2E2) + 0 (c3, g1))] 00 2 2 3 3 1: [eitx ( 1 + itcc1 (x - E) + itc (!1 (x - 3Ex) + hEx + h (x - 3Ex)) +~ (it)2 c2 (ci (x4 - 6x2E + 3E2) + d1E (x2 - E) + d2E2) + 0 (c3, x))] 1:00 eitx f1;\x)dx. The last equality follows by the integration-by-parts formula. Therefore, x~:h(Tn) has a density J1;\x). D Let us define a function G(x) so that 9o ) e-g5/2x G(x) = goN ( ../x +x -/2irx' Then, the first and the second derivatives of G(x) are ' 1 e-g5/2x G (x) = 2 v2n, 2. Swaption pricing in a log-normal LIBOR market model 54 and II 95 - X e-g5/2x G (x) = -42 v'2n. X 21fX Theorem 2.4.11. An asymptotic price at time t E [O, Tn] of a European payer swaption with strike K, 7 maturity Tn and accrual period M - n + l is, for c-+ 0, PS(e)(t) G (c2E) - 2c2c1Eg0c' (c2E) + 2c4ciE2g5G 11 (c2E) <5Bn,M(t) +c2E (c2g6 + c2c3E) G 1 (c2E) + · · · , (2.4.64) where and In particular, a price of an ATM European payer swaption, that is, when Sn,M(t) = K,, is (2.4.65) Proof. From equation (2.4.45), the asymptotic price of a payer swaption at t follows PS(e)(t) <5Bn,M(t) where a€ = go/c. Then, after some calculations, we achieve formula (2.4.64) D 2. Swaption pricing in a log-normal LIBOR market model 55 When E = 0, formula (2.4.64) reduces to p5(o)(t) + ( ) = (Sn,M(t) - K,) . Bn,M t When n = M the formulae in Theorem 2.4.11 reduces to an asymptotic formula for a caplet. Corollary 2.4.12. An asymptotic price at time t E [O, Tn] of a caplet with strike K, and maturity Tn is, for E ---+ 0, Cpl~e:\t) G (E2E) - 2E2c1EgoG' (E2E) + 2E4ciE2g5G" (E2E) <5Bn+l (t) +E2E (c2g5 + E2c3E) a' (E2E) + ... ' (2.4.66) where In particular, the price of an ATM caplet, that is, when Ln(t) = K,, is (2.4.67) 2.4.3 Approximate formulae In practice, we are interested in a situation when E > 0, so if we substitute (2.4.68) into formula (2.4.64) and truncate the higher order terms, we obtain an approximate formula for a European payer swaption. 2. Swaption pricing in a log-normal LIBOR market model 56 Theorem 2.4.13. An approximate price at time t E [O, Tn] of a European payer swaption with strike K, maturity Tn and accrual period M - n + l is (2.4.69) where M M Tn A= LL1 ({Ji (u1, v(t)) 1 cp1 (u1, v(t)) du1, i=n j=n t and and the constants c1, c2 and c3 are the same as in Theorem 2.4-11. In particular, when n = M we obtain an approximate formula for a caplet. Corollary 2.4.14. An approximate price at time t E [O, Tn] of a caplet with strike K and maturity Tn is Cpln(t) ~ G (A) - 2c1AgoG 1 (A)+ 2ci A2g5c'' (A) 6Bn+1(t) +A (c2g5 + c3A) c' (A), (2.4.70) where and the constants c1 , c2 and c3 are the same as in Corollary 2.4.12. 2. Swaption pricing in a log-normal LIBOR market model 57 We note, however, that formula (2.4. 70) only gives an approximate caplet price even though there is a closed form formula. We consider the following transformation so that when n = M the formula gives an exact caplet price. It is quite surprising that the numerical results discussed later show that the transformation improves the accuracy of the formula even when n =IM. Formula (2.4.69) may be approximated as PS(t) :::::l G (X) , (2.4.71) oBn,M(t) where On the other hand, if we apply the asymptotic method to the Black swap model, we obtain an approximate formula for a European payer swaption, (2.4. 72) where - ( )2 2 ( 1 1 2 1 2 ) I;B = Sn,M t CTBTn l - Sn,M(t/0 + 12Sn,M(t)2g0 - 12 CTBTn · The right hand sides of (2.4.71) and (2.4.72) coincide when A= E8 , that is, when ai Sn,Mt)2Tn ( 1+ (sn,:(t) - Zc,) 9o + (c, + 12S::(t)2 - Sn~:\tJ) gJ + (c, + 12S.~(t)2 ) A) + · · Then, we have the following approximate formula for a European payer swaption in the form of the Black swaption formula. 2. Swaption pricing in a log-normal LIBOR market model 58 Theorem 2.4.15. An approximate price at time t E [O, Tn] of a European payer swaption with strike K,, maturity Tn and accrual period M - n + 1 is PS(t) ( ) n,M(t) ~ Sn,M(t)N (h) - K,N h - CYB../T:, , (2.4.73) where h _ ln (Sn,M(t)/K,) + ½CY1Tn - CY3ffn ' and u1 Sn,Mt)'Tn ( 1+ (s.,:(t) -2c,) 9a + ( c, + 12s::(t )' - s.~:\ t)) 9g+ ( c, + 12s.~(t)') 0, When n = M, formula (2.4.73) reduces to a closed form caplet formula. It is quite interesting to note that if we use only A/ (Sn,M(t)2Tn) as CY1, we obtain an approximate formula introduced by Andersen and Andreasen (2000b). Lemma 2.4.16 (Andersen and Andreasen, 2000b). An approximate price at time t E [O, Tn] of a European payer swaption with strike K,, maturity Tn and accrual period M -n+ 1 ZS (2.4.74) where h = In (Sn,M(t)/ K,) + ½CY1/Tn a3ffn ' and A 2. Swaption pricing in a log-normal LIBOR market model 59 2.4.4 An alternative approach Recall that based on the forward LIBOR processes (2.4.41), the dynamics of the processes v?\t) can be written in the following way. For i = n, ... , Mandt::; u::; Tn, (2.4.75) where Using v?'\u), the forward swap rate is M si:~(u) = L v?)(u). (2.4.77) i=n Numerical experiments with a Monte Carlo simulation suggest that, when E is small, the distribution of si:~(Tn) is close to log-normal. Thus, considering a log transformation of si:~ (u) before applying the asymptotic expansion, we might expect a better approximation to a price for a European payer swaption. To make the process si:~(Tn) be close to Gaussian, we consider the following log transformation of si:~(u), and introduce a new state variable xi:~(u) so that (2.4. 78) Let us denote (2.4.79) and !::I .. ( (e)( )) _ 8'1/Ji (u,v(e)(u)) u'l/J,,1 U, V U - (€) (2.4.80) 8vi (u) 2. Swaption pricing in a log-normal LIBOR market model 60 Then, using Ito's lemma, the dynamics of xi:~(u) are (2.4.81) Theorem 2.4.17. The second order asymptotic expansion for xi:~(u) is, for E - 0, (2.4.82) e=O where ax(e) (u) n,M and 2 1 8 x(e)n,M (u) 2 &2 Proof. By a direct differentiation of xi:~( u) with respect to E, we obtain the result. For the details of the proof, we refer to the proof in Theorem 2.4.6. D Let us write xi:~(u) at time u = Tn as, for E - 0, (2.4.83) where 9o = Xn,M(t), 2. Swaption pricing in a log-normal LIBOR market model 61 and Notice that we split the second order term into a deterministic integral g!J and a stochastic integral gf Assuming a non-degeneracy condition that, (2.4.84) 91 is distributed as N (0, E), where M M Tn 1 E =LL1 1Pi (u1, v(t)) 1Pi (u1, v(t)) du 1 . (2.4.85) i=n j=n t Also, we can write g!J = -E/2. Let us define YJ~lt-(Tn) by (2.4.86) Theorem 2.4.18. An asymptotic expansion for the density function of YJ~it-(Tn) is, for E-> 0, (2.4.87) where M M M ) l lTn (lui 1 C E2 ~ ~ t t ~ 1/Jdu2, v(t)) vi(t) (o-i(u2) + 'TJi (u2, v(t))) du2 M 1 L 1Pk (u1, v(t)) 81/Ji,i (u1, v(t)) du 1• (2.4.88) k=n Proof. For the details of the proof, we refer to the proof in Theorem 2.4.10. D 2. Swaption pricing in a log-normal LIBOR market model 62 Using the asymptotic density function f~)(x), an asymptotic price of a European payer swaption follows immediately. Theorem 2.4.19. An asymptotic price at time t E [O, Tn] of a European payer swaption with strike ;,, 1 maturity Tn and accrual period M - n + l is, for c - 0, p5(e:)(t) Sn,M(t)N (h(e:)) - ;,,N ( h(e:) - c~) <5Bn,M(t) +c 2cSn,M(t)E ( -h(0 \p (h(e:)) + 2c~ (h(e:)) + c2EN (h(e:))) + · · · , (2.4.89) where Proof. From equation (2.4.45), the asymptotic price of a European payer swaption at t follows <5Bn,M(t)JEn,M ( (s~:~(Tn) - K,) +I Ft) <5Bn,M(t)JEn,M ( (eX~~'.w(Tn) - K,) +I Ft) <5Bn,M(t)JEn,M ( (exn,M(t)+e:2gf+e:YJ~1(Tn) - K,) +I Ft) <5Bn,M(t)JEn,M ( (sn,M(t)e_e2E/2+eY~~1(Tn) - K,) +I Ft) +00 <5Bn,M(t) 1-a, ( Sn,M(t)e-e:2E/2+e:x - K,) f~\x)dx, (2.4.90) 2 where a0 = ¾ (in Sn,:(o) - c ~). Then, after some calculations, we achieve the result. D In practice, we are interested in a situation when c > 0, so if we substitute (2.4.91) into formula (2.4.89) and truncate the higher order terms, we obtain an approximate formula for a European payer swaption. 2. Swaption pricing in a log-normal LIBOR market model 63 Theorem 2.4.20. An approximate price at time t E [O, Tn] of a European payer swaption with strike ,-.,, maturity Tn and accrual period M - n + 1 is PS(t) ~ Sn,M(t)N (h) - ,-.,N ( h - JA) rSBn,M(t) +cSn,M(t)A ( -hcp(h) + 2VA¢(h) + AN(h)) , (2.4.92) where ln (Sn,M(t)/ ,-.,) + ½A h= vA ' M M T, 1 E =LL 1n 'lpi (u1, v(t)) 'lpj (u1, v(t)) du1, i=n j=n t and the constant c is given by equation (2.4- 88). 2.5 Jarrow and Rudd method 2.5.1 Introduction As an alternative method to obtain an approximate formula for a European payer swaption, we apply a method introduced by Jarrow and Rudd (1982). The method is a technique which approximates a given probability distribution, F(x), called the true distribution, with an alternative distribution, A(x), called the approximating distribution. This kind of method is examined in the context of pricing Asian options and Basket options by Gen tle (1993), Huynh (1994), Milevsky and Posner (1998a, 1998b) and Vorst (1992). Assume 2. Swaption pricing in a log-normal LIBOR market model 64 that the distributions are continuous and their density functions exist, and denote them by dF(x)/dx = f(x) and dA(x)/dx = a(x). Let us employ the following notation: +00 1-oo xi f(x)dx, +00 1-oo (x - a1(F)/ f(x)dx, (2.5.93) +00 (F, t) 1-oo eitx f(x)dx, where i 2 = -1, ai(F) is the jth moment of distribution F, µi(F) is the jth central moment of distribution F, and (F, t) is the characteristic function of F. We assume that ai(F) exists for j ::; n. Given etn(F) exists, the first n - l cumulants (or semi-invariants) Ki(F) from j 1, ... , n - l also exist. These are defined by, for t - 0, log(F,t) = I:Kj(F)(i~t +o(tn-l). (2.5.94) i=l J. For reference, the first four cumulants are K1(F) = et1(F), K2(F) = µ2(F), K3(F) = µ3(F), K4(F) = µ4(F) - 3µ 2 (F)2 . (2.5.95) Analogous notation will be employed for the moments, cumulants and characteristic function of A, i.e., ai(A), µi(A), Ki(A), and (A, t). It is assumed that both ai(A) and di A(x)/dxi exist for j ::; m (where m can differ from n). The following series expansion for f(x) in terms of a(x) is given by Jarrow and Rudd (1982). Theorem 2.5.1 (Jarrow, Rudd, 1982). Letting N = inf(n, M), a series expansion for f(x) in terms of a(x) is N-1 · · (-1)1 d3 a(x) f(x) = a(x) + L Ei--,--di + E(x, N), (2.5.96) . 1 J. X J= 2. Swaption pricing in a log-normal LIBOR market model 65 where Ej, j = 0, ... , N - 1 are defined by N-l (it)j) N-l (it)j N l exp ( L (Kj(F) - Kj(A)) -.-, = L Ej-.-, + o (t - ) . j=l J j=O J The residual error, c(x, N), contains any remaining difference between the left-hand and right-hand sides of (2.5.96). Given arbitrary true and approximating distributions (where some moments may not exist), no general analytic bounds for this error E(x, N) as a function of N (the number of terms included) are available. Thus, in general no term is negligible compared with a preceding term. In the statistics literature, this technique is called the Edgeworth series expansion (see Kendall and Stuart (1977) and Kolassa (1997)). In general, the Edgeworth series expansions concentrate on using the standard normal as the approximating distribution, but Theorem 2.5.1 treats any arbitrary distribution A(x). For reference, the first four coefficients Ej are Eo 1, (2.5.97) E3 K3(F) - K3(A) + 3E1 (K2(F) - K2(A)) + Ef, E4 K4(F) - K4(A) + 4 (K3(F) - K3(A)) E1 + 3 (K2(F) - K2(A))2 +6E~ (K2(F) - K2(A)) +Et. (2.5.98) In particular, when K 1(A) = K 1(F), that is, their first moments are the same, equa tion (2.5.96) can be written as f(x) = (2.5.99) 2. Swaption pricing in a log-normal LIBOR market model 66 2.5.2 An approximate formula For our problem, the true distribution, F(x), is the conditional distribution of the random variable Sn,M(Tn) given vn(t), ... , VM(t). Except the first moment, all other higher mo ments of distribution F(x) are not easily available, but for the validity of the method we need the existence of moments of F(x). Lemma 2.5.2. All moments of distribution F(x) under the forward swap measure Jpm,M exist, that is, JEn,M (Sn,M(Tn)PI.Ft) < 00, for P = l, 2, · · · · (2.5.100) Proof. E"•M ( (t. v;(Tn)r F,) Jl",M ( (t.°'+1(T.)L;(Tn)r F,). Since O < ai+1(Tn) < 1, i = n, ... , M, (2.5.101) Recall that the dynamics of the processes Li (u), t :::; u :::; Tn, under ]pm,M are dLi(u) ( )' ( ) nM) Li(u) = ,\ u -r,i(u du+ dWu' , i=n, ... ,M, (2.5.102) where (2.5.103) Setting Amax= SUPi,u l..\i(u)I, 'r/min,i < 'r/i(u) < 'r/max,i, i = n, ... , M, 2. Swaption pricing in a log-normal LIBOR market model 67 where M 77min,i = -Amax I:(M - j) and 77max,i = (M - i)Amax· j=n Therefore Lmin,i(u) < Li(u) < Lmax,i(u), i = n, ... , M, where and Thus, the processes Li (u) are bounded from below and above by the log-normal processes. Therefore, It follows that Sn,M(Tn) has all moments under the forward swap measure JPn,M_ D Since Sn,M(Tn) is a JPn,M martingale, the first moment a 1(F) is Sn,M(t). Because the higher moments are not easily available, we only assume that an approximating distribution A(x) has the first moment a 1(A) = Sn,M(t), and all higher moments exist. A true price of a European payer swaption, PSF(t), is expanded in terms of an approximating price of a European payer swaption, PSA(t), and approximating density function a(x). Lemma 2.5.3. A true price at time t E [O, Tn] of a European payer swaption with strike 2. Swaption pricing in a log-normal LIBOR market model 68 "', maturity Tn and accrual period M - n + l is PSp(t) = (2.5.105) where PSA(t) = OBn,M(t) 1:= (x - "'t a(x)dx. Proof. From equation (2.3.27), oBn,M(t)En,M ( (Sn,M(Tn) - K,tl Ft) OBn,M(t) 1:= (x - "'t f(x)dx. Substitution of equation (2.5.99) gives the result immediately. D Numerical experiments with a Monte Carlo simulation suggest that the conditional dis tribution of the forward swap rate Sn,M(Tn) given Sn,M(t) is close to log-normal. Thus, a possible candidate for an approximating distribution A(x) is a log-normal distribution with its first moment o:1 (A) = Sn,M (t). To completely specify the approximating log-normal distribution A(x), we need to determine the second moment a2(A). If A(x) is log-normally distributed with o: 1(A) = Sn,M(t) and second moment o:2(A), equation (2.5.105) simplifies to the following equation. Lemma 2.5.4. A true price at time t E [O, Tn] of a European payer swaption with strike "', maturity Tn and accrual period M - n + l is PSp(t) = PS ( ) oB ( ) K2(F) - K2(A) ( ) _ oB ( ) K3(F) - K3(A) da(x) I A t + n,M t 2! a K, n,M t 3! dx x=K- OB ( ) K4 (F) - K4 (A) + 3 (K2(F) - K2(A)) d2a(x) I (K N) + n,M t 4! dx2 + c ' ' X=K (2.5.106) 2. Swaption pricing in a log-normal LIBOR market model 69 where PSA(t) = 6Bn,M(t) ( Sn,M(t)N (h) - KN ( h - JA)) , (2.5.107) where and Proof. Using the integration-by-parts formula, for j 2'. 2, r+oo (x - Kt dia(x) dx 1-oo dx1 +00 dia(x) 1 (x - K) --.-dx "' dx1 di-2a(x) I dxi-2 . X=K Also, a standard calculation yields equation (2.5.106). D Thus, our goal is to determine the best second moment a 2 (A) of A such that the ap proximating log-normal distribution A is the 'closest' to the true distribution F in a sense that fewer terms in the right-hand side of equation (2.5.106) give a good approximation to the true price of European payer swaptions. From equation (2.3.28), M Sn,M(Tn) = LVi(Tn), (2.5.108) i=n where processes vi(u), t ~ u ~ Tn, are i=n, ... ,M, (2.5.109) 2. Swaption pricing in a log-normal LIBOR market model 70 and M M M 8vj(u) 8vk(u) 'T/i (u, Vn+i(u), ... , VM(u)) = '°'~ -(-) \(u) - '°'~ '°'~ aj+1(u)-(-) ,\k(u). a· u ak u j=i+l 1 j=n k=j+l Since each 'T/i(u) is a complicated stochastic process, the second moment of Sn,M(Tn) given Sn,M(t) is not easily available. The process Sn,M(u) can be written as M dSn,M(u) = L dvi(u) i=n M L vi(u) (,\i(u) + 'TJi(u))' dw;,M i=n (2.5.110) As stated before, Sn,M(u) is close to a log-normal process, thus it implies that the process M vi(u) M M M vi(t) _ Using the approximated process for as(u), the second moment a 2 (A) is given by a2(A) JEn,M (Sn,M(Tn)21Ft) 2 (~ ~ Vi(t)vj(t) lTn ( ( ) _( ))' ( ( ) _ ) Sn,M(t) exp ~ f=:, Sn,M(t) 2 t Ai u + T/i u Aj u + T/j(u)) du . (2.5.115) Therefore, assuming that the approximating distribution A is log-normally distributed with a 1(A) = Sn,M(t) and a 2 (A) as (2.5.115), Lemma 2.5.4 leads to the following result. Theorem 2.5.5. A price at time t E [O, Tn] of a European payer swaption with strike K-, maturity Tn and accrual period M - n + 1 is PSp(t) = PS ( ) &B ( ) K2(F) - K2(A) ( ) _ &B ( ) K3 (F) - K 3 (A) da(x) I A t + n,M t 2! a K- n,M t 3! dx x=K.. 2 &B ( ) K 4 (F) - K 4 (A) + 3 (K2(F) - K2(A)) d2a(x) I (K N) + n,M t 4! d 2 + c ' ' X x=I'. (2.5.116) where PSA(t) = &Bn,M(t) ( Sn,M(t)N (h) - K-N ( h - JA)) ) (2.5.117) ln (Sn,M(t)/K-) + ½A h= ../A ) and 2.6 Monte Carlo simulation This section proposes an efficient Monte Carlo simulation method for pricing European payer swaptions in the log-normal LIBOR market model. A variance reduction method us ing a control variate is proposed to speed up the computation. We first explain how to find 2. Swaption pricing in a log-normal LIBOR market model 72 the swaption prices using a standard simulation method and Glasserman and Zhao's (2000) discretization method under the terminal measure. Then we present our discretization method under the forward swap measure. 2. 6.1 Monte Carlo under the terminal measure The swaption valuation formula (2.2.1) at t = 0 can be written under the forward measure jp>M+l as PS(O) (2.6.118) with Cn = 1, cj = -K,O, j = n + 1, ... , M, and CM+l = -(1 + K,0). With a standard Monte Carlo simulation method to evaluate expectation (2.6.118), we discretize the forward LIBOR rate processes ( ) _ ~ o>.j(t)' Lj(t) , M+1 dLi t - - j~l l + oLj(t) >.i(t)Li(t)dt + >.i(t) Li(t)dWt , i = n, ... , M, (2.6.119) to obtain an estimator Li(Tn) of Li(Tn). To discretize each process, we consider, for example, an Euler scheme for log Li(t) with even spacing tk+l - tk = fl.t, k = 0, 1, ... , t0 = 0, that is, Li(tk+l) = Li(tk) exp ( >.i(tk) 1 Li(tk) ( (P,i(tk) - }>-i(tk)Li(tk)) fl.t + vAflk+l)) , (2.6.120) with 2. Swaption pricing in a log-normal LIBOR market model 73 and 6, 6, ... , independent standard normal d-dimensional vectors. Having obtained an estimator .t(Tn), using (2.6.118), we obtain an estimator PS(O) of PS(0) by (2.6.121) Suppose an i.i.d. sequence .?S1(0), .?S2 (0), ... is generated by the standard Monte Carlo simulation. A natural estimator of PS(0) with NoSim replications is then the sample mean _ l NoSim A PS(0) = NoSim ~ PSm(0). (2.6.122) It can be shown that PS(0) converges to the true value PS(0) as b.t - 0 and NoSim - oo. For a fixed b.t > 0, however, Li(Tn) is biased in general, thus PS(0) does not converge to PS(0) as NoSim - oo. In addition, Glasserman and Zhao (2000) pointed out that the discretization method (2.6.120) does not allow the discretized discounted bond price processes (2.6.123) to be martingales. They introduced a new discretization method which allows the dis cretized discounted bond price processes to be martingales, and, more importantly, pro duces smaller bias than the standard method. Their idea is not to discretize the forward LIBOR rate processes directly, but to discretize the differences of discounted bond price processes. Their method is summarized in the next theorem. Theorem 2.6.1 (Glasserman and Zhao, 2000). In the log-normal LIBOR market model, define new processes Xi(t), i = l, ... , M, by M 1 Xi(t) = Li(t) IT (1 + 8Li(t)) = 6 (Fi(t) - Fi+1(t)), i = n, ... , M, (2.6.124) j=i+l 2. Swaption pricing in a log-normal LIBOR market model 74 where Bi(t) F;(t) = B ( )' M+l t Then Xi(t), i = n, ... , M, follow under the forward measure pM+l (2.6.125) Using Xi(t), Li(t), i = n, ... , M, are expressed as L-(t) _ Xi(t) (2.6.126) ' - 1 + 6Xi+1(t) + · · · + 6XM(t). The discretization of log Xi(t) produces positive martingale processes Xi(t) and thus positive processes L; (t). The discretized discounted bond price processes (2.6.127) are positive martingales. Glasserman and Zhao (2000) provided numerical evidence that the bias with this dis cretization scheme is smaller than that with the standard discredization method by com paring caplet prices. 2.6.2 Monte Carlo under the forward swap measure So far, we have discussed Monte Carlo simulation methods under the terminal measure to find European payer swaption prices. We now examine a Monte Carlo simulation method under the forward swap measure. At the same time, we present a variance reduction method based on a control variate technique for our simulation to speed up the calculation. Recall the European payer swaption price under the forward swap measure is (2.6.128) 2. Swaption pricing in a log-normal LIBOR market model 75 where M Sn,M(Tn) = ~ Vi(Tn). (2.6.129) i=n The processes vi (t) follow (2.6.130) From this expression, it is more natural to descretize the processes vi(t). Then an Euler scheme for logvi(t) with even spacing tk+l - tk = !:lt, k = 0, 1, ... , is i\(tk+1) = vi(tk) exp ( (Ai(tk) + iji(tk)) (-~ (Ai(tk) + iji(tk)) !:lt + JM~k+l)) , (2.6.131) where (2.6.132) and 6, 6, . . . are independent standard normal d-dimensional vectors. Since (2.6.133) the coefficient of each process vi(t) is linearly bounded and Lipschitz. Thus, vi converges weakly to vi as L:lt - 0. Our discretization method is similar to the discretization method in Theorem 2.6.1 developed by Glasserman and Zhao (2000). Each vi(t) can be written as 1 vi(t) = J (ai(t) - ai+1(t)), where the discounted bond price ai+l (t) is Bi+1(t) ai+1(t) = B (t). n,M This expression of vi(t) reminds us of equation (2.6.124), in which Xi(t) is given by the difference of discounted bond prices divided by 8. Furthermore, from the relations 2. Swaption pricing in a log-normal LIBOR market model 76 each ai+i(tk) is a linear combination of the processes vi(tk)- It follows that the discretized discounted bond price processes are all martingales since vi(tk) are martingales. We sum marize our discretization method in the following theorem. Theorem 2.6.2. With the discretization method (2.6.131), we obtain an estimatorSn,M(Tn) M Sn,M(Tn) = I: vi(Tn), (2.6.134) i=n and then an estimator PS(O) of PS(O) by PS(O) = oBn,M(O) ( Sn,M(Tn) - K,) +. (2.6.135) Suppose we generate an i.i.d. sequence ?S1 (0), ?S2 (0), .... An estimator of PS(O) with N oSim replications, _ l NoSim A PS(O) = N PSm(O), (2.6.136) os· zm L m=l converges to the true value PS(O) as b..t - 0 and NoSim - oo. Furthermore, the dis cretized discounted bond price processes cii+ 1 (tk) are all martingales. Sn,M(Tn) also remains a martingale. To implement Monte Carlo simulations efficiently, there are many variance reduction techniques; see for example Boyle, Broadie and Glasserman (1997). Among these tech niques, a control variate is one of the easiest and the most effective variance reduction A C methods. In our problem, as b..t - 0 and NoSim - oo, an unbiased estimator PS (0) of PS(O) is given by PS\o) = PS(O) - x + Cx, (2.6.137) where X is an unbiased estimator of a certain random variable X and Cx is the expected value of X. When PS(O) and X are highly positively correlated, we expect large variance 2. Swaption pricing in a log-normal LIBOR market model 77 reduction in PS\O). More generally, we may choose a fJ so that the variance of the right hand side of PSc(O) = PS(O) - f3 ( X - Cx) (2.6.138) is minimized. For this thesis, we fix f3 = 1. For the log-normal LIBOR market model, we may incorporate a control in the following way. The idea comes from the approximation considered in Section 2.5. Monte Carlo with a control variate: Consider a process M dwn,M(t) ~ Vi(O) ( ( ) - ( ))' n,M (2.6.139) Wn,M(t) ~ Sn,M(O) ,\; t + T/i t dWt , Wn,M(O) Sn,M(O), (2.6.140) where Then, Cx OBn,M(O)JEn,M (wn,M(Tn) - t;;t oBn,M(O) ( Sn,M(O)N (h) - t;;N (h - JA)), (2.6.141) where h = In (Sn,M(O)/ t;;) + ½A v'A ' and ~ ~ vi(O)vj(O) {Tn - I - A=~ f;;:, Sn,M(0) 2 Jo (,\i(u) + T/i(u)) (,\j(u) + T/j(u)) du. An unbiased estimator PSc(O) of PS(O) is obtained by (2.6.142) 2. Swaption pricing in a log-normal LIBOR market model 78 Notice that Cx is nothing but the approximate formula (2.5.117) at time t = 0. The next section on numerical results shows that this control variate works very well. Remark: It is not necessary to choose a control variate so that the underlying formalism is the same. The efficiency of a control variate is only influenced by the correlation level with the pricing variable and by the cost of its computation. Thus, a control variate built with another formalism can be equally effective. In fact, a control variate does not need to be a valid financial price at all. It is possible to violate arbitrage and still have a very effective control variate. 2. 7 Numerical results This section gives various numerical results comparing the Monte Carlo simulation methods and the approximate formulae for European payer swaption prices. The system we used throughout for our numerical experiments is a Silicon Graphics Origin2000 Server with eight 250 MHz MIPS RlO000 processors each with 4 Mb cache, 6 Gb RAM and 54 Gb of user disk. The programs are all written in C++, using the MIPSpro compiler. All codes were executed on a single processor. For matrix manipulation and random number generation, we used the free libraries developed by Robert Davies which can be obtained from http://webnz.com/robert/. We first demonstrate the efficiency of our simulation method (2.6.142). Using the control variate dramatically speeds up finding accurate swaption prices. We then examine the accuracy of our approximate swaption formulae (2.4.69) and (2.4.73) using simulation prices as benchmarks. We find that our approximate formulae provide more accuracy than other approximate formulae by Brace, Dun and Barton (1998), Brace, Gatarek and Musiela (1997) and Andersen and Andreasen (2000a). 2. Swaption pricing in a log-normal LIBOR market model 79 2. 7.1 Details of numerical examples Before presenting the numerical results, we provide several parameters for the numerical experiments. We set o = 0.5 and N + l = 40, corresponding to a twenty-year term structure of semiannual rates. Partially referring to Glasserman and Zhao (2000), for the log-normal LIBOR market model, we consider the following four scenarios: • Scenario A(LN): Li(0) 0.06, for i = 0, ... , N, Ai(u) 0.2, for i = 0, ... , N and OS u ST;. • Scenario B(LN): Li(0) log(a + bi), for i = 0, ... , N, so that L0 (0) = 0.05 and £ 39 (0) = 0.09, 0.17 + 0.002(i - j), j = 0, ... , i - 1, i = 1, ... , 39, and constant over the interval [Tj, Ti+ 1 ). • Scenario C(LN): log(a + bi), for i = 0, ... , N, so that £ 0 (0) = 0.05 and £ 39 (0) = 0.09, 0.25 - 0.002(i - j), j = 0, ... , i - 1, i = 1, ... , 39, and constant over the interval [Tj, Ti+1 ). • Scenario D(LN): log(a + bi), for i = 0, ... , N, so that £ 0 (0) = 0.05 and £ 39 (0) = 0.09, ( 0.22 - O.OOl(i - j), JO.OOl(i - j) - 0.1), j = 0, ... , i - 1, i = 1, ... , 39, and constant over the interval [Ti, Ti+1 ). Each volatility structure implies that the volatility depends only on the time to maturity of the forward LIBOR rates. Since for each scenario there exists a constant K < l such 2. Swaption pricing in a log-normal LIBOR market model 80 that sup (l>-n(u1)1 2 , ... , i>.M(u1)12) < K 2 , E < K < 1. An at-the-money (ATM) swaption has a strike rate K = Sn,M(0), an in-the-money (ITM) swaption has K < Sn,M(0), and an out-of-the-money (OTM) swaption has K > Sn,M(0). An xx y has an option maturity of x years and a swap length of y years. All swaption prices are reported in basis points, that is, the true value x 10000. 2. 7.2 Comparison of Monte Carlo methods We compare three simulation methods to compute European payer swaption prices: • Method I: Glasserman and Zhao's discretization method described in equation (2.6.121) and Theorem 2.6.1. • Method II: our discretization method described in Theorem 2.6.2. • Method III: Method II plus the use of the control variate described in equation (2.6.142). As Glasserman and Zhao (2000) has shown, the standard discretization method (2.6.120) is more biased than Method I, thus it is not examined in our experiments. First of all, we examine a one factor model using Scenario B(LN). Choosing flt = 0.5 (= <5) as a time step for the discretization, Figure 2.7.1 plots simulated 5 x 5 ATM European payer swaption prices against the number of simulations. We see that the plot for Method III is almost flat, showing that Method III, using the control variate, produces much more rapid convergence than the others that do not use the control variate. It also shows that swaption prices with Method I and Method II converge in a very similar way. Table 2.7.1 shows simulated swaption prices and their standard deviations for each method by increasing the number of simulations. It is obvious that the standard deviations when using the control (Method III) are significantly smaller than those of without the control (Method I and Method II). The standard deviation in Method I or Method II after 1 million simulations is still higher than that in Method III after 10,000 simulations. As 2. Swaption pricing in a log-normal LIBOR market model 81 expected the rate of convergence of the simulation is inversely proportional to the square root of the number of simulations, that is, increasing the number of simulations 100 times decreases the standard deviation by one tenth. With the fixed number of Monte Carlo paths, Method III gives much smaller standard deviations than Method I and Method II. However, since using the control variate introduces additional computational operations, it might not be so efficient in terms of the CPU time. Table 2. 7.2 compares CPU times by computing 5 x 5 and 10 x 10 swaption prices using each method. The number of Monte Carlo paths is fixed with 10,000. Comparing Method II and Method III, we find that the CPU time from additional computational operations when using the control variate is negligible. Although Method I computes the swaption prices slightly faster, considering the size of its standard deviations in Table 2.7.1, Method III would be the most efficient to compute accurate swaption prices. 380 375 - Method! 370 Method II - - Methodlll .g" 365 Q. C: .Q 360 ~ ~ 355 "C .;" 350 - - '3 -- E 340 335 330~~~-~--~-~--~-~-~--~-~-~ 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 Number of simulations Fig. 2. 7.1: Comparison of the Monte Carlo simulation methods for 5 x 5 ATM European payer swaption prices with Scenario B(LN). As another variance reduction technique for Monte Carlo simulations, antithetic sam pling is very popular because of its simplicity of the implementation. For comparison, 500,000 simulations with an antithetic sampling (for a total of 1,000,000 paths) using the three methods give Method I: 349.71(0.66), Method II: 349.70(0.56), Method III: 2. Swaption pricing in a log-normal LIBOR market model 82 349.845(0.007). Comparing these standard deviations with those in Table 2.7.1, we find that in our problem antithetic sampling is not as powerful in reducing the variance as the control variate method. Thus we will not use an antithetic sampling for further simulations. Observing that Method III converges very fast, Table 2.7.3 shows the effect of taking a finer discretization. Using Method III, one factor 5 x 5 ATM (K; = 0.0648) and 1 x 9 OTM (K; = 0.0801) European payer swaptions are computed with 1,000,000 simulations by taking D.t=0.5, 0.25, 0.125 and 0.00625, which correspond to Dot= fJ, fJ/2, fJ/4 and fJ/8 respectively. We find that the standard deviation does not decrease by taking a finer discretization. In general, taking a finer discretization has an effect of reducing a bias. For the 5 x 5 ATM swaption, it seems that even taking Dot = 0.5 the simulation price is not biased so much since taking much finer discretization does not change the simulation price. However, for the 1 x 9 OTM swaption, we observe that the simulation with Dot = 0.5 produces a significant bias. Taking a finer time step produces a less biased value. Table 2.7.4 shows various swaption prices including ITM and OTM swaptions. Using the control is effective in all cases. However, we note that the values of standard deviations in Method III do not so much depend on the values of strikes as those in Method I or Method II. Another interesting finding in Table 2. 7.4 is that it appears that Method I is more biased than Method II and Method III since some values are deviating from the values obtained from the other two methods. As a result, in a one factor model, we find that Method III is favorable in terms of both faster convergence and lower bias. Secondly, we present a comparison of the Mont Carlo simulation methods in a two factor model using Scenario D(LN). Table 2.7.5 displays the simulation results. We find similar profiles with the results for the one factor model. That is, Method III computes simulation prices with faster convergence and lower bias. Consequently, Method III is a much more efficient Monte Carlo method to obtain accurate prices than the other two simulation methods. That is, Method III produces 2. Swaption pricing in a log-normal LIBOR market model 83 accurate simulation prices for swaptions even with the smaller number of simulations and the coarser time step for discretization. No. of sim. Method I Method II Method III 10,000 353.22 (7.52) 352.48 (6.71) 349.794 (0.056) 100,000 351.16 (2.38) 350.57 (2.12) 349.813 (0.018) 1,000,000 349.74 (0.75) 349.75 (0.66) 349.844 (0.006) Tab. 2. 7.1: Comparison of the Monte Carlo simulation methods for 5 x 5 ATM European payer swaption prices with Scenario B(LN). We set flt= 0.5. CPU time (s) Swaption Method I Method II Method III 5 X 5 2.12 2.55 2.61 10 X 10 6.59 10.56 10.72 Tab. 2. 7.2: Comparison of CPU times for the Monte Carlo simulation methods for ATM European payer swaption prices with Scenario B(LN). The Monte Carlo is based on 10,000 paths with flt = 0.5. b.t Swaption 0.5 0.25 0.125 0.0625 5 x 5 (ATM) 349.844 (0.006) 349.851 (0.006) 349.847 (0.006) 349.850 (0.006) 1 x 9 (OTM) 22.684 (0.001) 22.515 (0.002) 22.431 (0.002) 22.391 (0.003) Tab. 2. 7.3: Effect of taking a finer time step for discretization using Method III for payer Euro pean swaption prices with Scenario B(LN). The Monte Carlo is based on 1,000,000 paths. 2. Swaption pricing in a log-normal LIBOR market model 84 Option maturity x Swap length Strike Method I Method II Method III 0.0401 (ITM) 1324.91 (0.83) 1324.93 (0.74) 1324.691 (0.003) 1 X 9 0.0601 (ATM) 294.86 (0.52) 294.92 (0.48) 295.158 (0.003) 0.0801 (OTM) 21.94 (0.15) 22.24 (0.13) 22.391 (0.003) 0.0425 (ITM) 1010.93 (1.08) 1010.90 (0.95) 1011.540 (0.007) 3 X 7 0.0625 (ATM) 389.00 (0.78) 388.94 (0.69) 389.460 (0.007) 0.0825 (OTM) 120.81 (0.47) 121.01 (0.41) 121.243 (0.007) 0.0448 (ITM) 721.19 (0.95) 721.37 (0.84) 722.304 (0.006) 5 X 5 0.0648 (ATM) 348.81 (0.75) 348.98 (0.66) 349.850 (0.006) 0.0848 (OTM) 157.09 (0.54) 157.29 (0.47) 158.022 (0.006) 0.0525 (ITM) 1675.6 (4.9) 1675.1 (2.1) 1675.25 (0.15) 5 X 15 0.0725 (ATM) 892.7 (4.1) 892.7 (1.6) 893.33 (0.15) 0.0925 (OTM) 446.2 (3.4) 446.6 (1.2) 447.31 (0.14) 0.0591 (ITM) 1099.8 (4.9) 1114.2 (1.8) 1116.05 (0.19) 10 X 10 0.0791 (ATM) 745.7 (4.5) 759.2 (1.6) 761.09 (0.19) 0.0991 (OTM) 504.9 (4.1) 517.7 (1.4) 519.56 (0.19) Tab. 2. 7.4: Comparison of the Monte Carlo simulation methods for European payer swaption prices with Scenario B(LN). The Monte Carlo is based on 1,000,000 paths with t::i..t = 0.0625. 2. Swaption pricing in a log-normal LIBOR market model 85 Option maturity x Swap length Strike Method I Method II Method III 0.0401 (ITM) 1327.08 (0.92) 1327.13 (0.82) 1327.865 (0.005) 1 X 9 0.0601 (ATM) 327.27 (0.59) 327.27 (0.54) 327.988 (0.005) 0.0801 (OTM) 35.42 (0.20) 35.79 (0.19) 35.878 (0.004) 0.0425 (ITM) 1029.48 (1.21) 1029.40 (1.05) 1029.766 (0.009) 3 X 7 0.0625 (ATM) 431.41 (0.91) 431.42 (0.79) 431.825 (0.008) 0.0825 (OTM) 156.51 (0.59) 156.71 (0.51) 156.983 (0.008) 0.0448 (ITM) 744.68 (1.09) 744.32 (0.94) 744.305 (0.007) 5 X 5 0.0648 (ATM) 387.92 (0.89) 387.62 (0. 76) 387.526 (0.007) 0.0848 (OTM) 194.53 (0.68) 194.36 (0.58) 194.268 (0.007) 0.0525 (ITM) 1668.5 (4.6) 1662.9 (2.1) 1663.06 (0.11) 5 X 15 0.0725 (ATM) 886.7 (3.9) 882.0 (1.7) 882.02 (0.11) 0.0925 (OTM) 447.5 (3.2) 444.1 (1.3) 443.80 (0.10) 0.0591 (ITM) 1096.4 (5.4) 1101.0 (1.8) 1103.80 (0.13) 10 X 10 0.0791 (ATM) 741.1 (5.0) 745.5 (1.6) 747.85 (0.13) 0.0991 (OTM) 502.3 (4.6) 506.5 (1.4) 508.45 (0.13) Tab. 2. 7.5: Comparison of the Monte Carlo simulation methods for European payer swaption prices with Scenario D(LN). The Monte Carlo is based on 1,000,000 paths with flt= 0.0625. 2. Swaption pricing in a log-normal LIBOR market model 86 2. 7.3 Comparison of approximate formulae In this section, we compare five approximate formulae for European payer swaption prices: • Fl: the approximate formula (2.2.4) introduced by Brace, Dun and Barton (1998). • F2: the approximate formula (2.4.74) introduced by Andersen and Andreasen (2000a). • F3: the approximate formula (2.2.3) derived by Brace, Gatarek and Musiela (1997). • F4: our approximate formula (2.4.69). • F5: our approximate formula (2.4.73). Although a closed form solution for European payer swaptions is not available, the previous section shows that the Monte Carlo simulation with the control variate gives an accurate price quite efficiently. Thus we use a Monte Carlo price as a benchmark to compare the accuracy of the approximate formulae. Table 2.7.6 and Table 2.7.8 compare the accuracy of approximate formulae Fl-F5. The error is the price from an approximate formula minus the Monte Carlo price. Monte Carlo prices are shown with their standard deviations. We set a time step for simulation b.t = 0.0625(= 6/8) and simulate 1,000,000 paths. For most of the swaptions, formula F5 is the most accurate among those five formulae, then formula F3 and F4 come next, then formula F2, and then formula Fl. For formula Fl and F2, their errors tend to increase as the strike rate increases or either maturity or swap length of the swaptions increases. Especially, for the 5 x 15 and 10 x 10 OTM swaptions, they produce significant errors. We note that when both initial term structure and volatility is flat (Scenario A(LN)), they produces the same errors since formula F2 reduces to formula Fl. However, when initial term structure and volatility are not flat, formula F2 produces more accurate prices than formula Fl. Formula F3 is reasonably accurate for most of the swaptions. Comparing our formulae F4 and F5, formula F4 is very accurate for the ATM swaptions, and often more accurate than formula F5. Both 2. Swaption pricing in a log-normal LIBOR market model 87 formulae tend to be less accurate when the strike rate is away from the ATM strike rate or when either the maturity or the swap length of the swaptions increases. However, for the non-ATM swaptions, formula F4 produces more errors than formula F5. In view of equation (2.4.65), formula F4 for ATM swaptions is very simple and the dependence of E is very clear. On the other hand, equation (2.4.65) shows that the dependence of Eon the formula for non-ATM swaptions is not clear. Hence the order of magnitude is not clear as well. The formula is no more asymptotic for non-ATM. Observing that formula F5 is more accurate for non-ATM swaptions, the results show the usefulness of the log transformation considered in Section 2.4.3. Table 2. 7. 7 displays CPU times of evaluating each formula to price 5 x 15 and 10 x 10 ATM swaptions. Formula F4 and F5 require more CPU time because we need to find the values of the constants in Lemma 2.4.9. The constants involve integrals of the volatility function. In our numerical examples, we use piece-wise constant volatility functions, so a triple integral involves three nested sums to evaluate the integrals. An alternative would be to choose different volatility functions so that the integrals can be evaluated analytically. 2. Swaption pricing in a log-normal LIBOR market model 88 Error Monte Carlo Swaption Strike Fl F2 F3 F4 F5 price 0.0401 (ITM) 0.227 -0.099 -0.041 -0.244 0.001 1324.691 (0.003) 1 X 9 0.0601 (ATM) 4.155 0.100 0.009 -0.004 -0.003 295.158 (0.003) 0.0801 (OTM) 1.929 0.473 0.235 0.230 -0.031 22.391 (0.003) 0.0425 (ITM) 0.960 -0.352 -0.181 -0.825 -0.026 1011.540 (0.007) 3 X 7 0.0625 (ATM) 3.463 0.270 0.037 -0.002 0.009 389.460 (0.007) 0.0825 (OTM) 3.645 1.087 0.530 0.642 0.058 121.243 (0.007) 0.0448 (ITM) 0.632 -0.187 -0.128 -0.697 -0.039 722.304 (0.006) 5 X 5 0.0648 (ATM) 1.652 0.212 0.023 -0.020 0.002 349.850 (0.006) 0.0848 (OTM) 2.037 0.690 0.315 0.553 0.067 158.022 (0.006) Tab. 2. 7.6: Errors (approximate formula - Monte Carlo) of the approximate formulae for Euro- pean payer swaption prices with Scenario B(LN). Using the control variate, the Monte Carlo is based on 1,000,000 paths with flt = 0.0625. CPU time (s) Swaption Fl F2 F3 F4 F5 5 X 5 0.018 0.019 0.020 0.076 0.076 10 X 10 0.139 0.141 0.144 1.136 1.136 Tab. 2. 7. 7: Comparison of CPU times for the approximate formulae for ATM European payer swaption prices with Scenario B(LN). 2. Swaption pricing in a log-normal LIBOR market model 89 Error Monte Carlo Scenario Swaption Strike Fl F2 F3 F4 F5 price 0.04 (ITM) 1.84 1.84 -0.70 -3.53 -0.68 1612.71 (0.03) 5 X 15 0.06 (ATM) 4.84 4.84 0.21 0.02 0.17 769.32 (0.03) 0.08 (OTM) 5.85 5.85 1.58 3.53 1.48 344.29 (0.03) A(LN) 0.04 (ITM) 1.70 1.70 -0.49 -3.21 -0.49 1017.81 (0.05) 10 X 10 0.06 (ATM) 3.67 3.67 0.39 0.06 0.37 609.61 (0.05) 0.08 (OTM) 5.00 5.00 1.62 3.36 1.57 368.40 (0.05) 0.0525 (ITM) 19.03 -2.72 -3.52 -3.80 -1.33 1675.25 (0.15) 5 X 15 0.0725 (ATM) 41.18 8.29 1.00 0.03 0.25 893.33 (0.15) 0.0925 (OTM) 51.83 19.66 8.41 4.10 2.40 447.31 (0.14) B(LN) 0.0591 (ITM) 11.37 1.89 -1.35 -3.61 -0.67 1116.05 (0.19) 10 X 10 0.0791 (ATM) 20.06 7.96 1.58 0.54 1.09 761.09 (0.19) 0.0991 (OTM) 26.93 14.16 5.71 4.90 3.28 519.56 (0.19) 0.0525 (ITM) 31.14 9.06 -4.94 -5.27 -1.47 1683.56 (0.04) 5 X 15 0.0725 (ATM) 45.89 13.02 1.25 0.02 0.34 919.01 (0.04) 0.0925 (OTM) 44.33 11.89 7.20 5.45 2.71 484.81 (0.04) C(LN) 0.0591 (ITM) 17.69 8.24 -2.13 -5.02 -1.04 1119.74 (0.08) 10 X 10 0.0791 (ATM) 23.65 11.61 1.54 0.19 0.94 770.26 (0.08) 0.0991 (OTM) 25.99 13.27 5.26 5.54 3.30 533.98 (0.08) 0.0525 (ITM) 24.50 3.49 -3.77 -4.08 -1.11 1663.06 (0.11) 5 X 15 0.0725 (ATM) 42.40 10.44 1.21 0.26 0.50 882.02 (0.11) 0.0925 (OTM) 45.43 14.27 6.89 4.77 2.63 443.80 (0.10) D(LN) 0.0591 (ITM) 14.03 4.83 -1.76 -4.35 -1.08 1103.80 (0.13) 10 X 10 0.0791 (ATM) 21.05 9.26 1.43 0.09 0.67 747.85 (0.13) 0.0991 (OTM) 25.12 12.71 5.06 4.72 2.85 508.45 (0.13) Tab. 2. 7.8: Errors (approximate formula - Monte Carlo) of the approximate formulae for Euro- pean payer swaption prices with the log-normal volatility. Using the control variate, the Monte Carlo is based on 1,000,000 paths with !::,,.t = 0.0625. 2. Swaption pricing in a log-normal LIBOR market model 90 2. 7.4 Other numerical results This section presents other numerical results for the asymptotic expansion method and J arrow and Rudd method. The asymptotic expansion method First, we examine the effect of taking a higher order expansion in the asymptotic expansion method, that is, we compare the resulting approximate pricing formula for European payer swaptions when we expand equation (2.4.51) to the first order, second order and third order. From equation (2.4.69), an approximate formula for a time O price of a European payer swaption with strike K,, maturity Tn and accrual period M - n + l is PS(O) ' 2 2 2 " ( 2 ) ' oB (O) ~ G (A) - 2c1AgoG (A)+ 2c1A g0 G (A)+ A c2g0 + c3A G (A). (2.7.143) n,M We split the formula into the three parts, each of which comes from the first, second or third order asymptotic expansion. • the first term (1st): oBn,M(O)G (A). • the second term (2nd): • the third term (3rd): oBn,M(O) (2c1A222" 9oG (A)+ A ( C29o2 + c3A )')G (A) . Table 2.7.9 shows the effect of taking higher order expansions for 5 x 5 European payer swaptions. We observe that including higher order terms improves the accuracy of the 2. Swaption pricing in a log-normal LIBOR market model 91 lst+2nd Monte Carlo Swaption Strike 1st lst+2nd +3rd price 0.0448 (ITM) 43.319 1.988 -0.697 722.304 (0.006) 5 X 5 0.0648 (ATM) 2.820 2.820 -0.020 349.850 (0.006) 0.0848 (OTM) -38.093 3.237 0.553 158.022 (0.006) Tab. 2.7.9: Errors (approximate formula-Monte Carlo) of the approximate formulae for European payer swaption prices with Scenario B(LN). Using the control variate, the Monte Carlo is based on 1,000,000 paths with At= 0.0625. approximate formula. Hence, considering a higher order expansion would be useful if we require more accuracy in the approximation. Note that since g0 = 0 for the ATM swaption, including the second term does not change the error. Secondly, we examine approximate formula (2.4.92), in which we applied a log transfor mation before applying the asymptotic expansion. From equation (2.4.92) an approximate formula for a time O price of a European payer swaption with strike K, 1 maturity Tn and accrual period M - n + I is pn~~~~) ~ Sn,M(0)N (h) - K,N ( h - JA) + cSn,M(0)A (-h¢ (h) + 2v'A¢ (h) + AN (h)). (2.7.144) We split the formula into the two parts, each of which comes from the first or second order asymptotic expansion. • the first term (1st): <5Bn,M(0) (sn,M(O)N (h) - K,N ( h - JA)). • the second term (2nd): <5Bn,M(0) ( cSn,M(0)A (-h¢ (h) + 2v'A¢ (h) + AN (h))). Table 2.7.10 shows the effect of taking higher order expansions for 5 x 5 European payer swaptions. We observe that including the second order terms does not necessarily 2. Swaption pricing in a Jog-normal LIBOR market model 92 improve the accuracy of the approximate formula. Hence, contrary to our expectation, using a log transformation before applying the asymptotic expansion does not improve the approximation. Swaption Strike 1st 2nd Monte Carlo 0.0448 (ITM) -0.187 -0.351 722.304 (0.006) 5 X 5 0.0648 (ATM) 0.212 -0.359 349.850 (0.006) 0.0848 (OTM) 0.690 -0.160 158.022 (0.006) Tab. 2. 7.10: Errors (approximate formula-Monte Carlo) of the approximate formulae for Euro pean payer swaption prices with Scenario B(LN). Using the control variate, the Monte Carlo is based on 1,000,000 paths with tit = 0.0625. 2. Swaption pricing in a log-normal LIBOR market model 93 Jarrow and Rudd method We now give some numerical results for the approximate formula developed in Section 2.5 using the Jarrow and Rudd method. From equation (2.5.116), a true price for a time 0 European payer swaption with strike ;,,, maturity Tn and accrual period M - n + 1 is PSp(0) = PS (0) + <5B (0) K2(F) - K2(A) ( ) _ <5B (0) K3(F) - K3(A) da(x) I A n,M 2! a K, n,M 3! dx X=K <5B (0) K 4 (F) - K 4 (A) + 3 (K2(F) - K2(A))2 d2a(x) I (K N) + n,M 4! dx2 + E ' ' X=K, (2.7.145) where PSA(0) = <5Bn,M(0) ( Sn,M(0)N (h) - ;,,N ( h - JA)) ' (2. 7.146) ln Sn,M(0)/;,, + ½A h= vi. . We call each term in formula (2. 7.145): • the first term (1st): PSA(0). • the second term (2nd): • the third term (3rd): -<5B (0) K3(F) - K3(A) da(x) I n,M 3! dx · x=K- • the fourth term (4th): 2. Swaption pricing in a log-normal LIBOR market model 94 Table 2.7.11 compares 5 x 5 European payer swaption prices when each term is added successively. The cumulants K 2 (F), K 3 (F), and K 4 (F) are estimated with Monte Carlo simulations. The differences of the second, third and fourth moment between the true and approximating distributions result in the adjustments of the approximations for the swaption prices. However, it is observed that it is not necessarily true that the adjustment improves the approximation. The ITM swaption price with all four terms produces a significant error. In general, this unfavorable property holds for the series expansion based on the Edgeworth series expansions except for a special case (see Kolassa (1997)). lst+2nd lst+2nd Strike 1st lst+2nd +3rd +3rd+4th Simulation 0.0448 (ITM) -0.187 -1.501 -0.301 12.325 722.304 (0.006) 0.0648 (ATM) 0.212 -0.894 -2.743 -1.116 349.850 (0.006) 0.0848 (OTM) 0.690 0.087 -1.457 -3.241 158.022 (0.006) Tab. 2. 7.11: Errors of the approximate formulae: 5 x 5 European payer swaption prices. The Monte Carlo is based on 1,000,000 paths with tlt = 0.0625. True moments are estimated as a2(F) = 5.01274 x 10-3 (8 x 10-8 ), a 3 (F) = 4.6267 x 10-4 (4 x 10-8 ) and a4(F) = 5.084 x 10-5 (2 x 10-8 ). 3. SWAPTION PRICING IN EXTENDED LIBOR MARKET MODELS 3.1 Introduction This chapter considers pricing European payer swaptions in two kinds of extended LIBOR market models. For the first extension, in US, UK, and Japanese markets caps correspond to rates compounded quarterly, while swaptions are semiannual. In German market caps are quarterly and swaptions annual. We deal with this problem by assuming log-normal volatility structure on the quarterly rates. For the second extension, in the previous chapter we have discussed a LIBOR market model with a log-normal volatility function, but this chapter considers LIBOR market models with more general volatility functions. 3.2 An r8-period log-normal LIBOR market model First of all we introduce an r or zero function roz(i, r) defined by . { r if i mod r = 0 roz i r (3.2.1) ( ' ) = 0 otherwise.. Given the 5-period forward LIBOR rates Li(t), i = 1, ... , N, the r5-period forward LIBOR rates Lt)(t), i = r, 2r, ... , that is, the forward LIBOR rates which matures at Tr, T2r, ... , are defined by r 1 + r5Lt\t) = IJ (1 + 5Li+j-i(t)), i = r, 2r, .... (3.2.2) j=l Then, providing the swap payoff occurs at 'n+r, i = n, n + r, n + 2r, ... , M - r + 1, the 3. Swaption pricing in extended LIBOR market models 96 rc5-period forward swap rate is Bn(t) - BM+1(t) s(r)n,M (t) 6Bt~(t) °I:~n Bj+1(t)Lj(t) (3.2.3) B(r)n,M (t) where M st~(t) = L roz(j + (r - l)n + 1, r)BH1(t). j=n A valuation formula for a European payer swaption under the terminal measure can be written as B(r) (T. ) + ) p5(rl(t) = 6B (t)JEN+l ( n,M n (s(r) (T,) _ t,,) :F, N+l BN+l (Tn) n,M n t · (3.2.4) Discounted bond price processes are (rl(t) _ Bi(t) a. - M i = n + l, ... , M + l. (3.2.5) ' "I:j=n roz(j + (r - l)n + 1, r)Bj+1 (t)' Then the processes at\t) follow dat\t) = (r)(t)' dWn,M,(r) ( r) ( ) T/, t , (3.2.6) ai t where M TJY)(t) = ,fif(t) - I:roz(j + (r - l)n + l,r)a)11(t),;11,M(t), j=n j £5 (r) ( ) "'(r) (t) = ~ vk t A (t) 1i,J ~ (r) ( ) k , k=i ak t and M dWt'M,(r) = dWtN+l - L roz(j + (r - l)n + 1, r)a}11(thJ11,N(t). j=n Let us define a new martingale measure, the rc5-period forward swap measure Jpm,M,(r). 3. Swaption pricing in extended LIBOR market models 97 Definition 3.2.1. We call a probability measure ]pm,M,(r) on (n, Fn) equivalent to JP'N+l, with the Radon-Nikodym derivative given by dJP'n,M,(r) I Bt~(t)BN+1(0) q>~~(t) = dJPN+l (3.2.7) Ft BN+1(t)Bt~(O) the n5-period forward swap measure for the settlement Tn. Furthermore, the process t M Wn,M,(r) - WN+l - ~ roz(y· + (r - l)n + 1 r)a(r) (s),y(r) (s)ds t - t 1~ , j+l ij+l,N , Vt E [O, Tn], 0 j=n (3.2.8) follows a standard d-dimensional Brownian motion under the r6-period forward swap mea sure wm,M,(r). Moreover, under this measure, discounted bond processes aY\t) are all martingales. The swaption pricing formula (3.2.4) is expressed under JP'n,M,(r) as (3.2.9) Application of Ito's lemma shows that the processes vY)(t) (= al11(t)Li(t)) under JP'n,N,(r) are dvY\t) _ (' ( ) (r)( ))' dWn,M,(r) · _ 1 M 1 (r) - /\i t + rJi t t , i - n + , ... , + . (3.2.10) vi (t) The asymptotic expansion method can be employed to derive an approximate pricing formula for European payer swaptions. If we approximate the volatility of the forward swap rate deterministically as in equation (2.5.114), we obtain the following approximate formula. Theorem 3.2.2. An approximate r6-period price at time t E [O, Tn] of a European payer swaption with strike ,-.,, maturity Tn and accrual period M - n + 1 is (3.2.11) 3. Swaption pricing in extended LIBOR market models 98 where ln (S~:~(t)/"') + ½A h= JX. , M OV(r)(t) M M OV(r\t) 1f\u) = L (~) Aj(u) - L L roz(j + (r - l)n + 1, r)a;11(t) (~) .\k(u). j=i+l aj (t) j=n k=j+l ak (t) Using Glasserman and Zhao's Monte Carlo simulation method we are able to price ro-period swaptions as well with the following algorithm. First, simulate basic &-period forward LIBOR rate Li(t). Then use the relationship (3.2.2) between the &-period forward LIBOR rate Li(Tn) and the ro-period forward LIBOR rate Lt)(Tn)- Then find a swaption price with the formula PS(O) = (3.2.12) with c;;l = 1, cyl = -roz(j + (r - l)n + 1, r)"'o, j = n + 1, ... , M, and ct~ 1 = -(1 + r"'o). Thero-periods swaption is also priced under the ro-period forward swap measure. Using the relations (3.2.13) and 1 ( at\t) ) Li(t) = 8 (r) - 1 , (3.2.14) ai+l (t) 3. Swaption pricing in extended LIBOR market models 99 we obtain (3.2.15) Also clearly M L roz(j + (r - l)n + 1, r)a;1I(t) = 1. (3.2.16) j=n Then (3.2.15) and (3.2.16) give a system of equations (r) ( ) 1 -1 0 0 0 0 an+I t Vn+2(r) ( t ) (r) ( ) 0 1 -1 0 0 0 an+2 t Vn+3(r) ( t ) =<5 -1 (r) ( ) 0 0 0 0 1 a<;} (t) VM+I t Dn+I Dn+2 Dn+3 DM-I DM-2 DM+I (r) ( ) I aM+I t J (3.2.17) where Di+I = roz(i + (r - l)n + 1, r). (3.2.18) Solving this gives (r) ( ) (r) ( ) RM-Rn RM - Rn+I RM -RM-I 1 Vn+2 t an+I t (r) ( ) (r) ( ) -Rn RM-Rn+l RM -RM-I 1 Vn+3 t an+2 t (r) ( ) -Rn -Rn+I RM -RM-I 1 Vn+4 t <5 RM a<;}(t) -Rn -Rn+I RM -RM-I 1 (r) ( ) (r) ( ) VM+l t aM+I t -Rn -Rn+I -RM-I 1 I J (3.2.19) where Ri = LDj+I· (3.2.20) j=n 3. Swaption pricing in extended LIBOR market models 100 This shows that we can discretize processes (3.2.10), hence the TO-periods swaption can be priced under the TO-period forward swap measure. We now look at some simulation results in the TO-period log-normal LIBOR market model. In these numerical experiments, the inputs are the initial term structure and volatility of the 3-month forward LIBOR rates, but outputs are swaption prices in which swap payoffs occur every 6 months referring to the 6-month forward LIBOR rates. We set o = 6.t = 0.25 and N + 1 = 40, corresponding to a ten-year term structure of quarterly rates. The initial term structure is given by Li(O) = log(a+bi), in which a and bare chosen so that L0 (0) = 0.05 and L39 (0) = 0.07. We only look at a one factor volatility model. The volatilities of the forward LIBOR rates are constant over the interval [Tj, TH1) with >.i(Tj) = 0.15 + 0.0025(i - j), j = 0, ... , i - 1, i = 1, ... , 39. (3.2.21) We use approximate formula (3.2.11) as a control variate. Method I is Glasserman and Zhao's discretization method under the terminal measure, Method II is our discretization method under the forward swap measure and Method III is Method II plus the use of the control variate. Table 3.2.1 and Table 3.2.2 give the Monte Carlo simulation results and show the effectiveness of using the control variate. No. of sim. Method I Method II Method III 100,000 371.6 (2.6) 372.1 (2.3) 372.977 (0.015) 1,000,000 372.35 (0.83) 372.63 (0.72) 372.939 (0.005) 10,000,000 372.95 (0.26) 372.98 (0.22) 372.934 (0.002) Tab. 3.2.1: Comparison of the Monte Carlo simulation methods for one factor 5 x 5 ATM Euro pean payer swaption prices with 8 = 0.25 but swap payoffs every 6 months. We set b.t = 0.25. 3. Swaption pricing in extended LIBOR market models 101 Option maturity x Swap length Strike Method I Method II Method III 0.0500 (ITM) 763.4 (2.5) 762.4 (2.2) 765.285 (0.022) 1 X 9 0.0607 (ATM) 311.5 (1.8) 311.4 (1.6) 313.493 (0.021) 0.0800 (OTM) 30.76 (0.56) 32.39 (0.53) 32.26 (0.019) 0.0500 (ITM) 792.9 (3.1) 791.1 (2.8) 787.641 (0.030) 2 X 8 0.0619 (ATM) 394.8 (2.4) 394.1 (2.1) 390.521 (0.029) 0.0800 (OTM) 107.7 (1.3) 109.1 (1.2) 107.005 (0.028) 0.0500 (ITM) 766.6 (3.4) 766.4 (2.9) 766.576 (0.028) 3 X 7 0.0631 (ATM) 414.1 (2.7) 414.5 (2.4) 414.358 (0.028) 0.0800 (OTM) 167.2 (1.8) 168.7 (1.6) 168.558 (0.027) 0.0500 (ITM) 632.2 (3.1) 632.5 (2.7) 632.913 (0.015) 5 X 5 0.0654 (ATM) 371.6 (2.6) 372.1 (2.3) 372.977 (0.015) 0.0800 (OTM) 218.1 (2.1) 218.9 (1.8) 220.142 (0.014) Tab. 3.2.2: Comparison of Monte Carlo simulation methods for one factor European payer swap- tion prices with 8 = 0.25 but swap payoffs every 6 months. Using the control variate, the Monte Carlo is based on 100,000 paths with flt= 0.25. 3. Swaption pricing in extended LIBOR market models 102 3.3 LIBOR market models with more general volatility functions Referring to equation (1.3.43), we assume that the arbitrage-free forward LIBOR rate process Li(t) under the terminal measure pN+l follows dLi(t) = - _t <5).il(t ~}~~?)) Ai(t)(i (Li(t)) dt + ).i(t)' (i (Li(t)) dwt+1 , (3.3.22) J=i+l J where Li(O) is deterministic and Ai is a bounded Rd-valued function. It is assumed that (i satisfies some conditions so that both coefficients of equation (3.3.22) satisfy Lipschitz and linear growth conditions. As in equation (1.3.43), (i can depend on Li(t), ... , LN(t), but for simplicity we assume that it depends only on Li(t). Carrying out a similar analysis to Section 2.3, we obtain the following formulation of our problem corresponding to equation (3.3.22). Lemma 3.3.1. A price at time t E [O, Tn] of a European payer swaption with strike ;,,, maturity Tn and accrual period M - n + 1 is (3.3.23) where M Sn,M(Tn) = L Vi(Tn)- (3.3.24) i=n The processes vi (u) for t ::; u ::; Tn solve (3.3.25) where Fort ::; u 1 ::; u and i = n, ... , M, 3. Swaption pricing in extended LIBOR market models 103 where and 3.3.1 An asymptotic expansion method An asymptotic swaption formula To apply an asymptotic method, we assume the arbitrage-free dynamics of the forward LIBOR rates Li€\u), t ~ u ~ Tn, associated with a small disturbance c « 1 follow where (3.3.27) Note that when we obtain the original dynamics of the forward LIBOR rate processes (3.3.22). We assume (i in equation (3.3.26) satisfies a certain condition so that Theorem 2.4.2 guarantees that 3. Swaption pricing in extended LIBOR market models 104 0 Lf ) ( u) is three times continuously differentiable with respect to E. Using L;"\ u), Using L;°) (u), for t ::; u ::; Tn the processes L(e)(u) ITM. (1 + oL(_e)(u)) (e) , J=i+l J i=n, ... ,M, (3.3.28) vi (u) = M M ( (e) ) ' I:j=n+l ITk=j 1 + oLj (u) + l can be written as i=n, ... ,M, (3.3.29) where V (e)( U1 ) -_ ( Vn(e)( U1 ) , ... , V M(c)( U1 )) . Fort ::; u1 ::; u and i = n, ... , M, where and An asymptotic price at time t E [O, Tn] of a European payer swaption with strike K, maturity Tn and accrual period M - n + l is (3.3.30) 3. Swaption pricing in extended LIBOR market models 105 where M s~:~(Tn) = L v;E\Tn)- i=n Relations between v Since each L~E\ u) is three times continuously differentiable with respect to E, it follows that each v?\u) is also three times continuously differentiable with respect to E. The derivation carried out in Section 2.4.2 for the log-normal LIBOR market model is applicable in the same way, and we obtain the following asymptotic formula for a European payer swaption. Theorem 3.3.2. An asymptotic price at time t E [O, Tn] of a European payer swaption with strike l'i,, maturity Tn and accrual period M - n + 1 is, for E - 0, PS(t:)(t) G (c 2E) - 2c2c1Egoc' (c 2E) + 2c4ciE 2g5G" (c 2E) 6Bn,M(t) +c 2E (C295 + c2c3E) G1 (c2E) + · · · , (3.3.32) where and In particular, a price of an ATM European payer swaption, that is, when Sn,M(t) = l'i,, is (3.3.33) 3. Swaption pricing in extended LIBOR market models 106 When c = 0, formula (3.3.32) reduces to p5(0l(t) + Bn,M(t) = (Sn,M(t) - /'i,) . When n = M the formulae in Theorem 3.3.2 reduces to an asymptotic formula for a caplet. Corollary 3.3.3. An asymptotic price at time t E [O, Tn] of a caplet with strike "' and maturity Tn is, for c---* 0, Cpl~\t) G (c 2E) - 2c2c1EgoG 1 (c2E) + 2c4dE2g5G" (c2E) 8Bn+1(t) +c2E (c2g5 + c2c3E) c' (c2E) + · · · , (3.3.34) where 1 (~ (Ln(t)) Ct = 2 (n (Ln(t))' and In particular, the price of an ATM caplet, that is, when Ln(t) = "', is (3.3.35) 3. Swaption pricing in extended LIBOR market models 107 Approximate swaption formulae In practice, we are interested in a situation when E > 0, so if we substitute E= (3.3.36) into formula (3.3.32) and truncate the higher order terms, we obtain an approximate formula for a European payer swaption. Theorem 3.3.4. An approximate price at time t E [O, Tn] of a European payer swaption with strike K, maturity Tn and accrual period M - n + 1 is (3.3.37) where M M T 1 A= LL 1n and the constants c1, c2 and c3 are the same as in Theorem 3.3.2. In particular, when n = M we obtain an approximate formula for a caplet. Corollary 3.3.5. An approximate price at time t E [O, Tn] of a caplet with strike K and maturity Tn is (3.3.38) 3. Swaption pricing in extended LIBOR market models 108 where and the constants c1, c2 and C3 are the same as in Corollary 3.3.3. A modifi.ed CEV LIBOR market model Andersen and Andreasen (2000b) introduced a CEV (constant elasticity of variance) 11- BOR market model. This case corresponds to (i (Li(t)) = Li(t)f3 in equation (3.3.22), where f3 is a positive constant. The model is practically more appealing than the log normal model, since it produces the volatility skew for both caplets and swaptions, which is often observed in the market (see Blyth and Uglum (1999) and Khuong-Huu (1999)). However, when f3 < 1, zero is an attainable and absorbing boundary for the process, and when f3 > 1 the process could explode. To avoid these problems, we introduce a modified CEV LIBOR market model. Before that, let us give the following theorem. Theorem 3.3.6. Let (i(x) = xgi(x), i = 1, ... , N, where gi are Lipschitz and bounded. Then for every initial condition (L1 (0), ... , LN(0)) such that Li(0) > 0, there exists a unique non-exploding path-wise solution to the equation _ <5>-.i(t)'Li(t)gi (Li(t)) () () ( ( )) ( )' () ( ( )) N+1 dLi ()t - - _L~ l + <5L-(t) Ai t Li t gi Li t dt + >-.i t Li t gi Li t dWt . J=t+l J (3.3.39) Moreover Li(t) > 0 for all t > 0 and i = 1, ... , N. Finally, Li(t) is an arbitrage-free forward LIBOR process under the terminal measure jp>N+l _ Proof. Let Xi(t) = log Li(t). Then, Ito's lemma gives 3. Swaption pricing in extended LIBOR market models 109 It is easy to see that both coefficients of this equation are bounded and Lipschitz. Hence, there exist non-explosive unique positive solutions for Li(t), i = 1, ... , N. D For a modified CEV LIBOR market model, we set (3.3.41) where h ( l. la - x) h ( x - a) hi(x,a) = h(x-a)+h(l.la-x)' h2 (x,a) = h(x-a)+h(l.la-x)' and h(~) = { e-1/f. ~ > 0 0 ~ ~ 0. By choosing a small positive number for a when f3 < 1, and a large positive number for a when f3 > 1, the function (i(x) satisfies the condition in Theorem 3.3.6. Thus, a unique positive solution for each Li(t) exists. Furthermore, since the function (i(x) satisfies the conditions in Theorem 2.4.2, it follows that the processes L?)(u), i = n, ... , M, in equa tion (3.3.26) have all continuous derivatives with respect to c. Then, formulae (3.3.37) and (3.3.38) give approximate prices for a European payer swaption and a caplet respec tively. Although, there is no closed formula for a caplet with the modified CEV volatility model, we present the following transformation based on the CEV swap market model. Formula (3.3.37) may be approximated as PS(t) ~ G (A)' (3.3.42) <5Bn,M(t) where On the other hand, if we apply the asymptotic method to the CEV swap model, 3. Swaption pricing in extended LIBOR market models 110 we obtain (3.3.43) where - - 2/3 2 ( (3 5(32 - 4(3 2 {32 - 2(3 2 ) I:cev - Sn,M(t) (JcevTn 1- Sn,M(t)go + 12Sn,M(t)29o - 12 (JcevTn · The right hand sides of (3.3.42) and (3.3.43) coincide when A.= Ecev, that is, when o-;.. = Sn,Mt) 213Tn (I+ (sn,~(t) -Zc,) 90 2 2 ( 7{3 + 4(3 2{3c1 ) 2 ( {3 - 2(3 ) ) + C2 + 12Sn,M(t)2 - Sn,M(t) 90 + C3 - 12Sn,M(t)2 A + ... · Theorem 3.3.7. An approximate price at time t E [O, Tn] of a European payer swaption with strike K, maturity Tn and accrual period M - n + l is, for (3 < l, PS(t) n,M(t) ~ Sn,M(t) (1 - Q(a, b + 2, c)) - KQ(c, b, a), (3.3.44) and for (3 > l, PS(t) n,M(t) ~ Sn,M(t) (1 - Q(c, -b, a)) - KQ(a, 2 - b, c), (3.3.45) where /'i,2(1-/3) 1 Li (t )2(1-/3) b l - (3' C fTl ' = = (1 - (3)2 (Jcev2 1 n and o-;., Sn,M t) 2~Tn ( I+ (Sn,~ ( t) - Zc,) Ya 2 2 ( 7{3 + 4(3 2{3c1 ) 2 ( {3 - 2/3 ) ) + C2 + 12Sn,M(t)2 - Sn,M(t) 90 + C3 - l2Sn,M(t) 2 A · Q(x, p, q) is defined by Q(x,p, q) = f ((½;f e-~) Pr (G(q + 2j, 2) :::; x), (3.3.46) . 0 J. 1= 3. Swaption pricing in extended LIBOR market models 111 where Pr (G(r, 2) ~ x) is the cumulative distribution function for a gamma distributed random variable with parameters r and 2, that is, Again, it is quite interesting to note that if we use only A/ (Sn,M(t) 213Tn) as a-~ev, we obtain an approximate formula introduced by Andersen and Andreasen (2000b). Theorem 3.3.8 (Andersen and Andreasen, 2000b). An approximate price at time t E [O, Tn] of a European payer swaption with strike K, maturity Tn and accrual period M - n + l is, for /3 < 1, PS(t) n,M(t) ~ Sn,M(t) (1 - Q(a, b + 2, c)) - ;.,Q(c, b, a), (3.3.47) and for /3 > 1, PS(t) n,M(t) ~ Sn,M(t) (1 - Q(c, -b, a)) - ;.,Q(a, 2 - b, c), (3.3.48) where ,..,2(1-/3) 1 Li(t)2(1-f3) a= (l - /3)2-2O" cev.L ,.,, n ' b=l-/3' c= (1 - /3)2-2,.,,'O"cev.Ln and -2 A O"cev = Sn,M (t)2f3T,n · 3.3.2 Monte Carlo simulation In this section, we propose an efficient Monte Carlo simulation method for pricing swaptions in LIBOR market models with more general volatility functions. We first explain how to find swaption prices using a standard simulation method under the terminal measure. Then we present our discretization method under the forward swap measure. 3. Swaption pricing in extended LIBOR market models 112 Monte Carlo under the terminal measure The European payer swaption pricing formula (2.2.1) at t = 0 can be written under the forward measure JP>M + 1 as follows. B (0)JEN+l ( Bn,M(Tn) (S (T,) _ K)+) PS(O) 6 N+l B (T, ) n,M n N+l n 8B (0)JEM+l ( Bn,M(Tn) (S (T, ) - K)+) M+l B (T, ) n,M n M+l n M M )+ BM+l(0)JEM+l ( ~ cj g(1 + 8Li(Tn)) + CM+l (3.3.49) with Cn = 1, cj = -K8, j = n + 1, ... , M, and CM+l = -(1 + K6). With a standard Monte Carlo simulation method to evaluate the expectation (3.3.49), we discretize the forward LIBOR rate processes dLi(t) = - _t 6.\\(f ~_i~;t)) .\i(t)(i(Li(t)) dt + .\i(t)' (; (Li(t)) dwt+1, i = n, ... , M, J=t+l J (3.3.50) to obtain an estimator Li(Tn) of Li(Tn)- To discretize each process, we consider, for example, if L;(t) > 0, an Euler scheme for log Li(t) with even spacing tk+l - tk = lit, k = 0, 1, ... , t0 = 0, that is, with 3. Swaption pricing in extended LIBOR market models 113 and 6, 6, ... , are independent standard normal d-dimensional vectors. Once obtaining an estimator Li(Tn), using (3.3.49), we obtain an estimator PS(O) of PS(0) by Ps(D) ~ BM+1(D) (t. C; fl (1 +oL(Tn)) + CM+l) + (3.3.52) Suppose an i.i.d. sequence ?S1(0), ?S2 (0), ... is generated by the standard Monte Carlo simulation. A natural estimator of PS(0) with NoSim replications is then the sample mean _ l NoSim , PS(0) = N PSm(0). (3.3.53) as· im L m=l It can be shown that PS(0) converges to the true value PS(0) as Lit - 0 and NoSim - oo. Monte Carlo under the forward swap measure So far, we discussed a Monte Carlo simulation method under the terminal measure to find a price of a European payer swaption. We now examine a Monte Carlo simulation method under the forward swap measure. At the same time, we present a variance reduction method based on a control variate technique for our simulation to speed up the calculation. Recall a European payer swaption price under the forward swap measure: (3.3.54) where M Sn,M(Tn) = L Vi(Tn), (3.3.55) i=n Processes vi (t) follow (3.3.56) From this expression, it is quite natural to discretize the processes vi(t). Then, if vi(t) > 0, an Euler scheme for logvi(t) with evenly spacing tk+l - tk = lit, k = 0, l, ... , is ·-(t ) = ·-(t) ('Pi(tk,v(tk))' (-!'Pi(tk,v(tk)) At+ ~tc )) V, k+l V, k exp v(tk) 2 v(tk) u V Llt<,,k+l , (3.3.57) 3. Swaption pricing in extended LIBOR market models 114 where 6, 6, ... are independent standard normal d-dimensional vectors. Our discretization method is similar to the discretization method in Theorem 2.6.1 developed by Glasserman and Zhao (2000). Each vi(t) can be written as where the discounted bond price ai+l ( t) is This expression of vi(t) reminds us equation (2.6.124), in which Xi(t) is given by the difference of discounted bond prices divided by o. Furthermore, from the relations a,+1(t,) = M _On+ 1 (-;f_y- n)V;(t,) +,t, (M + 1 - j)V;(t,) + ~), i = n, ... , M, each ai+ 1(tk) is a liner combination of the processes vi(tk)- It follows that the discte tized discounted bond price processes are all martingales since vi(tk) are martingales. We summarize our discretization method in the following theorem. Theorem 3.3.9. With the discretization method (3.3.57), we obtain an estimator Sn,M(Tn) M Sn,M(Tn) = L vi(Tn), (3.3.58) i=n and then an estimator PS(O) of PS(0) by PS(O) = oBn,M(0) ( Sn,M(Tn) - K) +. (3.3.59) Suppose we generate an i.i.d. sequence ?S1 (0), ?S2 (0), .... An estimator of PS(0) with N oSim replications, NoSim PS(0) N ~- PSm(O), (3.3.60) = o im L m=l 3. Swaption pricing in extended LIBOR market models 115 converges to the true value PS(O) as f:i.t _, 0 and NoSim -, oo. Furthermore, the dis retized discounted bond price processes °'i+1(tk) are all martingales. Sn,M(Tn) also remains a martingale. As for the log-normal LIBOR market model, we can use a control variate method to speed up the Monte Carlo simulation. For the modified CEV LIBOR market model, we may incorporate a control variate in the following way. Monte Carlo with a control variate (the modified CEV LIBOR market model): Consider a process M dwn,M(t) ~ vi(O) ( ( ) ( )f3-1 ( ( )))' n,M (3.3.61) Wn,M(t)/3 ~ Sn,M(0) 2f3 Ai t Li O + T/i t, V O dWt , Wn,M(O) Sn,M(O), (3.3.62) where Then, for /3 < l, Cx oBn,M(O)JEn,M (wn,M(Tn) - Kt oBn,M(O) (Sn,M(O) (1 - Q(a, b + 2, c)) - KQ(c, b, a)), (3.3.63) and for /3 > l, Cx oBn,M(O)JEn,M (wn,M(Tn) - Kt OBn,M(O) (Sn,M(O) (1 - Q(c, -b, a)) - KQ(a, 2 - b, c)), (3.3.64) where K2(1-/3) 1 Li(0)2(1-/3) b = l - /3' C = (1 - f3)2 (TcevTn2 ' 3. Swaption pricing in extended LIBOR market models 116 and 2 1 ~ ~ vi(O)vj(O) {Tn ( ( ) ( )/3-1 ( ( )))' ( ( ) ( )/3-1 ( ( ))) acev = Tn {:;: f=:i Sn,M(0) 2/3 Jo Ai U Li O + 'r/i u, V O Aj U Lj O + 'r/j u, V O du. An unbiased estimator PSc(O) of PS(0) is obtained by (3.3.65) The next section on numerical results shows that this control variate works very well. 3.3.3 Numerical results This section gives various numerical results comparing the Monte Carlo simulation methods and the approximate formulae for European payer swaption prices in the modified CEV LI BOR market model. We first demonstrate the efficiency of our simulation method (3.3.65). Using the control variate dramatically speeds up finding accurate swaption prices. We then examine the accuracy of our approximate swaption formulae (3.3.37) and (3.3.44) using simulation prices as benchmarks. We find that our approximate formulae provide more accuracy than the approximate formula by Andersen and Andreasen (2000b). Details of numerical examples Before presenting the numerical results, we provide several parameters for the numerical experiments. We set 5 = 0.5 and N + 1 = 40, corresponding to a twenty-year term structure of semiannual rates. We assume the LIBOR market model with the modified CEV volatility with parameters a= 0.001 and (3 = 0.5 in (3.3.41). For the modified CEV LIBOR market model, to obtain similar swaption prices to the log-normal LIBOR market model, we multiply the volatilities in the scenarios for the log normal model by 0.25, and we consider the following four scenarios: 3. Swaption pricing in extended LIBOR market models 117 • Scenario A(MCEV): Li(0) 0.06, for i = 0, ... , N, Ai(u) 0.05, for i = 0, ... ,N and 0 ~ u ~ Ti. • Scenario B(MCEV): Li(0) log(a + bi), for i = 0, ... , N, so that £ 0 (0) = 0.05 and £ 39 (0) = 0.09, 0.25(0.17+0.002(i-j)), j=0, ... ,i-1, i=l, ... ,39, and constant over the interval [T1, T1+1 ). • Scenario C(MCEV): Li(0) log(a + bi), for i = 0, ... , N, so that £ 0 (0) = 0.05 and £ 39(0) = 0.09, 0.25 (0.25 - 0.002(i - j)), j = 0, ... , i - 1, i = 1, ... , 39, and constant over the interval [T1, T1+1 ). • Scenario D(MCEV): Li(0) log(a + bi), for i = 0, ... , N, so that £ 0 (0) = 0.05 and £39(0) = 0.09, Ai(T1)1 0.25 ( 0.22 - 0.00l(i - j), Jo.OOl(i - j) - 0.1) , j = 0, ... , i - 1, i = 1, ... , 39, and constant over the interval [T1, TH1). Each volatility structure implies that the volatility depends only on the time to maturity of the forward LIBOR rates. An at-the-money (ATM) swaption has a strike rate K = Sn,M(0), an in-the-money (ITM) swaption has K < Sn,M(0), and an out-of-the-money (OTM) swap tion has K > Sn,M(0). An x x y has an option maturity of x years and a swap length of y years. All swaption prices are reported in basis points, that is, the true value x 10000. Figure 3.3.1 shows the volatility skew produced by 5 x 5 swaptions with scenario B(MCEV) and scenario C(MCEV). 3. Swaption pricing in extended LIBOR market models 118 0.28 ~---.------.---.,------.----.....------.------, 0.26 o Scenario B (MCEV) • Scenario C (MCEV) ~ 15 0.24 g oi E 0.22 0 ~ u., 0.2 'a. .E 0.18 0.16 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 Strike Fig. 3.3.1: Implied log-normal volatility for 5 x 5 year European payer swaption prices: an ATM swaption has r;, = 0.0648. Comparison of Monte Carlo methods We compare two Monte Carlo simulation methods to compute swaption prices: • Method II: our discretization method described in Theorem 3.3.9. • Method III: Method II plus the use of the control variate described in equation (3.3.65). First of all, we choose l:!.t = 0.5 ( = 8) as a time step for the discretization. Table 3.3.3 shows simulated swaption prices and their standard deviations for each method by increas ing the number of simulations. It is obvious that the standard deviations when using the control (Method III) are significantly smaller than those of without the control (Method II). The standard deviation in Method II after 100,000 simulations is still higher than that in Method III after 1,000 simulations. As expected the rate of convergence of the simulation is inversely proportional to the square root of the number of simulations, that is, increasing the number of simulations 100 times decreases the standard deviation by one tenth. Observing that Method III converges very fast, Table 3.3.4 shows the effect of taking a finer discretization. Using Method III, one factor 5 x 5 ATM(,-;;= 0.0648) and 1 x 9 OTM (K = 0.0801) European payer swaptions are computed with 100,000 simulations taking 3. Swaption pricing in extended LIBOR market models 119 ~t=0.5, 0.25, 0.125 and 0.00625, which correspond to ~t = <5, J/2, J/4 and J/8 respectively. We find that the standard deviation does not decrease by taking a finer discretization. In general, taking a finer discretization has the effect of reducing the bias. For the 5 x 5 ATM swaption, it seems that even taking ~t=0.5 the simulation price is not biased so much since taking much finer discretization does not change the simulation price. However, for the 1 x 9 OTM swaption, we observe that the simulation with ~t =0.5 produces a significant bias. Taking a finer time step produces a less biased value. Table 3.3.5 shows various swaption prices including ITM and OTM swaptions. Using the control is effective in all cases. However, we note that the values of standard deviations in Method III do not depend as much on the value of the strike as those in Method II. Consequently, as in the log-normal LIBOR market model, we find that using the control is very effective in the modified CEV LIBOR market model as well. Method III produces accurate simulation prices for European payer swaptions very efficiently. No. of sim. Method II Method III 1,000 352.63 (17.87) 343.952 (0.079) 10,000 346.31 (5.76) 343.976 (0.024) 100,000 344.23 (1.82) 343.984 (0.008) Tab. 3.3.3: Comparison of the Monte Carlo simulation methods for 5 x 5 ATM European payer swaption prices with Scenario B(MCEV). We set !1t = 0.5. ~t Swaption 0.5 0.25 0.125 0.0625 5 x 5 (ATM) 343.984 (0.008) 343.982 (0.008) 343.959 (0.008) 343.962 (0.008) 1 x 9 (OTM) 17.533 (0.003) 17.363 (0.004) 17.280 (0.005) 17.256 (0.005) Tab. 3.3.4: Effect of taking a finer time step for discretization for payer European swaption prices with Scenario B(MCEV). The Monte Carlo is based on 100,000 paths with Method III. 3. Swaption pricing in extended LIBOR market models 120 Option maturity x Swap length Strike Method II Method III 0.0401 (ITM) 1329.2 (2.3) 1328.434 (0.007) 1 X 9 0.0601 (ATM) 300.2 (1.5) 300.143 (0.007) 0.0801 (OTM) 17.34 (0.34) 17.256 (0.005) 0.0425 (ITM) 1030.1 (2.8) 1028.159 (0.012) 3 X 7 0.0625 (ATM) 390.5 (2.0) 389.189 (0.011) 0.0825 (OTM) 101.8 (1.0) 100.891 (0.010) 0.0448 (ITM) 737.7 (2.4) 737.691 (0.008) 5 X 5 0.0648 (ATM) 343.6 (1.8) 343.962 (0.008) 0.0848 (OTM) 131.9 (1.1) 132.229 (0.007) 0.0525 (ITM) 1674.0 (5.6) 1674.11 (0.17) 5 X 15 0.0725 (ATM) 826.3 (4.2) 827.16 (0.16) 0.0925 (OTM) 342.3 (2.8) 343.10 (0.15) 0.0591 (ITM) 1090.1 (4.4) 1090.66 (0.13) 10 X 10 0.0791 (ATM) 680.9 (3.7) 680.72 (0.12) 0.0991 (OTM) 401.5 (2.9) 400.32 (0.12) Tab. 3.3.5: Comparison of the simulation methods for European payer swaption prices with See- nario B(MCEV). The Monte Carlo is based on 100,000 paths with l!..t = 0.0625. 3. Swaption pricing in extended LIBOR market models 121 Comparison of approximate formulae In this section, we compare three approximate formulae for European payer swaption prices: • F2: the approximate formula (3.3.47) introduced by Andersen and Andreasen (2000b). • F4: our approximate formula (3.3.37). • F5: our approximate formula (3.3.44). Although a closed form solution for the swaptions is not available, the previous section shows that the Monte Carlo simulation with the control variate gives an accurate price quite efficiently. Thus we use the simulation price as a benchmark to compare the accuracy of the approximate formulae. Table 3.3.6 and Table 3.3.7 compare the accuracy of approximate formulae F2, F4 and F5. The error is a price of an approximate formula minus a price of simulation. Simulation prices are shown with their standard deviations. We set a time step for simulation b.t = 0.0625( = o/8) and simulate 100,000 paths. It is found that both of our formulae F4 and F5 are more accurate than formula F2, but there is no significant difference between formulae F4 and F5 like the log-normal volatility model. Their errors tend to increase as the strike rate is away from the ATM strike rate or either maturity or swap length of the swaptions increases. 3. Swaption pricing in extended LIBOR market models 122 Error Monte Carlo Swaption Strike F2 F4 F5 price 0.0401 (ITM) -0.142 -0.058 0.006 1328.434 (0.007) 1 X 9 0.0601 (ATM) 0.114 -0.023 -0.023 300.143 (0.007) 0.0801 (OTM) 0.319 0.010 -0.038 17.256 (0.005) 0.0425 (ITM) -0.302 0.081 -0.006 1028.159 (0.012) 3 X 7 0.0625 (ATM) 0.330 0.005 0.002 389.189 (0.011) 0.0825 (OTM) 0.853 -0.048 0.016 100.891 (0.010) 0.0448 (ITM) -0.124 0.203 -0.026 737.691 (0.008) 5 X 5 0.0648 (ATM) 0.245 0.006 -0.005 343.962 (0.008) 0.0848 (OTM) 0.550 -0.132 0.019 132.229 (0.007) Tab. 3.3.6: Errors (approximate formula - Monte Carlo) of the approximate formulae for Euro- pean payer swaption prices with Scenario B(MCEV). Using the control variate, the Monte Carlo is based on 100,000 paths with !:::,.t = 0.0625. 3. Swaption pricing in extended LIBOR market models 123 Error Monte Carlo Scenario Swaption Strike F2 F4 F5 price 0.04 (ITM) 2.83 -0.29 -0.41 1667.91 (0.05) 5 X 15 0.06 (ATM) 5.02 0.00 0.03 786.64 (0.04) 0.08 (OTM) 4.75 0.38 0.48 310.49 (0.04) A(MCEV) 0.04 (ITM) 2.42 0.59 -0.27 1070.43 (0.04) 10 X 10 0.06 (ATM) 3.72 0.22 0.18 624.19 (0.04) 0.08 (OTM) 4.01 0.10 0.61 340.01 (0.04) 0.0525 (ITM) -0.93 -0.62 -0.59 1674.11 (0.17) 5 X 15 0.0725 (ATM) 7.18 -0.23 -0.18 827.16 (0.16) 0.0925 (OTM) 13.00 0.27 0.28 343.10 (0.15) B(MCEV) 0.0591 (ITM) 2.16 0.03 -0.30 1090.66 (0.13) 10 X 10 0.0791 (ATM) 5.98 0.23 0.27 680.72 (0.12) 0.0991 (OTM) 8.84 0.46 0.75 400.32 (0.12) 0.0525 (ITM) 11.05 -0.48 0.01 1688.19 (0.08) 5 X 15 0.0725 (ATM) 9.71 -0.01 0.06 861.08 (0.08) 0.0925 (OTM) 4.07 0.52 0.23 384.13 (0.07) C(MCEV) 0.0591 (ITM) 7.59 0.00 -0.04 1097.09 (0.05) 10 X 10 0.0791 (ATM) 7.50 0.13 0.21 693.51 (0.05) 0.0991 (OTM) 5.82 0.36 0.49 417.30 (0.05) 0.0525 (ITM) 5.85 -0.29 0.00 1664. 75 (0.16) 5 X 15 0.0725 (ATM) 8.37 0.11 0.17 822.32 (0.15) 0.0925 (OTM) 7.00 0.47 0.31 345.92 (0.15) D(MCEV) 0.0591 (ITM) 5.12 0.08 -0.06 1080.10 (0.10) 10 X 10 0.0791 (ATM) 6.59 0.21 0.27 670.91 (0.10) 0.0991 (OTM) 6.68 0.41 0.58 393.53 (0.10) Tab. 3.3. 7: Errors (approximate formula - Monte Carlo) of the approximate formulae for Euro- pean payer swaption prices with modified CEV volatility. Using the control variate, the Monte Carlo is based on 100,000 paths with 1::!,,.t = 0.0625. 4. SWAPTION HEDGING IN THE LIBOR MARKET MODEL 4.1 Introduction We have discussed analytical and numerical pricing methods for European payer swaptions in the LIBOR market model. This chapter focuses on hedging methods for European payer swaptions in the LIBOR market model. A hedge is a position in an asset designed to remove the risk from another position in a different instrument. In our problem, we try to replicate a price process for a European payer swaption with a different instrument, for example, a finite set of bond price processes. First we present a continuous time hedging method which assumes that we can hedge continuously. However, this assumption is not realistic in practice. Thus secondly we present a discrete time hedging method. To carry out these hedging methods we need values so-called Greeks. The Greeks are the option's partial derivatives with respect to its input parameters. We discuss how to estimate the Greeks for European payer swaptions analytically and numerically. Then we give some numerical results for the accuracy of the estimates. 4.2 Continuous time hedging As shown in Jamshidian (1997), in the LIBOR market model, an arbitrage price process for a derivative can be synthetically replicated with a finite set of zero-coupon bonds. Theorem 4.2.1 (Jamshidian, 1997). The arbitrage price process C(t) is synthetically replicated with a self-financing portfolio of zero-coupon bonds, that is, N+l C(t) = L ac(t) ac(t) ) (¢i(t), ... ' Thus, by continuously rebalancing the portfolio, the value of the portfolio is exactly the same as the value of the derivative C(t) for all t until its maturity. Since a European payer swaption is path-independent, its price process PS(t) is replicated with bond price processes so that M+l 8PS(t) PS(t) = ~ aBi(t) Bi(t), Vt E [O, Tn]- (4.2.3) i=n 4.3 Discrete-time hedging In practice, it is impossible to perfectly replicate a derivative since we can rebalance the portfolio only discretely. Moreover, considering transaction costs, it is not desirable to re balance it very frequently. Various methods for discrete-time hedging have been proposed, for example, Hull (1997), Jarrow and Turnbull (2000), Taleb (1997) and Wilmott (1998). 4.3.1 Delta hedging Consider a portfolio II(t) which consists of a European payer swaption and zero-coupon bonds: II(t) = PS(t) + nn(t)Bn(t) + · · · + nM+1(t)BM+1(t), (4.3.4) where nn(t), ... , nM+i(t) are predictable process and represent the number of zero-coupon bonds Bn(t), ... , BM+1(t) at time t. Suppose that we choose nn(t), ... , nM+i(t) at time 0 (=To), and rebalance the portfolio at time T1, T2 , ... until the maturity of the swaption Tn. Delta of the swaption is the rate of change of the swaption's price with respect to the price of the zero-coupon bond. A delta hedging with a delta neutral position is to make the 4. Swaption hedging in the LIBOR market model 126 portfolio insensitive to small changes in the value of the zero-coupon bond by constructing the portfolio to have a zero delta at each time Tk, k = 0, ... , n - 1. In order to construct a delta neutral portfolio we require oII(Tk) = 0 for i = n, ... , M + 1 and k = 0, ... , n - 1. (4.3.5) 8Bi(Tk) This gives (4.3.6) In addition, at time 0 with these choices for ni(0), we realize that the portfolio is actually self-financing at time 0 since II(0) = 0 using equation (4.2.3). However, after that the portfolio is no longer self-financing. The gamma of the swaption is the rate of change of the swaption's delta with respect to the underlying zero-coupon bond. Then the gamma of the delta neutralized portfolio is a2rr(n) 82 PS(Tk) for i = n, ... , M + 1 and k = 0, ... , n - 1. (4.3.7) 8Bi(Tk) 2 8Bi(Tk) 2 To measure the sensitivities with respect to changes in volatility, we add parameters 0i in the volatilities Ai(t) and consider derivatives with respect to 0i. This implies that 8.\i(0i, t)/80i = 1 for some i (and all t) but 8.\i(0i, t)/80i = 0 for all j =/ i corresponds to a parallel shift in the volatility of ,\i(t). The vega of the swaption is the rate of change of the swaption's price with respect to 0i. Then the vega of the delta neutralized portfolio is for i = n, ... , M + 1 and k = 0, ... , n - 1. (4.3.8) 4.4 Estimation of Greeks To use delta hedging, we need to estimate deltas, and to analyze the sensitivities for the delta neutralized portfolio, we need to estimate gammas and vegas. However, closed form formulae for Greeks for European payer swaptions are not available. We need to estimate 4. Swaption hedging in the LIBOR market model 127 Greeks numerically or analytically. To estimate accurate Greeks numerically, we use a finite difference estimator with the Monte Carlo simulation method described in Section 2.6 and Section 3.3.2. To estimate a delta with respect to the ith initial bond price Bi(O), for example, we perturb Bi(O) to Bi(O) + h, for some small increment hand obtain an Monte Carlo estimator PS(Bi(O) + h). Using the same sequence of random numbers, we perturb Bi(O) to Bi(O) - h and obtain a Monte Carlo estimator PS(Bi(O) - h). Then, a delta is estimated by oPS(O) PS(Bi(O) + h) - PS(Bi(O) - h) (4.4.9) 8Bi(O) ~ 2h Similarly, gammas and vegas are estimated by 82 PS(O) PS(Bi(O) + h) - 2PS(Bi(O)) + PS(Bi(O) - h) (4.4.10) 8Bi(0)2 ~ h2 and oPS(O) PS(.Xi(O) + h) - PS(.Xi(O) - h) (4.4.11) 80i ~ 2h For the other Monte Carlo methods to estimate Greeks, we refer to Broadie and Glasser man (1996), Glasserman and Zhao (1999) and Fournie et al. (1999). Although the Monte Carlo gives accurate values for Greeks, the speed of convergence is very slow. To estimate approximate Greeks analytically, we can use our approximate formulae for European payer swaptions. Because it is very difficult to differentiate our approximate formulae with respect to an initial parameter, we estimate the derivative by a finite difference. To estimate a delta with respect to the ith initial bond price Bi(O), for example, we perturb Bi(O) to Bi(O)+h, for some small increment hand obtain an estimator PS(Bi(O) + h) using an approximate formula. Then we perturb Bi(O) to Bi(O) - h and obtain an estimator PS(Bi(O) - h). Then a delta is estimated by oPS(O) ~ PS(Bi(O) + h) - PS(Bi(O) - h) (4.4.12) 8Bi(O) ~ 2h 4. Swaption hedging in the LIBOR market model 128 Similarly, gammas and vegas are estimated by 82 PS(O) PS(Bi(O) + h) - 2PS(Bi(O)) + PS(Bi(O) - h) (4.4.13) 8Bi(O) 2 ::::::! h2 and ~ ~ oPS(O) PS(.\i(O) + h) - PS(.\i(O) - h) (4.4.14) 80i ::::::! 2h 4.5 Numerical results This section presents various numerical results comparing estimated Greeks by using the Monte Carlo and the approximate formulae. We use the same data set given in Section 2. 7 and Section 3.3.3. For the Monte Carlo, we use method (2.6.142) for the log-normal model and method (3.3.65) for the modified CEV model. We use the following five approximate formulae to estimate Greeks: • Fl: the approximate formula (2.2.4) introduced by Brace, Dun and Barton (1998). • F2: the approximate formula (2.4.74) for the log-normal model or the approxi mate formula (3.3.47) for the modified CEV model introduced by Andersen and Andreasen (2000b). • F3: the approximate formula (2.2.3) derived by Brace, Gatarek and Musiela (1997). • F4: our approximate formula (2.4.69) for the log-normal model or our approximate formula (3.3.37) for the modified CEV model. • F5: our approximate formula (2.4.73) for the log-normal model or our approximate formula (3.3.44) for the modified CEV model. From Table 4.5.1 to Table 4.5.3 show numerical results for 2 x 2 European payer swaptions in the log-normal LIBOR market model. They show Greeks estimated by the Monte Carlo and errors between Greeks estimated by the Monte Carlo and by the approximate formulae. 4. Swaption hedging in the LIBOR market model 129 The Monte Carlo Greeks are shown with their standard deviations. To estimate Greeks by the approximate formulae, we use h = 0.0001 for the increment to obtain accurate values. To estimate deltas and vegas by the Monte Carlo, we use h = 0.0001 and simulate 100,000 paths with flt = 0.0625 to obtain the values with less biased and smaller standard deviations. However, when estimating gammas by the Monte Carlo, choosing a smaller value for the increment h causes a larger standard deviation. Choosing a larger value for the increment h causes a larger bias. To balance between the standard deviation and the bias we use h = 0.001 and simulate 1,000,000 paths with flt = 0.0625. We first check if the self-financing condition at t = 0 is satisfied, that is, if equa tion (4.2.3) at t = 0 is satisfied. Using the Monte Carlo, the values of left hand side of equation (4.2.3), the values of swaptions, are 340.5705 (ITM), 94.1258 (ATM) and 14.7696 (OTM) with their standard deviations 0.0002. On the other hand, the values of initial zero-coupon bond prices, B4 (0), ... , B8 (0), are 0.903185, 0.879365, 0.855738, 0.832325 and 0.809144 respectively. Using the values in Table 4.5.1, the values of right hand side of equation (4.2.3), the values of portfolios of zero-coupon bonds, are 340.5735 (ITM), 94.1196 (ATM) and 14.7718 (OTM) respectively, which shows that the self-financing condition is satisfied. As for Greeks estimated by the approximate formulae, F5 is the most accurate, but our formula F4 produces more errors for non-ATM swaptions. Table 4.5.4 and Ta ble 4.5.5 display comparisons of estimated Greeks for various swaptions in the log-normal LIBOR market model and we see that F5 is the most accurate for most of the Greeks. Table 4.5.6 and Table 4.5.7 compare estimated Greeks for the modified CEV LIBOR market model. It is found that both of our formulae F4 and F5 are more accurate than formula F2, but there is no significant difference between formula F4 and F5. 4. Swaption hedging in the LIBOR market model 130 Error Monte Carlo Strike Delta Fl F2 F3 F4 F5 delta 8PS(0)/8B4 (0) -6.09 1.40 1.31 -6.28 0.01 9711.46 (0.07) 8PS(0)/8B5 (0) 3.92 -0.88 -0.83 0.25 0.02 -171.56 (0.05) 0.0357(ITM) 8PS(0)/8B6 (0) 3.87 -0.85 -0.84 0.25 0.01 -171.51 (0.05) 8PS(0)/8B1 (0) 3.83 -0.81 -0.85 0.26 0.02 -171.47 (0.05) 8PS(0)/8B8 (0) -5.50 1.13 1.19 5.95 -0.07 -9875.03 (0.08) 8PS(0)/8B4 (0) -40.91 -0.81 -0.52 -1.34 0.00 5525.02 (0.07) 8PS(0)/8B5 (0) 28.68 0.20 0.15 0.05 0.01 -142.79 (0.05) 0.0557(ATM) 8PS(0)/8B6 (0) 28.24 0.20 0.17 0.05 0.02 -142.35 (0.05) 8PS(0)/8B7 (0) 27.76 0.19 0.19 0.05 0.01 -141.87 (0.04) 8PS(0)/8Bs(0) -43.83 0.28 0.04 1.34 -0.03 -5599.17 (0.08) 8PS(0)/8B4 (0) -23.99 -3.48 -3.63 -8.99 0.06 1386.90 (0.06) 8PS(0)/8B5 (0) 18.49 2.75 2.61 0.12 -0.06 -46.11 (0.05) 0.0757(OTM) 8PS(0)/8B6 (0) 18.11 2.62 2.59 0.12 -0.05 -45.73 (0.04) 8PS(0)/8B7 (0) 17.70 2.47 2.55 0.12 -0.04 -45.32 (0.04) 8PS(0)/8B8 (0) -30.63 -4.40 -4.15 9.86 0.08 -1384.74 (0.07) Tab. 4.5.1: Errors (approximate formula - Monte Carlo) of the approximate formulae for deltas for 2 x 2 European payer swaption prices with Scenario B(LN). The finite difference is based on h = 0.0001. The Monte Carlo is based on 100,000 paths with tlt = 0.0625. 4. Swaption hedging in the LIBOR market model 131 Error Monte Carlo Strike Gamma Fl F2 F3 F4 F5 gamma {)2 PS(0)/8B4(0)2 964 -158 -158 846 -11 27578 (4) fP PS(0)/8B5(0) 2 326 -69 -66 18 1 -295 (1) 0.0357(ITM) 82 PS(0)/8B6(0) 2 339 -71 -70 19 1 -308 (1) 82 PS(0)/8B7 (0) 2 353 -72 -75 19 1 -322 (1) 8 2 PS(0)/8Bs(0)2 -762 86 83 917 -22 29975 (5) 8 2 PS(0)/8B4(0) 2 746 425 426 -9 0 166632 (8) 8 2 PS(0)/8B5(0)2 2260 16 16 -2 -2 -2127 (2) 0.0557(ATM) 8 2 PS(0)/8B6(0) 2 2349 18 17 -2 -2 -2216 (2) 8 2 PS(0)/8B7 (0) 2 2445 22 20 -2 -1 -2312 (2) 82 PS(0)/8Bs(0)2 894 -441 -434 20 23 175565 (8) 82 PS(0)/8B4(0)2 -2134 -88 -76 -1566 18 94030 (4) 8 2 PS(0)/8B5(0)2 1402 205 198 -20 -4 -1345 (1) 0.0757(OTM) 82 PS(0)/8B6(0) 2 1453 207 205 -21 -4 -1396 (1) 82 PS(0)/8B7 (0) 2 1508 211 214 -21 -3 -1451 (1) 82 PS(0)/8B8 (0) 2 3630 228 237 -1618 -3 95249 (4) Tab. 4.5.2: Errors (approximate formula - Monte Carlo) of the approximate formulae for gammas for 2 x 2 European payer swaption prices with Scenario B(LN). The finite difference is based on h = 0.0001 for the approximate formulae and h = 0.001 for the Monte Carlo. The Monte Carlo is based on 1,000,000 with i:::,,.t = 0.0625. 4. Swaption hedging in the LIBOR market model 132 Error Monte Carlo Strike Vega Fl F2 F3 F4 F5 vega oPS(0)/804 0.21 0.17 -0.08 -0.87 -0.01 22.19 (0.01) oPS(0)/805 0.11 0.05 -0.03 -0.86 0.00 22.11 (0.01) 0.0357(ITM) oPS(0)/806 -0.01 -0.08 -0.01 -0.85 0.00 22.03 (0.01) oPS(0)/801 -0.14 -0.20 0.07 -0.84 0.01 21.94 (0.01) oPS(0)/804 0.10 0.10 0.00 -0.01 0.00 133.22 (0.01) oPS(0)/005 0.15 0.05 0.00 0.00 0.00 132.09 (0.01) 0.0557(ATM) oPS(0)/806 0.11 -0.02 0.00 -0.01 0.00 130.93 (0.01) oPS(0)/801 0.01 -0.08 0.01 0.00 0.01 129. 73 (0.01) oPS(0)/004 -0.53 -0.60 -0.13 0.56 0.04 74.49 (0.01) oPS(0)/805 -0.04 -0.16 -0.02 0.53 0.01 73.39 (0.01) 0.0757(OTM) oPS(0)/806 0.41 0.27 0.08 0.51 -0.01 72.28 (0.01) oPS(0)/801 0.80 0.68 0.16 0.47 -0.03 71.17 (0.01) Tab. 4.5.3: Errors (approximate formula - Monte Carlo) of the approximate formulae for vegas for 2 x 2 European payer swaption prices with Scenario B(LN). The finite difference is based on h = 0.0001. The Monte Carlo is based on 100,000 paths with 1::!,,t = 0.0625. Error Monte Carlo Swaption Strike Greek Fl F2 F3 F4 F5 Greek {)P S(0) I 8B2(0) -18.5 4.2 2.9 2.4 -0.3 9885.4 (0.1) 0.0401(ITM) 82PS(0)/8B 2(0) 2 893 -136 -116 181 7 3819 (11) 8PS(0)/802 0.70 0.21 -0.17 -0.43 -0.01 7.26 (0.01) 8PS(0)/8B2(0) -175.8 -9.7 -4.8 -0.1 0.6 5488.1 (0.3) ~ 2 2 1 X 9 0.0601(ATM) 8 p S(0) I 8B2(0) 113 416 411 -34 -31 52907 (19) CJ:i 8PS(0)/802 0.29 0.30 0.01 -0.01 -0.01 95.62 (0.01) ~ "ti...... ,... 8PS(0)/8B2(0) -32.3 -2.7 -5.9 3.2 0.3 781.9 (0.2) § b- 2 2 Cb 0.0801(OTM) 8 PS(0)/8B2(0) -2303 -268 -248 -408 3 -20701 (19) 0... O.S. 8PS(0) I 802 -0.30 -1.40 -0.21 0.38 0.06 35.29 (0.01) ::i Oq..... ::i.,... b- 8PS(0)/8B6(0) -95.0 16.1 15.7 -21.0 0.0 9157.9 (0.2) Cb t-, 0.0425(ITM) 82PS(0)/8B 6(0) 2 2568 -125 -148 -240 8 14837 (6) ti3 8PS(0)/806 3.30 2.56 -0.87 -0.73 -0.16 62.64 (0.04) 0 ::i::i s 8PS(0)/8B6(0) -244.5 -12.0 -7.0 -3.6 -0.4 5811.1 (0.4) ~ @ 3 X 7 0.0625(ATM) 82PS(0)/8B 6(0) 2 1228 776 766 -41 -24 .,... 1.20 1.23 0.07 -0.03 0.01 158.39 (0.04) s 8PS(0)/806 0 0...... _Cb 8PS(0)/8B6(0) -194.3 -30.3 -30.9 -19.2 0.5 2602.3 (0.3) 0.0825( OTM) 82PS(0)/8B 6(0) 2 -2594 168 245 492 9 33619 (17) {)PS(0)/806 -3.35 -4.08 -0.78 0.79 0.34 131.55 (0.04) Tab. 4.5.4: Errors (approximate formula - Monte Carlo) of the approximate formulae for Greeks for European payer swaption prices with Scenario B(LN). The finite difference is based on h = 0.0001 for the approximate formulae, Monte Carlo deltas and Monte Carlo vegas and h = 0.005 for the Monte Carlo gammas. The Monte Carlo is based on 1,000,000 paths with flt = 0.0625. It; c..:, Error Monte Carlo Swaption Strike Greek Fl F2 F3 F4 F5 Greek 8PS(0)/8B10 (0) -507 40 52 -37 -7 8595 (1) 0.0525(ITM) 82 PS(0)/8B10 (0) 2 4220 422 359 -124 150 7922 (75) BPS(0)/8010 18.07 15.94 -2.76 -3.80 -1.36 111.56 (0.20) 8PS(0)/8B10 (0) -806 -76 -52 -30 -15 6496 (1) ~ 2 2 5 X 15 0.0725(ATM) 8 PS(0)/8B10(0) 2574 1715 1707 125 188 14294 (162) (J:l BPS(0)/8010 9.14 9.50 1.15 -0.13 0.17 185.23 (0.20) ~ 'tl ....<"i- 0 8PS(0)/8B10(0) -780 -142 -145 -23 -7 4398 (1) 1:1 b- 2 2 (t, 0.0925(OTM) 8 PS(0)/8B10 (0) -1468 1097 1322 123 -114 17090 (174) Ct. BPS(0)/8010 -6.08 -7.72 -1.02 4.07 2.32 197.15 (0.20) ~- ()-q 5· <"i- b- 8PS(0)/8B20 (0) -580 43 48 -55 -12 8218 (2) (t, t-< 0.0591(ITM) 82 PS(0)/8B20 (0) 2 5544 949 974 -181 155 10563 (92) Ea BPS(0)/8020 22.29 22.03 -1.54 -6.63 -1.27 154.74 (0.47) 0 ~ s 8PS(0)/8B20 (0) -827 -57 -51 -53 -18 6913 (2) 2 2 (t,~ 10 X 10 0.0791(ATM) 8 PS(0)/8B20 (0) -4962 2652 2628 -41 196 14664 (151) ""' 16.66 17.05 2.82 0.05 1.30 209.20 (0.46) BPS(0)/8020 s0 Ct...... (t, 8PS(0)/8B20 (0) -939 -150 -154 -34 -13 5673 (2) 0.0991(OTM) 82 PS(0)/8B20(0) 2 2466 2967 3024 -335 -228 17518 (180) BPS(0)/8020 4.79 4.74 2.80 7.18 4.41 233.67 (0.46) Tab. 4.5.5: Errors (approximate formula - Monte Carlo) of the approximate formulae for Greeks for European payer swaption prices with Scenario B(LN). The finite difference is based on h = 0.0001 for the approximate formulae, Monte Carlo deltas and Monte Carlo vegas and h = 0.005 for the Monte Carlo gammas. The Monte Carlo is based on 1,000,000 paths with /)..t = 0.0625. It; 14 Error Monte Carlo Swaption Strike Greek F2 F4 F5 Greek &P S(0) I &B2(0) 4.3 -1.5 -0.6 9782.1 (0.1) 0.0401(ITM) 82PS(0)/&B 2(0) 2 101 73 61 5088 (11) &PS(0)/802 0.98 -0.05 0.02 13.75 (0.02) &PS(0)/&B2(0) -8.5 1.2 1.1 5363.7 (0.2) ~ 2 2 1 X 9 0.0601(ATM) 8 PS(0)/&B2(0) 407 64 67 44829 (17) CF.) &PS(0)/802 0.36 0.05 0.04 104.80 (0.01) ~ 't,...... ,.,. 0 &PS(0)/&B2(0) 0.9 -0.7 0.3 612.7 (0.2) ::s i:l"' 2 2 (1) 0.0801(OTM) 8 PS(0)/&B2(0) -315 -47 -38 15355 (17) Cl.. O.S. &PS(0)/802 -1.94 0.04 -0.01 31.99 (0.02) ::s Oq..... ::s .,.,. i:l"' &PS(0)/&BG(0) 8.8 -3.2 -0.6 8894.6 (0.1) (1) 2 2 t-< 0.0425(ITM) 8 PS(0)/&B6 (0) 1503 100 352 9935 (7) ti"3 &PS(0)/806 5.35 0.08 0.03 80.04 (0.03) 0 ~ s &PS(0)/&BG(0) -12.7 1.0 0.1 5564.2 (0.3) (1)~ 3 X 7 0.0625(ATM) 82PS(0)/&B 6 (0) 2 1535 185 191 26337 (21) .,.,. 1.15 0.05 0.04 167.63 (0.03) s &PS(0)/806 0 Cl.. ~ &PS(0)/&BG(0) -15.5 -1.8 0.5 2178.0 (0.1) 0.0825(OTM) 82PS(0)/&B 6 (0)2 1753 211 -61 22958 (10) &PS(0)/806 -6.02 -0.04 -0.04 125.12 (0.03) Tab. 4.5.6: Errors (approximate formula - Monte Carlo) of the approximate formulae for Greeks for European payer swaption prices with Scenario B(MCEV). The finite difference is based on h = 0.0001 for the approximate formulae, Monte Carlo deltas and Monte Carlo vegas and h = 0.005 for the Monte Carlo gammas. The Monte Carlo is based on 1,000,000 paths with /j,,_f = 0.0625. I~ Cll Error Monte Carlo Swaption Strike Greek F2 F4 F5 Greek 8PS(0)/8B10 (0) 8.2 -3.3 -3.4 8448.8 (0.6) 0.0525(ITM) 82 PS(0)/8B10 (0) 2 2113 419 634 -4618 (87) 8PS(0)/8010 21.72 -0.64 0.01 119.60 (0.12) 8PS(0)/8B1o(0) -64.5 -3.7 -2.6 6141.1 (0.8) ~ 5 X 15 0.0725(ATM) 82 PS(0)/8B10 (0) 2 2129 414 406 -1371 (143) en 8PS(0)/8010 7.55 0.31 0.38 192.24 (0.12) ~ 'ti,,... g· 8PS(0)/8B10(0) -69.5 1.5 1.7 3620.7 (0.6) b- 2 2 ('t) 0.0925( OTM) 8 PS(0)/8B10 (0) 668 186 40 3654 (97) 0.. O.S. 8PS(0)/8010 -14.2 1.16 0.75 188.64 (0.12) ::i o-q.... ::i,,... b- 8PS(0)/8B20 (0) -12.6 -3.5 -11.1 7922.4 (0.7) ('t) t-< 0.0591(ITM) 82 PS(0)/8B20 (0) 2 6433 1289 1780 -14337(98) ti3 8PS(0)/8020 22.68 -0.21 0.55 156.91 (0.17) 0 ::i:; s 8PS(0)/8B20 (0) -67.9 -5.8 -8.4 6321.6 (0.8) ~ @ 10 X 10 0.0791(ATM) 82 p S(0) I 8B20(0) 2 5117 895 946 -11174 (119) ,,... 10.05 0.46 0.62 206.99 (0.16) s 8PS(0)/8020 0 0.. '..._('t) 8PS(0)/8B20 (0) -96.6 3.5 -2.4 4661.0 (0.7) 0.0991(OTM) 82 PS(0)/8B20 (0) 2 2763 263 63 -5419 (100) 8PS(0)/8020 -7.37 1.08 0.69 218.83 (0.16) Tab. 4.5. 7: Errors (approximate formula - Monte Carlo) of the approximate formulae for Greeks for European payer swaption prices with Scenario B(MCEV). The finite difference is based on h = 0.0001 for the approximate formulae, Monte Carlo deltas and Monte Carlo vegas and h = 0.005 for the Monte Carlo gammas. The Monte Carlo is based on 1,000,000 paths with b..t = 0.0625. I~ 0-., 4. Swaption hedging in the LIBOR market model 137 4.6 Conclusion This thesis has developed pricing and hedging methods for European payer swaptions in the LIBOR market model. The main achievement of this thesis is the development of a new approximate pricing formula for European payer swaptions. The formula not only prices European payer swaptions very accurately, but also it can estimate very accurate Greeks such as deltas, gammas and vegas for European payer swaptions. The formula is accurate even when the strike rate is not ATM and when either the maturity or the swap length is long. How to calibrate the formula to market caplet and swaption prices is the next issue to be studied for the practical use. For the development of the formula, the asymptotic expansion method has been used. As shown in this thesis, the effectiveness of the method indicates that it has a great potential as an approximation tool to solve many problems in Mathematical Finance. For example, it will be interesting to apply the method to stochastic volatility models. Another achievement of this thesis is the development of the efficient Monte Carlo sim ulation method to price European payer swaptions. 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