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Using the LIBOR Market Model to Price

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Using the LIBOR Market Model to Price

    Using the LIBOR Market Model to Price

    the Interest Rate Derivatives A Recombining Binomial Tree Methodology

    LMM 1 (Correspondence Author) Tian-Shyr Dai, 戴天時, National Chiao Tung University,

    Graduate Institute of Finance, Assistant Professor.(新竹市大學路1001 國立交通大學財

    1 務金融研究所,03-5712121#57054cameldai@mail.nctu.edu.tw )

    2 Huimin Chung, 鐘惠民, National Chiao Tung University, Graduate Institute of Finance,

    Professor.

    3Chun-Ju Ho, 何俊儒, National Chiao Tung University, Graduate Institute of Finance,

    Student.

    4Wei-Ting Wang, 王薇婷, National Chiao Tung University, Graduate Institute of Finance,

    Student.

     1 The author was supported in part by NSC grant 96-2416-H-009-025-MY2 and NCTU research grant for

    financial engineering and risk management project.

Using the LIBOR Market Model to Price the Interest Rate Derivatives

    A Recombining Binomial Tree Methodology

    LMM

    Abstract

LIBOR market model (LMM) is a complicated interest rate model and it is hard to be

    evaluated both analytically and numerically. Because of the non-Markov property of the

    LMM, a naively implemented tree model will not recombine. Thus the size of this naïve

    tree model will grow explosively and the tree cannot be efficiently evaluated by

    computers. This paper proposes a recombining LMM tree model by taking advantages

    of tree construction methodology proposed by Ho, Stapleton, and Subrahmanyam (HSS).

    We first rewrite the discrete mathematical model for LMM suggested by Poon and

    Stapleton. Then we derive the conditional means and the variances of the discrete

    forward rates which are important for the tree construction. Finally, our recombining

    trees for pricing interest rate derivatives are built by taking advantages of the tree

    construction methodology proposed by HSS. Numerical results illustrated in Section 5

    suggest that our method can produce convergent and accurate pricing results for interest

    rate derivatives.

    Keywords: term structure, LMM, recombining tree

     1

LMM(LIBOR market model)利率期限模型是一個很複雜的利率模型,要在此模型

    下推導評價封閉公式解或是數值評價方法都很困難。由於LMM模型具備非馬可夫的特性,建構利率樹時會發生節點無法重合的問題,而導致利率樹的節點個數呈

    爆炸性成長,使得電腦無法有效率地使用該利率樹進行評價。本篇論文利用Ho, Stapleton, and Subrahmanyam (HSS)介紹的造樹法來建構節點重合的LMM利率樹。我們首先改寫Poon and StapletonLMM建構的離散時間數學模型,接下來

    推導遠期利率的條件期望值和條件變異數—這些資料對利率樹的建構十分重要。

    最後我們利用HSS提供的建樹法建構利率樹,並用該利率樹評價利率衍生性金融

    商品。第五節的數值資料驗證我們的數值模型可提供精確的評價結果。

     2

1. Introduction

    Many traditional interest rate models are based on instantaneous short rates and instantaneous forward rates. However, these rates can not be observed from the real

    world markets; consequently, it is hard to calibrate these models to fit the real world

    markets. LIBOR market model (LMM) is recently widely accepted in practice because

    it is based on the forward LIBOR rate which can be observed from the real world

    markets. This model was first proposed by Brace, Gatarek, and Musiella (1997)

    (abbreviate as BGM). In their model, the forward LIBOR rate is assumed to follow a

    lognormal distribution process, which makes the theoretical pricing formula for the

    caplet consists with the pricing formula under the Black’s model (Black 1976).

    However, when implementing the LMM by a tree method, the tree will not recombine due to the non-Markov property of LMM. This non-recombining property

    makes the size of the tree grows explosively and thus the tree method is inefficient and

    2difficult to price. To address this problem, this paper adapts the HSS methodology

    proposed by Ho, Stapleton, and Subrahmanyam (1995) to construct a recombining

    binomial tree for LMM. By applying the HSS methodology into the LMM, the tree

    valuation method becomes feasible in pricing the interest rate derivatives.

    The tree method proposed in this paper makes us have not to rely on the Monte Carlo simulation because our tree-based method is more accurate and efficient. Besides,

    the tree method can deal with American-style features, such as early exercise or early

    redemption, which is an intractable problem in Monte Carlo simulation.

    The paper is structured as follows. Section 2 reviews important interest rate models. Section 3 introduces the market conventions about LMM and derives the drift of

    discrete-time version of LMM which follows the development in Poon and Stapleton

    (2005). In section 4, we introduce the HSS recombining node methodology (Ho et al.

    1995) into the discrete-time version of LMM which derived in section 3 and construct

    the pricing model. The numerical pricing results and the sensitive analyses in section 5 verify the correctness and robustness of our tree model. Finally, section 6 concludes the paper.

     2 Similarly, HJM model also has the non-Markov property and thus the tree for HJM grows explosively

    as mentioned in Hull (2006).

     3

2. Review of Interest Rate Models

In this section we introduce some important interest rate models that can be generally

    categorized into two categories: equilibrium models and no-arbitrage models.

    Equilibrium models usually start with assumptions about economic variables and derive

    a stochastic process for the short rate r. On the other hand, a no-arbitrage model makes

    the behaviors of interest rate exactly consist with the initial term structure of interest

    rates. We will introduce some important equilibrium models first.

Vasicek model, suggested in Vasicek (1977), assumes that the short rate process r(t)

    follows the Ornstein-Uhlenbeck process and has the following expression under the

    risk-neutral measure:

     drtrtdtdWt()(())()??????,

    ??where mean reversion rate , average interest level , and volatility are ?constants. Note that the short rate r(t) appears to be pulled back to long-run average

    interest level , which is called mean reversion property. Vasicek shows that the price ?

    at time t of a zero-coupon bond that pays $1 at time T can be expressed as

    ?BtTrt(,)()PtTAtTe(,)(,)? where

    ???()Tt1?eBtT(,)? ?

    2222((,))(/2)(,)BtTTtBtT???????AtT(,)exp[]??2??4. The drawback of Vasicek model is that the short rate could be negative. To improve this

    drawback, Cox, Ingersoll, and Ross (1985) propose CIR model which makes the short

    rate r(t) always non-negative. Under the risk neutral measure, r(t) follows the following process:

     drtrtdtrtdWt()(())()()??????,which also has the mean reversion property. Moreover, to make the short rate

    nonnegative, CIR model use a non-constant volatility to replace the constant ?r(t)volatility in Vasicek model. The zero-coupon bond price P(t,T) in CIR model can be expressed as follows:

    ?BtTrt(,)()PtTAtTe(,)(,)? ,where

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    ?()Tt?2(1)e?BtT(,)? ()Tt??()(1)2???e???,

    ()()/2????Tt22?e2/???AtT(,)[]? ()Tt??()(1)2e??????,

    22and . ?????2

Note that equilibrium models cannot exactly fit prevailing term structure of interest

    rates. Thus, no-arbitrage models are designed to calibrate prevailing term structure of

    interest rates. We first focus on instantaneous short rate models.

Ho and Lee (1986) propose the first no-arbitrage model. The short rate process of Ho-

    Lee model under the risk-neutral measure is as follows:

     drttdtdWt()()()????,where θ (t) is a function of time chosen to ensure that the model fits the initial term

    structure, and it can be expressed by the instantaneous forward rate as follows:

    2??()(0,)tftt?? t,

    ft(0,)where is the instantaneous forward rate for maturity t as seen at time zero and t

    the subscript t denotes a partial derivative with respect to t. Moreover, the price of the zero-coupon bond P(t,T) in Ho-Lee model can be expressed as

    ??rtTt()()PtTAtTe(,)(,)? where

    PT(0,)122??????ln(,)ln()(0,)().AtTTtfttTtPt(0,)2 Hull and White (1990) then provide a generalized version of the Vasicek model and it

    provides an exact fit to the prevailing term structure. The short rate process of the Hull-

    White model is

     drttrtdtdWt()[()()]()??????,

    ??where and are constants and the function of can be calculated from the ?()tinitial term structure as follows:

    2?2??t()(0,)(0,)(1)tftfte?????? t2?The zero-coupon bond price P(t,T) in Hull-White model has the same general form as in

    Vasicek model:

    ?BtTrt(,)()PtTAtTe(,)(,)? where

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    ???()Tt1?eBtT(,)? ?

    PT(0,)1222?????Ttt??????ln(,)ln(,)(0,)()(1)AtTBtTfteee 3Pt(0,)4?.

On the other hand, Heath, Jarrow and Morton (1992) model the stochastic process of

    the instantaneous forward rate to describe the evolution of the entire yield curve in

    continuous time. The instantaneous forward rate for the fixed maturity T under ftT(,)the risk-neutral measure is described as follows:

     dftTtTdttTdWt(,)(,)(,)()????

    where

    WtWtWt()((),,())? is a d-dimensional Brownian motion, 1d

    ???(,)((,),,(,))tTtTtT? is a vector of adapted processes, 1ddTT?????(,)(,)(,)(,)(,)tTtTtsdstTtsds??. ii?tt??i?1

    Given the dynamics of the instantaneous forward rate, the Ito’s lemma can be ftT(,)applied to obtain the dynamics of the zero-coupon bond price: PtT(,)

    TdPtTPtTrtdttsdWt(,)(,)[()((,))()]??? ?t,where can be expressed as follows: rt() tttrtfttftutusdsdustdWs()(,)(0,)(,)(,)(,)()??????? ???u00.

    Note that the short rate process in the HJM model is non-Markov and makes a rt()

    naively-implemented tree for simulating the short rate process a non-recombined tree.

    Besides, another drawback of the HJM model is that it is expressed in terms of

    instantaneous forward rates, which can not be directly observed in the market. Thus, it

    is difficult to calibrate the HJM model to price the actively traded instruments.

    To address the aforementioned problem, Brace, Gatarek, and Musiela (1997)

    suggest the BGM model that models the dynamics of the forward rates. However,

    Miltersen, Sandmann, and Sondermann (1997) discover this model independently, and

    Jamshidian (1997) also contributes significantly to its initial development. To reflect the

    contribution of multiple authors, many practitioners, including Rebonato(2002),

    renamed this model to LIBOR market model (LMM).

    There are two common versions of the LMM, one is the lognormal forward

    LIBOR model (LFM) for pricing caps and the other is the lognormal swap model (LSM)

    for pricing swaptions. The LFM assumes that the discrete forward LIBOR rate follows a

    lognormal distribution under its own numeraire, while the LSM assumes that the

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discrete forward swap rate follows a lognormal distribution under the swap numeraire.

    The two assumptions do not match theoretically, but lead to small discrepancies in

    calibrations using realistic parameterizations. The following derivations are based on

    the LFM.

    Unlike HJM that models, the instantaneous forward rate at timeas seen TftT(,)

    ftTT(;,)at time t the LFM models, the discrete forward rate seen at time t for the ,ii?1

    TTftTT(;,)period between time and time . follows a zero-drift stochastic ii?1ii?1process under its own forward measure:

    dftTT(;,)ii?1()()tdWt ??iiftTT(;,)ii?1,

    Ti?1dWt()Qwhere is a Brownian motion under the forward measure defined with i

    PtT(,)?()trespect to the numeraire asset , and measures the volatility of the i?1iforward rate process. Using Ito’s lemma, the stochastic process of the logarithm of the

    forward rate is given as follows:

    2??()tidftTTdttdWtln(;,)()()??? (1) 1iiii?2

    0??tTThe stochastic integral of equation (1) can be given as follows. For all , i

    2tt??()uiln(;,)ln(0;,)()()ftTTfTTduudWu???? (2) 11iiiiii??00??2

    ?()tSince the volatility function is deterministic, the logarithm of the forward i

    rate is normally distributed, implying that the forward rate is lognormally distributed.

    tT?LTTfTTT(,)(;,)?For , equation (2) implies that the future LIBOR rate is iiiiii??11

    also lognormally distributed. This explains why this model is called the lognormal

    forward LIBOR model. Thought each forward rate is lognormally distributed under its

    own forward measure, it is not lognormally distributed under other forward measure.

3. Market Conventions of the LMM and the Discrete-Time

    Version of the LMM

    In this section, we first introduce some market conventions of LMM and related

    interest rate instruments such as caplets and forward rate agreements (FRAs). Next, we

    restate some key results in the Poon and Stapleton (2005) and then re-derive generalized

    formulas for discrete-time version of the LMM that can be directly used in our tree

    construction procedure.

3.1 Market Conventions of the LMM

    LTT(,)???TTThe relationship between the discrete LIBOR rate for the term ii?1iii?1

    PTT(,)and the zero-coupon bond price is given as follows: ii?1

     7

    1(,) , (3) PTT?ii?11(,)?LTT?iii?1

    tTTTT??????where is the time line and is called the tenor or accrual 012ni

    TTfraction for the period to . ii?1

    ???TTThe time t discrete forward rate for the term is related to the price iii?1

    TTratio of two zero-coupon bonds maturing at times and as follows: ii?1

    PtT(,)i . (4) ???1(;,)ftTTiii?1PtT(,)i?1

    TThe forward rate converges to the future LIBOR rate at time , or: i

     lim(;,)(,)fTTLTT?? . (5) iiii??11?T?i

    We can rewrite equation (4) as follows:

    1. ftTTPtTPtTPtT??(;,)(,)[(,)(,)]iiiii???111?i

    Then, we define basic terms that are frequently used in the market as follows:

    FortTT(,,): the forward price at time t to invest a zero coupon bond matured at time 1n

    TTPtTPtT(,)/(,) at time and can be expressed as . n1n1ytT(,)T: the annual yield rate at time t to time and its relation with the zero coupon 11

    PtTytT(,)1/(1(,))???bond is givens as . 111

    ftTT(;,)TT: the forward rate at time t for the time period to and its relation nn?1nn?1

    with forward price of a zero coupon bond is given as

    FortTTftTT(;,)1/(1(;,))???. nnnnn??11

    After introducing these basic terms, we introduce a popular interest rate option

    Tan interest rate cap. A cap is composed of a series of caplets. For a -maturity caplet, ithe practitioners widely use the Black’s formula to obtain its value at time t as follows:

    caplettAPtTftTTNdKNd()(,)[(;,)()()]????? , (6) iiiii??1112

    where

    2ln((;,)/)()/2ftTTKTt???iiii?1d?,1?Tt?ii 2ln((;,)/)()/2ftTTKTt???iiii?1d?,2Tt??ii

     A: the notional value of the caplet,

    ?: the length of the interest rate reset interval as a proportion of a year, i

    PtT(,)T: the zero coupon bond price paying 1 unit at maturity date , i?1i?1

    K: the caplet strike price,

    ?: the Black implied volatility of the caplet, i

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    : the cumulative probability distribution function for a standardized N()?

    normal distribution.

    Furthermore, under the LIBOR basis, we can derive the same theoretical pricing

    equation for the caplet as equation (6) from the LFM model. Because both of LFM and

    Black’s model are assuming that the forward rate follows the lognormal distribution and

    we get the consistent results.

    Another instrument we illustrate here as a key to derive out the discrete-time

    version of the LMM is the forward rate agreement (FRA). A FRA is an agreement made

    at time t to exchange fixed-rate interest payments at a rate for variable rate K

    TTpayments, on a notional amount , for the loan period to equal to one year. Ann?1

    TThe settlement amount at time on a long FRA is n

    AyTTK((,))?nn?1FRAT()? , (7) n1(,)?yTTnn?1

    yTT(,)TTwhere is the annual yield at time to . At the time of the contract nn?1nn?1inception, a FRA is normally structured so that it has zero value. To avoid the arbitrage,

    ftTT(;,)the strike rate is set equal to the market forward rate . We denote the Knn?1

    FRAtT(,)value of the FRA at time t as which can be expressed as n

    AyTTftTT((,)(;,))?nnnn??11 . (8) FRAtTE(,)[]0??nt1(,)?yTTnn?1

3.2 The Discrete-Time Version of the LMM

    We first restate the most important results which are under the “risk-neutral” measure in

    the Poon and Stapleton (2005).

(A) For a zero-coupon bond price is given by

    PtTPtTEPTT(,)(,)((,))? , (9) ntn11

    or we can write

    PtT(,)n EPTTFortTT((,))(,,)??tnn11PtT(,)1.(B) The drift of the forward bond price is given by

    EForTTTFortTT[(,,)](,,)?tinin1

     (10) PtT(,)1 cov[(,,),(,)]??ForTTTPTTtinn11PtT(,)n.

    T(C) The drift of -period forward rate is obtained from the equation (8) and given by n

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