行政院國家科學委員會專題研究計畫 成果報告
跨通貨利率保證財務契約之評價：跨國 LIBOR 市場模型
研究成果報告(精簡版)
計 畫 類 別 ： 個別型 計 畫 編 號 ： NSC 98-2410-H-004-075- 執 行 期 間 ： 98 年 08 月 01 日至 99 年 07 月 31 日 執 行 單 位 ： 國立政治大學金融系 計 畫 主 持 人 ： 陳松男 計畫參與人員： 碩士班研究生-兼任助理人員：鍾明希 博士班研究生-兼任助理人員：謝宗佑 博士班研究生-兼任助理人員：陳宏銘 博士班研究生-兼任助理人員：李章益 處 理 方 式 ： 本計畫涉及專利或其他智慧財產權，2 年後可公開查詢中 華 民 國 99 年 06 月 22 日
行政院國家科學委員會補助專題研究計畫
■ 成 果 報 告
□期中進度報告
（計畫名稱）
跨通貨利率保證財務契約之評價：跨國LIBOR 市場模型
計畫類別：■ 個別型計畫
□ 整合型計畫
計畫編號：
NSC 98 2410 H 004 075
-執行期間：
98 年 8 月 1 日 至
99 年 7 月 31 日
計畫主持人：陳松男
共同主持人：
計畫參與人員：謝宗佑、李章益、陳宏銘、鍾明希
成果報告類型(依經費核定清單規定繳交)：
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精簡報告
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□赴國外出差或研習心得報告一份
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執行單位：國立政治大學
中
華
民
國
99
年
6
月
18
日
摘要
跨通貨利率保證契約(cross-currency interest rate guaranteed contracts; CIRGCs)， 為現實生活中常見之財務契約，如投資保證契約(GICs)、保險產品及退休金計劃… 等，此種契約保證投資人在特定期間所可獲得的最低報酬率。CIRGCs 連結之標的 物包含了不同幣別計價之資產與利率，以往文獻均只探討標的均為單一幣別資產的 情形，因此本研究針對 CIRGCs 進行評價與分析。
為能適切地評價 CIRGCs，本文使用跨通貨 LIBOR 市場模型(CLMM)進行評 價。CLMM 將原始之 LIBOR Market Model (LMM) 由單一國家經濟環境延伸至兩國 經濟環境，模型中並包含跨國之權益資產、匯率及利率之動態過程。此跨國模型適 用於 CIRGCs 的評價工作。 文中推導出 CIRGCs 之理論公平價格，除可供學術研究之參考，亦可供發行金 融機構與投資人於實務運用之參考。文中亦進行蒙地卡羅模擬 （Monte Carlo Simulation）以驗證模型理論解的準確性。 關鍵詞：利率保證，跨通貨，LIBOR 市場模型
Abstract
We derive the pricing formulae for the financial contracts, such as guaranteed investment contracts (GICs), life insurance contracts, pension plans, and others, with the guaranteed minimum rate of return set relative to a LIBOR interest rate. Further, we analyze the guaranteed contracts in which the asset that provides the underlying return for the contract and the guaranteed interest rate are denominated in different currencies, which is a common practice. The guaranteed contracts with the above characteristics are called “cross-currency interest rate guaranteed contracts”(CIRGCs).
To value CIRGCs, a cross-currency LIBOR market model is introduced. The LIBOR market model for a single-currency economy is extended to a cross-currency economy which incorporates the traded-asset prices and exchange rate processes into the model setting. The cross-currency LIBOR market model (CLMM) is suitable and applicable to pricing a variety of CIRGCs. The pricing formulas derived under the CLMM are more tractable and feasible for practice than those derived under the instantaneous short rate model or the HJM model.
Four different types of CIRGCs are priced in this article. In addition, Monte-Carlo simulation is also provided to evaluate the accuracy of the theoretical prices.
報告內容
1. Introduction
Many real-word financial contracts have embedded some sort of minimum rate of return guarantee. Examples of such contracts could be guaranteed investment contracts (GICs), life insurance contracts, pension plans, and index-linked bonds. This leads to a tremendous amount of money managed by life insurance companies and pension funds. As a result, a further analysis of rate of return guarantees is warranted.
There are a variety of guarantee designs in financial contracts with guaranteed return in practice. One class of these guarantees is absolute guarantees, where the minimum rate of return is set to be deterministic. The other is the so-called relative guarantees in the literature (Lindset, 2004), where the minimum guaranteed rate of return is linked to a stochastic asset such as an index, a reference portfolio, an interest rate, a specific asset traded in financial markets, etc.
The purpose of this research is to extend the previous analysis to set up a theoretical framework that analyzes the financial contracts of the guaranteed minimum rate of return set relative to a LIBOR interest rate. Further, we analyze the guaranteed contracts in which the asset that provides the underlying return for the contract and the guaranteed interest rate are denominated in different currencies, which is a common practice. The guaranteed contracts with the above characteristics are called “cross-currency interest rate guaranteed contracts” (CIRGCs, hereafter). This type of contracts is quite different from previously-studied contracts regarding the minimum rate of return guarantee since they all assume that underlying assets in guaranteed contracts are denominated in a single (domestic) currency.
The motivation of our paper is inspired by the scarcity in the researches regarding the relative guarantees, especially under stochastic interest rates. Previous research on valuing guarantees for life insurance products or pension funds has focused on absolute guarantees, which provide participants with a constant or predetermined minimum rate of return. The existing literature which analyzes absolute guarantees under the assumption of deterministic interest rate includes Brennan and Schwartz (1976), Boyle and Schwartz (1977), Boyle and Hardy (1997), and Grosen and Jorgensen (1997, 2000). Other researches conducted by adopting the Vasicek stochastic interest rate model (1977) include Persson and Aase (1997) and Hansen and Miltersen (2002). Miltersen and Persson (1999), Lindset (2003), and Bakken, Lindset and Olson (2006) adopt the Heath-Jarrow-Morton framework (HJM, 1992).
However, granting a deterministic guaranteed rate results in the inability to attract contract participants by a low guaranteed rate, while contract issuers bear financial burdens to
attract contract participants with a high guaranteed rate. Consequently, a stochastic guaranteed rate, such as rate of return guarantees set relative to an interest rate or the rate of return on a mutual fund, has become more popular in recent developments. Despite the popularity of relative rate of return guarantees, especially those issued in Latin America, the relevant research is significantly less in number than absolute guarantees. Only a few articles were written on the relative rate of return guarantees. Ekern and Persson (1996) investigated unit-linked life insurance contracts with different types of relative guarantees. Pennacchi (1999) valued both the absolute and the relative guarantee provided for Chilean and Uruguayan defined contribution pension plans by employing a contingent claim analysis. Both papers assumed that interest rate was deterministic. However, Lindset (2004) analyzed a wide range of different kinds of minimum guaranteed rates of return within the HJM framework. The guaranteed rate of return examined in the above papers was set relative to the rates of return on equity-market assets. Moreover, Yang, Yueh and Tang (2008) extended their analysis to study rate of return guarantees relative to a return measured by market realized δ-year spot rates. The guarantees they examined were applied to all contributions in the accumulation period of a pension plan under the HJM model.
To value the CIRGCs, the cross-currency LIBOR Market Model (CLMM), which is derived by Wu and Chen (2007) via extending an original (single-currency) LIBOR market model (LMM) to a cross-currency LMM (CLMM), is adopted in this research.
There are several incentives to use the CLMM. First, the guaranteed return contracts reflect the volatile nature of rates of return due to the fact that that market interest rates influence any rate of return process. A proper valuation model should consider the stochastic behavior of interest rates. We have to choose a suitable interest rate model for valuation. Pricing CIRGCs under the LMM is more tractable for practice and avoids the problems caused by using other interest rate models. The short rate models, such as the Vasicek model, the Cox, Ingersoll and Ross (CIR) model, and the HJM instantaneous forward rate model have been extensively used for pricing contingent claims. However, some problems should be noted for using the short rate modes or the HJM model: (1) The instantaneous short rate or the instantaneous forward rate is abstract and market-unobservable, and the underlying rate is continuously compounded, thus contradicting the market convention of being discretely compounded on the basis of the LIBOR rates. So the recovery of model parameters from market-observed data is a difficult and complicated task. (2) The pricing formulae of widely traded interest rate derivatives, such as caps, floors, swaptions, etc., based on the short rate models or the Gaussian HJM model are not consistent with market practice. This results in
some difficulties in the parameter calibration procedure. (3) As examined in Rogers (1996), Gaussian term structure models have an important theoretical limitation: the rates can attain negative values with positive probability, which may cause pricing errors in many cases.
Second, the “quanto-effect”should be considered for pricing CIRGCs since the CIRGCs are linked to cross-currency assets. To achieve the goal, the pricing models derived from the CLMM are more adequate and suitable for pricing CIRGCs. If the model setting degenerates to the single-currency case, the pricing CIRGCs model becomes the pricing model of the single-currency interest-rate guaranteed contract in the LMM framework.
Third, the equity-type asset should be studied since the equity-type asset is also included in CIRGCs. The dynamics of equity-type asset is incorporated in the CLMM framework. Under the CLMM, the market is arbitrage-free and complete and contingent claims can be priced by the risk-neutral valuation method.
Our article has several contributions to relative guarantee contracts. First, we use CLMM to derive the pricing formulae for the minimum return guaranteed contracts in which the guaranteed rate is set relative to the level of a stochastic LIBOR rate, which is different from the setting of the previous literatures based on continuous short rates or instantaneous forward rates. The interest rates used in the CLMM are consistent with conventional market quotes. As a result, all the model parameters can be easily obtained from market quotes, thus making the pricing formulae under the CLMM more tractable and feasible for practitioners.
Second, we analyze the cross-currency interest rate guarantee contracts which have not yet been studied in previous researches. The guaranteed contracts are often linked to cross-currency assets in practice. The interest rate guarantee embedded in cross-currency guaranteed contracts can be represented as an option which is equivalent to the quanto-type option in the finance literature. As a result, the quanto-effect will appear in the pricing formulae of CIRGCs. The pricing models derived are more general and suitable for pricing Quanto interest-rate guarantees.
Third, the derived pricing formulae can be directly applied to pricing both maturity guarantees and multi-period interest guarantees with an arbitrary guarantee period. The pricing formula given by Yang, Yueh and Tang (2008) is available only for the guarantee period of one year. A maturity guarantee is binding only at the contract expiration. The cash flows connected to maturity guarantees are closely related to those of European options. For multi-period guarantees, the contract period is divided into several subperiods. A binding guarantee is specified for each subperiod. Many life insurance contracts and guaranteed investment contracts (GIC) sold by investment banks, cf. e.g., Walker (1992), are examples of
multi-period guarantees. In addition, the derived pricing formulae of CIRGCs represent the general formulae for the interest rate guarantee under the CLMM. They can be applied to pricing the guarantees measured by the forward LIBOR rate and those measured by the spot rate which has been commonly used in the previous literature.
Fourth, using our pricing formulae is more efficient than adopting simulation, especially for those guaranteed contracts with long duration such as life insurance products and pension plans. The cross-currency interest rate guarantees embedded in contracts can be valued by recognizing their similarity to various Quanto types of “exotic”options. As a result, the pricing formulae of the CIRGCs within the CLMM framework can be derived via the martingale pricing method. Fifth, we examine the accuracy of the pricing formulae via Monte-Carlo simulation.
The remainder of this paper is organized as follows. Section 2 briefly describes the results of the CLMM extended by Wu and Chen (2007). In Section 3, four different types of CIRGCs are defined and their pricing formulae based on the CLMM are derived. In Section 4, the accuracy of the pricing formulae is examined via Monte-Carlo simulation. In Section 5, the results of the paper are concluded with a brief summary.
2. Arbitrage-Free Cross-Currency LIBOR Market Model
We briefly specify the results of the cross-currency LMM (CLMM) which is extended by Wu and Chen (2007). The CLMM will be utilized to price different types of CIRGOs in Section 3.
Assume that trading takes place continuously in time over an interval
0,, 0 . The uncertainty is described by the filtered spot martingale probability space
,F Q, , Ft t0 ,
where the filtration is generated by independent standard Brownian motions W t
W t1
,W2 t , ...,Wm
t
. Q denotes the domestic spot martingale probability measure. The filtration
Ft t 0 , which satisfies the usual hypotheses represents the flow ofinformation accruing to all the agents in the economy.1 The notations are given below with d for domestic and f for foreign:
,k
f t T = the kth country’sforward interestratecontracted attimet for instantaneous
borrowing and lending at time T with 0 t T , where k
d f, .
,k
P t T = the time t price of the kth country’szero coupon bond (ZCB)paying onedollar at time T.
k
k
r t = the kth country’srisk-free short rate at time t.
k t
=
0
exptr u duk
, the kth country’smoney marketaccountattimet with an initial value k
0 .1
X t = the spot exchange rate at t
0, for one unit of foreign currency expressed in terms of domestic currency.For some 0, T
0, and k
d f, , define the forward LIBOR rate process
L t Tk , ; 0 t T
as given by
,
1 , , k k k P t T L t T P t T exp
,
T k T f t u du
(2.1)Based on the insights of Harrison and Kreps (1979) and Amin and Jarrow (1991; AJ), Wu and Chen (2007) extended the LMM model to a cross-currency case and clarified some conditions of the LIBOR rate process. Under these conditions, the market is arbitrage-free and complete and contingent claims can be priced by the risk-neutral valuation method.2It is important to note that the term structure of interest rates is modeled by specifying the LIBOR rates dynamics, rather than the instantaneous forward rates dynamics. Their results are provided in the following proposition.
Proposition 2.1 THECLMMUNDERTHEMARTINGALEMEASURE
Under the domestic spot martingale measure, the processes of the forward LIBOR rates and the exchange rate are expressed as follows:
, , , , , d Ld Pd Ld d dL t T t T t T dt t T dW t L t T (2.2)
, , , , , f Lf Pf X Lf f dL t T t T t T t dt t T dW t L t T (2.3)
d d Sd d dS t r t dt t dW t S t (2.4)
, f f X Sf Sf f dS t r t t t dt t T dW t S t (2.5)
d
f
X
dX t r t r t dt t dW t X t (2.6)
, , , d d Pd d dP t T r t dt t T dW t P t T (2.7)
, , , , f f k Pf Pf f dP t T r t t t T dt t T dW t P t T (2.8)where t
0,T , T
0, and Pk
t T, , k
d f, , is defined below.
1 1 , , 0, , 0, 0, , 1 , 0 . T t k Lk Pk j k L t T j t T j t T T T t T L t T j otherwise
(2.9)Unlike the abstract short rates in the instantaneous short rate models or the instantaneous forward rates in the HJM model, the forward LIBOR rates in the CLMM are market-observable. Furthermore, the cap pricing formula in the CLMM framework is consistent with the Black formula which is widely used in market practice and makes the calibration procedure easier. As a result, the volatility Lk
t T, , k
d f, , can be invertedfrom the interest rate derivatives traded in the market and Pk
t T, and k
d f, can be calculated from equation (2.9).According to the bond volatility process (2.9),
0,
,
Pk t T t T
is stochastic rather than deterministic. To solve equation (2.2) and (2.3) for L T T , Wu and Chen (2007) fixk
, at initial time s and approximate Pk
t T, by
,s Pk t T given below:
1 1 , , 0, & 0, , 1 , 0 . T t k s Lk Pk j k L s T j t T j t T T t T L s T j otherwise
(2.10)where 0 s t T . Hence, the calendar time of the process
0,
,
k t T j
F t Tj in
(2.10) is frozen at its initial time s and the process
, , s Pk t s T t T becomes deterministic.
By substituting sPk
t T,
for Pk
t T,
into the drift terms of (2.2) and (2.3), the drift and the volatility terms become deterministic, so we can solve (2.2) and (2.3) and find the approximate distribution of L T Tk
, to be lognormal.The Wiener chaos order 0 approximation used in (2.10) is first utilized by BGM (1997) for pricing interest rate swaptions, developed further in Brace, Dun and Barton (1998), and formalized by Brace and Womersley (2000). It also appeared in Schlogl (2002). This approximation has been shown to be very accurate.
The cross-currency LIBOR market model is very suitable and useful for pricing many kinds of quanto interest-rate guarantees. In Section 3, four variants of the cross-currency interest-rate guaranteed contracts are priced based on the CLMM.
3. Valuation of Cross-Currency Interest Rate Guarantee Embedded in
Financial Contracts
In this section, we clarify each type of financial contracts with cross-currency interest rate guarantees which are embedded in financial contracts as options. Then we derive the pricing formulae of four different types of cross-currency interest rate guarantees and the guaranteed contracts based on the cross-currency LIBOR market model. Introduction and analysis of each guarantee are presented sequentially as follows.
3.1 Valuation of First-Type Cross-Currency Interest Rate Guarantee
We define the guaranteed contracts first and then represent the interest rate guarantee as an option.
Definition 3.1.1 A financial contract with the payoff specified in (3.1.1) is called a First-Type
Financial Contract with Cross-Currency Interest Rate Guarantee (FC1CIRG)
1 d f f , 1 d ,
FC T N MaxS T S T L T T (3.1.1)
where
d
N = notional principal of the contract, in units of domestic currency
f
S = the underlying foreign asset price at timeη,
0, , ,t T T
,d
L T T = the domestic T-matured LIBOR rates with a compounding period
,d
P t = the time t price of the domestic zero coupon bond (ZCB) paying one dollar at time λ,
T T,
.T = the start date of the guaranteed contract T = the expiry date of the guaranteed contract
x = Max x
, 0
1 FC T can be rewritten as
1 d f f 1 d , f f FC T N S T S T L T T S T S T (3.1.2)
1 , 1 ,
d d f f d N L T T S T S T L T T (3.1.3)Equation (3.1.2) shows the payoff as the uncertain amount Sf
T Sf T plus the maturity payoff of a put option written on the return of a reference foreign asset with a forward-start exercise price
1Ld
T T,
. Alternatively, equation (3.1.3) indicates the payoff as the sum of the guaranteed amount
1Ld
T T,
and the final payoff of a call option to purchase the return of the reference foreign asset for the price
1 Ld
T T,
that the exercise price, the guaranteed interest rate, is not decided at time t but is to be determined at future time T. For simplicity, the FC1CIRG in (3.1.2) will be used for later
analysis hereafter, which is employed in most guaranteed contracts in practice.
According to (3.1.2), we represent the interest rate guarantee embedded in the FC1CIRG
as an option below.
Definition 3.1.2 An option with the payoff specified in (3.1.4) is called a First-Type
Cross-Currency Interest Rate Guarantee Option (C1IRGO)
1 1 d , f f
C IRGO T L T T S T S T
, (3.1.4)
There are several points worth noting. First, the guarantee of a minimum return,
1 Ld T T,
, is set relative to the LIBOR rate, which is different from the setting of the previous literature that the interest rate is measured by continuous short rates or instantaneous forward rates. In addition, the LIBOR rate is quoted in markets. As a result, the CLMM is more appropriate for pricing C1IRGOs, and all the parameters in the pricing formula can be
easily obtained from market quotes, thus making the pricing formula more tractable and feasible for practitioners.
Second, we extend the analysis on the guaranteed contracts to the case where the underlying asset and the guaranteed interest rate are denominated in different currencies. An C1IRGO is an option on the foreign-currency underlying asset Sf T Sf T with the
domestic-currency exercise price
1 ,
d
L T T
, and its final payments are denominated in domestic currency without directly incurring exchange rate risk. The previous researches on the minimum return guarantees use the assumption that both the underlying asset and the guaranteed interest rate are denominated in a single (domestic) currency. In practice, the guarantees (options) are often linked to a cross-currency asset. This is equivalent to a quanto-type option in the financial literature. As a result, the quanto-effect will be reflected in the pricing model of CIRGOs which is more suitable for pricing Quanto interest-rate guarantees, and if the model setting degenerates to the single-currency case, it reduces to the pricing model of interest rate guarantees in the LMM framework.
Third, the interest rate guarantee is set to begin at some future date T, rather than at current time t, and lasts for δ periods. The “forward-start”exercise price of this option,
guarantee for a minimum return over the period T to T+δ is analogous to a “forward-start” option. Setting the guarantee as the “forward-start”type has a notable advantage that the derived pricing formula of the maturity-guarantee can be directly applied to pricing the multi-period interest guarantees. However, only the pricing formulae for maturity-guarantee are presented for the parsimony sake.3 In addition, the “forward-start”setting represents the general form of the interest guarantees. Specially, the guaranteed interest rate for the period t to t+δ(T=t) as measured by the spot rate in the previous literature can be obtained by setting
T=t. As a result, the pricing formula of CIRGOs represents the general formulae for the
interest rate guarantees under the CLMM and can be applied for pricing the guarantees measured by the spot LIBOR rates. Moreover, our formulae can be derived for arbitrary values of δ. In contrast, the formula of Yang, Yueh and Tang (2008) is available only for the special case where the interest rate guarantee is linked to the one-year spot rate, i.e. δ=1, which will be examined later in Theorem 3.3.2. In addition, the “forward-start”pricing formulae provide more flexibility in the product design of interest-rate guarantees in practice.
Fourth, the cross-currency interest rate guarantees embedded in contracts can be valued by recognizing their similarity to various Quanto types of “exotic”options, such as “forward start options”, “options to exchange one asset for another”, and “options on the maximum of two risky assets”. As a result, the pricing formulae of the CIRGCs within CLMM framework can be derived via the martingale pricing method.
The C1IRGO pricing formula is expressed in the following theorem, and the proof is
provided in Appendix A.
Theorem 3.1.1 The pricing formula of a C1IRGO with the final payoff as specified in (3.1.4)
is expressed as follows:
1 C IRGO t
11 12
12 11 , 1 , 1 , T T t T u du u du d d d f N P t T L t T N d L t T e N d (3.1.5) where
11
12
12 11 12 11 1 1 1 , 1 ln 1 , 2 , T T f t T d L t T u du u du V L t T d d d V V
2
2 2 1 13 14 T T t T V
u du
u du
11 t X t Pf t T, Pd t T, Pf t T, Pf t T,
12 f , d , f , f P P P X S t t t T t T t T t
13 t Pf t T, Pf t T, Pd t T, Pd t T,
14 f , f P S t t t T
, , , s Pk t k d f is defined as (2.10).The pricing equation (3.1.5) resembles the Margrabe (1978) type or the Black-type formula, but in the framework of the cross-currency LMM. Note that the terms, θ11(t) and
θ12(t), appearing in (3.1.5) represent the effects of the exchange rate on pricing, which is
induced by the fact that expected foreign cash flow is expressed under the domestic martingale measure and by the compound correlations between all the involved factors (the exchange rate and the domestic and foreign bonds).
Equation (3.1.5) can be used to price the market value of FC1CIRGs at time t, and the
pricing formula is given in the following theorem, and the proof is provided in Appendix A.
Theorem 3.1.2 The time t market value of FC1CIRGs with the final payoff as specified in
(3.1.1) is given as follows:
1 FC t
11 12
12 11 , 1 , 1 , T T t u du T u du d d d f N P t T L t T N d L t T e N d (3.1.6)Note that the advantage of adopting the cross-currency BGM model rather than other interest-rate models is that all the parameters in (3.1.5) and (3.1.6) can be easily obtained from market quotes, thus making the pricing formula more tractable and feasible for practitioners.
3.2 Valuation of Second-Type Cross-Currency Interest Rate Guarantee
Definition 3.2.1 A financial contract with the payoff specified in (3.2.1) is called a
Second-Type Financial Contract with Cross-Currency Interest Rate Guarantee (FC2CIRG)
2 d d d , 1 f ,
FC T N Max S T S T L T T (3.2.1)
d
S = the underlying domestic asset price at timeη,
0, , ,t T T
,f
L T T = the foreign T-matured LIBOR rate with a compounding period
Similar to FC1CIRG，the maturity payoff of FC2CIRG can be rewritten as follows.
2 d d d 1 f , d d
Definition 3.2.2 An option with the payoff specified in (3.2.3) is called a Second-Type
Cross-Currency Interest Rate Guarantee Option (C2IRGO),
2 1 f , d d
C IRGO T L T T S T S T (3.2.3)
The difference between C2IRGO and C1IRGO is that C2IRGO is written on the domestic
underlying asset, Sd
T Sd T , with the foreign exercise price, 1Lf
T T, . C2IRGObears much resemblance to C1IRGO as mentioned in the previous section.
Next, we begin with pricing the C2IRGO. The resulting formula of the C2IRGO is then
used to value FC2CIRGs. The C2IRGOs pricing formula is given in Theorem 3.2.1 below and
the proof is provided in Appendix B.
Theorem 3.2.1 The pricing formula of C2IRGOs with the final payoff as specified in (3.2.3) is
presented as follows:
21
2 , 1 , 22 1 , 21 T t u du d d f d C IRGO t N P t T L t T e N d L t T N d (3.2.4)
21 22 21 22 21 2 2 1 , 1 ln ( ) 1 , 2 , T d t f L t T u du V L t T d d d V V
2
2 2 2 22 23 T T t T V u du u du
21 t X t Pf t T, Pd t T, Pf t T, Pf t T,
22 t Pd t T, Pd t T, Pf t T, Pf t T,
23 d , d P S t t t T Similar to C1IRGOs, the effect of the exchange rate θ21(t) still appears in (3.2.4), although
the maturity payoff is denominated in domestic currency without directly incurring exchange rate risk. Note that the influence of the exchange rate on C2IRGOs lasts only from period t to
T while it lasts from t to T+δon C1IRGOs. In addition, the foreign-currency denominated
exercise price,
1Lf
T T,
, in C2IRGOs is stochastic from period t to T, but known overthe period from T to T+δ.In contrast, the counterpart asset, Sf
T Sf T , in C1IRGOs isstochastic from period t to T+δ,and hence the exchange rate impact on the C1IRGOs pricing
extends further to the period from T to T+δ.
in the following theorem. The proof is provided in Appendix B.
Theorem 3.2.2 The market value at time t of FC2CIRGs with the final payoff as specified in
(3.2.1) is expressed as follows:
21
2 , 1 , 22 1 , 21 T t u du d d f d FC t N P t T L t T e N d L t T N d (3.2.5)3.3 Valuation of Third-Type Cross-Currency Interest Rate Guarantee
Definition 3.3.1 A financial contract with the payoff specified in (3.3.1) is called a
Third-Type Financial Contract with Cross-Currency Interest Rate Guarantee (FC3CIRG)
3 d f f , 1 f ,
FC T N Max S T S T L T T (3.3.1)
Once again, the expiry payoff of the FC3CIRG is rewritten as follows.
3 d f f 1 f , f f
FC T N S T S T L T T S T S T (3.3.2) Based on (3.3.2), we define the option embedded in the FC3CIRG below.
Definition 3.3.2 An option with the payoff specified in (3.3.3) is called a Third-Type
Cross-Currency Interest Rate Guarantee Option (C3IRGO)
3 d 1 f , f f
C IRGO T N L T T S T S T , (3.3.3)
Different from C1IRGOs and C2IRGOs, an C3IRGO is an option written on the difference
between the return on the foreign underlying asset,Sf
T Sf T , and the foreign interest rate, 1 Lf
T T,
, for period t to T+δ,but the final payment is measured in domestic currency. The holders of this guaranteed contract also have the advantage of avoiding direct exchange rate risk.
Since the C3IRGO can be priced in a similar way as the C2IRGO, we omit the proof for
the sake of parsimony.4
Theorem 3.3.1 The pricing formula of C3IRGOs with the final payoff as specified in (3.3.3) is
presented as follows:
31
31 32
3 32 31 , 1 , 1 , T T T t t T u du u du u du d d f f C IRGO t N P t T L t T e N d L t T e N d (3.3.4) where
2 32 3 31 32 31 3 3 1 2 , T T u du V d d d V V
2 2 3 33 T T V u du
32 f , d , f , f P P P X S t t t T t T t T t
31 t X t Pf t T, Pd t T, Pf t T, Pf t T,
33 f , f P S t t t T Similarly, although the maturity payoff is measured in domestic currency without directly involving the exchange rate, the exchange rate impact still is presented in (3.3.4) as shown by
31 t and 32
t . Note that the exchange rate impact on C3IRGOs lasts for the whole periodfrom t to T+δ.The foreign-currency denominated interest rate,
1Lf
T T,
, in C3IGOs isstochastic from period t to T, but known over the period from T to T+δ,while the stochastic nature of the foreign-currency denominated asset, Sf
T Sf T , prevails over the whole period from t to T+δ.Asaresult,the exchange rateaffectstheC3IGOs pricing in a differentway over the intervals [t, T] and [T, T+δ].
Once again, (3.3.4) is used to price FC3IRGs at time t, and the pricing formula is given in
the following theorem.
Theorem 3.3.2 The market value at time t of FC3IRGs with the final payoff as specified in
(3.3.1) is expressed as follows:
33 31 32
3 , 1 , 32 1 , 31 T T T t t T u du u du u du d d f f FC t N P t T L t T e N d L t T e N d (3.3.5)Yang, Yueh and Tang (2008) have derived under the HJM framework the pricing formulae for interest rate guarantee options, which are written on the underlying difference between the return on a domestic asset and a domestic interest rate, denominated in domestic currency. However, their pricing formula can not be used for pricing the options which are linked to the cross-currency assets. In comparison with their pricing formula, the major difference between Theorem 3.3.2 and their formula lies in the fact that not only the “quanto-effect” is considered in Theorem 3.3.2, but also all the parameters in Theorem 3.3.2 can be extracted from market quotes, which makes our pricing formula more tractable and feasible for practitioners. Besides, their setting of the guaranteed interest rate measured by the spot rate is a special case of our types. Moreover, our formula can be derived for arbitrary values of δ. In addition, their formula derived under the HJM framework is available only for the special
case where the interest rate guarantee is linked to the one-year spot rate, i.e. δ=1, in the pricing of multi-period rate of return guarantee.
3.4 Valuation of Fourth-Type Cross-Currency Interest Rate Guarantee
Definition 3.4.1 A financial contract with the payoff specified in (3.4.1) is called a
Fourth-Type Financial Contract with Interest Rate Guarantee (FC4CIRG)
4 f f f , 1 f ,
FC T X T N Max S T S T L T T (3.4.1)
X T = the floating exchange rate at time T+δ expressed as the domestic currency
value of one unit of foreign currency.
f
N = notional principal of the contract, in units of foreign currency.
The expiry payoff of FC4CIRGs can be expressed as follows.
4 f f f 1 f , f f
FC T X TN S T S T L T T S T S T (3.4.2) We define the option embedded in this contract below.
Definition 3.4.2 An option with the payoff specified in (3.4.2) is called a Fourth-Type
Cross-Currency Interest Rate Guarantee Option (C4IRGO)
4 f 1 f , f f
C IRGO T X TN L T T S T S T , (3.4.3)
From the viewpoint of domestic investors, holding an C4IRGO acts much in the same
way as longing an option, whose payoff is based on the difference between the foreign interest rate and the return on the underlying foreign asset, both denominated in foreign currency. The foreign-currency payoff is converted via multiplying the floating exchange rate into the domestic-currency payoff. The structure of an C4IRGO is different from that of an
C3IRGO in that this option is directly affected by movements in the exchange rate. If the
exchange rate moves upward, a holder of this option may enhance profits from the exchange rate gain when the option is in the money at expiry.
Since the C4IRGO can be priced in a similar way as the C3IRGO, we omit the proof.5The
pricing formula of C4IRGOs is given below.
Theorem 3.4.1 The pricing formula of C4IRGOs with the final payoff as specified in (3.4.2) is
presented as follows:
4 f f , 1 f , 42 1 f , 41
41 , 42 41 4 1 2 d V d d V
2 2 4 4 T T V u du
4 f , f P S t t t T By observing (3.4.3), the option pricing formula is directly affected by unanticipated changes in the exchange rate since the expiry payoff is determined by the spot exchange rate at time T+δ.The pricing formula shows that the option can be first priced under the foreign forward martingale measure and then the foreign-currency fair price is converted via multiplying the time t spot exchange rate X(t) into the domestic-currency market fair value.
Equation (3.4.4) is used to price FC4IRGs, and the pricing formula is represented in the
following theorem.
Theorem 3.4.2 The market value at time t of FC4IRGs with the final payoff as specified in
(3.4.1) is expressed as follows:
4 f f , 1 f , 42 1 f , 41
FC t N X t P t T L t T N d L t T N d (3.4.5)
The above four different pricing formulae of cross-currency interest rate guarantees have been derived. In section 4, we are devoted to some numerical examples.
4. Numerical Analysis
In this section, we examine the accuracy of the derived pricing formulae via a comparison with Monte Carlo simulation. Some practical examples are given to examine the accuracy of the pricing formulae derived in the previous section and compare the results with Monte Carlo simulation. Based on actual 2-year market data, four types of FCIRGs with different guarantee periods are priced at the date, 2008/6/30, and the results are listed in Exhibit 1. The notional value is assumed to be $1. The simulation is based on 50,000 sample paths. The domestic country is the U.S. and the foreign country is the U.K in the examples. The domestic stock index is the Dow Jones Industrials and the foreign index is the FTSE index.
Exhibit 1 show the prices of four types of FCIRGs with δ=1. Observing the numerical results yields several notable points. First, the pricing formulae have been shown to be accurate and robust in comparison with Monte Carlo simulation for the recent market data. Second, the value of FCIRGs decreases with the longer start date T for each type of FCIRGs with a fixed guarantee periodδ. Third, the value of FC4IRGs is higher than those of the other
three FCIRGs since FC4IRG is directly affected by the spot exchange rate. Finally, using the
derived formulae is more efficient than adopting simulation for those guaranteed contracts with long duration such as life insurance products and pension plans.
Exhibit 1. The Prices of Four Types of FCCIRGs with δ=1 Year
FC1CIRG FC2CIRG (t,T,T+δ) FC MC SE FC MC SE (0,1,2) 104.9482% 105.0205% 0.0573% 105.5409% 105.4854% 0.0518% (0,2,3) 100.5584% 100.5033% 0.0530% 101.1014% 101.1056% 0.0516% (0,3,4) 96.1686% 96.1535% 0.0507% 96.6757% 96.6476% 0.0502% (0,4,5) 92.0138% 91.9323% 0.0476% 92.4900% 92.5183% 0.0490% (0,5,6) 88.1651% 88.2050% 0.0458% 88.6139% 88.5980% 0.0473% (0,10,11) 72.4778% 72.4630% 0.0366% 72.8369% 72.8375% 0.0392% (0,15,16) 61.3509% 61.3433% 0.0309% 61.6537% 61.6381% 0.0333% (0,20,21) 53.2200% 53.2309% 0.0270% 53.4808% 53.4672% 0.0288% (0,25,26) 47.0944% 47.1129% 0.0234% 47.3252% 47.3243% 0.0255% (0,30,31) 42.3865% 42.3714% 0.0212% 42.5952% 42.5597% 0.0228% FC3CIRG FC4CIRG (t,T,T+δ) FC MC SE FC MC SE (0,1,2) 106.0262% 105.9490% 0.0539% 201.8597% 201.6028% 0.1859% (0,2,3) 101.0910% 101.1032% 0.0521% 190.2855% 190.4313% 0.1873% (0,3,4) 96.4282% 96.4194% 0.0497% 180.2621% 180.2126% 0.1825% (0,4,5) 92.0806% 92.0704% 0.0473% 171.5524% 171.4857% 0.1785% (0,5,6) 88.0760% 88.0850% 0.0456% 164.0535% 164.0666% 0.1743% (0,10,11) 72.2113% 72.2085% 0.0372% 136.8541% 136.7752% 0.1616% (0,15,16) 61.1070% 61.1089% 0.0314% 118.9955% 118.8971% 0.1583% (0,20,21) 52.9729% 52.9796% 0.0271% 106.6352% 106.7059% 0.1616% (0,25,26) 46.8762% 46.8679% 0.0239% 97.8470% 97.7910% 0.1686% (0,30,31) 42.2142% 42.2089% 0.0217% 91.1130% 91.0398% 0.1795%
5. Conclusions
Four different types of CIRGOs and FCCIRGs have been developed via the cross-currency LMM. The guaranteed contracts with the underlying asset and the guaranteed interest rate denominated in different currencies have been analyzed, and the guaranteed rate is set relative to the level of the LIBOR rate. The pricing formulae derived are more consistent with market practice than those given in the previous researches. They can also be applied to both maturity-guarantees and multi-period guarantees with an arbitrary guarantee The abbreviations FC, MC and SE represent the results of the formula, Monte Carlo simulations, and the standard error, respectively. The current time, the start date, and the expiry date of the guaranteed contract are represented by t, T and T+δ, respectively.
period δ. The derived pricing formulae represent the general formulae of the Margrabe (1978) type or the Black type in the framework of the cross-currency LMM and are easy for practical implementation. In addition, the pricing formulae have been shown numerically to be very accurate as compared with Monte-Carlo simulation. Pricing the guaranteed contracts with the derived formulae can be executed more efficiently than by adopting simulation, especially for the guaranteed contracts with a long duration such as life insurance or pension plans. Thus, the pricing formulae of FCCIRGs derived under the cross-currency LIBOR market model are more tractable and feasible for practical implementation.
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計畫成果自評
We have completed the research program as the plan:
1. We derive the pricing formulae of CIRGCs by adopting the martingale method and the cross-currency LIBOR market model.
2. Monte Carlo simulation is provided to examine the accuracy of the CIRGCs pricing formulae.
3. The calibration of parameters is also discussed in the research.
4. The research has been written in English and submitted to the journal listed in
