評價連結隨機保證報酬率之保證價值 - 政大學術集成

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(1)國立政治大學金融研究所博士論文指導教授 : 陳松男博士政治大. 立. ‧ 國. 學 ‧. 評價連結隨機保證報酬率之保證價值 y. Nat. sit. n. al. er. io. Pricing Guarantees Linked to Stochastic Guaranteed Rates of Return Ch. engchi. i n U. v. 研究生: 謝宗佑中華民國九十九年十一月.

(2) Abstract We derive the pricing formulas for the guarantees embedded in defined contribution (DC) pension plans with the guaranteed minimum rate of return set relative to a LIBOR interest rate. The guaranteed rate associated with a stochastic LIBOR interest rate has not yet been studied in the relevant literature, particularly in the presence of stochastic interest rates. An extended LIBOR market model (LMM) is employed to price the interest rate guarantees embedded in DC pension plans under maturity and multi-period guarantees. The pricing formulas derived under the extended LMM are more tractable and feasible for practice than those derived under the. 治政 for practical implementation. Monte Carlo simulation is 大 provided to evaluate the accuracy of 立 the theoretical results. instantaneous short rate models or the HJM model. Calibration procedures are also discussed. ‧ 國. 學. Keywords: Guarantee, Guaranteed rate of return, Stochastic, LIBOR market model, Defined. ‧. contribution pension plans. n. er. io. sit. y. Nat. al. Ch. engchi. i n U. v.

(3) Contents 1 Introduction. 3. 2 Guarantees and Economic Model. 8. 2.1. DC plans with Guarantees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 2.2. LIBOR Market Model (LMM) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10. 2.3. The Extended LIBOR Market Model . . . . . . . . . . . . . . . . . . . . . . . . 13. 3 Valuation of Relative Interest Rate Guarantees. 8. 16. 政治大 Valuation of the Second-Type Guarantee (Multi-Period Guarantee) 立. 3.1. Valuation of the First-Type Guarantee (Maturity Guarantee) . . . . . . . . . . . 16. 3.2. . . . . . . . 19. ‧ 國. 學. 4 Calibration Procedure and Numerical Examples. 21. Calibration Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21. 4.2. Numerical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23. sit. y. Nat. 5 Conclusions. ‧. 4.1. n. al. er. io. Appendix A: Proof of Theorem 3.1. Ch. Appendix B: Proof of Theorem 3.2. engchi. Bibliography. i n U. v. 26 27 34 39. 2.

(4) 1. Introduction. In the latest development, a number of pension reforms have been completed by privatizing pension obligations. The retirement pension system has been gradually converted from traditional defined benefit (DB) pension plans to defined contribution (DC) plans since DB plans suffer from critical financial burdens. As a result, DC plans have become increasingly popular throughout the world. In addition, the retirement benefits of participants in a DB plan are warranted in terms of some pre-specified formulas. Hence, employers are exposed to the risk of poor investment performance of pension funds. In contrast, employees in a DC plan bear investment risk, because the retirement benefits rely on the performance of investment port-. 政治大. folios. The primary argument for converting from DB to DC is that employees in DC plans. 立. run a risk of experiencing poor investment performance, possibly leaving them with insufficient. ‧ 國. 學. wealth during their retirement years. Therefore, to reduce employees’ exposure to investment risk, some guarantees have been provided with DC plans and governments usually bear the. ‧. subsidy for these guarantees. Evaluating a guarantee is quite important for the government to budget for the implicit subsidy, because an inadequate estimation may lead to the suffering. y. Nat. er. io. plans is warranted.. sit. of critical financial problems. As a result, further analysis for the values of guarantees in DC. al. n. v i n C guarantees, where the minimum rate ofhreturn e n gis csethtoi beUdeterministic. The other is so-called. There is a variety of guarantee designs in DC plans. One class of these guarantees is absolute. relative guarantees in the literature (Lindset, 2004), where the minimum guaranteed rate of return is linked to a stochastic interest rate (called interest rate guarantee hereafter), or the rate of return on a reference portfolio, an index, a specific asset traded in financial markets, etc. Previous research on valuing guarantees for pension funds or life insurance products has focused on absolute guarantees, under which a constant or predetermined minimum rate of return is contributed to participants. The existing literature that 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). Persson and Aase (1997) and Hansen and Miltersen (2002) employ the Vasicek (1977) stochastic 3.

(5) interest rate model. Miltersen and Persson (1999), Lindset (2003), and Bakken et al. (2006) adopt the Heath-Jarrow-Morton framework (HJM, 1992). However, a problem arising from awarding a deterministic guaranteed rate is the inability to attract plan participants with a low guaranteed rate, while plan providers suffer from financial burdens with a high guaranteed rate. As a result, a stochastic guaranteed rate, which is set relative to an interest rate or the rate of return on a mutual fund, has become more popular in recent years.1 Despite the growing popularity of relative rate of return guarantees, the analysis devoted to relative guarantees is significantly less in number than absolute guarantees. Only a few papers were written on the relative rate of return guarantees. Ekern and Persson. 治政 Pennacchi (1999) evaluated both the absolute and the relative 大 guarantee provided for Chilean 立 and Uruguayan DC pension plans by using a contingent claim analysis. The interest rate was (1996) studied unit-linked life insurance contracts with different types of relative guarantees.. ‧ 國. 學. assumed to be deterministic in both articles. Nevertheless, Lindset (2004) utilized the HJM framework to value a wide range of different kinds of minimum guaranteed rates of return.. ‧. The guaranteed rate of return examined in the above three papers was set relative to the. y. Nat. rates of return on equity-market assets. Besides, Yang et al. (2008) studied rate of return. io. sit. guarantees relative to a return measured by market realized δ-year spot rates. The guarantees. n. al. er. they examined were applied to all contributions in the accumulation period of a pension plan under the HJM model.. Ch. engchi. i n U. v. There is little work focusing on relative guarantees involved with stochastic interest rates. Because the guaranteed return designs reflect the volatile nature of rates of return, a proper valuation model should consider the stochastic behavior of interest rates in accordance with the real economic environment. 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 applied to pricing contingent claims. But some problems associated with employing 1. Examples of such contracts can be observed in pension plans and unit-linked life insurance contracts. The countries which provide pension plans with a stochastic guaranteed rate include Chile, Colombia, Peru and Argentina (see e.g., Pennacchi, 1999; Lindset, 2004). Ekern and Persson (1996) analyse a number of unit-linked contracts with stochastic guaranteed rates. Exhibit 1 shows the statistics regarding the unit-linked products provided by the European Insurance and Reinsurance Federation (CEA). From the statistics, the European life insurance market in 2006 was characterized by a significant rise and percentage in the share of unit-linked contracts in total life premium.. 4.

(6) the short rate models or the HJM model should be noted. First, the instantaneous short rate or the instantaneous forward rate is abstract and marketunobservable, and the underlying rate is continuously compounded, which contradicts with the market convention of being discretely compounded on the basis of the LIBOR rates. So it is complicated and difficult to recover model parameters from market-observed data. Second, the pricing formulas of extensively 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 leads to some difficulties in parameter calibration. In addition, as examined in Rogers (1996), the rates under Gaussian term structure models can become negative with a. 政治大. Exhibit 1: Share of Unit-Linked Contracts in Total Life Premium. ‡. n. †. 2005. 2006. 2005. 11,610 294 1,592 84 199 3,058 542 5,400 12,661 8,521 69,377 2,311 171 196,346 140,203 1,663 20,488 12,471 22,472 7,442 74,714 17,847 293 25,800 23 25,355. 9,831 274 1,218 81 221 3,205 465 3,779 13,832 8,979 73,471 1,935 137 166,455 120,668 1,508 25,255 11,007 20,532 7,561 72,636 19,229 256 24,800 23 20,890. 84.6 72.8 58.9 57.1 55.2 52.3 52.0 46.1 44.5 41.7 39.5 34.0 29.4 25.2 24.7 23.8 20.0 18.1 15.1 12.4 12.1 8.1 4.8 ‡ n.a. n.a. n.a.. 82.6 72.8 44.3 52.1 54.5 41.3 42.9 36.9 40.6 37.1 35.9 27.3 23.0 24.0 20.2 16.7 25.5 12.7 11.5 8.2 10.8 11.1 2.8 23.8 14.7 n.a.. 26.0. 23.7. Ch. engchi. 660,937. 608,249. y. sit. io. CEA (Total). al. 2006. er. Nat. Luxembourg Cyprus Hungary Estonia Romania Finland Slovenia Poland Sweden Portugal Italy Greece Malta United Kingdom France Czech Republic Belgium Denmark Spain Norway Germany Switzerland Croatia The Netherlands Latvia Others. Share-UL(%). ‧. ‧ 國. Country. †. Life Premium (Euro million). 學. 立. Source: CEA. i n U. v. ”Share-UL” represents the share of unit-linked contracts in total life premium. ”n.a.” denotes ”not available”.. 5.

(7) positive probability, which may cause pricing errors. This research attempts to extend the previous analysis to set up a theoretical framework to analyze the guaranteed minimum return of DC plans set relative to a LIBOR interest rate. It is different from previous research on relative guarantees with guaranteed rate set to an instantaneous forward rate or an instantaneous short rate. This research employs the popular LIBOR Market Model (LMM) to value the interest rate guarantee embedded in a DC plan (IRGEIDCP). The LMM has been developed by Musiela and Rutkowski (1997), Miltersen et al. (1997), and Brace, Gatarek and Musiela (BGM) (Brace et al., 1997). Pricing interest rate guarantees under the LMM is more tractable for practice and avoids the problems exhibited in. 治政 the dynamics of interest rate and the return on a reference大 portfolio must be incorporated into 立 the valuation model. Hence, the LMM should be extended to capture the return process of a. the other interest rate models as mentioned earlier. Moreover, to value the IRGEIDCPs, both. ‧ 國. 學. reference portfolio for deriving the pricing formulas. As a result, the pricing models derived from the extended LMM are more adequate and suitable for pricing the IRGEIDCPs. In. ‧. addition, we derive the pricing formulas for the IRGEIDCPs under both maturity and multi-. y. Nat. period guarantees via the martingale pricing method. (The IRGEIDCPs under maturity and. io. sit. multi-period guarantees are, respectively, called the first-type and the second-type guarantees.). n. al. er. Rate-of-return guarantees are embedded in these two fundamentally different types. Maturity. i n U. v. guarantees are binding only at contract expiration. In contrast, the contract period of multi-. Ch. engchi. period guarantees 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. Our article has several contributions to the literature on relative guarantees, particularly in the presence of stochastic interest rates. First, we use the LMM to derive the pricing formulas for the IRGEIDCP with guaranteed rate set relative to the level of a stochastic LIBOR rate. The interest rates used in the LMM are consistent with conventional market quotes. Therefore, all model parameters can be easily obtained from market quotes, thus making the pricing formula under the LMM more tractable and feasible for practitioners. Second, we derive the pricing formulas of IRGEIDCPs under both maturity and multiperiod interest guarantees with an arbitrary guarantee period. In contrast, the pricing formula 6.

(8) of IRGEIDCPs under multi-period guarantees given by Yang et al. (2008) is based on the HJM framework and available only for the one-year guarantee period. Third, using our pricing formulas to value IRGEIDCPs is more efficient than adopting time-consuming simulation, since pension plans usually have a long duration. Fourth, the calibration procedure is provided for practical implementation and the accuracy of the pricing formulas is examined via Monte Carlo simulation. This article is structured as follows. Section 2 describes the structure of the DC plans under maturity and multi-period guarantees and represents the economic environment. In addition, the LMM and its extension are also introduced. In Section 3, the IRGEIDCPs under maturity. 治政 are derived. In Section 4, the calibration procedure for practical 大 implementation is provided 立 and the pricing result is examined via Monte Carlo simulation. In Section 5, the results of this. and multi-period guarantees are defined and their pricing formulas based on the extended LMM. ‧ 國. 學. paper are concluded with a brief summary.. ‧. n. er. io. sit. y. Nat. al. Ch. engchi. 7. i n U. v.

(9) 2. Guarantees and Economic Model. In subsection 2.1, we introduce the framework of DC pension plans with guaranteed rates set relative to LIBOR rates. In subsection 2.2, the LIBOR market model is briefly introduced and extended to incorporate a reference investment portfolio, thus leading to an extended LMM. The extended LMM is then used to price two different types of guarantees in Section 3.. 2.1. DC plans with Guarantees. We describe the DC pension plans with interest rate guarantees under maturity and multiperiod guarantees. In addition, the guaranteed rates of return of DC plans are set relative to LIBOR rates.. 立. 政治大. Assume that T0 , T1 , . . . , TN ∈ [0, τ ] with 0 ≤ t ≤ T0 ≤ T1 . . . ≤ TN ≤ τ . In accordance with. ‧ 國. 學. practice, we define δ = Ti − Ti−1 , i = 1, 2, . . . , N and T0 − t ≤ δ. An employee contributes a constant proportion of his salary to the pension fund in each period and the employee’s salary. ‧. is expected to increase each period. The payoff of a DC plan with an interest rate guarantee under the maturity guarantee is defined as follows.. sit. y. Nat. io. er. Definition 2.1. P FTN (I) is defined as the terminal account value of a DC plan with an interest rate guarantee under the maturity guarantee at the retirement date. The payoff of P FTN (I) is. al. n. given as follows.. P FTN (I) =. N X. Ch. (. m · w0 (1 + g)n−1 max. n=1. =. N X n=1. engchi. ( m · w0 (1 + g)n−1. " N −1 Y ¡ i=n−1. ". i n U. v. −1 ¢ NY STi+1 δ 1 + δL (Ti , Ti ) , STi i=n−1. #) (2.1.1). # #) N −1 Y S S T T i+1 i+1 ,0 + . max (1 + δLδ (Ti , Ti )) − S S T T i i i=n−1 i=n−1 i=n−1 " N −1 Y. N −1 Y. (2.1.2) where. 8.

(10) m. = the contribution rate (i.e. the rate of salary contributed to the pension fund).. w0. = the initial wage.. g Sη. = the wage growth rate in each period. = the underlying investment portfolio price at time η (S(η) is an alternative representation of Sη , i.e. Sη = S(η)).. STi+1 /STi. = the actual rate of return on the investment portfolio in period (Ti+1 −Ti ) = δ.. Lδ (Ti , Ti ). = the Ti -matured LIBOR rates with a compounding period δ.. The maturity guarantee is binding only at the contract expiration. The cash flow connected to the maturity guarantee is closely related to that of a European option. The account value. 政治大. of a pension fund depends on the contribution, the performance of an investment portfolio. 立. (STi+1 /STi ), the guaranteed rate (1 + δLδ (Ti , Ti )), and the working period. Note that the. ‧ 國. 學. guaranteed rate of a DC plan is set relative to a LIBOR interest rate. It is different from previous research on the relative guarantee with guaranteed rate set to an instantaneous forward. ‧. rate, an instantaneous short rate, or the return rate of an equity-type traded asset. Next, the payoff of a DC plan with an interest rate guarantee under the multi-period guar-. y. Nat. er. io. sit. antee is defined as follows.. Definition 2.2. P FTN (II) is defined as the terminal account value of a DC plan with an. n. al. Ch. i n U. v. interest rate guarantee under the multi-period guarantee. The payoff of P FTN (II) is given as follows.. P FTN (II) =. N X. ( m · w0 (1 + g)n−1. n=1. =. N X n=1. ( m · w0 (1 + g). engchi. n−1. N −1 Y. max. i=n−1. "Ã N −1 Y. µ. · ¡. ¢ ST 1 + δLδ (Ti , Ti ) , i+1 STi. ST max (1 + δL (Ti , Ti )), i+1 STi i=n−1 δ. ¶. N −1 Y. STi+1 − STi i=n−1. ¸) (2.1.3). !. N −1 Y. STi+1 + STi i=n−1. #). (2.1.4) The contract period is divided into several subperiods for multi-period guarantees. The contract specifies a binding guarantee for each subperiod. From Definition 2.1 and 2.2, both the actual return of the underlying reference investment portfolio (STi+1 /STi ) and the minimum 9. ..

(11) guaranteed rate of the δ-year spot rate (1 + δLδ (Ti , Ti )) at time Ti determine the payoffs of the guaranteed DC pension plans. In the next subsection, we introduce the economic environment and the LIBOR market model.. 2.2. LIBOR Market Model (LMM). We briefly introduce the LIBOR market model (LMM). BGM (1997) developed the LMM based on the HJM model. BGM used much the same mathematical framework as employed by HJM. Hence, we review the HJM economy as follows before we introduce the LMM.. 政治大. Assume that trading takes place on a continuous basis in the time interval [0, τ ], for some fixed horizon 0 < τ < ∞. The uncertainty is described by the filtered probability space. 立. (Ω, F, Q, {Ft }t∈[0,τ ] ). The filtration {Ft }t∈[0,τ ] is the Q-augmentation of the filtration generated. ‧ 國. 學. by independent standard Brownian motions W (t) = (W1 (t), W2 (t), . . . , Wd (t)). Q represents the spot martingale probability measure. The filtration {Ft }t∈[0,τ ] denotes the flow of informa-. ‧. tion accruing to all the agents in the economy. The notations are given below:. y. Nat. the forward interest rate contracted at time t for instantaneous borrowing. P (t, T ). and lending at time T with 0 ≤ t ≤ T ≤ τ . h R i T = exp − t f (t, u)du , the time t price of a zero coupon bond (ZCB) paying. n. al. er. sit. =. io. f (t, T ). Ch. one dollar at time T .. engchi. i n U. v. r(t). = f (t, t), the risk-free short rate at time t. hR i t β(t) = exp 0 r(u)du , the money market account at time t with an initial value β(0) = 1. The spot martingale measure is the probability measure with respect to the numeraire β(t). Under the spot martingale measure, the relative price of a zero coupon bond (ZCB), P (t, T )/β(t), is a martingale. The two dynamics under the spot martingale measure in the HJM model are summarized in the following proposition. 2.2.1. The Dynamics under the Spot Martingale Measure in the HJM Model (1992) 10.

(12) For any T ∈ [0, τ ], the dynamics of the forward rates and the ZCB prices under the spot martingale measure Q are given as follows: df (t, T ) = σ(t, T ) · σP (t, T )dt + σ(t, T ) · dW (t), dP (t, T ) = r(t)dt − σP (t, T ) · dW (t), P (t, T ). (2.2.1) (2.2.2). where σ(t, T ) = (σ1 (t, T ), . . . , σd (t, T )) is the forward rate volatility and σi : Ω×{(t, s) : 0 ≤ t ≤ RT s ≤ T } → R is jointly measurable, adopted and 0 |σi (u, T )|du < ∞ a.e. Q for i = 1, . . . , d. RT And σP (t, T ) = (σP 1 (t, T ), . . . , σP d (t, T )) is the volatility of ZCBs with σP i (t, T ) = t σi (t, u)du for i = 1, . . . , d for all T ∈ [0, τ ].2. 政治大. Based on the economy and the arbitrage-free results of the HJM model, the deriving pro-. 立. cedure of the LMM is briefly introduced hereunder.3 Note that the LIBOR rate dynamics,. ‧ 國. of interest rate.. 學. rather than the instantaneous forward rates dynamics, is used to establish the term structure. Nat. P (t, T ) 1 + δL (t, T ) = = exp P (t, T + δ). µZ. T +δ. ¶ f (t, u)du .. io. sit. δ. y. given by. ‧. For some δ > 0, T ∈ [0, τ ], define the forward LIBOR rate process {Lδ (t, T ) : 0 ≤ t ≤ T } as. T. (2.2.3). n. al. er. Under the measure Q, Lδ (t, T ) is assumed to have a lognormal volatility structure and its stochastic process is given by. Ch. engchi. i n U. v. dL (t, T ) = µL (t, T )dt + Lδ (t, T )γ(t, T ) · dW (t), δ. (2.2.4). where γ(·, T ) : [0, T ] → Rd is a deterministic, bounded, and piecewise continuous volatility function and µL (t, T ) : [0, T ] → R is some unclarified drift function. The drift term µL (t, T ) in the LIBOR rate process can be further determined and the dynamics in (2.2.4) can be written as follows: dLδ (t, T ) = Lδ (t, T )γ(t, T ) · σP (t, T + δ)dt + Lδ (t, T )γ(t, T ) · dW (t). 2. (2.2.5). See HJM (1992) for the regularity conditions on the drift and volatility terms and more details. A further description regarding the LMM can be found in advanced textbooks in finance, see, e.g. Svoboda (2004) and Musiela and Rutkowski (2005). 3. 11.

(13) The bond volatility σP (t, T ) must be specified to fit the arbitrage-free condition in HJM and is given as follows:  −1 bδ P (T −t)c  δLδ (t, T − kδ)   γ(t, T − kδ) t ∈ [0, T − δ]   1 + δLδ (t, T − kδ)  k=1 & T − δ > 0, σP (t, T ) =       0 otherwise,. (2.2.6). where bδ −1 (T − t)c denotes the greatest integer that is less than δ −1 (T − t). As a result, the evolution of the LIBOR rate under the martingale measure is given by (2.2.5) and σP (t, T + δ) is defined by (2.2.6).. 政治大. According to the bond volatility process (2.2.6), {σP (t, T + δ)}t∈[0,T +δ] is stochastic rather. 立. than deterministic. To solve equation (2.2.5) for the distribution of Lδ (T, T ), BGM (1997). ‧. ‧ 國. 學. approximated σP (t, T ) by σ ¯P (t, T ) at any fixed initial time s, and given by  −1 bδ P (T −t)c  δLδ (s, T − kδ)   γ(t, T − kδ) t ∈ [0, T − δ]   1 + δLδ (s, T − kδ)  k=1 & T − δ > 0, σ ¯P (t, T ) =       0 otherwise,. Nat. al. er. io. sit. y. (2.2.7). n. where 0 ≤ s ≤ t ≤ T ≤ τ . Hence, the calendar time of the process {Lδ (t, T )}t∈[s,T ] in. Ch. (2.2.7) is frozen at its initial time s and the process {¯ σP. engchi. i n (t, UT )}. v. t∈[s,T ]. becomes deterministic. By. substituting σ ¯P (t, T ) for σP (t, T ) into the drift term of (2.2.5), the drift and the volatility terms become deterministic, so we can solve (2.2.5) and find the approximate distribution of Lδ (T, T ) to be lognormal. This Wiener chaos order 0 approximation used in (2.2.7) is first utilized by BGM (1997) for pricing interest rate swaptions, developed further in Brace et al. (1998), and formalized by Brace and Womersley (2000). It also appeared in Schlögl (2002) and Wu and Chen (2007a, 2007b). With a striking difference from 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 LMM are market-observable. Moreover, the cap pricing formula in the LMM framework is consistent with the Black formula which is extensively employed in market practice and hence. 12.

(14) makes the calibration procedure easier. As a result, the volatility can be inverted from interest rate derivatives traded in markets and can be calculated from equation (2.2.6).. 2.3. The Extended LIBOR Market Model. To value interest rate guarantees embedded in DC plans, not only the dynamics of interest rates but also a reference portfolio’s dynamics must be incorporated into the pricing model. We begin with an extended HJM model. Wu and Chen (2007a, 2007b) extended the original HJM term structure economy to include an arbitrary number of risky assets and clarified some conditions, under which the market is. 政治大. arbitrage-free and complete and contingent claims can be priced by the martingale valuation method.. 立. Assume that the economic environment of the original HJM model and the notations are the. ‧ 國. 學. same as those presented in Subsection 2.2. Consider that there are d bonds of maturities Ti for i = 1, 2, ..., d. Based on the result of Wu and Chen (2007a, 2007b), the original HJM economy. ‧. can be expanded to include an additional risky asset, i.e. a reference portfolio S(t). Thus,. y. Nat. the new probability space is (Ω, F, Q, {Ft }t∈[0,τ ] ) where the filtration {Ft }t∈[0,τ ] is generated by. n. al. er. io. D = d + 1.. sit. the D-dimensional independent standard Brownian motion W (t) = (W1 (t), . . . , WD (t)), where. i n U. v. Under the measure Q, the dynamics of the reference portfolio S(t) is assumed to have a. Ch. engchi. lognormal volatility structure and its stochastic process is given by dS(t) = µS (t)dt + φ(t) · dW (t), S(t). (2.3.1). where µS (t) and φ(t) = (φ1 (t), . . . , φD (t)) satisfy some standard regular conditions.4 The dynamics of the instantaneous forward rates, the ZCB prices and the reference portfolio under the equivalent spot martingale measure can be represented in the following proposition. 2.3.1. The Dynamics under the Spot Martingale Measure For any T ∈ [0, τ ], the dynamics of the forward rates, the ZCB prices and the reference φi (t) : [0, τ ] → R is deterministic for i = 1, 2, . . . , D. µS (t) : [0, τ ] → R is adopted, jointly measurable and Rτ satisfies E[ 0 |µS (u)|2 du] < ∞. 4. 13.

(15) investment portfolio under the spot martingale measure Q are given as follows: df (t, T ) = σ(t, T ) · σP (t, T )dt + σ(t, T ) · dW (t), dP (t, T ) = r(t)dt − σP (t, T ) · dW (t), P (t, T ) dS(t) = r(t)dt + φ(t) · dW (t), S(t). (2.3.2) (2.3.3) (2.3.4). where σ(t, T ) : [0, τ ] → RD is the forward rate volatility and σP (t, T ) =. RT t. σ(t, u)du.5. Note that even if the reference portfolio dynamics is incorporated into the HJM model, the dynamics of the forward rates and the ZCB prices under the spot martingale measure remain unchanged as those in the HJM model. The drift term of S(t) is composed of the short rate. 政治大. r(t).. 立. By using the results of Wu and Chen (2007a, 2007b), an extended LMM which incorporates. ‧ 國. 學. the dynamics of the reference portfolio can be developed as follows. First, by employing (2.3.2), (2.3.3) and the approach for deriving the LMM as mentioned in Subsection 2.2, we can obtain. ‧. the LIBOR rate dynamics under the spot martingale measure to be the same as (2.2.5) and σP (t, T + δ) is still defined as (2.2.6). Next, by Proposition 2.2, the dynamics of the reference. y. Nat. sit. portfolio S(t) and the ZCBs under the spot martingale measure remain to be represented,. al. n. following proposition.. er. io. respectively, by (2.3.4) and (2.3.3). As a result, the extended LMM can be summarized in the. Ch. engchi. i n U. v. 2.3.2. The Dynamics under The Spot Martingale Measure in the Extended LIBOR Market Model For any T ∈ [0, τ ], the dynamics of the LIBOR rates, the ZCB prices and the reference investment portfolio under the spot martingale measure Q are given as follows: dLδ (t, T ) = γ(t, T ) · σP (t, T + δ)dt + γ(t, T ) · dW (t), Lδ (t, T ). (2.3.5). dP (t, T ) = r(t)dt − σP (t, T ) · dW (t), P (t, T ). (2.3.6). dS(t) = r(t)dt + φ(t) · dW (t), S(t). (2.3.7). where σP (t, T ) is defined as (2.2.6).6 5 6. See Amin and Jarrow (1991) and Wu and Chen (2007a, b) for more details. See Wu and Chen (2007a, b) for more details.. 14.

(16) The extended LMM is more suitable for pricing the IRGEIDCPs than other interest rate models, as indicated earlier. In Section 3, two variants of the IRGEIDCPs are priced based on the extended LMM.. 立. 政治大. ‧. ‧ 國. 學. n. er. io. sit. y. Nat. al. Ch. engchi. 15. i n U. v.

(17) 3. Valuation of Relative Interest Rate Guarantees. We first define each type of interest rate guarantees embedded in DC pension plans (IRGEIDCPs). Then we derive the pricing formula of each type of interest rate guarantees based on the extended LMM. Each type of guarantees is presented sequentially as follows.. 3.1. Valuation of the First-Type Guarantee (Maturity Guarantee). Assume that T0 , T1 , . . . , TN ∈ [0, τ ] with 0 ≤ t ≤ T0 ≤ T1 ≤ . . . ≤ TN ≤ τ and T0 − t ≤ δ. Definition 3.1. GTN (I) is defined as the terminal account value of the first-type guarantee (i.e.. 政治大. the interest rate guarantee embedded in DC plans under maturity guarantees) with the payoff specified as follows.. m · w0 (1 + g)n−1. Ã N −1 Y. !#) S Ti+1 max (1 + δLδ (Ti , Ti )) − ,0 STi i=n−1 i=n−1 N −1 Y. ‧. (n). m · w0 (1 + g)n−1 ψTN (I) (n). (3.1.2). y. Nat. n=1. (3.1.1). sit. =. N X. ". ‧ 國. n=1. (. 學. GTN (I) =. N X. 立. Definition 3.2. ψTN (I) is defined as the time TN value of the first-type guarantee for $1. io. n. al. er. contributed in the n-th year, n ≤ N , with the payoff specified as follows.. Ch. engchi. i n U. v. # S T (n) i+1 (1 + δLδ (Ti , Ti )) − ,0 ψTN (I) = max S T i i=n−1 i=n−1 # " N −1 Y S TN = max (1 + δLδ (Ti , Ti )) − ,0 STn−1 i=n−1 " N −1 Y. where. N −1 Y. (3.1.3). (3.1.4). N −1 Y. STi+1 STN = . STi STn−1 i=n−1. By observing Definition 3.1 and 3.2, the guarantee of a minimum return, (1 + δLδ (Ti , Ti )), is set relative to the LIBOR rate, which is different from the setting of the previous literature where the interest rate is measured by continuous short rates or instantaneous forward rates. As a result, the extended LMM is more appropriate to price interest rate guarantees. Besides, the interest rate guarantees embedded in defined contribution pension plans can be valued by 16.

(18) recognizing their similarity to various 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”. In addition, the pricing formulas of the guarantees within the extended LMM framework can be derived via the martingale pricing method. The pricing formula of the first-type guarantee is given in the following theorem, and the proof is provided in Appendix A. Theorem 3.1. The pricing formulas of the first-type guarantee with the final payoff as specified in Definition 3.1 and 3.2 are given as follows:. Gt (I) =. n−1. 0. (n) t. n=1. ³ ´ ³ ´ (n) (n) (n) (n) (n) ψt (I) = Z1,t N d1 − Z2,t N d2. ‧. (n). Z1,t = P (t, Tn−1 ) · exp(a(n)),. y. Nat. (n). (3.1.5). 學 sit. Z2,t = P (t, Tn−1 ),. io. #−1 "n−2 Y (1 + δLδ (t, Tk )) , P (t, Tn−1 ) = [1 + (T0 − t)L(t, T0 )]−1. n. al. er. where. ‧ 國. 立. 政m · w (1治 + g) 大 ψ (I). N X. a(n) =. Z. N −1 X i=n−1. +. Ti. Ch. k=0. engchi. i n U. v. γBi (u) · (γBi (u) − γAi (u))du. t. (N −1)−1 N −1 X X i=n−1 j=i+1. Z. Ti. (γAi (u) − γBi (u)) · (γAj (u) − γBj (u))du,. t. γAi (t) = [−¯ σP (t, Ti ) + σ ¯P (t, TN )], σP (t, Ti+1 ) + σ ¯P (t, TN )], γBi (t) = [−¯ N (·) = the cumulative normal probability, (n) d1. =. 2 a(n) + 21 V(n). V(n). (n). (n). , d2 = d1 − V(n) , V(n) =. 17. q 2 V(n) ,. (3.1.6).

(19) 2 = Var(X(n)) + Var(Y (n)) − 2Cov(X(n), Y (n)), V(n). Var(X(n)) =. Z. N −1 X. kγAi (u) − γBi (u)k2 du. t. i=n−1. +2. Ti. (N −1)−1 N −1 X X i=n−1 j=i+1. Z. Tn−1. Var(Y (n)) =. Z. Ti. ¡. ¢ ¡ ¢ γAi (u) − γBi (u) · γAj (u) − γBj (u) du,. t. Z. kγCn−1 (u)k2 du. TN. +. t. kγE (u)k2 du,. Tn−1. γCn−1 (t) = [−¯ σP (t, Tn−1 ) + σ ¯P (t, TN )], γE (t) = [φ(t) + σ ¯P (t, TN )], Cov(X(n), Y (n)) =. N −1 X. Z. i A. 立Z. +. t. N −1 X. Ti. i=n−1. Tn−1. ¡. i B. n−1 C (u)du. ¢ γAi (u) − γBi (u) · γE (u)du. 學. ‧ 國. i=n−1. ¡ 治 ¢ 政 γ (u) − γ (u) · γ 大. Tn−1. σ ¯P (t, ·) is defined as (2.2.7). L(t, T0 ) is the simply-compounded spot interest rate prevailing at. ‧. time t for the maturity T0 and P (t, T0 ) = [1 + (T0 − t)L(t, T0 )]−1 .. y. Nat. By observation, pricing equation (3.1.6) bears resemblances to the Margrabe (1978) type or. io. sit. the Black-type formula but in the framework of the extended LMM. The advantage of adopting. n. al. er. the extended LMM model over other interest-rate models is that all the parameters in (3.1.6). i n U. v. can be easily obtained from market quotes, thus making the pricing formula more tractable and feasible for practitioners.. Ch. engchi. The value of the first-type guarantee associated with mortality in the LMM framework is further considered in the following Corollary 3.1. Corollary 3.1. Assume that the financial market and the employee’s mortality risk are independent. For an employee aged α, the time t value of the first-type guarantee with the time TN (n) payoff ψ˜TN (I) can also be priced under the extended LMM framework and is given below í ´ ³ h ³ (n) (n) (n) (n) (n) (3.1.7) ψ˜t (I) = θTαN · Z1,t N d1 − Z2,t N d2. where θTαN is the survival probability that an employee aged α remains alive after TN years, P rob(Rα > TN ). The random variable Rα denotes the remaining life time of an α-year-old employee. 18.

(20) 3.2. Valuation of the Second-Type Guarantee (Multi-Period Guarantee). In this subsection, we derive the value of the IRGEIDCP under a multi-period guarantee. We begin with defining the guarantee as follows. Definition 3.3. GTN (II) is defined as the terminal account value of the second-type guarantee (i.e. the interest rate guarantee embedded in DC plans under multi-period guarantees) with the payoff specified below.. GTN (II) =. N X. ( m · w0 (1 + g)n−1 ·. "Ã N −1 Y. ST max (1 + δLδ (Ti , Ti )), i+1 STi i=n−1. 政治大. n=1. (n). m · w0 (1 + g)n−1 ψTN (II). ‧ 國. n=1. 立. ¶!. STN − STn−1. #). (3.2.1). 學. =. N X. µ. (3.2.2). (n). ‧. Definition 3.4. ψTN (II) is defined as the time TN value of the second-type guarantee for $1. (3.2.3). er. io. sit. y. Nat. contributed in the n-th year, n ≤ N , with the payoff specified as follows. "Ã N −1 # µ ¶! Y S S T T (n) N ψTN (II) = max (1 + δLδ (Ti , Ti )), i+1 − S S T T n−1 i i=n−1. al. n. v i n C h expiration, whileUthe contract period of a multi-period guarantee is binding only at the contract engchi. The major difference between the first-type and the second-type guarantee is that a maturity. guarantee is divided into several subperiods, where a binding guarantee is specified for each subperiod. The pricing formula of the second-type guarantee is given in Theorem 3.2 below and the proof is provided in Appendix B. Theorem 3.2. The pricing formulas of the second-type guarantees with the final payoff as specified in Definition 3.3 and 3.4 are expressed as follows: Gt (II) =. N X. (n). m · w0 (1 + g)n−1 ψt (II). n=1 (n). ψt (II) = P (t, Tn−1 ). " N −1 Y i=n−1. 19. (3.2.4). # 2N (di3 ) − 1. (3.2.5).

(21) where di3 =. Vi2 2 Z Ti+1. Vi2 =. kγEi (u)k2 du. Ti. γEi (t) = [φ(t) + σ ¯P (t, Ti+1 )] Yang et al. (2008) derived under the HJM framework the pricing formulas for the IRGEIDCPs. In comparison with our pricing formulas, all the parameters in Theorem 3.1 and 3.2 can be extracted from market quotes, making our pricing formula more tractable and feasible for practitioners. Moreover, our formulas can be applied to any arbitrary guarantee period δ.. 政治大. In contrast, their formula is available only for a special case where the interest rate guaran-. 立. tee is linked to the one-year spot rate, i.e. δ = 1, in their pricing of the IRGEIDCPs under. ‧ 國. 學. multi-period guarantees.. The value of the second-type guarantee associated with mortality in the LMM framework. ‧. is further considered in the following corollary.. sit. y. Nat. Corollary 3.2. Assume that the financial market and the employee’s mortality risk are independent. For an employee aged α, the time t value of the second-type guarantee with the time. io. n. al. er. (n) TN payoff ψ˜TN (II) is given below. (n) ψ˜t (II). Ch. engchi. = θTαN · P (t, Tn−1 ). i n U. " N −1 Y. v. #. 2N (di3 ) − 1. (3.2.6). i=n−1. where θTαN is the survival probability as defined in Corollary 3.1. The above two different pricing formulas of interest rate guarantees have been derived. Section 4 is devoted to some practical issues regarding a calibration procedure and numerical examples.. 20.

(22) 4. Calibration Procedure and Numerical Examples. We first provide a calibration procedure for practical implementation and then examine the accuracy of the derived pricing formulas via a comparison with Monte Carlo simulation.. 4.1. Calibration Procedure. With the pricing formulas for caps and floors consistent with the popular Black formula (1976), the extended LMM is easier for calibration. Wu and Chen (2007a, 2007b) introduced the mechanism presented by Rebonato (1999) to engage in a simultaneous calibration of the extended LIBOR market model to the percentage volatilities and the correlation matrix of the underly-. 政治大. ing forward LIBOR rates and an equity-type asset (which is assumed to be a stock index to. 立. replace a reference portfolio for numerical analysis hereafter and could be stocks, mutual funds. ‧ 國. 學. or reference portfolios in practice). We briefly introduce it below.7. Assume that there are N − 1 forward LIBOR rates and a stock index in an m-factor frame-. ‧. work. The steps to calibrate the model parameters are briefly presented below: First, as given in Brigo and Mercurio (2001), the forward LIBOR rates, Lδ (t, ·), are assumed. y. Nat. sit. to have a piecewise-constant instantaneous total volatility structure depending only on the time. n. al. er. io. to maturity (i.e., Vij = vi−j ). The instantaneous total volatility υi−j applied to each period for. i n U. v. each rate as shown in Exhibit 2 can be stripped from market data. A detailed computational process is presented in Hull (2003).. Ch. engchi. In addition, the stock index S(t) is also assumed to have piecewise-constant instantaneous total volatility structures. The instantaneous total volatilities, ηi , applied to each period for the stock index as shown in Exhibit 3 can be calculated from the prices of on-the-run options in markets. Because the duration of options on the stock index is usually shorter than one year, the implied (or historical) volatilities of the underlying stock index are used and the term structures of volatilities are assumed to be flat (i.e., ηs (t) = ηs for t ∈ (T0 , TN ] ). 7. See Rebonato (1999) and Wu and Chen (2007a, 2007b) for more details.. 21.

(23) Exhibit 2: Instantaneous Volatilities of Lδ (t, ·) Instant. Total Vol.. Time t ∈ (T0 , T1 ]. (T1 , T2 ]. (T2 , T3 ]. ···. (TN −2 , TN −1 ]. Fwd. Rate: Lδ (t, T1 ). V1,1 = v0. Dead. Dead. ···. Dead. Lδ (t, T2 ) .. .. V2,1 = v1. V2,2 = v0. Dead. ···. Dead. ···. ···. ···. ···. ···. Lδ (t, TN −1 ). VN −1,1 = vN −2. VN −1,2 = vN −3. VN −1,3 = vN −4. ···. VN −1,N −1 = v0. Exhibit 3: Instantaneous Volatilities of the Stock Index Time t ∈ (T0 , T1 ]. Instant. Total Vol.. (T1 , T2 ]. (T2 , T3 ]. ···. (TN −1 , TN ]. V = η V = η ··· V = η 政治大 Second, the historical price data of the forward LIBOR rates and the stock index are used 立 VS1 = η1. S(t). S2. 2. S3. 3. SN. N. to derive a full-rank N × N instantaneous-correlation matrix Ψ such that Ψ = KΛK 0 where. ‧ 國. 學. K is a real orthogonal matrix and Λ is a diagonal matrix. Let A ≡ KΛ1/2 and thus AA0 = Ψ. Then, a suitable m-rank matrix B can be found such that the m-rank matrix ΨB = BB 0 can. ‧. be used to mimic the market correlation matrix Ψ, where m ≤ N .. Nat. al. n. bi,k =. sit. io.    cos θi,k Πk−1 j=1 sin θi,j if k = 1, 2, ..., m − 1,. er. computed by. y. By following Rebonato (1999), a suitable matrix B can be found with the ith row of B. i n if k = m, Ch engchi U.   Πk−1 sin θi,j j=1. v. for i = 1, 2, ..., N . θˆ is estimated by solving the optimization problem min θ. N X. 2 |ΨB i,j − Ψi,j |. i,j=1. ˆ can be found such that ΨB (= B ˆ Bˆ 0 ) is and thereby substituting θˆ into B, a suitable matrix B an approximate correlation matrix for Ψ. ˆ can be used to distribute the instantaneous total volatilities to each Finally, the matrix B Brownian motion at each period for the stock index and each LIBOR rate without changing the amount of the instantaneous total volatility. That is, ³ ´ ˆ 1), B(i, ˆ 2), . . . , B(i, ˆ m) = (γ1 (t, Ti ), γ2 (t, Ti ), . . . , γm (t, Ti )) , Vi,j B(i, ³ ´ ˆ ˆ ˆ ηj B(N, 1), B(N, 2), . . . , B(N, m) = (φ1 (t), φ2 (t), . . . , φm (t)) , 22.

(24) where i = 1, 2, ..., N − 1 and t ∈ (Tj−1 , Tj ], for each j = 1, 2, ..., N . ˆ the individual instantaneous volatility applied to each BrowVia the distributing matrix B, nian motion at each period for each process can be derived and used to calculate the prices of the interest rate guarantees derived in Theorem 3.1 and 3.2.. 4.2. Numerical Analysis. Some practical examples are given to examine the accuracy of the pricing formulas derived in the previous section by comparing the results with Monte Carlo simulation. Based on actual 2year market data,8 two types of interest rate guarantees with different guarantee periods (δ = 1. 政治大. year and δ = 0.5 year) are priced on the date, 2008/8/29, and the results are listed in Exhibit 4 and 5. The simulation is based on 50,000 sample paths. The FTSE index is used to replace. 立. the reference portfolio for the numerical purpose. An employee is assumed to start working at. ‧ 國. 學. the age of thirty. To ease the comparison and analysis, we assume that the contribution rate is 10% and the wage growth rate is 0%. The initial wage is assumed to be £10 in the case of. ‧. δ = 1 and £5 in the case of δ = 0.5.. y. Nat. Observing the numerical results yields several notable points. First, the pricing formulas. io. sit. have been shown to be accurate and robust in comparison with Monte Carlo simulation for the. n. al. er. recent market data. Second, Exhibit 5 shows that our formulas can be applied to arbitrary. i n U. v. values of δ (other than δ = 1). The formula of Yang et al. (2008) is available only for a. Ch. engchi. special case where the interest rate guarantee is linked to the one-year spot rate, i.e. δ = 1. Third, the second-type guarantee is more expensive than the first-type guarantee in both cases of δ = 1 and δ = 0.5. With a longer working period, the cost difference is becoming steadily more significant. The effect of higher guaranteed rates in some periods can be alleviated by lower guaranteed rates in other periods for the first-type guarantee. Such alleviation does not work for the second-type guarantee. Finally, using the derived formulas is more efficient than adopting time-consuming simulation for those guarantees with long duration. 8. All data are drawn and computed from the DataStream database. All the market data associated with the stock index and cap volatilities in the U.K. markets and the initial forward LIBOR rates are available upon request from the authors.. 23.

(25) Exhibit 4: The Price of Two Types of Guarantees with δ = 1 Year First-Type Guarantee Gt (I). Second-Type Guarantee Gt (II). Working. CS. MC. Difference. CS. MC. Difference. Period(TN ). (A). (B). (C). (D). (E). (F). 10. 1.2726. 1.2697. 0.0030. 3.6499. 3.6584. −0.0085. 15. 2.2732. 2.2819. −0.0087. 9.3536. 9.3601. −0.0066. 20. 3.3531. 3.3593. −0.0062. 18.9293. 18.9340. −0.0046. 25. 4.4799. 4.4856. −0.0058. 33.9819. 33.9753. 0.0065. 30. 5.6374. 5.6279. 56.8763. 0.0006. 35. 6.8158. 40. 8.0085. 6.8149 立. 政 0.0095治 56.8770 大 91.0664. 91.0712. −0.0048. −0.0022. 141.5686. 141.5592. 0.0094. ‧ 國. 學. *. 8.0107. 0.0009. CS and MC represent, respectively, the results of the formula and Monte Carlo simulations. (C)=(A)-(B) and (F)=(D)-(E).. ‧. Period(TN ). (A). 10. 1.3202. a(B) l C h. 15. 2.3028. 2.2991. 0.0037. 20. 3.3577. 3.3582. 25. 4.4562. 30. n. *. MC. Difference (C). CS. sit. CS. io. Working. Second-Type Guarantee Gt (II) MC. Difference. (E). (F). er. Nat. First-Type Guarantee Gt (I). y. Exhibit 5: The Price of Two Types of Guarantees with δ = 0.5 Year. i n U. (D). e−0.0043 n g c h i 6.3388. v. 6.3449. −0.0061. 17.2580. 17.2523. 0.0058. −0.0005. 37.8752. 37.8721. 0.0031. 4.4473. 0.0089. 74.7767. 74.7850. −0.0084. 5.5841. 5.5854. −0.0013. 139.2294. 139.2259. 0.0035. 35. 6.7321. 6.7349. −0.0028. 250.4363. 250.4292. 0.0072. 40. 7.8940. 7.8898. 0.0042. 441.0898. 441.0974. −0.0076. 1.3245. (C)=(A)-(B) and (F)=(D)-(E).. The values of two types of guarantees associated with mortality in the LMM framework are studied in Exhibit 6. The effect of mortality for both males and females are investigated. 24.

(26) by adopting the UK standard tables for annuitant and pensioner populations for the period 1991-1994 proposed by the CMI Bureau (1999).9 By observing Exhibit 4 and 6, the values of both guarantees are decreased with the mortality effect. The effect is slightly significant for males since the survival probability is slightly lower for males than females of the same age. Exhibit 6: The Prices of Two Types of Guarantees Associated with Mortality ( δ = 1 Year). Male. Female. Working. Gt (I). Gt (II). θT30N. Gt (I). Gt (II). θT30N. Period(TN ). (A). (B). (C). (D). (E). (F). 10. 1.2650. 3.6280. 15. 2.2512. 0.9903. 2.2625. 9.3096. 0.9953. 20. 3.3035. 立9.2629 18.6492. 0.9852. 3.3303. 18.8006. 0.9932. 4.3791. 33.2173. 0.9775. 4.4355. 33.6454. 學. 0.9901. 5.4441. 54.9261. 0.9657. 5.5523. 56.0181. 0.9849. 6.4457. 86.1215. 0.9457. 6.6461. 88.7988. 0.9751. Nat. 128.5160. 0.9078. 7.6433 135.1131. 0.9544. io. The symbol θT30N represents the survival probability that an employee who begins working at the age of thirty remains alive after TN years. (A) = (C) × (A) in Exhibit 3, (B) = (C) × (D) in Exhibit 3, (D) = (F) × (A) in Exhibit 3 and (E) = (F) × (D) in Exhibit 3.. n. al. er. *. 7.2701. y. 40. ‧. 35. 0.9970. sit. 30. ‧ 國. 25. 政0.9940治1.2688大 3.6390. Ch. engchi. 9. i n U. v. See Appendix G in Yang et al. (2008) for more details regarding the methodology of constructing the mortality table by CMI Bureau (1999).. 25.

(27) 5. Conclusions. Two different types of IRGEIDCPs have been developed via the extended LMM. The guaranteed rate in DC plans is set relative to the LIBOR rate. The pricing formulas derived are more consistent with market practice than those given in previous researches. The formulas of IRGEIDCPs under maturity and multi-period guarantees can be applied to any arbitrary guarantee period δ. The derived pricing formulas are similar to the Margrabe (1978) type or the Black formula in the extended LMM framework and thus are easy for practical implementation. In addition, the pricing formulas have been shown numerically to be sufficiently accurate as compared with Monte Carlo simulation. The IRGEIDCPs pricing with the derived. 政治大. formulas can be executed more efficiently than time-consuming simulation since the guaranteed. 立. DC pension plans usually have a long duration. Thus, the pricing formulas of IRGEIDCPs de-. ‧. ‧ 國. io. sit. y. Nat. n. al. er. implementation.. 學. rived under the extended LIBOR market model are more tractable and feasible for practical. Ch. engchi. 26. i n U. v.

(28) Appendix A: Proof of Theorem 3.1 A lemma is first presented and then employed to price Theorem 3.1. Lemma 1. Given that X and Y are normal random variables with mean zero and variances σ 2 (·), the following identity holds: · µ ¶ µ ¶¸+ σy2 σx2 E K1 exp X − − K2 exp Y − = K1 N (d) − K2 N (d − V ) 2 2 where (z)+ = max(z, 0), d =. ln(K1 /K2 )+ 12 V 2 V. and V 2 is the variance of X − Y .. Proof. See Amin and Jarrow (1991, p324) for details.. 政治大. Proof of Equation (3.1.5) and (3.1.6) in Theorem 3.1. 立. 學. ‧ 國. By applying the martingale pricing method, the market value of the first-type guarantees at time t, 0 ≤ t ≤ T0 ≤ T1 ≤ . . . ≤ TN , is derived as follows:. ‧. o n R TN r(u)du − t GTN (I)|Ft , Gt (I) = E e Q. Nat. y. (A.1). er. io. sit. (where E∗ {·|Fη } = E∗η {·} , r(t) = rt and β(t) = βt .). Substituting GTN (I) as shown in (3.1.2) into (A.1), we know " N #) ( X R TN (n) (A.1) = EQ e− t ru du m · w0 (1 + g)n−1 ψTN (I) t. n. al. Ch. engchi. i n U. v. (A.2). n=1. =. N X. o n R TN − t ru du (n) ψ (I) m · w0 (1 + g)n−1 EQ e t TN. (A.3). n=1. =. N X. (n). m · w0 (1 + g)n−1 ψt (I),. (A.4). n=1 (n). where EQ η (·) denotes the expectation under the spot martingale measure Q and ψt (I) is derived as follows. n R TN o (n) − t ru du (n) e ψ (I) ψt (I) = EQ t TN ½ ¾ P (TN , TN )/P (t, TN ) (n) Q = Et P (t, TN )ψTN (I) βTN /βt n o (n) = P (t, TN )ETt N ψTN (I) . 27. (A.5) (A.6) (A.7).

(29) ETη N (·) denotes the expectation under the forward martingale measure QTN (with respect to the numeraire P (t, TN )) defined by the Radon-Nikod´ ym derivative. dQTN dQ. =. P (TN ,TN )/P (t,TN ) 10 . βTN /βt. (n). By inserting the definition of ψTN (I) as shown in (3.1.4) into (A.7), (A.7) can be shown to be " #+  −1   NY ¡ ¢ STN (A.7) = P (t, TN )ETt N 1 + δLδ (Ti , Ti ) − (A.8)   S T n−1 i=n−1  +             −1   NY P (Ti , Ti ) STN    TN = P (t, TN )Et − , (A.9)    P (Ti , Ti+1 ) STn−1     i=n−1     {z } | {z }     | (A-2)  (A-1) where. 立. N −1. P (Ti , Ti ) STN i i , (A-1) = and (A-2) = . P (Ti , Ti+1 ) P (T , T ) S i i+1 T n−1 i=n−1. 學. ‧ 國. 1 + δLδ (Ti , Ti ) =. 政治大 Y P (T , T ). We then solve (A-1) and (A-2), respectively.. ‧. The dynamics of P (Ti , Ti )/P (Ti , Ti+1 ), i = n − 1, . . . , N − 1 and STN /STn−1 are determined. n. al. Ch. er. io. P (Ti , Ti ) P (Ti , Ti )/P (Ti , TN ) Ai (Ti ) = , ≡ i P (Ti , Ti+1 ) P (Ti , Ti+1 )/P (Ti , TN ) B (Ti ). sit. y. Nat. below.. iv n ) U E(T. STN STN /P (TN , TN ) P (Tn−1 , Tn−1 N) = ≡ C n−1 (Tn−1 ). STn−1 STn−1 /P (Tn−1 , TN ) P (Tn−1 , TN ) E(Tn−1 ). engchi. (A.10). (A.11). We define each variable at time t as follows. Ai (t) = P (t, Ti )/P (t, TN ), i = n − 1, . . . , N − 1,. (A.12). B i (t) = P (t, Ti+1 )/P (t, TN ), i = n − 1, . . . , N − 1,. (A.13). C n−1 (t) = P (t, Tn−1 )/P (t, TN ), E(t) = St /P (t, TN ).. (A.14) (A.15). By employing Proposition 2.3 and Itô’s Lemma and substituting σ ¯P (t, ·) defined in (2.2.7) for σP (t, ·), the dynamics of (A.12)∼(A.15) under the forward measure QTN can be obtained. Under 10. See Shreve (2004) for details on the changing-numeraire mechanism.. 28.

(30) the forward measure QTN , the random variables defined from (A.12) to (A.15) are martingales. Their dynamics can be written as follows. . . dAi (t)   σP (t, Ti ) + σ = −¯ ¯P (t, TN ) · dWtTN = γAi (t) · dWtTN , i {z } | A (t) i (t) γA. . (A.16). . dB i (t)   σP (t, Ti+1 ) + σ = −¯ ¯P (t, TN ) · dWtTN = γBi (t) · dWtTN , i {z } | B (t). (A.17). i (t) γB. . 政治大 . 立. ‧ 國. n−1 γC (t). . 學. dC n−1 (t)   = −¯ σ (t, Tn−1 ) + σ ¯ (t, TN ) · dWtTN = γCn−1 (t) · dWtTN , {z P } | P C n−1 (t) . ‧. dE(t)   = φ(t) + σ ¯P (t, TN ) · dWtTN = γE (t) · dWtTN . {z } | E(t). y. Nat. γE(t). n. er. io. al. sit. (where dW TN (t) = dWtTN .). Ch. engchi. 29. i n U. v. (A.18). (A.19).

(31) Solving the stochastic differential equations from (A.16) to (A.19), we obtain: µ. ¶ Z Z Ti 1 Ti i 2 i TN A (Ti ) = A (t) exp − kγA (u)k du + γA (u) · dWu , (A.20) 2 t t ¶ µ Z Z Ti 1 Ti i 2 TN i i i , (A.21) kγB (u)k du + γB (u) · dWu B (Ti ) = B (t) exp − 2 t t µ ¶ Z Z Tn−1 P (t, Tn−1 ) 1 Tn−1 n−1 n−1 n−1 2 TN C (Tn−1 ) = exp − kγC (u)k du + γC (u) · dWu , P (t, TN ) 2 t t i. i. (A.22) ³ R ´ R TN 1 TN 2 TN E(t) exp − kγ (u)k du + γ (u) · dW E E u 2 t t E(TN ) ³ R ´ = R T T E(Tn−1 ) E(t) exp − 21 t n−1 kγE (u)k2 du + t n−1 γE (u) · dWuTN ¶ µ Z Z TN 1 TN 2 TN kγE (u)k du + γE (u) · dWu , = exp − 2 Tn−1 Tn−1. 立. (A.23). 學. ¢ Ai (Ti ) P (Ti , Ti ) 1 + δLδ (Ti , Ti ) = = i P (Ti , Ti+1 ) B (Ti ) ½ Z ¤ 1 Ti £ i P (t, Ti ) kγA (u)k2 − kγBi (u)k2 du exp − = P (t, Ti+1 ) 2 t ¾ Z Ti £ i ¤ i TN + γA (u) − γB (u) · dWu .. ‧. ‧ 國. ¡. 政治大. t. y. Nat. N −1 Y ¢ 1 + δL (Ti , Ti ) =. al. P (Ti , Ti ) P (Ti , Ti+1 ) i=n−1. n. δ. i=n−1 N −1 Y. C h( e n1 gXc ·hZi U ¡ N −1. P (t, Ti ) = · exp − P (t, Ti+1 ) 2 i=n−1 i=n−1 N −1 X. +. i=n−1. Z. Ti. Ti. er. io. (A-1) =. ¡. sit. By using (A.24), (A-1) can be derived as follows. N −1 Y. (A.24). v ni. kγAi (u)k2. (A.25). −. kγBi (u)k2. ¢. ¸ du. t. ¤ £ i γA (u) − γBi (u) · dWuTN. ) .. (A.26). t. Define Z. Ti. Xi =. £ i ¤ γA (u) − γBi (u) · dWuTN , i = n − 1, . . . , N − 1,. (A.27). t. X(n) =. N −1 X. Xi .. (A.28). i=n−1. Then, we know that. 30.

(32) µZ. ¡. Ti. Var(Xi ) = Var. γAi (u). −. γBi (u). ¢. ¶ ·. dWuTN. Ti. Cov(Xi , Xj ) = Cov. ¡. −. γAi (u). γBi (u). ¢. Z. Ti. =. ¡. Tj. dWuTN ,. t. ° i ° °γA (u) − γBi (u)°2 du,. t. Z ·. Ti. =. t. µZ. Z ¡. γAj (u). −. ¢. γBj (u). (A.29). ¶ ·. dWuTN. t. ¢ ¡ ¢ γAi (u) − γBi (u) · γAj (u) − γBj (u) du, j = i + 1,. (A.30). t N −1 X. Var(X(n)) =. (N −1)−1 N −1 X X. Var(Xi ) + 2. i=n−1. Z. N −1 X. =. i=n−1. +2. Cov(Xi , Xj ). i=n−1 j=i+1. ° i ° °γA (u) − γBi (u)°2 du. Ti. 治¢¡ 政 ¡ ¢ γ (u) − γ (u) ·大 γ (u) − γ (u) du.. t. (N −1)−1 N −1 X X. 立. i=n−1 j=i+1. Z. Ti. i A. j A. i B. j B. (A.31). t. ‧. ‧ 國. 學. By combining equations from (A.27) to (A.31) with (A.26), (A.26) can be written as (A.32). · ¸ 1 P (t, Tn−1 ) (A.26) = exp [a(n)] · exp X(n) − Var(X(n)) , (A.32) P (t, TN ) 2 | {z } (n). K1. sit. y. Nat. where. n. al. er. io. N −1 Y P (t, Tn−1 ) P (t, Ti ) = , P (t, TN ) P (t, T ) i+1 i=n−1 ¸ N −1 ·Z Ti ¡ i ¢ 1 X 1 2 i 2 kγA (u)k − kγB (u)k du + Var(X(n)) a(n) = − 2 i=n−1 t 2. =. N −1 X. ·Z. Ti. γBi (u). ·. ¡. engchi. γBi (u). −. γAi (u). ¢. v. ¸ du. t. i=n−1. +. Ch. i n U. (A.33). (N −1)−1 (N −1) ·Z Ti X X i=n−1 j=i+1. ¡. γAi (u). −. ¸ ¢ ¡ j ¢ j · γA (u) − γB (u) du .. γBi (u). (A.34). t. Next, (A-2) can be obtained by adopting (A.22) and (A.23) below. STN E(TN ) n−1 = C (Tn−1 ) STn−1 E(Tn−1 ) ½ ·Z Tn−1 ¸ Z TN P (t, Tn−1 ) 1 n−1 2 2 = exp − kγC (u)k du + kγE (u)k du P (t, TN ) 2 t Tn−1 ·Z Tn−1 ¸¾ Z TN n−1 TN TN + γC (u) · dWu + γE (u) · dWu .. (A-2) =. t. Tn−1. 31. (A.35). (A.36).

(33) Define Z. Tn−1. γCn−1 (u) · dWuTN ,. Y1 =. (A.37). t. Z. TN. Y2 =. γE (u) · dWuTN ,. (A.38). Tn−1. Y (n) =. 2 X. Yi .. (A.39). i=1. As a result, Z. Tn−1. Var(Y1 ) =. kγCn−1 (u)k2 du,. (A.40). t. Z. TN. Var(Y2 ) =. kγ (u)k du, 治政 Z大 E. Tn−1 Z Tn−1. 立. Var(Y (n)) =. 2. kγCn−1 (u)k2 du. TN. +. (A.41) kγE (u)k2 du.. (A.42). Tn−1. t. ‧ 國. 學. By combining equations from (A.37) to (A.42) with (A.36), (A.36) can be written as (A.43). · ¸ 1 P (t, Tn−1 ) exp Y (n) − Var(Y (n)) . (A.36) = P (t, TN ) 2 | {z }. ‧. (A.43). (n). y. sit. Nat. K2. n. al. er. io. By the results of (A-1) and (A-2) as shown in (A.32) and (A.43), (A.9) can be represented as " #+  −1   NY STN P (Ti , Ti ) (n) − ψt (I) = P (t, TN )ETt N  P (Ti , Ti+1 ) STn−1  i=n−1 ½· µ ¶ 1 (n) TN = P (t, TN )Et K1 exp X(n) − Var(X(n)) 2 µ ¶¸+ ) 1 (n) −K2 exp Y (n) − Var(Y (n)) . (A.44) 2. Ch. engchi. i n U. v. Applying Lemma 1, we know ½. ´ P (t, T ) ³ ´ ³ P (t, Tn−1 ) n−1 (n) (n) exp[a(n)]N d1 − N d2 (A.44) = P (t, TN ) P (t, TN ) P (t, TN ) ´ ´ ³ ³ (n) (n) (n) (n) = Z1,t N d1 − Z2,t N d2 ,. 32. ¾ (A.45) (A.46).

(34) where (n). (A.47). (n). (A.48). Z1,t = P (t, Tn−1 ) exp[a(n)], Z2,t = P (t, Tn−1 ),. "n−2 #−1 Y P (t, Tn−1 ) = [1 + (T0 − t)L(t, T0 )]−1 (1 + δLδ (t, Tk )) , k=0 (n) d1. (A.49). q 2 , = V(n). (A.50). = Var (X(n)) + Var (Y (n)) − 2Cov (X(n), Y (n)) .. (A.51). 2 a(n) + 21 V(n). =. ,. V(n). (n) d2. (n) d1. =. − V(n) , V(n). 2 V(n) = Var (X(n) − Y (n)). 政治大. Ti. Cov. Nat. +. Ti. ¡. Tn−1. t. ¡. γAi (u). −. ¢. γBi (u). N −1 X i=n−1. Z. Tn−1. γCn−1 (u). , Z. ·. dWuTN. ¢ γAi (u) − γBi (u) · γCn−1 (u)du. al. Ti. TN. ,. γE (u) ·. ¸ ·. dWuTN ¸. dWuTN. Tn−1. n. i=n−1. ·. Z. dWuTN. t. io. =. ¢. y. ·Z. Cov. Z. −. γBi (u). t. i=n−1 N −1 X. γAi (u). i=n−1 j=1. ‧. N −1 X. ¡. j=1. t. i=n−1. +. i=n−1. ·Z. ¡. Ch. sit. N −1 X. er. =. ‧ 國. 立. 學. Cov (X(n), Y (n)) can be derived below. Ã N −1 ! 2 N −1 X 2 X X X Cov (X(n), Y (n)) = Cov Xi , Yj = Cov (Xi , Xj ). i n U. ¢ γAi (u) − γBi (u) · γE (u)du.. Tn−1. engchi. v. (A.52). P (t, Tn−1 ) can be derived as follows. "n−2 # Y P (t, Tk ) 1 1 = P (t, Tn−1 ) P (t, T0 ) k=0 P (t, Tk+1 ). (A.53). According to (A.53), we can obtain (A.49) after rearrangement. Note that the time differences, (T0 − t) and δ = Tn − Tn−1 , are measured as year fraction between two dates. L(t, T0 ) is the simply-compounded spot interest rate prevailing at time t for the maturity T0 .. Therefore, Theorem 3.1 has been derived.. 33.

(35) Appendix B: Proof of Theorem 3.2 Proof of Equation (3.2.4) and (3.2.5) in Theorem 3.2 By applying the martingale pricing method, the market value of the second-type guarantees at time t, 0 ≤ t ≤ T0 ≤ T1 ≤ . . . ≤ TN , is derived as follows: n R TN o ru du − t Gt (II) = E e GTN (II)|Ft . Q. (B.1). Substituting GTN (II) as shown in (3.2.2) into (B.1), we know ( " N #) R TN X (n) Q (B.1) = Et e− t ru du m · w0 (1 + g)n−1 ψTN (II) o n R TN ru du (n) − t ψ (II) m · w0 (1 + g)n−1 EQ e t TN (n). m · w0 (1 + g)n−1 ψt (II).. (B.4). max. io. i=n−1. ¢ ST 1 + δL (Ti , Ti ) , i+1 STi δ. n. al. · ¡. Hence,. Ch. (n). sit. =. N −1 Y. ¸ (B.5). er. Nat. (n) MTN (II). y. n=1. Define. (B.3). ‧. ‧ 國 =. n=1 N X. 立. (B.2). n=1. 學. =. N X. 政治大. i n U. engchi (n). ψTN (II) = MTN (II) −. v. STN STn−1. (B.6). (n). Therefore, ψt (II) can be derived below. n R TN o (n) − t ru du (n) e ψ (II) ψt (II) = EQ t TN ¾ ½ R n R TN o T (n) Q Q − t ru du − t N ru du STN = Et e MTN (II) − Et e STn−1. (B.7) (B.8). (n). According to the definition of MTN (II), we know n R TN o (n) − t ru du EQ e M (II) = EQ t t TN. ( e−. R TN t. ru du. " N −1 Y i=n−1. 34. #) CTi+1. (B.9).

(36) where CTi+1. · ¡. ¸ ¢ STi+1 = max 1 + δL (Ti , Ti ) , STi ¸ · ¡ ¢ STi+1 + STi+1 δ + , i = n − 1, . . . , N − 1. = 1 + δL (Ti , Ti ) − STi STi δ. (B.10) (B.11). By “The Law of Iterated Conditional Expectation ” in Duffie (1988), (B.12) and (B.13) can be obtained as follows. (B.9) =. ½ RT ¾ −1 n R Tn−1 o NY − T i+1 ru du Q − t ru du e ETi e i CTi+1 i=n−1 | {z } (B-1). EQ t. = P (t, T ). 立. i 3. 學. n R Tn−1 o − t ru du e = P (t, Tn−1 ), ¾ ½ RT − T i+1 ru du i EQ CTi+1 = 2N (di3 ), e Ti EQ t. io. Vi2 =. kγEi (u)k2 du,. n. Ti. Ch. (B.16). sit. Vi Z Ti+1. q Vi2 ,. y. 1 = V i , Vi = 2. (B.15). (B.17). er. 1 2 V 2 i. (B.14). ‧. Nat. di3 =. al. (B.13). i=n−1. ‧ 國. where. 政 2N (d ),治大. N −1 Y. (B.12). i n U. v. γEi (t) = [φ(t) + σ ¯P (t, Ti+1 )] .. engchi. (B.18). We solve (B-1) as follows. ½ (B-1) =. EQ Ti. =. EQ Ti. −. e ½. R Ti+1 Ti. ¾ ru du. CTi+1. P (Ti+1 , Ti+1 )/P (Ti , Ti+1 ) P (Ti , Ti+1 )CTi+1 βTi+1 /βTi T. = P (Ti , Ti+1 )ETi+1 {CTi+1 }. i. ¾ (B.19) (B.20). where ETη i+1 (·) denotes the expectation under the forward martingale measure QTi+1 defined by the Radon-Nikod´ ym derivative. dQTi+1 dQ. =. P (Ti+1 ,Ti+1 )/P (Ti ,Ti+1 ) . βTi+1 /βTi. 35.

(37) T. By using (B.11) , ETi+1 {CTi+1 } can be derived below. i (· ¸+ ) ¾ ½ ¡ ¢ S STi+1 T Ti+1 Ti+1 Ti+1 i+1 δ ETi {CTi+1 } = ETi 1 + δL (Ti , Ti ) − + ETi , STi STi {z } | {z } | (B-1b) (B-1a) ¡ ¢ i) where 1 + δLδ (Ti , Ti ) = PP(T(Ti ,Ti ,Ti+1 . ). (B.21). We then solve (B-1a) and (B-1b), respectively. ·µ (B-1a) =. ST P (Ti , Ti ) − i+1 P (Ti , Ti+1 ) STi. ¶. ¸. ½. · Iα , α =. ST P (Ti , Ti ) > i+1 P (Ti , Ti+1 ) STi. 政  1,治if 大 is an indicator function with  0, otherwise. 立. P (Ti ,Ti ) P (Ti ,Ti+1 ). P (Ti ,Ti ) P (Ti ,Ti+1 ). STi+1 S Ti. are determined below.. ‧. y. n. a lC (t) = P (t, T )/P (t, T ), i v n Ch U E (t) =eS n /Pg (t,cT h i). i. i. (B.23) (B.24). sit. er. io. We define each variable at time t as follows. i. (B.22). STi+1 . S Ti. P (Ti , Ti ) ≡ C i (Ti ), P (Ti , Ti+1 ) ST /P (Ti+1 , Ti+1 ) P (Ti , Ti ) E i (Ti+1 ) i ≡ C (Ti ). = i+1 STi /P (Ti , Ti+1 ) P (Ti , Ti+1 ) E i (Ti ). Nat. STi+1 STi. and. >. ¾. 學. The dynamics of. ‧ 國. Iα. T ETi+1 i. t. i+1. i+1. (B.25) (B.26). By employing Proposition 2.3 and Itô’s Lemma and substituting σ ¯P (t, ·) defined in (2.2.7) for σP (t, ·), the dynamics of (B.25) and (B.26) under the forward measure QTi+1 can be obtained as given below. Under the forward measure QTi+1 , the random variables defined in (B.25) and (B.26) are martingales, and their dynamics can be written as follows. . . i. dC (t)   T T = −¯ σP (t, Ti ) + σ ¯P (t, Ti+1 ) · dWt i+1 = γCi (t) · dWt i+1 , i | {z } C (t) . i (t) γC. (B.27). . dE i (t)   T T = φ(t) + σ ¯P (t, Ti+1 ) · dWt i+1 = γEi (t) · dWt i+1 . | {z } E i (t) i (t) γE. 36. (B.28).

(38) Solving the stochastic differential equations (B.28), we obtain: ³ R ´ R Ti+1 i Ti+1 1 Ti+1 i 2 i E(t) exp − kγ (u)k du + γ (u) · dW u E E 2 t t E (Ti+1 ) ³ R ´ = R i T T T E (Ti ) E(t) exp − 12 t i kγEi (u)k2 du + t i γEi (u) · dWu i+1 µ ¶ Z Z Ti+1 1 Ti+1 i 2 i Ti+1 = exp − kγE (u)k du + γE (u) · dWu , 2 Ti Ti STi+1 E i (Ti+1 ) i = C (Ti ) STi E i (Ti ) µ ¶ Z Z Ti+1 P (Ti , Ti ) 1 Ti+1 i 2 i Ti+1 = exp − kγE (u)k du + γE (u) · dWu . P (Ti , Ti+1 ) 2 Ti Ti Hence, P (Ti , Ti ) Ti+1 P (B.22) = P (Ti , Ti+1 ) r. 立. 政治 ¶大 −E. ST P (Ti , Ti ) > i+1 P (Ti , Ti+1 ) STi. µ. Ti+1 Ti. ¶ STi+1 · Iα , STi. (B.30). (B.31). denotes the probability under the forward martingale measure QTi+1 .. 學. ‧ 國. where. PTr i+1 (·). µ. (B.29). n. al. di3. =. Ch. Vi2 =. 1 2 V 2 i. er. io. sit. y. Nat. where. ‧. By inserting (B.30) into PTr i+1 (·) , the probability can be obtained after rearrangement as follows: µ ¶ STi+1 P (Ti , Ti ) Ti+1 Pr > = N (di3 ), (B.32) P (Ti , Ti+1 ) STi. 1 = Vi , 2. Vi Z Ti+1. i n U. v. e n gkγc (u)k h i du, i E. 2. (B.33) (B.34). Ti. q Vi =. Vi2 .. Using (B.30), we know ¶ µ ¶ µ STi+1 P (Ti , Ti ) dRi Ti+1 Ti+1 ETi · Iα = E · Iα STi P (Ti , Ti+1 ) Ti dQTi+1 µ ¶ STi+1 P (Ti , Ti ) P (Ti , Ti ) Ri = P > P (Ti , Ti+1 ) r P (Ti , Ti+1 ) STi. (B.35). (B.36) (B.37). i where PR r (·) denotes the probability under the martingale measure Ri which is defined by the. Radon-Nikod´ ym derivative µ ¶ Z Z Ti+1 1 Ti+1 i dRi 2 i Ti+1 = exp − kγE (u)k du + γE (u) · dWu . dQTi+1 2 Ti Ti 37. (B.38).

(39) From the Radon-Nikod´ ym derivative. dRi dQTi+1. , we know that. T. dWt i+1 = dWtRi + γEi (t)dt. (B.39). Under the measure Ri , we obtain (B.40) by substituing (B.39) into (B.30) µ Z Ti+1 ¶ Z Ti+1 STi+1 P (Ti , Ti ) 1 i 2 i Ri = exp kγE (u)k du + γE (u) · dWu . STi P (Ti , Ti+1 ) 2 Ti Ti. (B.40). i By inserting (B.40) into PR r (·) , the probability can be obtained after rearrangement as follows: ¶ µ STi+1 P (Ti , Ti ) Ri > = N (− di3 ). (B.41) Pr P (Ti , Ti+1 ) STi. By combing (B.22), (B.31), (B.32), (B.37) with (B.41), (B-1a) can be obtained below. (B-1a) =. ½. (B-1b) =. T ETi+1 i. STi+1 STi. ¾ =. P (Ti , Ti ) . P (Ti , Ti+1 ). 學. T. (B.21) = ETi+1 {CTi+1 } = (B-1a) + (B-1b) i. Nat. P (Ti , Ti ) P (Ti , Ti ) P (Ti , Ti ) N (di3 ) − N (− di3 ) + P (Ti , Ti+1 ) P (Ti , Ti+1 ) P (Ti , Ti+1 ) P (Ti , Ti ) =2 N (di3 ). P (Ti , Ti+1 ). n. al. er. io. sit. y. =. Ch. And we obtain (B-1) as shown in (B.46).. engchi T. (B.42). (B.43). ‧. ‧ 國. 政治大. 立. From (B.30), we obtain. Hence,. P (Ti , Ti ) P (Ti , Ti ) N (di3 ) − N (− di3 ). P (Ti , Ti+1 ) P (Ti , Ti+1 ). i n U. (B.44) (B.45). v. (B-1) = P (Ti , Ti+1 )ETi+1 {CTi+1 } = 2N (di3 ). i Besides, (B.47) can be obtained by using (A.36). ¾ ½ ¾ ½ R T STN P (t, Tn−1 ) Q TN − t N ru du STN = P (t, Tn−1 ). Et e = P (t, TN )Et = P (t, TN ) STn−1 STn−1 P (t, TN ). (B.46). (B.47). Inserting (B.13) and (B.47) into (B.8), we derive the result as follows. ¾ ½ R n R TN o T (n) (n) Q Q − t ru du − t N ru du STN MTN (II) − Et e ψt (II) = Et e STn−1 = P (t, Tn−1 ). N −1 Y. 2N (di3 ) − P (t, Tn−1 ). i=n−1. = P (t, Tn−1 ). " N −1 Y. # 2N (di3 ). i=n−1. Therefore, the proof of Theorem 3.2 is completed. 38. −1 .. (B.48).

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