The Valuation of Bermudan Guaranteed Return Contracts

Transcription

1 The Valuation of Bermudan Guaranteed Return Contracts Steven Simon 1 November K.U.Leuven and Ente Luigi Einaudi

2 Abstract A guaranteed or minimum return can be found in different financial products, e.g. guaranteed investment contracts (GIC s issued by investment banks and life-insurance contracts. We consider the so-called multi-period or compounding guaranteed return contracts. With such a contract a minimum return is guaranteed over a series of sub-periods, making the pay-off at maturity path-dependent. We first derive some important features of the European style contract in a general Heath-Jarrow- Morton framework. Secondly, we analyse the effect of adding a Bermudan put feature to this type contract and we derive some interesting properties about the optimal excersice behavior. Keywords: Guaranteed return contracts, Surrender feature, Optimal Exercise Decision, Heath- Jarrow-Morton term-structure model.

3 1 Introduction Several financial products exist that guarantee a minimum return to investors, e.g. guaranteed investment contracts (GIC s issued by investment banks and life-insurance contracts. Such guaranteedreturn contracts can be divided in two classes: maturity guarantees and periodic guarantees (or compounding guarantees. With a maturity guarantee the minimum return is guaranteed over the entire maturity of the contract, i.e. a minimum pay-off is guaranteed at the maturity date. Such a contract is essentially a covered European call (put option. And as such, results on prices of the call (put options can be used to price contracts withe a maturity guarantee. Contracts with a periodic guarantee provide investors with a minimum return over a series of sub-periods. This results in a path-dependent pay-off at maturity. Guaranteed return contracts have been studied by several others, see for instance Miltersen and Persson (1999 and references therein. However, so far little has been done on incorporating a surrender option (Bermudan put option into guaranteed return contracts. Grosen and Jørgensen (1997 model a contract with a maturity guarantee which contains an early surrender feature and Grosen and Jørgensen (2002 include a barrier-option feature in a contract with a maturity guarantee. We analyze the effect of including an early surrender feature in a periodic (compounding guaranteed return contract. The main results are derived in a general Heath-Jarrow-Morton framework; and some expressions for prices are obtained for both deterministic and stochastic interest rates. The remainder of the paper is organized as follows. The next section contains a brief discussion of the Heath-Jarrow-Morton option-pricing framework. Section 3 introduces discusses guaranteed return contracts. Compounding (periodic guaranteed return contracts are analyzed in more detail in Section 4. In Section 5 the effect of adding a specific type of surrender feature to guaranteed return contracts is analyzed. Finally, Section 6 contains the conclusions. 2 The Model Basics Heath et al. (1992 take the instantaneous forward rate as the primitive of their model. The continuously compounded forward rates f(t, s are defined by: P (t, T = e T t f(t,sds. (1 The forward rates are assumed to exist for 0 t s τ. In the HJM-model one then assumes that for any fixed s [0, τ] the forward rate f(t, s satisfies the following equation for all t [0, s] : 1

4 f(t, s f(0, s = t α(v, s, ωdv t 0 0 σ i (v, s, ωdw i (v, (2 where {W i (t} N t=0 are N independent Brownian motions defined on a given filtered probability space and the processes α(v, s, ω and σ i (v, s, ω (i = 1...N satisfy certain regulatory conditions. As Heath et al.(1992 point out, the above conditions impose no other significant economic restrictions on the forward rate processes but that they have continuous sample paths and are jointly driven by a finite number of random shocks. Heat et al. (1992 show that under some, fairly weak, regulatory conditions there exists a unique equivalent martingale measure under which the dynamics of the forward rates become: f(t, s f(0, s = t σ i (v, s, ω t 0 v t 0 σ i (v, y, ω dy dv σ i (v, s, ωdw i (v. (3 As such the risk-neutral dynamics of the forward rates are completely determined by the volatility processes σ i (v, s, ω of the risk-natural (physical dynamics. For the dynamics of the spot rate one obtains: r(t = f(t, t = f(0, t t 0 t σ i (v, t, ω t 0 v σ i (v, y, ω dy dv σ i (v, t, ωdw i (v, (4 From formulas 3 and 4 it follows that the forward rate and spot rate processes will in general not be Markovian. This is clearly seen in case of the spot rate by differentiating equation 4, which yields: dr(t = f(0, t t t [ N σ i (t, t, ω 0 t v σ i (v, y, ω dy t 0 σ i (v, t, ω 2 dv ] σ i (v, t, ω dw i (v dt σ i (v, t, ωdw i (v (5 t All of the three terms in the drift term in the above equation depend on the history of the process. As a result, the spot rate will, in general, be non-markovian. A similar analysis applies to the dynamics of the forward rate processes. In order for the term structure model determined by the volatility processes to have the Markov property extra conditions will have to be imposed. See for instance 2

5 Jeffrey (1995 for the conditions in the one-factor case. Furthermore, it is well known that for the model to be Gaussian the volatility processes will need to be deterministic, σ i (v, s, ω σ i (v, s (i = 1...N. For the time being we will not make any assumptions about whether or not the term structure model is Gaussian or Markovian. Nor will we specify the number of driving factors. At this point we do not make any assumptions about the underlying asset S, the portfolio underlying the life insurance contracts, other than that it is a traded asset and that the risk-neutral dynamics are given by: L ds(t S(t = r(tdt η i (t, ωdw i (t, (6 with {W i (t} L i=n1 L N additional independent Brownian motions. Here, we introduce three filtrations that will be used in the paper. With (G t t we refer to the filtration generated by the N Brownian motions that drive the term-structure and with (H t t we indicate the filtration generated by the L N additional Brownian motions which only contribute to the dynamics of S(t. As such, for each t the two filtrations G t and H t are independent. Finally, we define for each t: F t = σ (G t, H t. 3 Compounding Guaranteed Return Contracts In this section we give a short discussion of compounding guaranteed return (CGR contracts. Some analytic results are given in case the underlying asset is a stock and interest rates are deterministic. 3.1 Contract Specifications Consider a portfolio or underlying asset S. In this paper we consider contracts of which the pay-off at maturity is given by: ( n [ ] e rg(ti ti 1 S(ti (1 β S(t i 1 er G(t i t i 1 determined by n 1 so-called contract dates : t 0,..., t n 1, t n. Such a contract provides a minimum return r G over each of the sub-periods. In return for this guarantee, the policyholder only receives the fraction (1 β of the excess return for each period that the guaranteed return is not in the money. This type of guarantee is for instance very common in life insurance contracts. A special case is the one-period version or maturity guarantee of which the pay-off is: (7 3

6 Let us introduce the following notation: [ ] e rg(t2 t1 S(t2 (1 β S(t 1 er G(t 2 t 1 (8 Then, δ i is given by: δ i = ti t i 1 ( r(t 1 2 As such equation 7 is equivalent with: n N1 δ i = S(t i S(t i 1. η 2 i (t, ω dt N1 ti t i 1 η i (t, ωdw i (t ( e rg(ti ti 1 (1 β [e ] δi e r G(t i t i 1 The first treatment of the maturity guarantee can be found in Brennan and Schwartz (1976. The multi-period version of the contract has been studied by Person and Aase (1997,Grosen and Jørgensen (2000, Hansen and Miltersen (2000 and Miltersen and Person (2000. Two useful concepts in this context are the reserve value and the prospective value of a CGR contract, as given by the two following definitions. Definition 1 For a compounding guaranteed return contract with contract dates: t 0,..., t n the reserve value D(t i at a contract date t i is given by: i k=1 ( [ ] e rg(t k t k 1 (1 β e δ k e r G(t k t k 1 From this definition one sees that the reserve value at a contract date t i is in fact the amount to which the return guarantee applies from t i onwards. Definition 2 For a compounding guaranteed return contract with contract dates: t 0,..., t n the prospective value P V (t i at a contract date t i is given by: with P (t i the value of the contract at t i. P (t i D(t i Note that the prospective value of a CGR contract at a contract date t i is given by the value of a CGR contract defined by the n i 1 contract dates t i,..., t n of the given contract and with all other features equal to those of the given contract. The following corollary is an immediate result of the definition of a CGR contract as given by equation 7 and the definition of prospective value. (9 4

7 Corollary 1 For a compounding guaranteed return contract with contract dates: t 0,..., t n the prospective value P V (t i at any contract date t i is F ti -measurable. 3.2 Valuation in a Gaussian Framework Here we restrict the dynamics of the term structure of interest rates and the underlying asset to be Gaussian in order to be able to derive closed form expressions Valuation under Black-Merton-Scholes Assumptions In this section closed-form formulas are given for the price of a CGR contract under the assumptions underlying the Black-Scholes-Merton formula, see Black and Scholes (1973 and Merton (1973. Consider a risky asset S(t defined on a filtered probability space (Ω, P, (F t t where the filtration (F t t is generated by the process S(t. Furthermore, we assume that S(t satisfies: ds(t S(t = µdt σ db(t. (10 and that there exists a riskless money market account in which money can be invested at a spot rate r. The next two propositions give the value and the delta of the contract at different points in time before maturity. Proposition 1 Under the BSM assumptions the price at any of the contract dates t i (i = 0,..., n 1 the value of the contract as described above is given by: where for all j = i,..., n the factor c j is given by: n D(t i P V (t i = D(t i c j (11 j=i c j = (1 β (αn(d j e (r r G(t j1 t j N ( d j σ t j1 t j e (r r G(t j1 t j (12 with: d j = ln [ αe r G(t j1 t j ] ( r σ 2 /2 (t j1 t j σ t j1 t j. (13 5

8 Proof : Since the different ratios S(t i /S(t i 1 are independent of one another the above result is a straightforward application of the results on prices of European call options in Black and Scholes (1973 and Merton (1973. Note that the price of the contract does not depend on the value of the underlying asset S, and therefore the delta is equal to zero. In case all sub-periods are of equal length one obtains the following elegant version of Proposition 1. Corollary 2 In case t i t i 1 is independent of i, say t i t i 1 = τ one obtains that the value of the contract at any of the reserve dates: t 0,..., t n 1, is given by: { D(t n i e (r rgτ (1 β (αn(d e (r rgτ N ( d σ τ } n i (14 with: d = (r r G σ 2 /2τ ln(α σ. (15 τ Allowing for Stochastic Interest Rates Here we allow for stochastic interest rates, but we do assume that all diffusion coefficients σ i and η j of the dynamics of the term structure of interest rates and the underlying asset are deterministic. We give the result for a two-period CGR contract derived by Miltersen and Persson (1999. The results for multi-period CGR contracts are obtained by Lindset (2001. Let us denote the return on the money market account over the period (t i 1, t i by β i. That is: β i = ti t i 1 r(udu. Then at t 0 conditional on F t0 the vector (β 1, β 2, δ 1, δ 2 has a multi-variate normal distribution with mean given by: µ = ln P (t 0, t 1 σ 2 β 1 /2 ln F 2 σ 2 β 2 /2 c ln P (t 0, t 1 σ 2 β 1 /2 σ 2 δ 1 2 ln F 2 σ 2 β 2 /2 c σ 2 δ 2 /2 with 6

9 F 2 = P (t 0, t 2 P (t 0, t 1 is the t 1 - forward price at time t 0 of a zero-coupon bond with maturity t 2. The variance-covariance matrix of (β 1, β 2, δ 1, δ 2 is: Σ = σ 2 β 1 c σ 2 β 1 θ 1 c c σ 2 β 2 c θ 2 σ 2 β 2 θ 3 σ 2 β 1 θ 1 c θ 2 σ 2 δ 1 c θ 2 c σ 2 β 2 θ 3 c θ 2 σ 2 δ 2 Formulas for σ 2 β 1, σ 2 β 2, σ 2 δ 1, σ 2 δ 2, c, θ 1, θ 2 and θ 3 are given in Appendix A. Proposition 2 The price at t 0 of a two-period CGR contract is: N 2 ( a 1, b 1, ρ F 2 e r G(t 2 t 1 ρσ 2 δ 1 σ 2 δ 2 N 2 ( a 2, b 2, ρ P (t 0, t 1 e r G(t 1 t 0 N 2 (a 3, b 3, ρ(16 P (t 0, t 2 e r G(t 2 t 0 N 2 (a 4, b 4, ρ, where: a 1 = r Gt 1 t 0 ln P (t 0, t 1 σ 2 δ 1 /2 σ δ1, a 2 = a 1 ρσ δ2, a 3 = a 1 σ δ1, a 4 = a 1 ρσ δ2 σ δ1, b 1 = r G(t 2 t 1 ln F 2 σ 2 δ 2 /2 σ δ2 ρσ δ1, b 2 = b 1 σ δ2 b 3 = b 1 ρσ δ1, b 4 = b 1 ρσ δ1 σ δ2, ρ = c θ 2 σ δ1 σ δ2. From Proposition 1, one sees that under the Black-Scholes-Merton (BSM assumptions the value of the contract at any of the contract dates t 0,..., t n 1 does not depend on the value of the underlying. From the above proposition one sees that this property still holds in a stochastic interest rate environment if the dynamics of the underlying asset and the term structure are both Gaussian. This Value Independence property is quite interesting in its own right, and has important implications for contracts with a surrender feature, as will be discussed in section 5. Therefore, we analyze this feature in more detail in the next section. 4 The Value-Independence Property In the previous section we found that the value of a CGR contract with as underlying asset a stock is independent of the value of the underlying in case the term structure is deterministic or Gaussian. In this section we will establish the sufficient and necessary conditions on the dynamics of the underlying 7

10 asset and the term structure of interest rates for this property to hold. Therefore, we make no a priori assumptions about the dynamics of the term-structure of interest rates or about the dynamics of the underlying portfolio, other than those made in Section 2. We start by giving a definition of the Value-Independence property in this more general setting. Definition 3 For an asset S(t the Value-Independence property holds if and only if for all r G 0 and all t 0, t 1 with t 0 < t 1 the conditional expected value: is G t0 -measurable. E Qt 1 [ [S(t1 ] S(t 0 er G(t 1 t 0 ] F t 0 (17 This definition seems to apply only to one-period CGR contracts. However, in the remainder of this section it will be shown that there is a strong link between one-period and multi-period CGR contracts, such that the above definition applies to both of them. The next proposition shows that the independence property holds for maturity guarantee contracts if and only if the dynamics of the underlying asset S(t meet a specific condition. Proposition 3 The two following statements are equivalent: 1. For all t 1 > t 0 and all r G the conditional expectation: [ [S(t1 ] ] E Qt 1 S(t 0 er G(t 1 t 0 F t 0 (18 is G t0 -measurable. 2. For all t 1 > t 0 under the F t1 -forward measure the ratio S(t 1 /S(t 0 is independent of the filtration H t0. Proof : If 1 then 2 The value of the option is given by: [ [ ] ] P (t 0, t 1 E Qt 1 e rg(t1 t0 S(t1 (1 β S(t 0 er G(t 1 t 0 F t 0 [ [S(t1 ] = e rg(t1 t0 P (t 0, t 1 (1 βp (t 0, t 1 E Qt 1 S(t 0 er G(t 1 t 0 ] F t 0 (19 8

11 with Q t1 the t 1 -Forward measure. Next, consider the following well known property of a stop-loss transform, the proof which is straightforward. P(X K = 1 with X a random variable defined on a probability space (Ω, F, P. ( E P [X K] K, (20 Since the value of the contract at time t 0 is G t0 -measurable, so will be the derivative of this value with respect to r G. Using the above equation, this derivative is given by: ( [ ] (1 βe r S(t1 G P (t 0, t 1 Q t1 S(t 0 > er G(t 1 t 0 F t 0 The fact that the above expression is G t1 -measurable implies the same property for the conditional [ probability Q S(t1 t1 F t0 ]. If 2 then 1 S(t 0 er G Assume that under the F t1 -forward measure the distribution (at t 0 of the ratio S(t 1 /S(t 0 is G t0 -measurable. Then statement 1 follows immediately from the next equality: (21 = E Qt 1 K [ [S(t1 ] S(t 0 er G(t 1 t 0 (x e r G(t 1 t 0 x ] F t 0 ( [ S(t1 Q t1 ] S(t 0 er G(t 1 t 0 F t 0 dx. (22 Note that in case the underlying asset is Markovian, the condition that S(t 1 /S(t 0 is independent of the filtration H t0 reduces to the condition that S(t 1 /S(t 0 is independent of S(t 0. That is, the holding period return on the underlying asset is independent of its value at the start of the period. The following proposition creates a strong link between maturity guarantees and periodic guaranteed return contracts. More precisely, it shows that Definition 3 is indeed an unambiguous definition of the Value Independence Property for both one-period and multi-period CGR contracts. Proposition 4 Given 0 β < 1. If for all τ 0 and τ 1 with τ 1 > τ 0 and all r G the value at τ 0 of a guaranteed-return contract with pay-off at time τ 0 given by: [ ] S(τ e rg(τ 1 τ 0 1 (1 β S(τ 0 er G(τ 1 τ 0 9 (23

12 is G τ 0 -measurable, then for any CGR contract with pay-off at maturity date t n given by ( n [ ] e rg(ti ti 1 S(ti (1 β S(t i 1 er G(t i t i 1 (t 0 <... < t n (24 the prospective value P V (t i at a contract date t i is G ti -measurable. Proof : Let us assume that the proposition holds for contracts with a number of sub-periods equal to or less than n 1. We will now prove that the proposition also holds for contracts with n sub-periods. Note that the prospective value at any future contract date t i of the n-period CGR contract is equal to the value of a CGR contract with n i sub-periods and inception date t i. Therefore, all that remains to be proven is that at inception, the market price of the n-period CGR contract is G t0 -measurable. The value at t 0 of the contract with n sub-periods is given by: [ n ( [ ] P (t 0, t n E Qtn e rg(ti ti 1 S(ti (1 β S(t i 1 er G(t i t i 1 = P (t 0, t 1 E Qtn [ E Q tn [D(t1 P V (t 1 F t1 ] Ft0 ] = P (t 0, t 1 E Qtn [ D(t1 E Qtn [P V (t 1 F t1 ] F t0 ], (25 F t 0 ] with Q tn the t n -Forward measure. The first equality follows from Corollary 1, the second equality holds because RV (t 1 is F t1 -measurable. By assumption the prospective value of the n-period CGR contract at t 1 is G t1 -measurable. From Proposition 3 one has that RV (t 1 is independent from H t0. Since H t0 and G t0 are independent, one obtains that: [ P (t 0, t 1 E Qtn D(t1 E Qtn [P V (t 1 F t1 ] ] Ft0 = [ P (t 0, t 1 E Qtn D(t1 E Qtn [P V (t 1 G t1 ] ] Ft0 = P (t 0, t 1 E Qtn [D(t 1 P V (t 1 G t0 ]. (26 Therefore, the value of the n-period CGR contract at t 0 is G t0 -measurable. The next proposition gives the necessary and sufficient condition for the independence Value- Independence property to hold. 10

13 Proposition 5 For all t 1 > t 0 one has that under the F t1 -forward measure the ratio S(t 1 /S(t 0 is independent of G t0 if and only if for all i = 1,..., N 1 the coefficient η i (t, ω is adapted to the filtration G t. Proof : Proof of Sufficiency Let us first give the dynamics of P (t, t 1, the price of the zero-coupon bond with maturity date t 1 under the equivalent martingale measure Q implied by the money-market account as numéraire. Under this measure, the dynamics of P (t, t 1 are: dp (t, t 1 = r(tdt ξ i (t, ωd W i (t, (27 with ξ i (t, ω (i = 1,..., N adapted to G t and with W i (t (i = 1,..., N 1 independent Brownian motions under Q. Now, assume that η i (t, ω is adapted to the filtration G t for all i = 1,..., N 1. Then, we have: = ln (S(t 1 /S(t 0 ( t1 r(t η i (ω, tξ i (t, ω t 0 L t1 (η i (ω, t 2 dt t 0 L η i (ω, td W i (t, (28 with W i (t (i = N 1,..., L L N additional independent Brownian motions (under the measure under Q. Since all ξ i (t, ω s and η i (t, ω s are adapted to the filtration(g t t one has that: P (t 0, t 1 E Qt 1 [ [S(t1 ] S(t 0 er G(t 1 t 0 ] F t 0 is G t0 -measurable. Using an argument similar to that in the first part of the proof of Proposition 3 this leads to the required result. (29 Proof of Necessity Assume that for all t 0 and t 1 with t 0 < t 1 the ratio S(t 1 /S(t 0 is independent of H t0. Since for any i = 1,..., N 1 the increment W i (t 1 W i (t 0 is independent of F t0, it follows from the Role of Independence property of conditional expectations that E Qt 1 [ln (S(t 1 /S(t 0 (W i (t 1 W i (t 0 F t0 ] is G t0 -measurable. Observe that the dynamics of ln(s(t under the t 1 - forward measure Q t1 are given by: 11

14 ( d ln S(t = r(t As such we have: L η i (t, ω 2 η i (t, ωξ i (t, ω dt L η i (t, ωd W i (t. (30 = = E Qt 1 [(ln (S(t1 ln (S(t 0 (W i (t 1 W i (t 0 F t0 ] L [ t1 t1 ] E Qt 1 η i (t, ωdw j (t dw i (t F t 0 j=1 t1 t 0 t 0 t 0 E Qt1 [ηi (t, ω F t0 ] dt. (31 The first equality follows from the fact that dtdw i (t = 0. The second equality follows from the Ito-isometry and the fact that dw i (tdw j (t = δ ij dt. As such, we have that t 1 t 0 E Qt 1 [η i (t, ω F t0 ] dt (i = 1,..., N 1 is G t0 -measurable. Taking the derivative with respect to t 1 preserves this property. Therefore, the same property holds for the value of ( lim t t1 E Qt 2 [η i (t, ω F t1 ]. Taking the limit for t 1 t 0 again preserves this property. That is, the value of: ( lim lim E Qt 1 [ηi (t, ω F t0 ] t 1 t 0 t t 1 (32 is G t0 -measurable. However, because η i (t, ω (i = 1,..., N 1 is RCLL we have that: Hence, η i (t 0, ω is G t0 -measurable ( lim lim E Qt 2 [ηi (t, ω F t0 ] = η i (t 0, ω a.s. (33 t 1 t 0 t t 1 From the analysis in this section one sees that whether or not the Value Independence property holds is completely determined by the dynamics of the underlying asset; the dynamics of the term structure of interest rates have no bearing on this. On the other hand, the condition on the dynamics of the underlying are rather strong. The Value Independence property is only obtained if the underlying asset follows an exponential Brownian motion. 12

15 5 Including the Surrender Feature 5.1 General Discussion A natural way to incorporate a surrender feature in a CGR contract with contract dates t 0, t 1,..., t n 1, t n would be to give investors the possibility to terminate the contract at any of the contract dates t 1,...t n 1 in exchange for the reserve value D(t i. Where for Bermudan CGR contracts the reserve value is defined by Definition 1. Note that if two CGR contracts only differ from one another in the fact that one is European and the other has a surrender feature as described above, the two contracts will have the same reserve values at each of their contract dates. For a CGR contract with such an embedded surrender feature one has the following result. Proposition 6 If β = 0 then it is never optimal to surrender a CGR contract prior to maturity Proof : Investing the reserve value D(t i in the underlying asset results in a pay-off at maturity equal to: However, the pay-off of the contract at maturity is given by: D(t k n i=k1 D(t k S(t n S(t k. (34 { } S(ti max S(t i 1, e(ti ti 1r G. (35 Since the pay-off given by equation 35 dominates that of equation 34 the contract is always worth more than its reserve value D(t k. Therefore, it can not be optimal to surrender the contract prior to maturity. From the above proposition one sees that in case β = 0 a Bermudan CGR contract is identical to a CGR contract without a surrender feature. However, if 0 < β < 1 then Bermudan and European CGR contracts are in general no longer equivalent. The next proposition characterizes the optimal surrender decision at a contract date t i in terms of the prospective value P V (t i. The concept of prospective value for Bermudan CGR contracts is identical to that of the European style contract as given by Definition 2. Note that at each of the contract dates the prospective value of a Bermudan CGR contract will in general be greater than that of its European counterpart. 13

16 Proposition 7 If a CGR contract with n sub-periods can be surrendered at any of its contract dates t i (i = 1,..., t n 1 in return for the reserve value D(t i, surrendering the contract at a contract date t i is optimal if and only if P V (t i < 1. Proof : The result follows immediately from the fact that on the one hand surrender at a contract date t i is optimal if and only if the value of the contract P (t i is smaller than the reserve value D(t i and that on the other we have that: P (t i = D(t i P V (t i. Proposition 7 leads to the following result. Proposition 8 If the Value-Independence property holds, then the prospective value at a contract date t i (t i < t n of a Bermudan CGR contract is G ti -measurable, and as such the optimal surrender decision is a stopping-time with respect to the filtration (G t t. Proof : We first prove that the proposition holds at t n 1. At t n 1 the value of the contract is: max {D(t n 1, D(t n 1 P V (t n 1 } = D(t n 1 max {1, P V (t n 1 }, (36 with P V (t n 1 given by: [ ] P (t n 1, t n E [e Qtn r G(t n t n 1 S(tn (1 β S(t n 1 er G(t n t n 1 ] F t n 1, (37 with Q tn the t n -forward measure and P (t n 1, t n the price at t n 1 of a zero-coupon bond with maturity date t n. From the above equation one sees that P V (t n 1 is G tn 1 -measurable. Hence the exercise decision is determined by the set {ω Ω P V (t n 1 (ω 1}, which is G tn 1 -measurable. Given a contract date t i (< t n 1. Assume that the proposition holds for all contract dates t j with: t i < t j < t n. We now prove that the proposition also holds at t i. At t i the value of the contract is given by: 14

17 { } max D(t i, P (t i, t i1 E Qt i1 [D(ti1 max {1, P V (t i1 } F ti ] { } = D(t i max 1, P (t i, t i1 E Qt i1 [πi (ω max {1, P V (t i1 } F ti ] (38 with: π i (ω = ( [ ] e r G(t i1 t i S(ti1 (1 β e r G(t i1 t i. (39 S(t i Note that if the Value-Independence property holds, then both π i (ω and P V (t i1 are independent of H ti. As such, P V (t i is G ti -measurable. From Proposition 8 one sees that if the Value Independence property holds, the optimal exercise decision for a Bermudan CGR contract will be determined by the evolution of the term structure of interest rates. Note that this does not imply that the exercise decision will be independent from the evolution of the underlying asset. As in general, the underlying asset will not be independent from the term structure. 5.2 Valuation in a Gaussian Framework As in Section 3.2, we restrict the dynamics of the term structure of interest rates and the underlying asset to be Gaussian in order to be able to derive closed form expressions Valuation under the Black-Scholes-Merton Assumptions From Proposition 7 one sees that whether or not the surrender option is binding at a future contract date t k (t k < T is already known at inception since the term structure of interest rates is deterministic and hence H t = {φ, Ω} for all t. This implies that if the contract can be surrendered before maturity, the optimal surrender date is already known at inception. The next proposition describes the optimal exercise decision for a CGR contract under those assumptions. Proposition 9 Under these assumptions one has: 1. Let us start at the maturity date t n. Going back in time, the first contract date at which the surrender option is in the money is the contract date t n i determined by the smallest non-zero integer i for which: 15

18 where for j = 1,..., i the coefficient c n j is given by equation 12. i c n j > 1, (40 j=1 2. Given a contract date t n k at which the reserve requirement is binding. The next (earlier reserve date t n i (i > k at which he surrender option is in the money is the contract date t n i determined by the smallest integer i > k for which: i c n j > 1, (41 where for j = k 1,..., i the coefficient c i is given by equation 12. j=k Proof : We will first prove equation 40. Let us assume that for a given reserve date t n i there is no reserve date between t n i and t n at which the minimum-value requirement is binding. For instance because i = 1. Then the continuation value of the contract at t n i as given by equation 11 is equal to: i D(t n i c n j. (42 j=1 Since the required minimum value is equal to D(t n i, the minimum-value requirement will be binding if and only if: i c n j > 1. (43 j=1 We will now prove equation 41. Let a reserve date t k be given. If the last known reserve date at which the minimum-value requirement is binding is t n i, i < k. Then the continuation value of the contract at t n k is equal to: k D(t n i c n j. (44 j=i1 Since the required minimum value is again equal to D(t n i, the minimum-value requirement will be binding at t n i if and only if: k j=i1 c n j > 1. (45 16

19 In case all sub-periods are of equal length, say t n i1 t n i = τ, one has the following simplification of Proposition 9. Corollary 3 For a contract as discussed above and with: t n i1 t n i = τ for i = 1,..., n, the surrender option is in the money at a given surrender date t n j, j = 1,..., n 1 if and only if: where the coefficient d is given by ( e (r rgτ (1 β αn(d e (r rgτ N (d > 1, (46 d = ln [αe rgτ ] ( r σ 2 /2 τ σ. (47 τ From the above corollary we see that the condition that has to be met for surrender option to be in the money at a given reserve date t n j, j = 1,..., n 1 does not depend on t n j. That is, either the Bermudan reserve requirement is binding at all reserve dates or at non. This implies that for such a contract with sub-periods of equal length either the Bermudan reserve requirement is already binding at the first reserve date or it never is. For a contract that can be surrendered by the policyholder in exchange for the reserve value this leads to the result that the policyholder will surrender at the first surrender/reserve date or he never will Valuation in a Gaussian HJM-Framework As in Section we assume that all diffusion coefficients σ i and η j of the dynamics of the term structure of interest rates and the underlying asset are deterministic. We consider the case that the contract has two sub-periods. The analysis in this section will use 5 random variables: β 1, β 2, δ 1, δ 2 and ln P (t 1, t 2. Conditional on F t0 the vector ( β 1, β 2, δ 1, δ 2, ln P (t 1, t 2 has a multi-variate normal distribution with mean given by: and covariance matrix: µ = ln P (t 0, t 1 σ 2 β 1 /2 ln F 2 σ 2 β 2 /2 c ln P (t 0, t 1 σ 2 β 1 /2 σ 2 δ 1 /2 ln F 2 σ 2 β 2 /2 c σ 2 δ 2 /2 ln F 2 σ 2 β 1 /2 k (48 17

20 with: Σ = σ 2 β 1 c σ 2 β 1 θ 1 c σ P,β1 c σ 2 β 2 c θ 2 σ 2 β 2 θ 3 σ P,β2 σ 2 β 1 θ 1 c θ 2 σ 2 δ 1 c θ 2 σ P,δ1 c σ 2 β 2 θ 3 c θ 2 σ 2 δ 2 σ P,δ2 σ P,β1 σ P,β2 σ P,δ1 σ P,δ2 σ 2 P, (49 σ 2 P = σ 2 β 1 2q 1 q 2 σ P,β1 = σ 2 β 1 q 1 σ P,δ1 = σ 2 β 1 κ λ q 1 σ P,β2 = σ P,δ2 = c q 1 q 2 and: k = q 1 = q 2 = κ = λ = t1 [ t2 t 0 v t1 ( t1 t 0 u t1 ( t2 t 0 v t1 ( t2 t 0 u t1 ( t1 t 0 u ( σ i (v, u u v ] σ i (v, xdx du dv ( t2 σ i (u, vdv σ i (u, vdv u σ i (u, vdv 2 du σ i (u, vdv η i (udu σ i (u, vdv η i (udu du and with σ 2 β 1, σ 2 β 2, σ 2 δ 1, σ 2 δ 2, c, θ 1, θ 2 and θ 3 as in Section Before giving the main result for the 2-period contract we first give two lemmas that will prove useful later on. Lemma 1 At t 0 the value of a one-period CGR contract is given by: ( rg (t 0 t 1 ln P (t 0, t 1 σ 2 δ P V (t 0 = (1 βn 1 /2 σ δ1 [ ( ] rg P (t 0, t 1 e r G(t 1 t 0 (t 1 t 0 ln P (t 0, t 1 σ 2 δ β (1 βn 1 /2, (50 Proof : Straightforward application of results on European call options that can be found in Merton (1973 and Amin and Jarrow (1992. σ δ1 18

21 Lemma 2 For a one-period CGR with 0 < β < 1 there exists a unique number P such that: P (t 0 1 P (t 0, t 1 P, with the convention that P 0 if P (t 0 > 1 a.s. and P if P (t 0 1 a.s. Proof : For this lemma to hold, it is sufficient to prove that the value of a one-period contract P (t 0 is increasing in P (t 0, t 1. Notice that the expression for the price of a one-period CGR contract in the Gaussian framework as given by equation 50 has the same structure as that for the price of such a contract under the BSM assumptions. As such, using well-known results on the Greeks of European options under the BSM assumptions, one easily obtains that: [ ( ] P (t 0 P (t 0, t 1 = rg er G(t 1 t 0 (t 1 t 0 ln P (t 0, t 1 σ 2 δ (1 βn 1 /2 β 0. σ δ1 Note that it follows from Proposition 6 that P = 0 whenever β = 0. We are now ready to prove the following result for the price of a 2-period CGR contract. Proposition 10 At t 0 the value of the two-period Bermudan CGR is given by 1 : P (t 0 = (1 β 2 κ 1 N 3 ( α 1, β 1, γ 1 ; 0, Σ 1 β(1 βp (t 0, t 1 κ 2 N 3 ( α 2, β 2, γ 2 ; 0, Σ 1 (51 β(1 βf 2 κ 3 e r G(t 2 t 1 N 3 ( α 3, β 3, γ 3 ; 0, Σ 1 β 2 P (t 0, t 2 e r G(t 2 t 0 N 3 ( α 4, β 4, γ 4 ; 0, Σ 1 (1 βe r G(t 1 t 0 P (t 0, t 1 κ 2 N 3 (α 2, β 2, γ 2 ; 0, Σ 3 βp (t 0, t 2 e r G(t 2 t 0 N 3 (α 4, β 4, γ 4 ; 0, Σ 3 (1 βe r G(t 2 t 1 F 2 κ 3 N 3 ( α 3, β 3, γ 3 ; 0, Σ 2 βp (t 0, t 2 e r G(t 2 t 0 N 3 ( α 4, β 4, γ 4 ; 0, Σ 2 P (t 0, t 2 e r G(t 2 t 0 N 3 (α 4, β 4, γ 4 ; 0, Σ 4 (1 βκ 4 N 2 (p 1, q 1 ; 0, Σ 5 e r G(t 1 t 0 P (t 0, t 1 [N (q 3 (1 βn 2 (p 2, q 2 ; 0, Σ 5 ] The different coefficients and matrices are given in the appendix. 1 N k (x 1,..., x k ; µ, Σ indicates the k-variate normal probability determined by the upper bounds x 1,..., x k, the vector of means µ and variance-covariance matrix Σ. 19

22 Proof : See Appendix C. Note that because of Proposition 6 the right-hand side of equation 51 collapses in the right-hand side of equation 16 when β = 0. 6 Conclusions In this paper we have demonstrated that whether or not the Value Independence property holds for CGR contracts is completely determined by the dynamics of the underlying asset, the choice of dynamics of the term structure of interest rates not having any effect. Moreover, we showed that if the Value Independence property holds for a given underlying asset or portfolio, then the optimal surrender decision of a Bermudan style CGR contract is a stopping time with respect to (the filtration generated by the term structure of interest rates. With respect to the valuation of Bermudan CGR contracts this paper seems to suggest the following. If both the underlying asset (portfolio and the term structure have Gaussian dynamics, then closed form expressions for the prices of Bermudan CGR contracts can be obtained, as illustrated by Proposition 9. However, for non-trivial numbers of sub-periods the evaluation of these expressions becomes very time-consuming or even computationally infeasible. For a European style CGR contract withy n sub-periods, one needs to evaluate 2 n n-dimensional multivariate normal probabilities; see for instance Lindset (2001. For Bermudan style contracts both the number of probabilities and the dimension of the multivariate distributions involved increase compared to European style contracts. Hence, for higher numbers of sub-periods it might be computationally more efficient to use a fully numerical approach. Since a CGR contract with an embedded surrender option is a Bermudan style contract with a path dependent pay-off, the Least Squares Monte Carlo method of Longstaff and Schwartz (2001 seems to be a well suited candidate. References K. AMIN and R. JARROW (1992, Pricing Options on Risky Assets in a Stochastic Interest Rate Economy, Mathematical Finance 2, M. BLACK and M. SCHOLES (1973, The Pricing of Options and Corporate Liabilities, Journal of 20

23 Political Economy 81, M.J. BRENNAN and E.S. SCHWARTZ (1976, The Pricing of Equity-linked Life Insurance Policies with an Asset Value Guarantee, Journal of Financial Economics 3, A. GROSEN and P. L. JØRGENSEN (1997, Valuation of early Exercisable Interest Rate Guarantees, The Journal of Risk and Insurance 64, A. GROSEN and P. L. JØRGENSEN (2000, Fair valuation of life insurance liabilities: The impact of interest rate guarantees, surrender options, and bonus policies, Insurance Mathematics And Economics 26, A. GROSEN and P. L. JØRGENSEN (2002, Life Insurance Liabilities at Market Values: An Analysis of Insolvency Risk, Bonus Policy, and Regulatory Intervention rules in a Barrier Option Framework, Journal of Risk and Insurance 69, D. HEATH, R. JARROW and A. MORTON (1992, Bond pricing and the term structure of interest rates; a new methodology for contingent claim valuation, Econometrica 60, A. JEFFREY (1994, Single factor Heath-Jarrow-Morton Term Structure Models Based on Markov Spot Interest Rate Dynamics, Journal of Financial and Quantitative Analysis 30, S. LINDSET (2001, Pricing of Rate of Return Guarantees on Multi-period Assets,Working Paper, Norwegian School of economics and Business Administration. F. A. LONGSTAFF and E.S. SCHWARTZ (2001, Valuing American Options by Simulation: A Simple Least-Squares Approach, Review of Financial Studies 14, R.C. MERTON (1973, Theory of Rational Option Pricing, Bell Journal of Economics and Management Science K.R. MILTERSEN and S. PERSSON (1999, Pricing Rate of Return Guarantees in a Heath-Jarrow- Morton Framework, Insurance: Mathematics and Economics 25, K.R. MILTERSEN and S. PERSSON (2000, Guaranteed Investment Contracts: Distributed and Undistributed Excess Return, Working Paper, Department of Accounting, Finance and Law, Odense University, University of Southern Denmark, Odense, Denmark. S.A. PERSSON and K.K. AASE (1997, Valuation of the Minimum Guaranteed Return Embedded in Life Insurance Products, The Journal of Risk and Insurance 64,

24 Appendix A The following formulas can be found in Miltersen and Persson (1999. where: σ 2 β 1 = σ 2 β 2 = c = t1 ( ti k=1 0 v t1 ( t2 k=1 0 t 1 t1 ( t1 k=1 σ 2 δ 1 = σ 2 β 1 σ 2 d 1 2γ 1, σ 2 δ 1 = σ 2 β 2 σ 2 d 2 2θ 3, 0 v σ k (v, udu 2 dv, 2 σ k (v, udu dv k=1 t2 ( t2 t 1 ( t2 σ k (v, udu σ k (v, udu t 1 v dv, σ k (v, udu 2 dv, and: σ 2 d 1 = σ 2 d 2 = L t1 k=1 t 0 t2 L k=1 t 1 δ 2 k(vdv δ 2 k(vdv θ 1 = θ 2 = θ 3 = t1 k=1 0 t1 k=1 0 t2 k=1 0 t1 δ k (v σ k (v, ududv, v t2 δ k (v σ k (v, ududv, t 1 t2 δ k (v σ k (v, ududv. v Appendix B α 1 = r G (t 1 t 0 ln P (t 0, t 1 σ 2 β 1 /2 σ 2 δ 1 /2 θ 1, β 1 = r G (t 2 t 1 ln F 2 σ 2 β 2 /2 σ 2 δ 2 /2 θ 3 θ 2, γ 1 = ln P ln F 2 σ 2 β 1 /2 k κ λ, α 2 = r G (t 1 t 0 ln P (t 0, t 1 σ 2 β 1 /2 σ 2 δ 1 /2 θ 1 22

25 β 2 = r G (t 2 t 1 ln F 2 σ 2 β 2 /2 σ 2 δ 2 /2 θ 3 γ 2 = ln P ln F 2 σ 2 β 1 /2 k q 1 α 3 = r G (t 1 t 0 ln P (t 0, t 1 σ 2 β 1 /2 σ 2 δ 1 /2 θ 1 θ 2 c, β 3 = r G (t 2 t 1 ln F 2 σ 2 β 2 /2 σ 2 δ 2 /2 θ 2 θ 3 c γ 3 = ln P ln F 2 σ 2 β 1 /2 k c q 1 q 2 κ λ α 4 = r G (t 1 t 0 ln P (t 0, t 1 σ 2 β 1 /2 σ 2 δ 1 /2 θ 1 θ 2 c, β 4 = r G (t 2 t 1 ln F 2 σ 2 β 2 /2 σ 2 δ 2 /2 θ 3 γ 4 = ln P ln F 2 σ 2 β 1 /2 k q 1 p 1 = r G (t 1 t 0 ln P (t 0, t 1 σ 2 β 1 /2 σ 2 δ 1 /2 θ 1 q 1 = ln P ln F 2 σ 2 β 1 /2 k κ λ p 2 = r G (t 1 t 0 ln P (t 0, t 1 σ 2 β 1 /2 σ 2 δ 1 /2 θ 1 q 2 = ln P ln F 2 σ 2 β 1 /2 k q 1 q 3 = ln P ln F 2 σ 2 β 1 /2 k q 1 σ P κ 1 = e (σ2 β 1 /2σ 2 β 2 /2 (θ 1θ 3 κ 2 = e σ2 β /2 θ 3 2 κ 3 = e σ2 β /2 c θ 1 1 θ 2 κ 4 = e σ2 β 1 /2 θ 1 The five matrices Σ 1,..., Σ 5 are given by: Σ 1 = Σ 2 = σ 2 δ 1 c θ 2 σ P,δ1 c θ 2 σ 2 δ 2 σ P,δ2 σ P,δ1 σ P,δ2 σ 2 P σ 2 δ 1 c θ 2 σ P,δ1 c θ 2 σ 2 δ 2 σ P,δ2 σ P,δ1 σ P,δ2 σ 2 P 23

26 Σ 3 = Σ 4 = σ 2 δ 1 c θ 2 σ P,δ1 c θ 2 σ 2 δ 2 σ P,δ2 σ P,δ1 σ P,δ2 σ 2 P σ 2 δ 1 c θ 2 σ P,δ1 c θ 2 σ 2 δ 2 σ P,δ2 σ P,δ1 σ P,δ2 σ 2 P and: Appendix C [ σ 2 Σ 5 = δ1 σ P,δ1 σ P,δ1 σ 2 P ]. Before we start the proof of Proposition 10 we state the following useful result on the multivariate normal density from Lindset (2001. Lemma 3 For a multivariate normal random variable X with expectation µ, variance-covariance matrix Σ and density φ (X; µ, Σ, one has that: e m X φ (X; µ, Σ = e m µ 1 2 m Σm φ (X; µ Σm, Σ We are now ready to give the proof of Proposition 9. Proof : The value at t 0 is given by: P (t 0 = E Q t 0 [ e β 1D(t1 max 1, P V (t 1 ] = E Q [ ] β 1D(t1 I {P (t1,t 2 P } E Q [ β 1D(t1 I {P (t1,t 2>P }P V (t 1 ] = E Q t 0 [ e β 1D(t1 ; P (t 1, t 2 P ] E Q t 0 [ e β 1e β 2D(t2 ; P (t 1, t 2 > P ], with P such that P V (t 1 1 P (t 1, t 2 P and where for i = 0, 1 the operator E Q t i indicates that the expectation is conditional on F ti. For the last equality we have used that e β 1, D(t1 and I {P (t1,t 2>P } are all F t1 -measurable. One obtains: E Q t 0 [ e β 1D(t1 ; P (t 1, t 2 P ] = E Q t 0 [ e β 1D(t1 ; P (t 1, t 2 P, δ 1 > r G (t 1 t 0 ] (52 E Q t 0 [ e β 1D(t1 ; P (t 1, t 2 P, δ 1 r G (t 1 t 0 ] 24

27 and: E Q [ β 1D(t2 ; P (t 1, t 2 > P ] = E Q [ β 1e β 2D(t2 ; δ 1 > r G (t 1 t 0, δ 2 > r G (t 2 t 1, P (t 1, t 2 > P ] (53 E Q [ β 1e β 2D(t2 ; δ 1 > r G (t 1 t 0, δ 2 r G (t 2 t 1, P (t 1, t 2 > P ] E Q [ β 1e β 2D(t2 ; δ 1 r G (t 1 t 0, δ 2 > r G (t 2 t 1, P (t 1, t 2 > P ] E Q [ β 1e β 2D(t2 ; δ 1 r G (t 1 t 0, δ 2 r G (t 2 t 1, P (t 1, t 2 > P ]. In order to complete the proof we need to evaluate the two expected values on the right-hand side of equation 52 and the four expected values on the right-hand side of equation 53. We restrict ourselves to the evaluation of the first expected value on the right-hand side of equation 53; the other cases are completely analogue. E Q [ β 1e β 2D(t2 ; δ 1 > r G (t 1 t 0, δ 2 > r G (t 2 t 1, P (t 1, t 2 > P ] = { (1 β 2 E Q [ β 1e β 2e δ 1 e δ2 ; δ 1 r G (t 1 t 0, δ 2 r G (t 2 t 1, P (t 1, t 2 > P ] (54 e r G(t 1 t 0 E Q [ β 1e β 2e δ 2 ; δ 1 r G (t 1 t 0, δ 2 r G (t 2 t 1, P (t 1, t 2 > P ] e r G(t 2 t 1 E Q [ β 1e β 2e δ 1 ; δ 1 r G (t 1 t 0, δ 2 r G (t 2 t 1, P (t 1, t 2 > P ] e r G(t 2 t 0 E Q [ β 1e β 2; δ1 r G (t 1 t 0, δ 2 r G (t 2 t 1, P (t 1, t 2 > P ]} { (1 βe r G(t 2 t 1 E Q [ β 1e β 2e δ 1 ; δ 1 r G (t 1 t 0, δ 2 r G (t 2 t 1, P (t 1, t 2 > P ] e r G(t 2 t 1 E Q [ β 1e β 2e δ 1 ; δ 1 r G (t 1 t 0, δ 2 r G (t 2 t 1, P (t 1, t 2 > P ]} { (1 βe r G(t 1 t 0 E Q [ β 1e β 2e δ 2 ; δ 1 r G (t 1 t 0, δ 2 r G (t 2 t 1, P (t 1, t 2 > P ] e r G(t 2 t 1 E Q [ β 1e β 2e δ 1 ; δ 1 r G (t 1 t 0, δ 2 r G (t 2 t 1, P (t 1, t 2 > P ]} e r G(t 2 t 0 E Q [ β 1e β 2; δ1 r G (t 1 t 0, δ 2 r G (t 2 t 1, P (t 1, t 2 > P ] Again, we limit ourselves to valuation of the first expected value on the right-hand side of the above equation, the other cases being analogue. = E Q t 0 [ e β 1e β 2e δ 1 e δ2 ; δ 1 r G (t 1 t 0, δ 2 r G (t 2 t 1, P (t 1, t 2 > P ] dx 1 dx 2 r G (t 1 t 0 r G (t 2 t 1 dx 3 dx 4 dx 5 e m x φ(x; µ, Σ, (55 ln P 25

28 with µ and Σ given by equations 48 and 49 respectively and with m given by: m = (1, 1, 1, 1, 0. Using Lemma 3 and some symmetry properties of the multivariate normal distribution one sees that the above expected value is equal to: e m µ 1 2 m Σm = e m µ 1 2 m Σm dx 1 dx 2 r G (t 1 t 0 dx 3 r G (t 2 t 1 dx 4 rg (t 1 t 0 rg (t 2 t 1 dx 1 dx 2 dx 3 with C a diagonal matrix with diagonal given by: ln P dx 5 φ(x; µ Σm, Σ ln P dx 4 dx 5 φ(x; ν, C ΣC, (1, 1, 1, 1, 1 and ν given by: ν = C(µ Σm. As such, the above expected value is equal to: e m µ 1 2 m Σm N ( r G (t 1 t 0 ν 3, r G (t 2 t 1 ν 4, ln P ν 5 ; 0, C ΣC = κ 1 N ( α 1, β 1, γ 1 ; 0, Σ 1. 26