Valuation of Caps and Swaptions under a Stochastic String Model

Transcription

1 Valuation of Caps and Swaptions under a Stochastic String Model June 1, 2013 Abstract We develop a Gaussian stochastic string model that provides closed-form expressions for the prices of caps and swaptions that, under certain conditions, reduce to Black 1976) formulas. We also propose a stochastic string LIBOR market model that generalizes the models of Brace et al. 1997) and Longstaff et al. 2001a) and allows us to obtain the cap price calculated with the Black 1976) formula for caps. This model can be approximated by one that is compatible with the previous Gaussian models. We show that one of the assumptions of Longstaff et al. 2001a) is incompatible with our model and, then, we obtain a possible explanation for some problems related to the relative valuation of caps and swaptions. We attain the observational equivalence of Kerkhof and Pelsser 2002) and we prove that, under the stochastic string approach, the models proposed by Brace et al. 1997) and Longstaff et al. 2001a) are more parsimonious than previously stated in the original papers. We provide a number of new results, such as: i) a closed-form expression for the price of a portfolio option that is a multi-factor extension of the result obtained by Jamshidian 1989) and ii) an analytical formula for swaption prices. Keywords: Stochastic string model, LIBOR market model, Black formulas, caps, swaptions. JEL classification: C02, G12. 1

2 1 Introduction During the last years, the market of interest rate derivatives has experienced an uninterrupted increase from an average monthly trading volume of 2,177 in 2005 to 6,729 in 2012 ISDA 2012a)). 1 The dynamism of this market and its high trading volume $341.2 trillions of total notional outstanding in 2012, ISDA 2012b)) have created a great interest among researchers. We can mention as specially fruitful the theoretical and empirical analysis of caps and swaptions. 2 See Brace and Musiela 1994a, b), Brace et al. 1997), Jamshidian 1997), Navas 1999), De Jong et al. 2001, 2004), Longstaff et al. 2001a, b), Collin-Dufresne and Goldstein 2002a, b), Klaasen et al. 2003), Eberlein and Kluge 2006), among others. Recently, Bueno-Guerrero et al. 2013a, b) have enlarged considerably the stochastic string modeling for the term structure of interest rates TSIR) initiated by Santa-Clara and Sornette 2001). The main goal of this paper is to apply the stochastic string framework to price caps and swaptions and to shed light on three fundamental issues: a) the use of Black 1976) formulas by practitioners, b) the relative valuation of caps and swaptions, as proposed in Longstaff, Santa-Clara and Schwartz 2001a) LSS from now on), and c) the observational equivalence obtained by Kerkhof and Pelsser 2002). In the industry, practitioners typically use the Black 1976) formula for pricing options on futures to value caps and swaptions given their implied volatilities. However, it is well known that this approach is inconsistent see, for instance, Björk 2004) for a more detailed discussion). Several theoretical papers have tried to explain this behavior Goldys et al. 1994), Musiela 1994), Miltersen et al. 1994, 1997), Sandmann and Sondermann 1997)) and the culmination of these efforts arrives with the LIBOR market model of Brace et al. 1997) BGM from now on). Almost simultaneously, Jamshidian 1997) re-obtains by other means the BGM model and applies his methodology to the valuation of swap derivatives, obtaining the so called swap market model SMM). The main achievement of the BGM model consists of obtaining the Black formula for caplets within the Heath, Jarrow, and Morton 1992) HJM, from now on) scheme. This is achieved by imposing the log-normality of the LIBOR rate through a relationship between its volatility and the HJM volatilities. In a similar way, dealing with the forward swap rate, the Black 1976) formula for swaptions in the swap market model is obtained. We pose three fundamental objections to market models. First, these models do not obtain the Black formula within a general framework for interest rates. Instead, they impose ad hoc the log-normality of forward rates to obtain the desired results. Second, although the main idea underlying market models is that the Black formulas reflect the 1 Except a decrease in the period that could be due to the financial crisis. 2 For an introduction to these derivatives see, for instance, Longstaff et al. 2001a). 2

3 market view on caps and swaptions prices, the use of these formulas in the market is just a way of determining the price of the derivative through its implied volatility as emphasized by LSS). Thus, it does not imply that market participants perceive the Black 1976) model as the most appropriate one for caps and swaptions. Thirdly, although each market model justifies the use of the Black formula for each derivative, both models BGM and SMM) are incompatible see Jamshidian 1997)). Hence, using Black formulas for caps and swaptions is logically inconsistent. We aim to apply stochastic strings for modeling the term structure of interest rates, as discussed by Bueno-Guerrero et al. 2013a, b), and pricing interest rate derivatives. We consider two alternative ways. The first one is a Gaussian model where the cap valuation formula is the infinitedimensional extension of that obtained in Gaussian HJM models. The Black s formula is obtained as an approximation when the day-count fraction tends to infinity. Similarly, an exact swaption pricing formula is derived, which can be approximated by Black formula for swaptions assuming that covariances between bond prices are equal. Our second alternative leads us to the formulation of a stochastic string LIBOR market model that provides an exact Black formula. The relative valuation of caps and swaptions is the second problem we are interested in. As we shall see later, a cap can be seen as a portfolio of bond put options and a swaption is equivalent to a a put option on a bond portfolio. Obviously, the valuation of any portfolio is affected by the correlations between its constituents. Then, as shown by Merton 1973) and emphasized by LSS, the relation between prices of caps and swaptions is driven by the correlation structure between forward rates. As we will see later, using different assumptions, LSS price caps with real swaption prices to obtain information on this correlation structure. They obtain that: a) the hypothesis that the real cap prices equate the values implied by the swaptions market is rejected for all the maturities and b) caps are undervalued with respect to swaptions. Several solutions to this problem have been proposed in the literature. LSS suggest to include a time-varying covariance structure in the model. Following this idea and based on empirical evidence, Collin-Dufresne and Goldstein 2002a) propose a random field model with stochastic volatility and correlation and they obtain closed-form expressions for cap prices. They also develop some efficient methods for pricing swaptions. We will be able to solve the problem of the relative valuation of caps and swaptions maintaining time-independent covariances. We will provide closed-form expressions and we will show that one of the assumptions in LSS is inconsistent with our framework. Hence, the problem of the relative valuation of caps and swaptions could arise from a bad specification of the LSS model. Our third issue of interest is related to the number of parameters to be estimated in the BGM 3

4 and LSS models. LSS argued that we need to estimate just nn 1)/2 parameters in a model with n forward rates as this model is totally determined by the forward covariance matrix. In contrast, in a n-factor forward model as BGM), we need to estimate n 2 parameters n parameters for each forward rate). Hence, the LSS model is more parsimonious than the BGM one. However, in a posterior paper, Kerkhof and Pelsser 2002) show that both models are observationally equivalent and, then, require the same number of parameters to be estimated. In more detail, these authors obtain nk kk 1)/2 parameters for k-factor models. We will show that the observational equivalence is still true in our framework but the number of parameters to estimate is k, implying that our model is the most parsimonious one in almost all the cases. The remaining of the paper is as follows. Section 2 includes the main results of the stochastic string modeling and applies this method to price European options on bonds and bond portfolios. We present a type of covariance function between shocks to the forward curve, with important properties, introduced by Bueno-Guerrero et al. 2013b). For this covariance function, the shortterm interest rate follows the Hull and White 1990) dynamics and, then, we can generalize the pricing expressions for portfolio options obtained in Jamshidian 1989) to any number of factors. Section 3 provides a closed-form expression for the cap price that is the infinite-dimensional generalization of that obtained in Gaussian HJM models. We show that a certain approximation of this closed-form expression constitutes the Black 1976) formula commonly used in the industry. In a similar way, we obtain analytical expressions for swaption prices, generalizing those obtained by Brace and Musiela 1994b) and Musiela and Rutkowski 1995). Another approximation allows us to recover the Black formula for swaptions. This Section ends applying the generalization of the Jamshidian 1989) formula for pricing portfolio options to the pricing of swaptions. Section 4 presents a stochastic string model for the LIBOR forward rate that allows us to recover explicitly the Black formula for caps. This model nests the BGM model and an approximated one that is compatible with Gaussian models. We show that the swap forward rate cannot follow a lognormal distribution eliminating the possibility of including the swap market model). Section 5 analyzes in detail the assumptions of LSS from our modeling point of view and tries to explain the problem of the relative valuation of caps and swaptions. We find that the assumption related to the equality of the factors for the historical and implied covariances is wrong. Then, the problem of the relative valuation of caps and swaptions can arise because of a bad specification of the LSS model. Section 6 proves that the observational equivalence between the LSS and BGM models remains true in our setup. We also show that, within the stochastic string framework the LSS and BGM models become more parsimonious in almost all the cases. Section 7 summarizes the main conclusions. Two final technical Appendices include the expressions of the approximated stochastic string LIBOR market model and the proofs, respectively. 4

5 2 Preliminary results Bueno-Guerrero et al. 2013a) assume the following dynamics for the instantaneous forward rate in the Musiela 1993) parameterization df t, x) = α t, x) dt σ t, x) dz t, x) 1) where Z t, x) is the so called stochastic string process. Several assumptions on this process and on the filtration of the probability space jointly with a martingale representation property lead to the no-arbitrage dynamics [ f t, x) df t, x) = x x dyr t x, y) dt σ t, x) d Z t, x) 2) where d Z t, x) is the stochastic string shock under the equivalent martingale measure Q. Moreover, where R t x, y) c t, x, y) σ t, x) σ t, y) = cov [df t, x), df t, y) dt c t, x, y) = d [Z, x), Z, y) t = corr [dz t, x), dz t, y) 4) dt is the correlation function between stochastic string shocks. Taking σ t, x) and c t, x, y) as deterministic Gaussian model), Bueno-Guerrero et al. 2013b) provide the following result for the pricing of European-type derivatives whose payoff function is homogeneous of degree one. Theorem 1 Bueno-Guerrero et al. 2013b)) Consider the following assumptions: a) The final payoff of a contingent claim at time T 0 is given by C [T 0, {P T0 } = max Φ {P T0 }, 0) where Φ {P T0 } is a homogeneous function of degree one in the set of bond prices {P T0 } = {P T 0, T 0 ),..., P T 0, T N )} where P T 0, T i ) denotes the price at time T 0 of the bond maturing at time T i, T 0 T 1 T N. b) The correlation matrix M with elements M ij s, T 0 ) = deterministic and non-singular, with ij s, T 0 ) cov s [ln P T 0, T i ), ln P T 0, T j ) = T0 Then, the price at time s of this contingent claim is given by t=s 3) ij s,t 0 ), i, j = 1,..., N is ii s,t 0 ) jj s,t 0 ) [ Ti t Tj t dt dydur t u, y) y=t 0 t u=t 0 t 5) C [s, {P s } = d N xg x 1,..., x N ; M) [ ) Φ P s, T 0 ), P s, T 1 ) 11 x e ,..., P s, T N ) NN x e N 1 2 NN 6) 5

6 where 1 g x 1,..., x N ; M) = exp 1 2π) N 2 M N x i M 1 ) ij x j i,j=1 is the density function of a multivariate normal random variable. The following result Bueno-Guerrero et al. 2013a, b)) arises directly from the previous Theorem. Corollary 1 Consider a European call option that matures at time T 0 with strike K written on a zero-coupon bond that matures at time T > T 0. The price at time t of this option is given by Call K t, T 0, T ) = P t, T ) N d 1 ) KP t, T 0 ) N d 2 ) 7) where N ) denotes the distribution function of a standard normal random variable with ) ln P t,t ) KP t,t 0 ) 1 2 Ω t, T 0, T ) d 1 = Ω t, T0, T ) d 2 = d 1 Ω t, T 0, T ) and Ω t, T 0, T ) = T0 v=t [ T v T v dv dydwr v u, y) y=t 0 v w=t 0 v 8) Using the identity x K) K x) = x K, another consequence of Theorem 1 is the put-call parity. Corollary 2 Consider two European call / put) options that mature at time T 0 with strike K written on a zero-coupon bond that matures at T > T 0. The prices at time t of both options are denoted as Call K t, T 0, T ) and Put K t, T 0, T ), respectively, and are related by the equality Call K t, T 0, T ) Put K t, T 0, T ) = P t, T ) KP t, T 0 ) Applying Corollary 2 and the properties of the distribution function for a standard normal variable allows us to obtain explicitly the put price, as stated in the next Corollary. Corollary 3 Consider a European put option that matures at time T 0 with strike K written on a zero-coupon bond that matures at T > T 0. The price at time t of this option is given by Put K t, T 0, T ) = KP t, T 0 ) N d 2 ) P t, T ) N d 1 ) 9) 6

7 The pricing of swaptions requires pricing a European put option on a bond portfolio. This price is obtained as a corollary of the following Theorem that prices a European call portfolio option. Theorem 2 Consider a European call option that matures at time T 0 with strike K written on the cash-flows C 1,..., C n received at times T 1,..., T n. The price at time t of this option is given by [ ) Call K [t, {P t } = d n x C i P t, T i ) g x 1 1i 11,, x n ni nn ; M KP t, T 0 ) g x 1,..., x n ; M) 10) Proof: See Appendix B. Remark 1 The previous Theorem for n = 1 and C 1 = 1 provides expression 7) for the price of a European call on a zero-coupon bond. A change of variable leads to the following version of Theorem 2. Theorem 3 Consider a European call option that matures at time T 0 with strike K written on the cash-flows C 1,..., C n received at times T 1,..., T n. The price at time t of this option is given by [ Call K [t, {P t } = d n y C i P t, T i ) g y 1 γ 1i,..., y n γ ni ; I n ) KP t, T 0 ) g y 1,..., y n ; I n ) with γ i = γ 1i,..., γ ni ) verifying γ i γ j = ij. Proof: See Appendix B. Similarly to Theorems 2 and 3, we now provide put option prices. Theorem 4 Consider a European put option that matures at time T 0 with strike K written on the cash-flows C 1,..., C n received at times T 1,..., T n. The price at time t of this option is given by [ ) Put K [t, {P t } = d n x KP t, T 0 ) g x 1,..., x n ; M) C i P t, T i ) g x 1 1i 11,, x n ni nn ; M or, alternatively, Put K [t, {P t } = d n y [ KP t, T 0 ) g y 1,..., y n ; I n ) with γ i = γ 1i,..., γ ni ) verifying γ i γ j = ij. C i P t, T i ) g y 1 γ 1i,..., y n γ ni ; I n ) 11) 7

8 Remark 2 Expression 11) is formally identical to that in Theorem 3.1 in Brace and Musiela 1994a). The only difference is that, in our expression, the covariance is given by ij t, T 0 ) = T0 s=t ds k=0 [ Ti s dyσ k) s, y) y=t 0 s [ Tj s duσ k) s, u) y=t 0 s where { σ k) s, y) } is the volatilities set in an infinite-dimensional HJM model. Moreover, in k=0 Brace and Musiela 1994a), the sum in 12) includes just a finite number of terms, as it corresponds to a multi-factor HJM model see Bueno-Guerrero et al. 2013b)). Previous expressions for portfolio options are quite computationally cumbersome as they involve solving multiple integrals, as many as bonds included in the portfolio. Jamshidian 1989) avoids this problem introducing what is known as the Jamshidian s trick: in one-factor models, portfolio options can be interpreted as a portfolio of options with appropriate strikes) on the same underlying bonds. We will use a very concrete type of covariance function, proposed by Bueno-Guerrero et al. 2013b) to show that, in our framework, we can extend the Jamshidian s idea to models with any number of factors. In this way, we can obtain a simple expression for the price of a portfolio option avoiding the need of solving complex numerical integrals. We start by presenting a Proposition that shows an important property of the covariance in Bueno-Guerrero et al. 2013b) in its time-independent version. Proposition 1 Under the stochastic string model given by the homogeneous covariance R n) x, y) = e τxy) k=0 12) λ k L k x)l k y), 0 < τ < 1, n N 13) 2 with λ k λ k1 > 0, where L n are Laguerre polynomials L n x) = 1 dn n! ex dx e x x n ), the shortterm interest rate rt) follows the dynamics specified in Hull and White 1990) under the n equivalent martingale measure. That is: with drt) = [ω t) κr t) dt σd Zt, 0) 14) κ = n k=0 λ k k τ) n k=0 λ k σ = n [ t ω t) = κ k=0 λ k du u=0 f u, x) x r 0) f 0, x) x t u=0 du 2 f u, x) x 2 σ 2 t 8

9 Proof: See Appendix B. Remark 3 According to this Proposition, all the multi-factor HJM models obtained with the covariance 13) are markovian and allow us to obtain bond prices given by with P r t), t, T ) = e At,T ) Bt,T )rt) 15) A t, T ) = 1 T T 2 σ2 dsb 2 s, T ) dsω s) B s, T ) s=t s=t 1 e κt t) B t, T ) = κ It is important to note that if we had considered time-dependent coefficients in 13) only the model with n = 0 would be Markovian see Bueno-Guerrero et al. 2013b)). Using the previous results, the Jamshidian s trick and expression 7) for the price of a European bond call, straightforward algebra leads to the final call price. Proposition 2 Consider a European call and a European put option with maturity T 0 and strike K written on the cash-flows C 1,..., C n received at times T 1,..., T n. Then, if R t x, y) = R m) x, y) the price of these options is given by Call m) K [t, {P t} = Put m) K [t, {P t} = C i [P t, T i ) N d m) 1,i C i [K i P t, T 0 ) N ) K i P t, T 0 ) N d m) 2,i ) P t, T i ) N d m) 2,i ) d m) 1,i ) where ) 1 2 Ω m) t, T 0, T i ) ln P t,ti ) d m) K i P t,t 0 ) 1,i = Ω m) t, T 0, T i ) d m) 2,i = d m) 1,i Ω m) t, T 0, T i ) with Ω m) t, T 0, T i ) = T0 v=t [ Ti v Ti v dv dydwr m) w, y) y=t 0 v w=t 0 v where K i = P r, T 0, T i ) and r solves n C ip r, T 0, T i ) = K, for P r, t, T ) given by 15). Proof: See Appendix B. 9

10 3 Valuation of caps and swaptions under the Gaussian stochastic string model From now on we consider the interval [T 0, T n and the partition {T j = T 0 δj} n j=1, δ = Tn T 0 n. A cap that starts at time T 0 and ends at time T n on a $1 principal is constituted by a set of n contingent { } n claims, Cpl Tj 1,T j t) with maturities T j, named caplets, whose payoff function is given by j=1 Cpl Tj 1,T j T j ) = δ [L T j 1 ) K 16) where K denotes the cap rate and L T j 1 ) is the LIBOR rate determined at time T j 1 for the period [T j 1, T j and defined by P T j 1, T j ) = [1 δl T j 1 ) 1 17) As is well known Hull and White 1990)), the caplet Cpl Tj 1,T j T j ) can be interpreted as 1 δk) European puts maturing at T j 1 with strike 1 δk) 1 on a $1 face value discount bond that matures at time T j. Using the expression for the European put price see 9)) and summing the caplet prices we obtain the cap price, as stated in the following result. Proposition 3 The price at time t of a cap is given by Capt) = n j=1 Cpl T j 1,T j t) where Cpl Tj 1,T j t) = P t, T j 1 ) N h j,1 t)) 1 δk) P t, T j ) N h j,2 t)) 18) with h j,1 t) = 1δK)P t,tj ) ln P t,t j 1 ) Ωt,Tj 1,T j ) ) 1 2 Ωt,T j 1,T j ), h j,2 t) = h j,1 t) Ω t, T j 1, T j ) 19) Remark 4 Expression 18) for the cap price appeared previously in Chu 1996). If we consider the following approximation see Bueno-Guerrero et al. 2013b)) [ Tj 1 Tj Ω t, T j 1, T j ) = Ω n) v t, T j 1, T j ) = dv v=t y=t j 1 v Tj v w=t j 1 v dydw we recover the cap price in Gaussian HJM models see Brace and Musiela 1994a,b)). Corollary 4 If δ >>, an approximated expression for the cap price is given as Cap B t) = δ l=0 σ l) HJM,t y) σl) HJM,t w) 20) P t, T j ) [ L t, T j 1 ) N h B j,1t) ) KN h B j,2 t) ) 21) j=1 10

11 with h B j,1t) = ln Lt,T j 1) ln K 1 2 Ωt,T j 1,T j ), h B Ωt,Tj 1,T j ) j,2t) = h B j,1t) Ω t, T j 1, T j ) where L t, T j 1 ) denotes the process for the LIBOR rate, defined as 1 δl t, T j 1 ) = P t, T j 1) P t, T j ) 22) Proof: See Appendix B. Remark 5 Expression 21) is the infinite-dimensional extension of that obtained in the LIBOR market model by Brace et al. 1997) and corresponds to the Black formula for caps see, for instance, Björk 2004)). We consider now a fixed payer swaption with strike K, $1 principal, and maturing at time T 0. At maturity T 0, its holder has the option to enter into a swap with settlement times {T j } n j=1, with T 0 < T 1 < < T n, T j T j 1 = δ where he has to pay the fixed amount Kδ and receives δl T j 1 ). It is well known Chance 2003)) that the fixed payer swaption can be interpreted as a European put with strike $1 on a sequence of cash-flows C 1,..., C n obtained at times T 1,, T n with C j = δk for j = 1,..., n 1 and C n = 1 δk. Therefore, we can apply Theorem 4 to obtain the following result. Proposition 4 The price at time t of a fixed payer swaption with strike K that matures at time T 0 is given by Swn t) = d n x [ P t, T 0 ) g x 1,..., x n ; M) ) C i P t, T i ) g x 1 1i 11,, x n ni nn ; M with C j = δk for j = 1,..., n 1 and C n = 1 δk. Equivalently, this price can be written as [ Swn t) = d n y P t, T 0 ) g y 1,..., y n ; I n ) C i P t, T i ) g y 1 γ 1i,..., y n γ ni ; I n ) 23) 24) with γ i = γ 1i,..., γ ni ) verifying γ i γ j = ij. Remark 6 The finite-dimensional version of expression 24) can be found in Brace and Musiela 1994b) and Musiela and Rutkowsky 1995). 11

12 Corollary 5 If ij = <<, i, j = 1,..., n, an approximated expression for the price of a fixed payer swaption is with where Swn B t) = δ is the forward swap rate. Proof: See Appendix B. P t, T i ) [ κ t, T 0, n) N g1 B t, T 0 ) ) KN g2 B t, T 0 ) ) 25) g B 1 t, T 0 ) = ln κt,t 0,n) ln K 1 2, g B 2 t, T 0 ) = g B 1 t, T 0 ) κ t, T 0, n) = [P t, T 0 ) P t, T n ) δ 1 P t, T i )) 26) Remark 7 Expression 25) coincides with the Black formula for swaptions see, for instance, Björk 2004)). To end this Section, applying Proposition 2 to the pricing of swaptions provides the following result that, to our knowledge, is new in the literature. Proposition 5 If we consider R t x, y) = R m) x, y), the price at time t of a fixed payer swaption with strike K that matures at time T 0 is given by Swn m) t) = C i [K i P t, T 0 ) N where C j = δk for j = 1,..., n 1, C n = 1 δk, and ln d m) 1,i = P t,ti ) K i P t,t 0 ) d m) 2,i = d m) 1,i d m) 2,i ) P t, T i ) N ) 1 2 Ω m) t, T 0, T i ) Ω m) t, T 0, T i ) Ω m) t, T 0, T i ) d m) 1,i with T0 [ Ti Ω m) v Ti v t, T 0, T i ) = dv dydwr m) w, y) v=t y=t 0 v w=t 0 v where K i = P r, T 0, T i ) and r solves n C ip r, T 0, T i ) = 1, for P r, t, T ) given by 15). It is important to note the savings in computational cost of expression 27) relative to expressions 23) or 24). Moreover, our expression is not the result of a mathematical approximation but it is exact until the desired order. 3 multi-factor HJM model Bueno-Guerrero et al. 2013b)). ) 27) This order is determined by the number of factors considered in a 3 Approximate expressions for the price of a swaption appear, for instance, in Brace and Musiela 1994b), Brace et al. 1997), Pang 1999), Collin-Dufresne and Goldstein 2002b), and Han 2007). 12

13 4 The stochastic string market model In this Section we introduce a Libor market model under the stochastic string framework that nests the BGM and LSS models. Bueno-Guerrero et al. 2013a) provide the following expression for the dynamics of the bond price ratio d [P t, ν) /P t, τ) P t, ν) /P t, τ) = dt dt τ t τ t τ t ν t dxdyr t x, y) dxdyr t x, y) τ t ν t d Z t, y) dyσ t, y) d Z t, y) dyσ t, y) and this relationship between stochastic string shocks under the martingale equivalent measure, d Z t, y), and under the τ-forward measure, d Z τ t, y) d Z τ t, y) = d Z t, y) dt τ t u=0 28) duc t, u, y) σ t, u) 29) Making ν = T j 1, τ = T j and joining 28) and 29), we obtain the following expression under the T j -forward measure d [P t, T j 1 ) /P t, T j ) P t, T j 1 ) /P t, T j ) = Tj t y=t j 1 t d Z T j t, y) dyσ t, y) 30) Differentiating 22) we obtain dl t, T j 1 ) L t, T j 1 ) = 1 δl t, T j 1) δl t, T j 1 ) d [P t, T j 1 ) /P t, T j ) P t, T j 1 ) /P t, T j ) 31) that, jointly with 30), provides the stochastic string dynamics for the LIBOR rate dl t, T j 1 ) L t, T j 1 ) = 1 δl t, T j 1) δl t, T j 1 ) Tj t y=t j 1 t d Z T j t, y) dyσ t, y) 32) Our LIBOR market model is completely general as we have not made any assumptions on the volatility σ t, y) yet. From now on, we will consider two alternative volatility structures that will lead to two concrete models within our framework. 4.1 The stochastic string LIBOR market model Expression 32) can be rewritten as dl t, T j 1 ) L t, T j 1 ) = Tj t y=t j 1 t d Z T j t, y) dyσ t, y) 33) with σ t, y) = 1δLt,T j 1) δlt,t j 1 ) σ t, y). If we take a deterministic σ t, y), we obtain a lognormal stochastic string dynamics for the LIBOR forward rate that will be named stochastic string LIBOR market 13

14 model. From now on, functions that depend on σ will be overlined to indicate that we have replaced σ by σ in all the volatilities. As an application we will obtain again explicitly expression 21) that corresponds to the Black 1976) formula for the cap price. Using the stochastic exponential in 33), we have { L t, T j 1 ) t Tj s L 0, T j 1 ) = exp d Z T j s, y) dyσ s, y) 1 s=0 y=t j 1 s 2 Making t = T j 1 in 34), we obtain { L T j 1, T j 1 ) Tj 1 Tj s = exp d L 0, T j 1 ) Z T j s, y) dyσ s, y) 1 s=0 y=t j 1 s 2 t s=0 Tj s Joining expressions 34) and 35) and taking logarithms, we obtain ) L Tj 1, T j 1 ) Tj 1 Tj s ln = d L t, T j 1 ) Z T j s, y) dyσ s, y) 1 s=t y=t j 1 s 2 s=t x=t j 1 s Tj s Tj 1 Tj s s=0 y=t j 1 s x=t j 1 s Tj 1 Tj s x=t j 1 s Tj s dsdxdyr s x, y) y=t j 1 s Tj s y=t j 1 s 34) } dsdxdyr s x, y) 35) dsdxdyr s x, y) Then, under the T j -forward measure, Q T j, L T j 1, T j 1 ) follows a conditional lognormal distribution with E QT j [ln L T j 1, T j 1 ) F t = ln L t, T j 1 ) 1 2 Ω t, T j 1, T j ) V ar Q T j [ln L T j 1, T j 1 ) F t = Tj 1 Tj s s=t x=t j 1 s Tj s y=t j 1 s Hence, the caplet price is given by Cpl Tj 1,T j t) = E Q e ) T j s=t dsrs) δ [L T j 1 ) K F = δp t, T j ) E QT j ) [L Tj 1 ) K F t [ ) = δp t, T j ) L t, T j 1 ) N h B j,1 t) KN where we have moved from measure Q to measure Q T j dsdxdyr s x, y) = Ω t, T j 1, T j ) ) h B j,2 t) 36) Brace and Musiela 1994a)) and we have applied the properties of the lognormal distribution. Adding the caplet prices takes us to expression 21). An interesting particular case of 33) arises with the process d Z T j t, x) = i=0 σ i) HJM,t x) σt, x) where σ i) HJM,t x) are the volatilities of a multi-factor HJM model with ) σ 2 t, x) = σ i) 2 HJM,t x) i=0 14 d W T j i t) 37) }

15 This process was proposed originally by Pang 1999). Bueno-Guerrero et al. 2013b) show that, under certain regularity conditions, this process is an admissible stochastic string shock. Replacing it into 33) we obtain with γ t, T t) = dl t, T j 1 ) L t, T j 1 ) = γ t, T j 1 t) d W T j t) 38) 1 δl t, T ) δl t, T ) T tδ y=t t dyσ0) HJM,t y). T tδ y=t t dyσn) HJM,t y) 39) Expression 38) with deterministic γ equates exactly expression 3.6) in Brace et al. 1997). Hence, we have shown explicitly that the BGM model is nested in our framework. Moreover, this construction can be done in infinite-dimensional terms just taking the appropriate regularity conditions for the volatilities Bueno-Guerrero et al. 2013b)). However, obtaining a lognormal model for the LIBOR rate that is compatible with HJM, the BGM model, implies that the HJM volatilities must be state dependent see 39)). 4 This fact suppose the disadvantage, extensible to the stochastic string LIBOR model see 33)), of the incompatibility of the BGM model with the Gaussian ones, with all their important results see Section 2). The next Subsection proposes a way of avoiding this incompatibility at the cost of losing the exactness of the model and the absence of arbitrage. 4.2 An approximated stochastic string LIBOR model If we take σ t, y) as deterministic in expression 32), we will work under the Gaussian stochastic string modeling and we will obtain the same results for pricing caps as in Section 2. As we said before, the Gaussian modeling is incompatible with Black formula, but we will see that we can obtain it as an approximate result. Thus, we can explain the use of this formula in the market as an approximation) with the advantages of the Gaussian framework discussed in the previous Subsection. as As in Corollary 4, the approximation consists of taking δ >>. In this case, 32) can be rewritten dl t, T j 1 ) Tj t L t, T j 1 ) = d Z T j t, y) dyσ t, y) 40) y=t j 1 t As σ is deterministic, we recover the lognormality of the LIBOR rate although in a approximated way. Note that, although useful for pricing caps, this approximation is conceptually inappropriate as the 4 This problem is shared with other previous market models included in the HJM scheme. See, for instance, Miltersen et al. 1997). 15

16 usual values for δ are 0.25, 0.5, or 1. Moreover, this approximation implies arbitrage opportunities. In fact, taking into account 22), the approximation 1δLt,T j 1) δlt,t j 1 ) 1 is equivalent to P t, T j ) = 0. 5 It is important to remark that 33) and 40) are formally identical and, then, will provide similar results. However, the results obtained from 33) will be exact and incompatible with the Gaussian models while those obtained from 40) are approximated and compatible with the Gaussian models. From now on we will work just with the exact stochastic string LIBOR market model see 33)) and relegate to the Appendix A all the expressions corresponding to the approximated model obtained from 40). For instance, expression 54) in this Appendix includes the caplet price in the approximated model and is formally identical to 36) but replacing h B j, t) by h B j, t). Thus, from the point of view of our modeling, we can see the Black formula for cap prices alternatively as a) an exact expression incompatible with Gaussian models or b) an approximation with arbitrage opportunities) of expression 18), valid in Gaussian stochastic string models. 4.3 The dynamics of the forward swap rate We now analyze whether we could extend the swap market model Jamshidian 1997)) to the stochastic string setting in order to recover the Black formula for swaptions that is used in the market. Or, at least, recover this formula through an approximation similar to that previously used. The key point of the swap market model Björk 2004)) consists of proposing a lognormal dynamics for the forward swap rate κ t, T 0, n) under the measure Q X with numeraire X t) = n P t, T i). In the stochastic string setup and working as in expression 30), we obtain d [P t, T j 1 ) /X t) P t, T j 1 ) /X t) = 1 X t) P t, T j ) j=1 Tj t d Z X t, y) dyσ t, y) Tj 1 t d Z X t, y) dyσt, y) where d Z X t, y) = d Z t, y) dt n Xt) P t, T i) T i t duc t, u, y) σ t, u) is the stochastic string shock under Q X. 6 Applying this expression to the forward swap rate, expression 26), we obtain the following result. Proposition 6 In the stochastic string model, the dynamics of the forward swap rate under Q X is 5 Alternatively, note that δ >> implies T j >> and, then, P t, T j) 0. Regarding this point, see also the conditions imposed on bond prices in Bueno-Guerrero et al. 2013a). 6 Making n = 1 in this expression gives d Z T 1 t, y) = d Z t, y)dt T 1 t duc t, u, y) σ t, u), a relationship previously obtained by Bueno-Guerrero et al. 2013a). 16

17 given by dκ t, T 0, n) κ t, T 0, n) = 1 X t) P t, T j ) j=1 Tj t d Z X t, y) dyσ t, y) 1 P t, T 0 ) P t, T n ) [ Tn t P t, T n ) d Z X t, y)dyσt, y) P t, T 0 ) T0 t d Z X t, y)dyσt, y) This Proposition shows that the forward swap rate does not follow a lognormal distribution except in the simplest case n = 1) where, making κ t, T 0, 1) = L t, T 0 ), we recover the exact dynamics 32) of the LIBOR rate. 5 The problem of the relative valuation of caps and swaptions This Section analyzes the LSS model from the point of view of the stochastic string modeling. Our aim is to provide a theoretical solution to the problem of the relative valuation of caps and swaptions, i.e., the caps mispricing with information extracted from the swaptions market. We start by analyzing the assumptions of the LSS model related to this issue. We will see that some of their assumptions do not hold in our model. Assumption 1 The dynamics of the LIBOR rate is given by dl t, T j 1 ) L t, T j 1 ) = α j 1 t) dt σ j 1 t) d Z j 1 t), j = 1,, n 41) where α j 1 is undetermined, σ j 1 is deterministic, and Z j 1 are correlated, and specific for each forward rate, Brownian motions under Q. If we rewrite the exact dynamics of the LIBOR rate see 33)) under the equivalent martingale measure and we make and Tj t y=t j 1 t 1 δl t, T j 1 ) δl t, T j 1 ) d Z t, y) dyσ t, y) σ j 1 t) d Z j 1 t) 42) Tj t y=t j 1 t T j t u=0 we recover the LIBOR dynamics 41) of the previous assumption. dydur t u, y) α j 1 t) 43) 17

18 Applying the correlation condition on Z j 1 and the identification 42), we get and, then, σ 2 i 1 t) = Ti t [ dt = d Zi 1 ), Z i 1 ) = t x=t i 1 t T i t y=t i 1 t dydur t x, y) = dt Ti t T i t σ 2 i 1 t) dydur t x, y) x=t i 1 t y=t i 1 t 1 δl t, Tj 1 ) δl t, T j 1 ) ) 2 Ti t l=0 dxσ l) HJM,t x) x=t i 1 t 44) where we have applied that R n) t x, y) = n l=0 σl) HJM,t x) σl) HJM,t y) is the n-order approximation to R t in the stochastic string modeling Bueno-Guerrero et al. 2013b)). Hence, we have identified the volatilities σ i t) in 41) as a function of the multi-factor HJM volatilities σ l) HJM,t x). 7 Then, the dynamics 41) of the LSS model is compatible with the exact model for the LIBOR rate see 33)) and, applying Section 4.1, its multi-factor HJM reduction is also compatible with the BGM model. Moreover, doing the necessary changes in 42)-43), it is also compatible with the approximated model see Appendix A). Assumption 2 The dynamics of the bond price is given by where dp t) = r t) P t) dt J 1 t) σ t) L t) d Z t) 45) P t) = P t, T 1 ),..., P t, T n 1 )) σ t) L t) d Z t) = σ 0 t) L t, T 0 ) d Z 0 t),..., σ n 2 t) L t, T n 2 ) d Z n 2 t) and J t) is the Jacobian matrix with J ii t) = 1 δ cases. P t, T i 1 ) P 2 t, T i ), J i,i 1t) = J t) = [L t, T 0),..., L t, T n 2 ) [P t, T 1 ),..., P t, T n 1 ) 1 δp t, T i ) ) ) 2 for i = 1,, n 1, and zero in the remaining The introduction of the short-term interest rate in the drift of 45) is really the way of imposing the no-arbitrage condition as LSS just applies the Itô s rule to obtain the diffusion term because α j 1 t) is unknown. 8 However, with our approximation, we can apply this rule to 41) and check if 45) is right. 7 Appropriate regularity conditions allow us to extend this analysis to infinite-dimensional HJM volatilities. 8 Note that this no-arbitrage condition does not eliminate the trivial arbitrage possibility previously discussed, when considering δ >>. 18

19 From the definition of the LIBOR rate process see 22)) we obtain P t, T j 1 ) = j 2 k=0 P t, T 0 ) [1 δl t, T k ) So we can apply the Itô s rule to P t, T j 1 ) in the form dp t, T j 1 ) = j 2 i=0 P t, T j 1 ) L t, T i ) dl t, T i) 1 2 j 2 i,l=0, j = 2,..., n 2 P t, T j 1 ) L t, T i ) L t, T l ) d [L, T i), L, T l ) t P t, T j 1) P t, T 0 ) dp t, T j 2 2 P t, T j 1 ) 0) L t, T i=0 i ) P t, T 0 ) d [L, T i), P, T 0 ) t 1 2 P t, T j 1 ) 2 P 2 t, T 0 ) d [P, T 0), P, T 0 ) t For the two first terms of dp t,t j 1) P t,t j 1 ), working with the dynamics 41), using 22) and the symmetry of R t x, y) and some algebra leads to Tj 1 t T0 t y=t 0 t u=0 j 2 dydur t u, y)dt For the remaining terms of dp t,t j 1) P t,t j 1 ), substituting 42) in 41) we have i=0 δp t, T i1 ) L t, T i ) σ i t) d P t, T i ) Z i t) 46) dl t, T j 1 ) Tj t L t, T j 1 ) = α j 1 t) dt d Z t, y) dyσ t, y) y=t j 1 t that jointly with the dynamics of the bond return in the stochastic string modeling Bueno-Guerrero et al. 2013a)) leads to the contribution r t) dt T0 t dp t, T 0 ) P t, T 0 ) d Z t, y) dyσ t, y) Adding up 46) and 47) we obtain [ dp t, T j 1 ) P t, T j 1 ) = r t) 1 2 j 2 i=0 = r t) dt T0 t Tj 1 t T0 t x=t 0 t T0 t T0 t d Z t, y) dyσ t, y) dxdyr t x, y)dt 1 2 dxdyr t x, y) dt δp t, T i1 ) L t, T i ) σ i t) d P t, T i ) Z i t) T0 t T0 t T0 t d Z t, y) dyσ t, y) dxdyr t x, y)dt 47) 19

20 Finally, as in LSS footnote 12), we assume that for y T 0 t) the volatility σ t, y) vanishes and so R t x, y)). Taking into account that [ J 1 t) ij = δ P t, T i) P t, T j ) if j i P t, T j 1 ) 0 if j > i we get dp t) = r t) P t) dt J 1 t) σ t) L t) d Z t) that equates expression 45) of Assumption 2. Assumption 3 The factors that generate the historical covariance matrix also generate the implied covariance matrix, but the implied variances of such factors can differ from their historical values. 9 If we take Σ ij t) = 1 dl t, dt cov Ti 1 ) L t, T i 1 ), dl t, T ) j 1) L t, T j 1 ) as the historical covariance matrix and use the expression 33) of the exact LIBOR model, we obtain Σ ij t) = Ti t x=t i 1 t Tj t y=t j 1 t 48) dxdyr t x, y) 49) Bueno-Guerrero et al. 2013b) show that, under very general conditions, 10 we can write R t x, y) = λ t,k f t,k x)f t,k y) k=0 where λ t,k and f t,k are, respectively, the eigenvalues and eigenvectors of an integral operator associated to R t. Taking into account that λ t,k > 0 and λ t,k1 < λ t,k, we can approximate R t x, y) taking a finite number of terms. Using this approximation jointly with the time homogeneity following LSS) and replacing in 49), we obtain Σ ij t) = 1 δl t, T i 1) 1 δl t, T j 1 ) Ti t Tj t dxdy λ k f k x)f k y) δl t, T i 1 ) δl t, T j 1 ) x=t i 1 t y=t j 1 t k=0 = 1 δl t, T i 1) 1 δl t, T j 1 ) ) Ti t ) Tj t λ k dxf k x) dxf k y) δl t, T i 1 ) δl t, T j 1 ) x=t i 1 t x=t j 1 t k=0 50) 9 To simplify the calculations, LSS considered the historical correlation matrix, verifying that the obtained factors are almost identical to those achieved with the covariance matrix. For a comparative analysis of both matrices, see Kerkhof and Pelsser 2002). 10 Basically, R t must be a Hilbert-Schmidt integral kernel in L ) 2 p R for each t. 20

21 Furthermore, assuming that the real swaption price is given by the exact in our Gaussian modeling) expression 23), we have that the covariance implied in this price corresponding to Σ ij is Θ ij t, T 0 ) = t ij t, T 0 ) = = Ti t λ k k=0 x=t 0 t Ti t x=t 0 t Tj t y=t 0 t dxdyr x, y) ) Tj t ) dxf k x) dyf k y) y=t 0 t ) This covariance is the same that could be obtained taking cov dp t,ti ) P t,t i ), dp t,t j) P t,t j ) in 45). This agrees with the LSS procedure, consisting in taking the implied covariance obtained from Assumption 3 to generate sample paths for the dynamics 45). This covariance is robust with respect to the model specification, being the covariance between changes in the log-prices identical in the stochastic string Gaussian, exact LIBOR, and approximated LIBOR models. Hence, the result to be obtained will not depend on the model that really describes the swaption price. It is easy to check that a necessary condition for Assumption 3 to be verified is the commutativity of the implied and historical covariance matrices, that is, ΣΘ = ΘΣ ) However, this is not the case, as can be seen by considering expressions 50)-51) and applying some algebra. This is true even for the approximated model. Therefore, Assumption 3 is incompatible with our model. However, it is important to note that, although the historical and implied covariance matrices are diagonalized by different eigenvectors, both matrices are determined by f k, although these values are not obtained by the LSS procedure. The importance of these inner factors is emphasized by Roncoroni et al. 2010) that assigned them the name shape factors and by Bueno-Guerrero et al. 2013b) who show the relation with the factors obtained from a Principal Component Analysis. The knowledge of {λ i, f i } k i=0 is equivalent to the knowledge of R k) x, y) that drives the dynamics of the interest rates in a k 1)-factor model. Assumption 4 The variance used in the Black s 1976) formula to price a caplet is the average variance for the corresponding LIBOR rate. According to our exact model, the variance associated to a caplet in the Black s 1976) formula is given by or, alternatively, Ω t, T j 1, T j ) = 11 See LSS, footnote 17, p Tj 1 v=t dv Tj v Ω t, T j 1, T j ) = x=t j 1 v Tj 1 v=t Tj v y=t j 1 v dydurx, y) dvσ 2 j 1 v) 52) ) 21

22 If we take into account that LSS assumes constant variances for each interval, 52) can be interpreted as the time average of the variances between changes in LIBOR rates. We have just seen that Assumption 3 of LSS does not hold in the stochastic string framework, which could explain the problem of the relative valuation of caps and swaptions. A possible way of avoiding the need of making this Assumption and maintaining the remaining ones in LSS consists of taking a k 1)-factor parametric form of the covariance R x, y) with good properties for instance, R k) in 13)) and estimating λ 0,, λ k from swaptions prices. 12 With these values we can build Ω and apply the Black formula. It is important to mention that, given the generality of the stochastic string modeling, there exist other alternatives to perform the relative valuation of caps and swaptions. For example, we could use the Gaussian scheme of Section 3 or we could estimate the eigenvalues directly from the TSIR. However, these possibilities involve an empirical study that is beyond the scope of this paper. 6 The observational equivalence problem As mentioned earlier, LSS argue that their model is more parsimonious than the multi-factor models for the LIBOR forward rate. However, Kerkhof and Pelsser 2002) reject this argument and show that both models must estimate the same number of parameters. The key point in their argument is that the covariance matrices of the log-changes in the forward rates are equal in the LSS and the BGM models. We will show that this is still true in our framework, validating then the observational equivalence. Kerkhof and Pelsser obtain the aforementioned covariance matrix for the LSS model that, with our notation and eliminating the functional dependence, can be written as Σ LSS ij = ρ ij σ i 1 σ j 1 53) [ where ρ ij = d Zi 1 ), Z j 1 ) /dt. In our model, using 42), we have that t ρ ij = 1 σ i 1 σ j 1 Ti t x=t i 1 t Tj t y=t j 1 t dxdyr t x, y) Replacing this expression in 53), we get Ti Σ LSS t Tj t ij = dxdyr t x, y) x=t i 1 t y=t j 1 t that coincides with the expression obtained for the LSS model in our framework, see 49). 12 See Roncoroni et al. 2010) for more details on the estimation of the eigenvalues. 22

23 We will obtain now explicitly the value of Σ for the BGM model under our setup. expressions 38)-39), we get Σ BGM ij = 1 [ dt cov dlt,ti 1 ) Lt,T i 1 ), dlt,t j 1) Lt,T j 1 ) Ti = δlt,t t Tj t i 1) δlt,t j 1 ) 1δLt,T i 1 ) 1δLt,T j 1 ) x=t i 1 t y=t j 1 t Ti t Tj t = x=t i 1 t y=t j 1 t dxdyr k) t x, y) dxdyr k) t x, y) Using where R k) t x, y) = k l=0 σl) HJM,t x)σl) HJM,t y) is an approximation of order n to the covariance R t x, y). Then, in concordance with Kerkhof and Pelsser 2002), we see that the covariances of the LSS and BGM models coincide if we consider a factorial version of the LSS model with the same number of factors as the BGM model. This coincidence remains even for the LSS and BGM models in our approximated framework. Then, we can conclude that our framework maintains the observational equivalence between the LSS and BGM models and, then, both models need to estimate the same number of parameters. However, considering both models under the stochastic string modeling modifies the number of parameters to estimate, making these models more parsimonious. The reason is that, as we have seen, our framework is completely defined in a k 1)-factor approximation by {λ i, f i } k i=0. The eigenfunctions f i usually belong to a uniparametric family. For instance, in the case of the covariance R, we have f i x) = L i x) e τx and the parameter τ is usually estimated a priori and maintained fixed see Roncoroni et al. 2010) and Bueno-Guerrero et al. 2013b)). Hence, in general, in a k-factor approximation, we need to estimate just k parameters, a value independent of the number of forward rates, n. Then, the stochastic string approximation is more parsimonious if k < nk kk 1)/2, the number of parameters obtained by Kerkhof and Pelsser 2002). This is equivalent to k < 2n 1, an inequality that is verified in most the cases. 7 Conclusions This paper has applied recent developments in the stochastic string modeling to value caps and swaptions. We have obtained closed-form expressions for caps and swaptions prices and we have shown how these expressions reduce to Black formulas under certain approximations. We have also developed a stochastic string LIBOR market model that nests the LSS and BGM models. With the same approximation applied previously to caps, we have obtained an approximated model that is compatible with the Black formula for caps. Under our framework, we have shown that the LSS model is not well specified, which could explain the problem of the relative valuation of caps and swaptions. We have proposed a possible 23

24 solution to this problem. We have also corroborated the observational equivalence of the LSS and BGM models obtained by Kerkhof and Pelsser 2002). Additionally, the number of parameters to estimate is smaller than what was stated by these authors in almost all the cases. The generality of the stochastic string framework allows us to consider several concrete models, namely, Gaussian and exact or approximated) lognormal ones. Hence, a possible line of future research could be to analyze empirically which model fits better to caps and swaptions market prices. Two possible alternatives are: a) a Gaussian model compatible with deterministic HJM volatilities and with an approximated Black formula for caps or b) a model that is compatible with lognormal LIBOR rates and state-dependent HJM volatilities) and exact Black formula. Regarding this issue, we can mention the existence of certain indirect empirical evidence in favor of Gaussian models BGM, De Jong et al. 2004), Han 2007).). Another extension of the paper would be to develop a setting that allows for the correct relative valuation of caps and swaptions. This could be done, for instance, by rebuilding the LSS model correcting Assumption 3 as indicated in Section 5. Finally, we are studying the implications that some concrete new results obtained in this paper the closed-form expressions for the prices of portfolio options and swaptions) will have for future developments in this field. 24

25 Appendix A This Appendix provides the main expressions corresponding to the approximated stochastic string LIBOR model that is obtained making δ >> in the exact stochastic string LIBOR model. Dynamics of the LIBOR rate: dl t, T j 1 ) Tj t L t, T j 1 ) = d Z T j t, y) dyσ t, y), y=t j 1 t deterministic σ Caplet price: Cpl Tj 1,T j t) = δp t, T j ) [ L t, T j 1 ) N h B j,1t) ) KN h B j,2t) ) 54) BGM dynamics: with deterministic γ t, T t) given by dl t, T j 1 ) L t, T j 1 ) = γ t, T j 1 t) d W T j t) T tδ y=t t dyσ0) HJM,t y). T tδ y=t t dyσn) HJM,t y) LSS dynamics: with dl t, T j 1 ) L t, T j 1 ) = α j 1 t) dt σ j 1 t) d Z j 1 t) α j 1 t) σ j 1 t)d Z j 1 t) Tj t y=t j 1 t Tj t y=t j 1 t T j t u=0 dydur t u, y) d Zt, y)dyσt, y) LSS volatility: σ 2 i 1 t) = Ti t T i t x=t i 1 t y=t i 1 t dydur t x, y) = Ti t dxσ l) HJM,t x) x=t i 1 t l=0 ) 2 Bond return in the LSS model: j 2 dp t, T j 1 ) P t, T j 1 ) = r t) dt σ i t) d Z i t) i=0 25

26 Historic covariance matrix: Σ ij t) = Implied covariance matrix: Appendix B Θ ij t, T 0 ) = Ti t x=t i 1 t Ti t x=t 0 t Tj t y=t j 1 t Tj t y=t 0 t dxdyr x, y) dxdyr x, y) Proof of Theorem 2 The pay-off at time T 0 from the option is [ Call K [T 0, {P T0 } = C i P T 0, T i ) K [ = C i P T 0, T i ) KP T 0, T 0 ) = [Φ {P T0 } with Φ homogeneous of degree one. Applying Theorem 1 we get [ Call K [t, {P t } = d n xg x 1,..., x n ; M) C i P t, T i ) ii x e i 1 2 ii KP t, T 0 ) = [ d n x C i P t, T i ) g KP t, T 0 ) g x 1,..., x n ; M) x 1 1i 11,, x n ni nn ; M ) where the last equation arises from applying the identity g x 1,..., x n ; M) ii x e i 1 ) 2 ii = g x 1 1i 11,, x n ni nn ; M 55) Proof of Theorem 3 Consider the matrix Q = q ij ) that verifies M = Q Q and define y = Q ) 1 x. We will have d n xg x 1,..., x n ; M) = d n yg y 1,..., y n ; I n ). Using Theorem 2 and 55) we can write [ Call K [t, {P t } = d n n yg y 1,..., y n ; I n ) C i P t, T i ) ii e k=1 q kiy k 1 2 ii KP t, T 0 ) In addition, applying n k=1 q2 ki = M ii = 1, it is easy to obtain the identity g y 1 γ 1i,..., y n γ ni ; I n ) = g y 1,..., y n ; I n ) e ii n k=1 q kiy k 1 2 ii 56) 26