Jump-telegraph models for the short rate: pricing and convexity adjustments of zero coupon bonds
aa r X i v : . [ q -f i n . M F ] J a n Jump-telegraph models for the short rate: pricing andconvexity adjustments of zero coupon bonds
Oscar L´opez ∗ , Gerardo E. Oleaga † , and Alejandra S´anchez ‡ Universidad Nacional de Colombia, Carrera 45 No. 26-85, Bogot´a, Colombia Departamento de An´alisis Matem´atico y Matem´atica Aplicada & Instituto de Matem´aticaInterdisciplinar, Universidad Complutense de Madrid, Spain
January 11, 2019
Abstract
In this article, we consider a Markov-modulated model with jumps for short ratedynamics. We obtain closed formulas for the term structure and forward rates using theproperties of the jump-telegraph process and the expectation hypothesis. The results arecompared with the numerical solution of the corresponding partial differential equation.
Keywords : Jump-telegraph process, Markov-modulated model, term structure, zero couponbond, short rate model, convexity adjustment.
Regime changes associated to unexpected events in the economy provide an important fieldof study in quantitative finance. In the dynamics of the interest rates, the impact of thosechanges appears in the form of stochastic jumps. A well-known mathematical tool to modelthese types of events is provided by the family of Markov-modulated processes (see [7], [10]).Among them, the so-called telegraphic processes with jumps are of particular importance fora number of reasons (see [6], [8]). First, they capture some of the stylized facts reported inthe literature (see [13], [5] and the references therein); next, closed formulas for the primarydescriptors (mean, variance, moment generating functions) are available; therefore, they aregood candidates to model interest rate dynamics, providing explicit results for fixed incomeinstruments.In this work, we obtain formulas for the term structure modeled by the short rate dynamicunder jump-telegraphic Markov-modulated processes. The models proposed (not necessarilyaffine) are such that the drift, jump size, and jump intensity depend on a continuous timeMarkov chain with two states. In our approach, jumps are due to a sudden change in theeconomic regime, that is, a switch in the market environment. For the applications considered,we require the integral of this process, whose primary characteristics are not known (see ∗ [email protected] † [email protected] ‡ [email protected] rational expectation hypothesis is assumed. Thisimplies that the forward instantaneous rate can be computed as the expected value of thefuture short rate. This approximation, also known as a convexity adjustment (see [12]), is keyto obtaining a closed formula for zero coupon bonds. This result is subsequently validatedusing a different approach. Under the no-arbitrage hypothesis, we provide the system ofPartial Differential Equations (PDEs) that determine the price of the zero coupon bonds.A numerical approximation of the solutions to this system shows a close correspondence ofboth methods. In the second approach, the expectation hypothesis is not used; therefore, theconvexity adjustment can be estimated.This article is organized as follows. In the following section, the jump-telegraph modelsare defined; some of their properties are shown and some results connected to equivalentmeasures and martingales are proven. In Section 3 a general model for the short rate evolutionis presented, involving a Markov-modulated process with jumps. The change of measure andan extension of the Feynman–Kaˇc theorem are analyzed. In Section 4, we provide analyticformulas for the forward rates in four particular cases of the general model. In Section 5,we provide formulas for the zero coupon bonds, as well as numerical approximations for thePDE system obtained for each example in Section 4. Conclusions are provided in the lastsection. We present herein the jump-telegraph process and explore two types of technical properties.The first group (Propositions 2.1 and 2.2) is connected with its primary parameters andmoment generating functions. These are required to provide an approximating formula forthe forward rate in Section 4. The second group of results (Theorem 2.1 and its corollary)will be used to provide a characterization of the equivalent probability measures in Section 3.This is performed through a change of measure process with specific martingale properties.Let
T > , F , {F t } t ∈ [0 ,T ] , P ) a filtered probability space underthe typical hypotheses. In this space, we consider a Markov chain in continuous time ε = { ε ( t ) } t ∈ [0 ,T ] , with state space { , } and infinitesimal generator given by Λ := (cid:16) − λ λ λ − λ (cid:17) , λ , λ >
0. Denoted by { τ n } n ≥ the switching times of the Markov chain ε and defining τ := 0,it is known that the inter-arrival times { τ n − τ n − } n ≥ are independent, exponential randomvariables, with P { τ n − τ n − > t | ε ( τ n − ) = i } = e − λ i t , i ∈ { , } .Let N = { N t } t ∈ [0 ,T ] be the process that counts the number of state switches of the chain ε , defined as N t := X n ≥ { τ n ≤ t } , N = 0 .N is a non-homogeneous Poisson process with stochastic intensity given by { λ ε ( t ) } t ∈ [0 ,T ] , aspecial case of the so-called doubly stochastic Poisson processes see [2].Let c , c , h , and h be real numbers such that c = c , and h , h = 0, and ε j = ε ( τ j − )is the value of the Markov chain ε just before the j -th change at time τ j . We define thejump-telegraph process Y = { Y t } t ∈ [0 ,T ] as the sum Y t := X t + J t = Z t c ε ( s ) d s + N t X j =1 h ε j , X = { X t } t ∈ [0 ,T ] is known as an asymmetric telegraph process [9], and J = { J t } t ∈ [0 ,T ] is a pure jump process [8].By fixing the initial state ε (0) = i ∈ { , } , we have the following equality in the distri-bution: Y t D = c i t { t<τ } + (cid:2) c i τ + h i + e Y t − τ (cid:3) { t>τ } , (1)for any t >
0, where e Y = { e Y t } t ∈ [0 ,T ] is a jump-telegraph process independent of Y , driven bythe same parameters, but starting from the opposite initial state 1 − i .We denote by p i ( x, t ) the following density functions: p i ( x, t ) := P i { Y t ∈ d x } d x , i = 0 , , where P i {·} = P {· | ε (0) = i } . That is, for any Borel set ∆, ∆ ⊂ R , R ∆ p i ( x, t )d x = P i { Y t ∈ ∆ } .By (1) and the total probability theorem, the functions p i ( x, t ) satisfy the following systemof integral equations on [0 , T ] × R p i ( x, t ) = e − λ i t δ ( x − c i t ) + Z t p − i ( x − c i s − h i , t − s ) λ i e − λ i s d s, i = 0 , , where δ ( · ) is Dirac’s delta function.We denote by E i [ · ] = E [ · | ε (0) = i ], i = 0 , ε . Proposition 2.1 ([6] Section 4.1.2) . For any t >
0, the conditional expectations m i ( t ) := E i [ Y t ], i = 0 , Y are given by m i ( t ) = 12 λ (cid:20) ( λ d + λ d ) t + ( − i λ i ( d − d ) (cid:18) − e − λt λ (cid:19)(cid:21) , (2)where 2 λ := λ + λ and d i = c i + λ i h i , i = 0 , Proposition 2.2 ([8] Theorem 3.1) . For any z ∈ R and t > , the moment generatingfunctions φ i ( z, t ) := E i [e zY t ], i = 0 , Y are given by φ i ( z, t ) = e t ( cz − λ ) cosh (cid:16) t √ D (cid:17) + ( − i (cid:16) az − κ + ( − i λ i e zh i (cid:17) sinh (cid:16) t √ D (cid:17) √ D , (3)where D = ( az − κ ) + λ λ e zH , 2 c = c + c , 2 a = c − c , 2 κ = λ − λ and H = h + h .We define the filtration F = {F t } t ∈ [0 ,T ] generated together by the Markov chain and thePoisson process: F t = σ ( ε ( s ) , s ∈ [0 , t ]) ∨ σ ( N s , s ∈ [0 , t ])The following two results are key to obtaining a set of equivalent measures (see Proposition1.7.1.1 in [4]). 3 heorem 2.1. The following processes are F -martingales Z t := N t X j =1 h ε j − Z t h ε ( s ) λ ε ( s ) d s, (4) E t ( Z ) := exp (cid:18) − Z t h ε ( s ) λ ε ( s ) d s (cid:19) N t Y j =1 (1 + h ε j ) , for h , h > − . (5)Here, E t ( · ) denotes the stochastic (Dol´eans-Dade) exponential (see, e.g., [4], Section 9.4.3). Proof.
Observe that Z is a jump-telegraph process with c i = − h i λ i , i = 0 ,
1. Subsequently,by Proposition 2.1, E i [ Z t ] = 0, i = 0 ,
1. Let s , 0 ≤ s ≤ t be fixed. Let i ∈ { , } be the valueof ε at time s and let k ∈ N be the value of N at time s . By the strong Markov property, wehave the following conditional identities in the distribution ε ( s + u ) (cid:12)(cid:12) { ε ( s )= i } D = ˜ ε ( u ) (cid:12)(cid:12) { ˜ ε (0)= i } , N s + u (cid:12)(cid:12) { ε ( s )= i } D = N s + e N u (cid:12)(cid:12) { ˜ ε (0)= i } , u ≥ ,τ k + j (cid:12)(cid:12) { ε ( s )= i } D = ˜ τ j (cid:12)(cid:12) { ˜ ε (0)= i } , j = 1 , , . . . (6)where ˜ ε , e N and { ˜ τ j } are copies of the processes ε , N , and { τ j } , independents of F s . Subse-quently, using the zero conditional property of Z and (6), we obtain E [ Z t − Z s | F s ] = E i e N t − s X j =1 h ˜ ε j − Z t − s h ˜ ε ( u ) λ ˜ ε ( u ) d u = 0 , and the first part follows.The jump-telegraph process is defined as follows: b Z t = N t X j =1 log(1 + h ε j ) − Z t h ε ( s ) λ ε ( s ) d s, Subsequently, we have E t ( Z ) = e b Z t and by Proposition 2.2 we obtain E i [e b Z t ] = 1, i = 0 , E (cid:2) e b Z t − b Z s (cid:12)(cid:12) F s (cid:3) = E i exp e N t − s X j =1 log(1 + h ˜ ε j ) − Z t − s h ˜ ε ( u ) λ ˜ ε ( u ) d u = 1 , and the desired result is obtained. Corollary 2.1.
The following processes are F -martingales M t := N t − Z t λ ε ( s ) d s, (7) L θt := exp (cid:18)Z t (1 − θ ε ( s ) ) λ ε ( s ) d s (cid:19) N t Y j =1 θ ε j , for θ , θ > . (8)4 roof. For the first part, notice that if we use h = h = 1, by (4) the result is obtained.For the second part, let e Z t = N t X j =1 ( θ ε j − − Z t ( θ ε ( s ) − λ ε ( s ) d s. Therefore, L θt = E t ( e Z ); subsequently, by (4) and (5), the process L θ is a P -martingale with E i [ L θt ] = 1, i = 0 , For each i ∈ { , } , let µ i : Ω × R → R , σ i : Ω × R → R ≥ and η i : Ω × R → R \ { } aremeasurable functions; we model the dynamics of the short interest rate byd r t = µ ε ( t ) ( r t )d t + σ ε ( t ) ( r t )d W t + η ε ( t − ) ( r t − )d N t , (9)where µ denotes the mean return rate, σ the volatility, and η the amplitude of the jumpsof the short rate whenever a switch occurs in the Markov chain ε . Here, W = { W t } t ∈ [0 ,T ] is a standard Wiener process, independent of the Markov chain ε and hence of the Poissonprocess N .We define the filtration F = {F t } t ∈ [0 ,T ] now generated by the Markov chain, Poissonprocess, and Wiener process: F t = σ ( ε ( s ) , s ∈ [0 , t ]) ∨ σ ( W s , s ∈ [0 , t ]) ∨ σ ( N s , s ∈ [0 , t ]) . The conditions that guarantee a unique strong solution of (9) for each initial value, can befound for instance in [14], Proposition 2.1. Finally, the dynamics of the bank account is givenby d B t = r t B t d t. Let M = { M t } t ∈ [0 ,T ] be the martingale associated to the Poisson process N (see Corollary2.1). Let us define a set of equivalent measures using the product of two Girsanov transforms.First, we define the change of measure process for the Wiener process W . Hence, we let L ψ = { L ψt } t ∈ [0 ,T ] be the process defined by L ψt := exp (cid:18) − Z t ψ ε ( s ) d W s − Z t ψ ε ( s ) d s (cid:19) . (10)where ψ , ψ ∈ R . It is obvious that { ψ ε ( t ) } t ∈ [0 ,T ] satisfies Novikov’s condition; therefore, L ψ is a F -martingale with E [ L ψT ] = 1.We now consider the martingale L θ = { L θt } t ∈ [0 ,T ] defined by (8). Subsequently, theprocess L = { L t } t ∈ [0 ,T ] defined by L t := L ψt · L θt is an F -martingale with E [ L T ] = 1 (see [4]Section 10.3.1). Therefore, we define a set of equivalent measures using the Radon–Nikodymderivative d Q d P (cid:12)(cid:12)(cid:12) F t := L t , t ∈ [0 , T ]. 5 roposition 3.1. Under measure Q , the process W Q t := W t − Z t ψ ε ( s ) d s, t ∈ [0 , T ] , (11)is a Wiener process, and M Q t := M t − Z t (cid:0) − θ ε ( s ) (cid:1) λ ε ( s ) d s = N t − Z t θ ε ( s ) λ ε ( s ) d s, (12)is a martingale. Proof.
For the first part, considering (10), applying the Girsanov theorem to the Wienerprocess, and using the independence of W and N , we immediately obtain that (11) is aWiener process under Q .For the second part, using integration by parts and the independence of W and N , weobtain d( M Q t L θt ) = M Q t − d L θt + L θt − d M Q t + d[ M Q , L θ ] t = M Q t − d L θt + L θt − d M Q t + L θt − ( θ ε ( t − ) − N t = M Q t − L θt − ( θ ε ( t − ) − M t + L t − d M t + L θt − ( θ ε ( t − ) − M t = (cid:0) M Q t − ( θ ε ( t − ) −
1) + θ ε ( t − ) (cid:1) L θt − d M t . Thus, the process M Q L θ is a P -martingale and the process M Q is a Q -martingale.Notice that, because of (12), the process N under measure Q is a Poisson process withintensities λ Q = θ λ and λ Q = θ λ . Substituting in (9), the dynamics of the short rate inthe measure Q is given byd r t = (cid:0) µ ε ( t ) ( r t ) + σ ε ( t ) ( r t ) ψ ε ( t ) + η ε ( t ) ( r t ) θ ε ( t ) λ ε ( t ) (cid:1) d t + σ ε ( t ) ( r t )d W Q t + η ε ( t − ) ( r t − )d M Q t (13)The particular structure of the short rate model proposed in (9) requires a special formof the Feynman–Kaˇc formula. To the best of our knowledge, such a formula is not availablefor this type of process in the current literature. Theorem 3.1.
We denote by F ( t, ε ( t ) , r t ) := F ε ( t ) ( t, r t ) the price of a zero coupon bond withmaturity T . Consider the coupled Cauchy problem: ∂F i ∂t ( t, x ) + L F i ( t, x ) = xF i ( t, x ) , ( t, x ) ∈ [0 , T ) × R , i = 0 , ,F ( T, x ) = F ( T, x ) = 1 , (14)where L is the operator defined by L F i ( t, x ) := (cid:0) µ i ( x )+ ψ i σ i ( x ) (cid:1) ∂F i ∂x ( t, x )+ 12 σ i ( x ) ∂ F i ∂x ( t, x ) + λ Q i (cid:2) F − i ( t, x + η i ( x )) − F i ( t, x ) (cid:3) . If F i ( · , · ), i = 0 , F ε ( t ) ( t, r t ) = E Q h e − R Tt r s d s | F t i , (15)where the short rate process r satisfies the stochastic differential equation (13).6 roof. By adapting the Itˆo formula to the process F ε ( t ) ( t, r t ) (see for instance [4], Section10.2.2), we obtain F ε ( T ) ( T, r T ) = F ε ( t ) ( t, r t ) + Z Tt σ ε ( s ) ( r s ) ∂F ε ( s ) ∂x ( s, r s )d W Q s + Z Tt (cid:2) F − ε ( s ) (cid:0) s, r s − + η ε ( s ) ( r s − ) (cid:1) − F ε ( s ) ( s, r s − ) (cid:3) d M Q s + Z Tt (cid:20) ∂F ε ( s ) ∂t ( s, r s ) + L F ε ( s ) ( s, r s ) (cid:21) d s Using (14), the last integral is equal to R Tt r s F ε ( s ) ( s, r s )d s . Consider now the process Z = { Z t } t ∈ [0 ,T ] , defined as Z t := e − R t r s d s F ε ( t ) ( t, r t ). By applying the product rule we obtain: Z T = e − R t r s d s F ε ( s ) ( t, r t ) + Z Tt e − R s r u d u σ ε ( s ) ( r s ) ∂F ε ( s ) ∂x ( s, r s )d W Q s + Z Tt e − R s r u d u (cid:2) F − ε ( s ) (cid:0) s, r s − + η ε ( s ) ( r s − ) (cid:1) − F ε ( s ) ( s, r s − ) (cid:3) d M Q s . With the appropriate integrability conditions, the process Z is a martingale and we finallyobtain (15).Notice that (15) provides the price of a zero coupon bond when the equivalent measure Q is selected as a risk neutral measure, once the free parameters θ i and Ψ i for i = 0 , The unbiased expectation hypothesis postulates that, in an efficient market, the instantaneousforward rate at time t with maturity T must be equal to the expected future spot rate, thatis f ε ( t ) ( t, T, r t ) = E Q [ r T | F t ] . Because the short rate is not deterministic, these values do notcoincide, and their difference is a measure known as the convexity adjustment, see [1], Section26.4 and [3]. We exploit this hypothesis and the jump-telegraph model to obtain a suitableanalytical approximation of the forward rate.
By adapting the classical Merton model for the short rate, our first model is established bythe following stochastic differential equation:d r t = µ ε ( t ) d t + η ε ( t − ) d N t . Notice that in this case, we have σ i = 0 for i = 0 ,
1; therefore, under the equivalent measure Q , the dynamics of the short rate are maintained; however, the intensities of the process N are λ Q i = θ i λ i . The solution is given by r t = r + X t + J t , where X t = Z t µ ε ( s ) d s y J t = Z t η ε ( s − ) d N s = N t X j =1 η ε j . t , t ∈ [0 , T ], we have that E Q [ r T | F t ] = E Q (cid:20) r t + Z Tt µ ε ( s ) d s + Z Tt η ε ( s − ) d N s (cid:12)(cid:12)(cid:12) F t (cid:21) (16)By the Markov property and applying the distributional equalities in (6), we can write (16)as follows: E Q [ r T | F t ] = r t + E Q i "Z T − t µ ˜ ε ( s ) d s + e N T − t X j =1 η ˜ ε j = r t + E Q i h e X T − t + e J T − t i = r t + E Q i h e Y T − t i . (17)By Proposition 2.1, we have that the conditional expectation is given by E Q i h e Y T − t i =12 λ Q (cid:20) ( λ Q d + λ Q d )( T − t ) + ( − i λ Q i ( d − d ) (cid:18) − exp( − λ Q ( T − t ))2 λ Q (cid:19)(cid:21) , (18)for i = 0 ,
1, where 2 λ Q = λ Q + λ Q y d i = µ i + λ Q i η i , i = 0 , By adapting the classical Dothan model for the short rate, our second model is given by thefollowing stochastic differential equation that we term the jump-telegraph Dothan’s model :d r t = r t − (cid:0) µ ε ( t ) d t + η ε ( t − ) d N t (cid:1) . As in the previous example, the dynamic under Q is conserved, and only the intensities ofthe process N are affected. It is possible to write the solution in terms of the stochasticexponential as follows: r t = r E t ( X + J ) = r exp (cid:18)Z t µ ε ( s ) d s (cid:19) N t Y j =1 (cid:0) η ε j (cid:1) = r exp (cid:18)Z t µ ε ( s ) d s + Z t log (cid:0) η ε ( s − ) (cid:1) d N s (cid:19) . Let b J t = P Nj =1 log (cid:0) η ε j (cid:1) ; subsequently, r t = r exp (cid:16) X t + b J t (cid:17) . So that, for each t ∈ [0 , T ]we have that r T = r t exp (cid:18)Z Tt µ ε ( s ) ds + Z Tt log (cid:0) η ε ( s − ) (cid:1) dN s (cid:19) Subsequently, the expected value is as follows: E Q [ r T | F t ] = r t E Q (cid:20) exp (cid:18)Z Tt µ ε ( s ) ds + Z Tt log (cid:0) η ε ( s − ) (cid:1)(cid:19) dN s | F t (cid:21)
8y the Markov property and (6), we can write E Q [ r T | F t ] = r t E Q i exp Z T − t µ e ε ( s ) ds + e N T − t X j =1 log (1 + η e ε ) = r t E Q i h exp (cid:16) e X T − t + e J T − t (cid:17)i = r t E Q i h exp (cid:16) e Y T − t (cid:17)i . (19)By Proposition 2.2, we find that the exponential E Q i h exp (cid:16) e Y T − t (cid:17)i = exp (( T − t )( ζ − λ Q )) h cosh (cid:16) ( T − t ) √ D (cid:17) +( − i (cid:0) χ − ν + ( − i λ Q i (1 + η i ) (cid:1) sinh (cid:16) ( T − t ) √ D (cid:17) √ D (20)for i = 0 ,
1, where 2 ζ = µ + µ , 2 χ = µ − µ , 2 ν = λ Q − λ Q , D = ( χ − ν ) + λ Q λ Q (1 + η ) (1 + η ). Now, we add a diffusion term to the jump-telegraph Merton model:d r t = µ ε ( t ) d t + σ ε ( t ) d W t + η ε ( t − ) d N t . Under measure Q , this is given byd r t = ( µ ε ( t ) + σ ε ( t ) ψ ε ( t ) )d t + σ ε ( t ) d W Q t + η ε ( t − ) d N t The solution is now written as follows: r t = r + X t + Z t + J t , where X t = Z t (cid:0) µ ε ( s ) + σ ε ( s ) ψ ε ( s ) (cid:1) d s, Z t = Z t σ ε ( s ) d W Q s and J t = Z t η ε ( s − ) d N s = N t X j =1 η ε j . Following similar steps as in the non-diffusive case, we obtain E Q [ r T | F t ] = r t + E Q i h e X T − t + e Z T − t + e J T − t i = r t + E Q i h e X T − t + e J T − t i = r t + E Q i h e Y T − t i , (21)because E Q i [ e Z T − t ] = 0 for i = 0 ,
1. The last expected value exhibits the same functionalformula as (18), replacing d i by ˜ d i = µ i + σ i ψ i + λ Q i η i , i = 0 , .4 Jump-telegraph diffusion Dothan model By adding a diffusion term to the jump-telegraph Dothan Model, we obtain the followingstochastic evolution for the short rate:d r t = r t − (cid:0) µ ε ( t ) d t + σ ε ( t ) d W t + η ε ( t − ) d N t (cid:1) which, under measure Q , is written as follows:d r t = r t − (cid:0) ( µ ε ( t ) + σ ε ( t ) ψ ε ( t ) )d t + σ ε ( t ) d W Q t + η ε ( t − ) d N t (cid:1) The solution is written in terms of the stochastic exponential as follows: r t = r E t ( X + Z + J )= r exp (cid:18)Z t ( µ ε ( s ) + σ ε ( s ) ψ ε ( s ) )d s + Z t σ ε ( s ) d W s − Z t σ ε ( s ) d s (cid:19) N t Y j =1 (cid:0) η ε j (cid:1) = r exp (cid:18)Z t (cid:18) µ ε ( s ) + σ ε ( s ) ψ ε ( s ) − σ ε ( s ) (cid:19) d s + Z t σ ε ( s ) d W s + Z t log(1 + η ε ( s − ) )d N s (cid:19) = r exp (cid:16) b X t + Z t + b J t (cid:17) . To establish a useful formula for the expected value of the future spot rate, we require that σ = σ = σ . Following similar steps as those performed for the non-diffusive case, we obtain E Q [ r T | F t ] = r t E Q i h exp (cid:16) e X T − t + e Z T − t + e J T − t (cid:17)i = r t E Q i h exp (cid:16) e Y T − t (cid:17)i E Q i h exp (cid:16) e Z T − t (cid:17)i = r t E Q i h exp (cid:16) e Y T − t (cid:17)i exp (cid:18) σ T − t ) (cid:19) . (22)Again, in this case, the expected value exhibits the same functional form as that in (20),replacing µ i by ˜ µ i = µ i + σψ − σ / i = 0 , We will calculate the price of a zero coupon bond for each model of the previous section bytwo methods: first, using standard numerical methods for the solution of the Cauchy problemin (14); second, using the unbiased expectation hypothesis, together with the relationship F ε ( t ) ( t, r t ) = exp (cid:18) − Z Tt f ε ( t ) ( t, s, r t )d s (cid:19) . (23) In the jump-telegraph Merton model the system (14) reduces to ∂F i ∂t ( t, x ) + µ i ∂F i ∂x ( t, x ) + λ Q i [ F − i ( t, x + η i ) − F i ( t, x )] = xF i ( t, x ) , i = 0 , F ( T, x ) = F ( T, x ) = 1 . (cid:18) − Z Tt E Q [ r s | F t ]d s (cid:19) = exp (cid:18) − Z Tt r t d s − Z Tt E Q i [ e Y s − t ]d s (cid:19) = exp (cid:0) r t C ( t, T ) + D i ( t, T ) (cid:1) . (24)Here, C ( t, T ) = − ( T − t ) , and D i ( t, T ) = − λ Q "(cid:0) λ Q d + λ Q d (cid:1) ( T − t ) − i λ Q i ( d − d ) λ Q ( T − t ) + e − λ Q ( T − t ) − λ Q · λ Q ! . Observe that, under the unbiased expectations hypothesis, the bond price exhibits an affinestructure.
Numerical results:
Data provided for the following parameter values: µ = − . , µ =0 . , λ Q = 1 , λ Q = 2 , η = 0 . , η = − . Initial rate: ODE Expectation
Maturity F F F F . . . . . . . . Zero coupon bond prices: Jump-telegraph Merton model.
In the jump-telegraph Dothan case, the system (14) reduces to ∂F i ∂t ( t, x ) + xµ i ∂F i ∂x ( t, x ) + λ Q i [ F − i ( t, x (1 + η i )) − F i ( t, x )] = xF i ( t, x ) , i = 0 , ,F ( T, x ) = F ( T, x ) = 1 . The system is solved numerically using a finite difference up-wind scheme for the transportterms.By (23), (19), and (20), the price of the bond, using the approximation provided by theexpectation hypothesis, is given byexp (cid:18) − Z Tt E [ r s | F t ] ds (cid:19) = exp (cid:18) − r t Z Tt E Q i h exp (cid:0) e Y s − t (cid:1)i d s (cid:19) = exp (cid:0) − r t (cid:2) G ( t, T ) + E i H ( t, T ) (cid:3)(cid:1) , G ( t, T ) = e ( T − t )( ζ − λ Q ) h ( ζ − λ Q ) cosh (cid:0) ( T − t ) √ D (cid:1) − √ D sinh (cid:0) ( T − t ) √ D (cid:1)i − ( ζ − λ Q )( ζ − λ Q ) − D ,E i = ( − i χ − ν + ( − i λ Q i (1 + η i ) √ D , i = 0 , , and H ( t, T ) = e ( T − t )( ζ − λ Q ) h ( ζ − λ Q ) sinh (cid:0) ( T − t ) √ D (cid:1) − √ D cosh (cid:0) ( T − t ) √ D (cid:1)i + √ D ( ζ − λ Q ) − D .
Numerical results:
The solution is provided for the following parameter values: µ = − . , µ = 0 . , λ Q = 1 . , λ Q = 2 . , η = 0 . , η = − . Initial rate: Finite Differences Expectation
Maturity F F F F . . . . . . . . . . . . . . . . Zero coupon bond prices: Jump-telegraph Dothan model.
In the case of a jump-telegraph Merton model with diffusion, the system (14) reduces to ∂F i ∂t ( t, x ) + ( µ i + ψ i σ i ) ∂F i ∂x ( t, x ) + σ i ∂ F i ∂x ( t, x ) + λ Q i [ F − i ( t, x + η i ) − F i ( t, x )] = xF i ( t, x ) ,i = 0 , ,F ( T, x ) = F ( T, x ) = 1 . (25)This coupled system can be transformed in a system of two ordinary differential equations.The effect of λ Q in the first term is negligible for long maturities. The opposite occurs withvolatility.To obtain the zero coupon bond prices under the expectation hypothesis, we applied thefact that the expected value E [ Z t ] is equal to zero. The calculations are similar to thoseperformed in the jump-telegraph Merton model, and the bond price matches that in (24) butwith ˜ d i = µ i + σ i ψ i + η i λ Q i , in place of d i , i = 0 , Numerical results : Data provided for the following parameter values: µ = − . , µ =0 . , ψ = 0 . , ψ = 1 . , σ = 0 . , σ = 0 . , λ Q = 1 . , λ Q = 2 . , η = 0 . , η = − . nitial rate: ODE Expectation
Maturity F F F F Zero coupon bond prices: Jump-telegraph diffusion Merton model.
Under this model, the system (14) for the prices of the zero-coupon bonds is given by ∂F i ∂t ( t, x ) + ( µ i + ψ i σ i ) x ∂F i ∂x ( t, x ) + σ i x ∂ F i ∂x ( t, x ) + λ Q i [ F − i ( t, x (1 + η i )) − F i ( t, x )]= xF i ( t, x ) , i = 0 , ,F ( T, x ) = F ( T, x ) = 1 . (26)The system is solved numerically by an implicit, up-wind finite difference scheme. Moreover,from (23) and (22), we obtain that the price of the bond under the expectation hypothesiscan be approximated byexp (cid:18) − Z Tt E [ r s | F t ] ds (cid:19) = exp (cid:18) − r t Z Tt E Q i h exp (cid:16) e Y s − t (cid:17)i exp (cid:18) σ s − t ) (cid:19) d s (cid:19) = exp (cid:16) − r t (cid:2) e G ( t, T ) + E i e H ( t, T ) (cid:3)(cid:17) , where e G ( t, T ) = e ( T − t )( ζ − λ Q + σ / h ( ζ − λ Q + σ /
2) cosh (cid:0) ( T − t ) √ D (cid:1) − √ D sinh (cid:0) ( T − t ) √ D (cid:1)i − ( ζ − λ Q + σ / ζ − λ Q + σ / − D , e H ( t, T ) = e ( T − t )( ζ − λ Q + σ / h ( ζ − λ Q + σ /
2) sinh (cid:0) ( T − t ) √ D (cid:1) − √ D cosh (cid:0) ( T − t ) √ D (cid:1)i + √ D ( ζ − λ Q + σ / − D .
Here E is defined in Section 5.2, and ζ , χ , ν , D are given in (20) with ˜ µ i = µ i + σψ − σ / µ i , i = 0 , Numerical results : The solution is provided for the following parameter values: µ = − . , µ = 0 . , ψ = ψ = 1 . , σ = σ = 0 . , λ Q = 1 . , λ Q = 2 . , η = 0 . , η = − . Initial rate: Finite differences Expectation
Maturity F F F F . . . . . . . . . . . . . . . . Zero coupon bond prices: Jump-telegraph diffusion Dothan model. Conclusions
We herein presented new results in two main directions. First, we used the properties ofthe telegraph processes with jumps to prove a Feynman–Kaˇc representation theorem forzero coupon bond pricing. Meanwhile, an analytical solution was obtained for differentdynamics of the short rate under telegraph processes with jumps. To obtain this solution,the unbiased expectation hypothesis was assumed. Furthermore, it was shown that the costof assuming this hypothesis was low, because the numerical results for the Cauchy problemwere sufficiently close to the analytical approximations.To our best knowledge, none have incorporated these types of processes for fixed incomemodeling, nor analytical formulas to approximate the prices of zero coupon bonds when thedynamics of the short rate allow for jumps and regime switches. Owing to the significantinterest in the effect of these aspects on the yield curve, we believe that this work will bebeneficial.A natural extension of this work is the search of analytical formulas for more robustmodels that include, for instance, mean reversion (CIR, Hull–White) and the valuation ofmore complex fixed income instruments.
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