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Multivariate European option pricing in a Markov-modulated Lévy framework
Journal of Computational and Applied Mathematics 317 (2017) 171–187
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Multivariate European option pricing in a Markov-modulated Lévy framework Griselda Deelstra, Matthieu Simon ∗ Université libre de Bruxelles (ULB), Département de Mathématique, Campus de la Plaine C.P. 210, B-1050 Bruxelles, Belgium
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Article history: Received 10 February 2016 JEL classification: C02 G13 C63 D52 F31 Keywords: Regime-switching Esscher transform Exchange options Quanto options
abstract This paper studies the pricing of some multivariate European options, namely Exchange options and Quanto options, when the risky assets involved are modelled by MarkovModulated Lévy Processes (MMLPs). Pricing formulae are based upon the characteristic exponents by using the well known FFT methodology. We study these pricing issues both under a risk neutral martingale measure and the historical measure. The dependence between the asset’s components is incorporated in the joint characteristic function of the MMLPs. As an example, we concentrate upon a regime-switching version of the model of Ballotta et al. (2015) in which the dependence structure is introduced in a flexible way. Several numerical examples are provided to illustrate our results. © 2016 Elsevier B.V. All rights reserved.
1. Introduction It has already been recognized for a long time that the classical Black–Scholes model has limitations when pricing options. However, models based on the exponential Lévy processes are not able as well to capture all empirical features of the market because of the independence and stationarity of the Lévy process increments, as pointed out e.g. in [1]. A natural way to relax the stationarity and independence of the increments is to allow the process to change its parameters at certain times to reflect e.g. the fact that there exist business cycles in the market. This leads to the so called regime-switching models where a Markov process serves to describe parameter changes. The regime-switching framework has been largely developed in the literature, although most of the articles on the subject are dedicated to univariate options. The prices of the risky asset involved in univariate options are often modelled as the exponential of a Markov-modulated Brownian motion (MMBM). Buffington and Elliot [2] found the Black–Scholes equations in the two states regime-switching setting and used it to price vanilla and American options. In [3], a partial differential equations approach is developed to provide an algorithm allowing to price numerically Asian options and lookback options. More recently, Zhu et al. [4] deal with the Fourier Transform method to derive a closed-form formula to price vanilla options in the MMBM setting. We refer to Elliott et al. [5] for a good overview of the literature and financial applications of regimeswitching models. In this important paper, the authors introduce moreover a regime-switching Esscher transform in order to determine an equivalent martingale measure, since the market described by a MMBM is incomplete in general. Markov-modulated Lévy processes (MMLPs) have also been used to price univariate options. In [1], the authors consider the two state case and suppose that the risky asset is a Variance Gamma process in each phase, under a risk neutral
∗
Corresponding author. E-mail addresses: [email protected] (G. Deelstra), [email protected] (M. Simon).
http://dx.doi.org/10.1016/j.cam.2016.11.040 0377-0427/© 2016 Elsevier B.V. All rights reserved.
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measure. They calculate the corresponding characteristic function to estimate the parameters and price the options. Elliott and Osakwe [6] generalize this paper by considering a pure jump switching process with an arbitrary number of states. A more general approach is followed in [7], where the author prices vanilla and exotic options when the price of the risky asset is the exponential of a one-dimensional MMLP. A transition in the Markov process can also induce a jump in the asset price. The author starts under the risk neutral measure and employs the Carr–Madan formula (see [8]) to price vanilla options and gives numerical procedures to deal with exotic options. Regime-switching models further have shown to be useful in the study of currency options. In [9], the authors perform the valuation of European and American currency options in the context of MMBMs for the FX rate. They start from the real world measure and then employ the Esscher change of measure introduced in [5] to determine a risk neutral measure and explicit pricing formulae under that measure. For European option pricing, Bo et al. [10] generalize this model by adding jumps with lognormal distributions. The main focus of this paper is to concentrate on multivariate options in a regime-switching framework. In this context, Yoon et al. [11] start from a risk neutral measure and propose an analytic valuation method based on the occupation times to price European-type multivariate contingent claims. In particular, they price quanto options and exchange options when the risky asset is modelled by a two-dimensional geometric MMBM with two phases. Chen et al. [12] concentrate upon European quanto options when the forward interest rates are supposed to follow a regime-switching HJM model, the stock prices by a regime-switching jump–diffusion and the spot FX rate is supposed to be into the Black–Scholes setting. They adapt the approach in [5] in order to obtain a risk neutral measure. In these papers, dependence between the different assets is easily described by the correlation between the Brownian motions showing up in the MMBM parts. In this paper, we study three types of European options under the hypothesis that the risky assets involved are general Markov-Modulated Lévy Processes (MMLPs) modulated by a Markov chain with an arbitrary number of states. We begin by revisiting the pricing of European call options, for which we give some original examples in order to illustrate the large variety of Markov models that can be used to describe a given regime-switching situation, and to show how these models can lead to different option prices. Next, we apply the introduced methods into the pricing of European exchange and quanto options, which is the main contribution of this paper. As these options deal with two-dimensional MMLPs, the dependence between the two components has to be taken into account. We first provide pricing formulae that are valid whatever the correlation structure between the components. Then, we generalize the model introduced in [13] to the regime-switching case. As in [14], the dependence structure is then incorporated in a flexible way, and the pricing formulae can be easily implemented. We choose the Variance Gamma case to provide numerical illustrations. We derive the option prices by using the Fast Fourier Transform as presented in [8], and therefore calculate the necessary characteristic exponents. As usual, the pricing analysis is performed under a risk neutral measure. However it appears sometimes more interesting to start from the historical probability, e.g. for asset–liability management reasons or in order to estimate the Markov chain parameters (see e.g. [15]). The last section of this paper is therefore devoted to the derivation of the pricing formulae starting from a real world model. Since the financial market is incomplete, the risk-neutral measure is not uniquely determined. We follow the Esscher transform approach, introduced in [16] and adapted to the regime-switching case in [5], to define a risk neutral measure. We state conditions that ensure the existence of such a measure in our framework and then we give the link with the pricing formulae obtained in the preceding sections. The paper is organized as follows: Section 2 is devoted to vanilla options. Exchange options and quanto options are considered in Section 3. In Section 4, we concentrate upon the model under the real world measure and use a regimeswitching Esscher transform to obtain the dynamics in the risk-neutral world. 2. Markov models and vanilla call options In this section, we consider the pricing of a European call option when the price of the risky asset {S (t )} is modelled under a risk-neutral measure Q by the exponential of a one-dimensional Markov-Modulated Lévy Process. Although the pricing formulae presented here are either not new or represent straightforward generalizations of some known results (see for example [1,6,7]), we present the derivation methods since we will use them later when dealing with multivariate options. Moreover, we will give some original regime-switching examples that can be treated in this Markovian framework. To formulate the model we define the following processes:
• A Markov process {M (t ) | t ∈ R+ } on the state space {1, 2, . . . , N } and determined by the generator Q ∈ RN ×N and the initial vector p ∈ RN , • N independent Lévy processes {Yj (t ) | t ∈ R+ } j = 1, 2, . . . , N determined under the measure Q by the characteristic functions:
EQ eiuYj (t ) = e−Φj (u)t ,
u ∈ C.
(2.1)
Based on these processes, we define the MMLP {X (t ) | t ∈ R+ } as follows: X (0) = 0 and ∀t > 0: dX (t ) =
N
dYj (t ) 1M (t )=j .
j =1
Intuitively, when the Markov process M is in state j, the process X behaves like the Lévy process Yj .
(2.2)
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We start by recalling a well known result. It is proved in the Appendix in a d-dimensional setting, under the assumption that the expectations in Eqs. (2.3) and (A.2) below are finite. Lemma 2.1. Consider the MMLP X defined in (2.2) and a Markov-modulated drift process C (t ) =
N
t 0
c (s)ds where c (t ) =
j=0 cj 1M (t )=j . Then, ∀a ∈ C and ∀t ≥ 0:
EQ eC (t )+aX (t ) = pe(Q −A)t 1,
(2.3)
where 1 is a vector with each component equal to 1 and A is the diagonal matrix such that Ajj = Φj (−ia) − cj with Φj defined in (2.1). Under the measure Q, we assume that the price of the risky asset S has the form S (t ) = S (0) eΛ(t )+X (t ) ,
(2.4)
where the drift process Λ(t ) is defined as
Λ(t ) =
t
µ(s) ds with µ(t ) = 0
N
µj 1M (t )=j ,
j =0
where the coefficients µj are constant. We further denote the risk-free interest rate by r (t ) and assume that it also depends on the state occupied by the Markov process: r (t ) =
N
rj 1M (t )=j ,
j =0
where the coefficients rj are constant. In the following, we also need the integrated interest rate process, which we denote by U (t ): U (t ) =
t
r (s) ds. 0
We further introduce the vectorial notations: r = (r1 , r2 , . . . , rN ) , µ = (µ1 , µ2 , . . . , µN ) , 8(u) = (Φ1 (u), Φ2 (u), . . . , ΦN (u)) .
(2.5)
Since Q is a risk-neutral probability, the vector µ should be chosen in a particular way: Lemma 2.2. In order to ensure that the process e−U (t ) S (t )
t
is a Q-martingale, the vector µ should be determined by
µ = r + 8(−i).
(2.6)
Proof. In order to check the Q-martingality of e−U (t ) S (t ), we verify whether
EQ e−U (t )+Λ(t )+X (t ) = 1 ∀t .
(2.7)
The left hand side of this last equality follows easily from Lemma 2.1 with a = 1 and C (t ) = Λ(t ) − U (t ):
EQ e−U (t )+Λ(t )+X (t ) = pe(Q −A)t 1
with A = diag (r − µ + 8(−i)). Since peQt 1 = 1 ∀t, Eq. (2.7) can be rewritten as pe(Q −A)t 1 = peQt 1. The only solution of this equation is A = 0, so that we get the announced relation. Our first aim is to determine the price V (S , K , T ) of a European call option with maturity T and strike K : V (S , K , T ) = EQ e−U (T ) (S (T ) − K )+ .
(2.8)
Theorem 2.3. The price of the European call option at time t = 0 is given by V (S , K , T ) =
e−α k
π
∞
e−iv k a(v) e(α+1+iv)s0 pe(Q −G(α,v))T 1 dv,
(2.9)
0
where α is a damping factor,
−1
a(v) = (α 2 + α − v 2 ) + i(2αv + v)
,
G(α, v) = diag (8(v − i(α + 1)) − (α + 1 + iv)8(−i) − (α + iv)r ) , and s0 = ln(S (0)), k = ln(K ).
(2.10) (2.11)
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Proof. To calculate the right hand side of (2.8), we use the well known Carr–Madan formula (see [8]). We put s0 = ln(S (0)) and k = ln(K ) and write V (S , K , T ) =
∞
−∞
∞
e−u es − ek f (s, u) ds du,
k
where f (s, u) is the density function of the vector (s0 + Λ(T ) + X (T ), U (T )). Therefore, V (S , K , T ) =
e−α k
π
∞
e−iv k a(v) e(α+1+iv)s0 EQ e−U (T )+(α+1+iv)(X (T )+Λ(T )) dv
(2.12)
0
−1
where a(v) = (α 2 + α − v 2 ) + i(2αv + v) and α > 0 is chosen in such a way that the last integral exists. The expectation in (2.12) can be calculated by using Lemma 2.1 with a = α + 1 + iv and C (t ) = (α + 1 + iv)Λ(t ) − U (t ). The price of the European call option then easily follows.
Formula (2.9) allows the numerical computation of the prices in an efficient way thanks to e.g. the Fast Fourier Transform method (see [8]). For completeness, we also determine the price of the European call option at an arbitrary time t ≤ T , that is V (S , K , T , t ) = EQ e−U (T ) (S (T ) − K )+ | Ft
K = e−U (t ) S (t )EQ e−(U (T )−U (t )) e(X (T )−X (t ))+(Λ(T )−Λ(t )) − | Ft , S (t ) + where Ft is the σ -algebra generated by the history of the Markov process and by the MMLP X up to time t. Now, conditionally on the value of S (t ), U (t ), Λ(t ) and the state j occupied by the Markov process at time t, the processes X (T )−X (t ), U (T )−U (t ) and Λ(T ) − Λ(t ) are independent of Ft and have the same law as the processes X (T − t ), U (T − t ) and Λ(T − t ) given that M (0) = j. So, when denoting S = S (t ), U = U (t ) and M (t ) = j,
V (S , K , T , t ) = e−U EQ e−U (T −t ) S˜ (T − t ) − K
+
| M (0) = j ,
X (t )+Λ(t )
with S˜ (t ) = Se . The last formula corresponds to the price of a European call option with risky asset S˜ (t ), strike K and maturity T − t. Theorem 2.3 therefore immediately leads to the following corollary: Corollary 2.4. If at time t ≤ T , S (t ) = S, U (t ) = U and M (t ) = j, then e−α k V (S , K , T , t ) = e−U
∞
π
e−iv k a(v) e(α+1+iv)s ej e(Q −G(α,v))(T −t ) 1 dv,
(2.13)
0
with ej the jth vector of the usual basis of RN , with s = ln(S ) and k = ln(K ), and where a(v) and G(α, v) are given by (2.10) and (2.11). Examples The examples presented here aim to illustrate how the behaviour of the Markov process M influences the option prices in different ways, and how we can build some Markovian regime-switching models that differ from the classical Markovian setting. We consider the particular case when S (t ) is modelled as in (2.4) and X (t ) is assumed to be a Markov-modulated Merton jump–diffusion process. In particular, we suppose that there are N different phases and that the vector of characteristic exponents 8(u) is given by (2.5) with
8j (u) = u2 σj2 + λj 1 − e
− 21 u2 τj2
j = 1, . . . , N ,
so that the jumps in each phase have zero mean. In all examples, we will set S (0) = 100 and K = 90. We denote by λ, σ and τ the vectors containing the parameter values of the model in each phase. Example 1. We first consider a model with three phases A, B and C . The phase A corresponds to a normal behaviour of the market, when the volatilities are low. The phase C corresponds to a bad state of the market (e.g. with many jumps and high volatilities). Phase B corresponds to an intermediate state when there is a risk to fall in phase C but there also is a possibility that the market comes back into phase A. A first way to define a Markov process {M (t )} that controls the transitions between the three phases is by choosing the following classical rules:
• When M1 is in phase A, it stays in A during an exponential period of time with parameter α . Then the process goes to phase B with probability 1 − q1 or to phase C with probability q1 .
G. Deelstra, M. Simon / Journal of Computational and Applied Mathematics 317 (2017) 171–187
175
Fig. 1. Prices for different initial vectors.
• When M1 is in phase B, it stays in B during an exponential period of time with parameter β . Then the process goes to phase C with probability q2 or to phase A with probability 1 − q2 . • When M1 is in phase C , it stays in C during an exponential period of time with parameter γ before going to phase A. This leads to a first model (Model 1), where we choose the following parameters:
−α (1 − q1 )α q1 α 0.01 0 0.02 0 −β q2 β , Q = (1 − q2 )β r = 0.01 , λ= 2 , σ = 0.1 , τ = 0.1 , γ 0 −γ 0.01 10 0.25 0.25 and p = p1 p2 p3 . In this example, we focus on the effect of the initial distribution on the prices. We will consider the
basis vectors of R3 as initial vectors, as well as the stationary vector of Q . The latter is a reasonable choice when the initial vector is unknown. Fig. 1 shows the prices as a function of the maturity date T when the parameter values are α = 2.5, β = 5, γ = 4, q1 = 0.8 and q2 = 0.4. We note that the initial state vector influences the prices for short maturities but becomes less and less important as T increases. This is natural because the Markov process M converges to its stationary distribution for large maturities. Example 2. We now consider again the model defined in Example 1, but this time the Markov process {M ′ (t )} that controls the transitions between the three phases is chosen so that the time spent during a sojourn in phase A or B depends on the next phase to be visited. More precisely:
• When M ′ is in phase A, two situations are possible: with probability 1 − q1 the process stays in A during a period exponentially distributed with parameter α1 and then goes to B. With probability q1 the process stays in A during a period exponentially distributed with parameter α2 and then goes to C . • When M ′ is in phase B, two situations are possible: with probability 1 − q2 the process stays in B during a period exponentially distributed with parameter β1 and then goes to A. With probability q2 the process stays in B during a period exponentially distributed with parameter β2 and then goes to C . • When M ′ is in phase C , it stays in C during a period exponentially distributed with parameter γ before going to phase A. The Markov process {M ′ (t )} has then five states because we need to duplicate the phases A and B in order to include the different possible sojourn times. This yields the following generator:
−α1
0 Q ′ = (1 − q1 )β1 0 (1 − q1 )γ and p′ = p1 (1 − q1 )
r1 r1 r ′ = r2 , r 2 r3
(1 − q2 )α1
0
−α2
q1 β1 0 q1 γ
p1 q1
0
−β1 0 0
p2 (1 − q2 )
λ1 λ1 λ′ = λ2 , λ 2 λ3
q2 α1 0 0
0
α2 0 , β2 −γ
−β2 0 p2 q2
p3 . The other parameters are given by
σ1 σ1 σ ′ = σ2 , σ 2 σ3
τ1 τ1 τ ′ = τ2 . τ 2 τ3
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G. Deelstra, M. Simon / Journal of Computational and Applied Mathematics 317 (2017) 171–187
Fig. 2. Prices for the different models.
In Fig. 2, we fix α = 0.5, β = 2, γ = 1, q1 = 0.3, q2 = 0.2 and p1 = 1 and we compare the prices obtained when the modulating process is M as in Example 1 (Model 1) with the ones obtained when the modulating process is M ′ (Model 2) in three situations. First, for Model 2.1, we set α1 = α2 = α , β1 = 5β and we choose β2 equal to
β2 = q2 /((1/β) − (1/(β1 )) ∗ (1 − q2 )),
(2.14)
so that the average duration of a visit in state B is 1/β as in Model 1. For Model 2.2, we take β1 = β2 = β , α1 = 2α and
α2 = 1/((1/α) − (1/(α 1)) ∗ (1 − q1 )).
(2.15)
In this way, the average duration of a visit in state A is 1/α as in Model 1. Finally, in Model 2.3 we take α1 = 2α , β1 = 5β and we choose β2 and α2 as in (2.14) and (2.15). Even if the four models presented in Fig. 2 have, by construction, the same mean sojourn time in each phase and the same transition probabilities, we observe that the prices can be different, especially when the maturity date is large enough. Example 3. As a last example in this section, we start from a model with two states A and B only and with the parameters
−α α , β −β
Q =
r =
0.05 , 0.02
λ=
0 , 10
σ=
0.05 , 0.2
τ=
0 , 0.2
and p = 1 0 . This is a classical Markov-switching situation in which the sojourn times in phase A (respectively B) are exponentially distributed with parameter α (respectively β ). In particular, the duration of a visit to phase A (B) has a mean 1/α (1/β ) and a variance 1/α 2 (1/β 2 ). We want to construct an analogous model where the sojourn times in A and B have the same mean durations but where they are now distributed as Erlang random variables. This leads to a model where a sojourn time in phase A (phase B) has an Erlang distribution with parameters n and nα (nβ ) for some n ∈ N0 . In this new situation the mean sojourn times in A and B remain unchanged but the variances become respectively 1/(nα 2 ) and 1/(nβ 2 ). As n increases, the variances of the sojourn times therefore decrease to zero. We will compare the option prices obtained in the two models. The model with Erlang sojourn times can be built by decomposing the two phases A and B into n successive states. A visit in state A (respectively B) is then translated by the successive visits to n states in which the sojourn time is exponentially distributed with parameter nα (respectively nβ ). The generator of the model with Erlang sojourn time is given by the 2n × 2n matrix
−n α 0 . .. 0 Q′ = 0 0 .. . nβ
nα −n α .. .
0 nα
··· ···
0 0 0
0 0 0
··· ··· ···
0
0
···
.. .
.. .
0 0
.. .
.. .
−nα 0 0
.. .
0
0 0
.. . nα −n β
0 0
.. .
0 0
··· ···
.. .
0
0 0 nβ
··· ··· ···
0
0
0
···
.. .
0 0 0
,
.. .
0 nβ −nβ
.. .
0 0
.. .
.. .
−nβ
G. Deelstra, M. Simon / Journal of Computational and Applied Mathematics 317 (2017) 171–187
(a) Prices for different values of n.
177
(b) Prices with or without regime switching.
Fig. 3. Option prices for different Erlang models.
and the other parameters are given by the following vectors (with n + n components): r′ =
r1 1 , r2 1
λ′ =
λ1 1 , λ2 1
σ′ =
σ1 1 , σ2 1
τ′ =
τ1 1 , τ2 1
p2 0 · · · 0 . and p′ = p1 0 · · · 0 Fig. 3(a) compares the option prices obtained for different values of n when α = 1, β = 1.5 and p1 = 1. We observe that the prices obtained when n > 1 oscillate around the curve associated with n = 1, and that the amplitude of the oscillations decreases with the number of transitions in the Markov Process. The intersections between the curves seem to correspond approximately to the transition instants of the Markov process. Fig. 3(b) compares the option prices obtained for n = 1 and n = 30 with the prices obtained where there is no regime switching and the parameters are the ones associated with the first phase or with the second one. It provides an interpretation of the oscillations: when n is large, the transition times in the Markov process are nearly deterministic. The prices associated with n show then a behaviour similar to the case where the parameters of the model change at deterministic instants. 3. Multivariate option pricing 3.1. Exchange options In this section, we concentrate upon exchange options in a MMLP framework and we derive prices by using the methods presented in the previous section. We recall that an analytic valuation method for exchange options was already proposed in [11] in the case of MMBMs, but as far as we know, no results are known in the MMLP setting. Whereas the dependence between the price of the two assets involved in an exchange option is easily modelled in the case of Brownian motions, specifying a dependence structure for general Lévy processes is a more challenging problem. In this section, we start by providing an option pricing formula which applies for arbitrary dependence structures as it is based upon the joined characteristic exponent of the two-dimensional MMLP X . We are thus interested in the evaluation of the price V (S1 , S2 , T ) at time 0 of an exchange option with maturity T : V (S1 , S2 , T ) = EQ e−U (T ) (S2 (T ) − S1 (T ))+ ,
(3.1)
where the two-dimensional process of asset prices S (t ) = (S1 (t ), S2 (t )) has the dynamics S1 (t ) = S1 (0)eΛ1 (t )+X1 (t )
and S2 (t ) = S2 (0)eΛ2 (t )+X2 (t ) ,
with X (t ) = (X1 (t ), X2 (t )) a two-dimensional MMLP controlled by a Markov process {M (t ) | t ∈ R+ } defined as before by a generator Q and an initial vector p. We assume that X (0) = 0 and, when M = j, X has the same behaviour as the two-dimensional Lévy process Yj determined by the characteristic exponent Φj (u):
EQ ei⟨u,Yj (t )⟩ = e−Φj (u)t ,
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with u = (u1 , u2 ). The processes Λ1 (t ) and Λ2 (t ) denote some drift processes:
Λ1 (t ) =
t
µ(s) ds,
µ(t ) =
0
Λ2 (t ) =
µj 1M (t )=j ,
(3.2)
j =0 t
N
ν(s) ds,
ν(t ) =
0
N
νj 1M (t )=j
(3.3)
j=0
where the coefficients µj and νj are constant. As in Section 1, U (t ) is the integral up to time t of the interest rate process: r (t ) =
N
rj 1M (t )=j ,
U (t ) =
t
r (s) ds. 0
j =0
Using the notations r = (r1 , r2 , . . . , rN ) , µ = (µ1 , µ2 , . . . , µN ) , ν = (ν1 , ν2 , . . . , νN ) , 8(u) = (Φ1 (u), Φ2 (u), . . . , ΦN (u)) , an analogous result as in Lemma 2.2 shows that the vectors µ and ν need to be defined in the following way:
µ = r + 8(−i, 0), ν = r + 8(0, −i).
(3.4) (3.5)
In order to evaluate (3.1), the following change of measure turns out to be useful (see e.g. [17]):
˜ dQ
e −U ( T ) S 1 ( T )
=
dQ
EQ
1
= e−U (T ) S1 (T ). S1 (0) 1 (T )
(3.6)
e−U (T ) S
Using this change of measure in (3.1) leads to:
EQ e−U (t ) (S2 (T ) − S1 (T ))+ = EQ˜
S1 (0) S1 (T )
(S2 (T ) − S1 (T ))+ = S1 (0) EQ˜ (H (T ) − 1)+ ,
S (0)
˜ , the problem boils down to the evaluation of where H (t ) = S2 (0) e(Λ2 (t )+X2 (t ))−(Λ1 (t )+X1 (t )) . Therefore, under the measure Q 1 the European Call Option price when H (t ) is the risky asset, T is the maturity, the strike is equal to 1 and the risk-free interest rate equals 0. We can therefore use the results of Section 2 to calculate V (S1 , S2 , T ): Theorem 3.1. The price of an exchange option as in (3.1) is given by V (S1 , S2 , T ) =
e−α k
∞
π
(1) (2) e−iv k a(v) e−(α+iv)s0 +(α+1+iv)s0 pe(Q −G(α,v))T 1 dv
,
(3.7)
k=0+
0
where G(α, v) = diag (8 (−v + iα, v − i(α + 1)) + (α + iv)8(−i, 0) − (α + 1 + iv)8(0, −i)) , (1)
(2)
and a(v) is given in (2.10), s0 = ln(S1 (0)), s0 = ln(S2 (0)). Proof. As we deal with European options, Eq. (2.12) holds modulo the fact that the process S is replaced by the process H. Adapting (2.12) to H yields the following expression for V (S1 , S2 , T ): S1 (0)
e−α k
π
∞
e−iv k a(v)e(α+1+iv)(ln(S2 (0))−ln(S1 (0))) EQ˜ e(α+1+iv)(−X1 (T )−Λ1 (T )+X2 (T )+Λ2 (T )) dv.
0
The above expectation can be calculated by the change of measure (3.6) under the original measure Q:
EQ˜ e(α+1+iv)(−X1 (T )−Λ1 (T )+X2 (T )+Λ2 (T )) = EQ e−U (T )−(α+iv)(X1 (T )+Λ1 (T ))+(α+1+iv)(X2 (T )+Λ2 (T )) ,
which can be rewritten by applying Lemma A.1 with d = 2, C (t ) = (α + 1 + iv)Λ2 (t ) − (α + iv)Λ1 (t ) − U (t ) and a = [−(α + iv)α + 1 + iv]. Using Eqs. (3.4) and (3.5) leads then to the announced result.
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179
Table 1 Parameter values.
a1 a2
Phase 1
Phase 2
Phase 3
−1.2
1 0
−1 −1
1
Z1 Z2 Z3
k
θ
σ
k
θ
σ
k
θ
σ
0.005 0.08 0.07
−0.02 0.15 −0.1
0.11 0.1 0.1
0.01 0.05 0.05
0.6 0.05 −0.5
0.03 0.15 0.1
0.12 0.1 0.05
0.2 0.15 0.1
0.11 0.1 0.1
Example 4. This example is inspired by Ballotta and Bonfiglioli [14]. Indeed, we concentrate upon a two-dimensional MMLP in which the dependence between the two components in each phase follows from the assumption of a common systematic component, besides the idiosyncratic parts Z1 (t ) and Z2 (t ): X1 (t ) = Z1 (t ) + a1 (t )Z3 (t ), X2 (t ) = Z2 (t ) + a2 (t )Z3 (t ),
(3.8)
with al ( t ) =
N
alj (t )1M (t )=j
(l = 1, 2)
j =1
some loading factors influencing the dependence between the two components of X . In this model, Z1 , Z2 and Z3 are assumed to be three independent one-dimensional MMLPs. As explained in [14], this factor construction implies a model with a flexible correlation structure which can be easily implemented since the characteristic function can be determined in a straightforward way: Lemma 3.2. The jth component Φj (u1 , u2 ) of the vector 8(u1 , u2 ) giving the characteristic exponents of X in phase j is determined by:
Φj (u1 , u2 ) = ϕ1j (u1 ) + ϕ2j (u2 ) + ϕ3j (a1j u1 + a2j u2 ), where ϕlj is the characteristic exponent of the process Zl when M = j. For a numerical experiment we suppose that when the Markov process M is in phase j, the process Zl behaves as a Variance Gamma process Ylj with parameters klj , θlj and σlj . The characteristic exponent ϕlj (u) of Ylj is well known:
ϕlj (u) =
1 klj
ln 1 − iuklj θlj +
1 2
u2 klj σlj2
(l = 1, 2, . . . , N , j = 1, 2, . . . , N ).
The goal of this example is to describe a regime-switching model which can switch between N = 3 phases with different correlation structures for X . The parameters for the Markov process are chosen to be equal to
−3 Q =
2 2
3 −3 3
0 1 , −5
p = p1
p2
p3 .
Table 1 gives the parameter values of the processes in (3.8) in the different phases, whereas the corresponding means, variances and correlations are stated in Table 2. Remark that phase 1 and phase 3 correspond respectively to a negative and positive correlation between X1 and X2 . Phase 2 corresponds to an uncorrelated situation. Fig. 4 compares the option prices related to this model (MMLP) with the prices when there is no regime-switching and the parameters are the ones of the first phase (LP 1), the second phase (LP 2) or the third phase (LP 3). The initial values of the two risky assets are S1 (0) = 90 and S2 (0) = 100, and the initial vector is given by p2 = 1. We observe that the last three models are relatively close to each other while the model LP 1 yields much higher prices, which can be explained by the assumption of a negative correlation between the two assets. As expected, the prices related to the model with regime-switching are always situated between the ones related to the models without switching. 3.2. Quanto options This section is concerned with the evaluation of quanto future options. Quanto options have already been explored in the setting of MMBM. Yoon et al. [11] provide some analytic valuation method under a risk neutral measure, Chen et al. [12] consider a more general MMBM model since they include a regime-switching HJM interest rate model. In this section, we generalize the model of Ballotta et al. [13] to a MMLP framework.
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G. Deelstra, M. Simon / Journal of Computational and Applied Mathematics 317 (2017) 171–187 Table 2 Means, variances and correlations.
E [X1 ] Var [X1 ] E [X2 ] Var [X2 ] Corr [X1 , X2 ]
Phase 1
Phase 2
Phase 3
0.1 0.027 0.05 0.023 −0.516
0.1 0.027 0.05 0.023 0
0.1 0.027 0.05 0.023 0.421
Fig. 4. Prices for the four models.
As before, we suppose that the market can switch between N phases 1, 2, . . . , N and that the phase transitions are controlled by a Markov process {M (t ) | t ∈ R+ } defined by a generator Q and an initial vector p. We consider that the spot FX rate between the foreign currency (f ) and the domestic currency (d) at time t, denoted by Xd|f (t ), is quoted as the amount of currency f per unit of currency d. We also consider an index S (t ) traded in the foreign currency. We denote by rf (t ) and rd (t ) resp. the interest rates in the foreign and the domestic market. We assume that these rates are constant in each phase of the market: rf ( t ) =
N
rf ,j 1M (t )=j ,
rd ( t ) =
N
j =0
rd,j 1M (t )=j ,
j =0
with rf ,j , rd,j ∈ R+ , and we introduce the notations: Uf (t ) =
t
rf (s) ds and Ud (t ) = 0
t
rd (s) ds. 0
We suppose that the processes S (t ) and Xd|f (t ) have the following form under the risk-neutral measure Qf defined in the foreign market: S (t ) = S (0)eΛ1 (t )+X1 (t )
and
Xd|f (t ) = Xd|f (0)eΛ2 (t )+X2 (t ) ,
(3.9)
where X (t ) = (X1 (t ), X2 (t )) is a two-dimensional MMLP controlled by {M (t )} such that X (0) = 0 and, when M = j, X has (1)
the same behaviour as the Lévy process Yj = Yj
, Yj(2) determined by the characteristic exponent Φj (u):
EQf ei⟨u,Yj (t )⟩ = e−Φj (u)t ,
and where the drift processes Λ1 (t ) and Λ2 (t ) are defined as in Eqs. (3.2) and (3.3). As before, we will use some vectorial notations: rf = rf ,1 , rf ,2 , . . . , rf ,N , rd = rd,1 , rd,2 , . . . , rd,N , µ = (µ1 , µ2 , . . . , µN ), ν = (ν1 , ν2 , . . . , νN ), 8(u) = (Φ1 (u), Φ2 (u), . . . , ΦN (u)). We also put Rf = diag(rf ) and Rd = diag(rd ). As in Lemma 2.2, the condition that the processes S (t ) and Xd|f (t ) are properly defined under a risk-neutral measure Qf , that is e−Uf (t ) S (t ) and e−Uf (t )+Ud (t ) Xd|f (t ) are Qf -martingales, implies the choice of µ and ν:
µ = rf + 8(−i, 0) and ν = rf − rd + 8(0, −i).
(3.10)
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It is well known that it is convenient to switch from the risk neutral foreign world to the risk neutral domestic market by defining the following change of measure from Qf to Qd :
Ud (t ) Xd|f (t ) = e , Uf (t ) dQf Ft e Xd|f (0)
dQd
(3.11)
where Ft is the σ -algebra generated by the history of the Markov process and by the MMLP X up to time t. The aim of this section is to derive the arbitrage free price of a European quanto call option with maturity date T1 on the quanto future of the index S, expressed in units of domestic currency, with maturity T2 and strike K , that is:
V FTd1 (T2 ), K , T1 = Ed e−Ud (T1 ) FTd1 (T2 ) − K
+
,
(3.12)
where Ed stands under the measure Qd , T2 is the maturity of the option contract and, for T1 < T2 , for the expectation FTd1 (T2 ) = Ed S (T2 ) | FT1 is the quanto future with underlying index S, under the assumption that the applied FX rate between the two currencies is set to 1 (d/f) (see e.g. [18]). This price can be reformulated in a simpler way by applying the change of measure (3.11): Lemma 3.3. FTd1 (T2 ) = S (T1 ) eM (T1 ) e(Q +Rf +∆)(T2 −T1 ) 1,
(3.13)
where eM (T1 ) is a random unitary vector in RN with the jth component equal to 1 if and only if M (T1 ) = j, and where
∆ = diag (8(−i, 0) + 8(0, −i) − 8(−i, −i)) .
(3.14)
Proof. In order to determine FTd1 (T2 ), we apply the change of measure (3.11): FTd1 (T2 ) = S (0)Ed eΛ1 (T2 )+X1 (T2 ) | FT1
Ef eA(T2 )+X1 (T2 )+X2 (T2 ) | FT1
= S (0)
Ef eB(T2 )+X2 (T2 ) | FT1
= S (0)eA(T1 )−B(T1 )+X1 (T1 )
Ef e(A(T2 )−A(T1 ))+(X1 (T2 )−X1 (T1 ))+(X2 (T2 )−X2 (T1 )) | FT1
Ef e(B(T2 )−B(T1 ))+(X2 (T2 )−X2 (T1 )) | FT1
,
where A(t ) = Λ1 (t ) + Ud (t ) − Uf (t ) + Λ2 (t ) and B(t ) = Ud (t ) − Uf (t ) + Λ2 (t ). Conditionally on the state j occupied by the Markov process at time T1 , the processes X (T2 ) − X (T1 ), A(T2 ) − A(T1 ) and B(T2 ) − B(T1 ) are independent of FT1 and have the same law as the processes X (T2 − T1 ), A(T2 − T1 ) and B(T2 − T1 ) given that M (0) = j. Therefore, if M (T1 ) = j:
Ef eA(T2 −T1 )+X1 (T2 −T1 )+X2 (T2 −T1 ) | M (0) = j
FTd1
(T2 ) = S (T1 )
Ef eB(T2 −T1 )+X2 (T2 −T1 ) | M (0) = j
.
By the use of Lemma A.1, assuming that M (T1 ) = j and using (3.10), it is easily found that
Ef eA(T2 −T1 )+X1 (T2 −T1 )+X2 (T2 −T1 ) | M (0) = j = ej e(Q +Rf +∆)(T2 −T1 ) 1
with ∆ as in (3.14), and
Ef eB(T2 −T1 )+X2 (T2 −T1 ) | M (0) = j = ej eQ (T2 −T1 ) 1 = 1.
This implies the announced result.
Remark. In a similar way, it can be shown that the quanto future with underlying index S under the measure Qf , that is f
FT1 (T2 ) = Ef S (T2 ) | FT1 , can be expressed as
FT1 (T2 ) = S (T1 ) eM (T1 ) e(Q +Rf )(T2 −T1 ) 1, f
which is the regime-switching generalization of Equation (10) in [13]. Note that the expression for FTd1 (T2 ) is the same as the f
one for FT1 (T2 ), except that the diagonal matrix ∆ is added in the exponential. This justifies the naming of quanto adjustment for ∆. When N = 1, and therefore no regime-switching, the matrix ∆ becomes the quanto adjustment q obtained in [13].
Lemma 3.3 is particularly useful for reformulating the price of V FTd1 (T2 ), K , T1 following lemma.
in (3.12), as can be observed in the
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Lemma 3.4. V FTd1 (T2 ), K , T1 =
N
(j)
Qadj Ed e−Ud (T1 ) S (T1 ) − Kj + 1M (T1 )=j ,
(3.15)
j =1
(j)
Kj = K /Qadj
where
Qadj = ej e(Q +Rf +∆)(T2 −T1 ) 1. (j)
and
(3.16)
Proof. Using Lemma 3.3 leads immediately to the result:
Ed e−Ud (T1 ) FTd1 (T2 ) − K
=
N
+
Ed e−Ud (T1 ) S (T1 )eM (T1 ) e(Q +Rf +∆)(T2 −T1 ) 1 − K
j =1
=
N
+
ej e(Q +Rf +∆)(T2 −T1 ) 1 Ed e−Ud (T1 ) S (T1 ) −
j =1
1M (T1 )=j
K
ej e(Q +Rf +∆)(T2 −T1 ) 1 +
1M (T1 )=j
.
As a result, the price of a quanto option can be seen as the sum of European call options upon the index S and strikes Kj and therefore, the results of Section 2 imply the following theorem: Theorem 3.5. The price of a quanto call option upon the quanto future on the index S with strike K and maturity date T1 is determined by V
FTd1
(T2 ), K , T1 =
N
(j)
Qadj
j =1
e−αj kj
∞
π
e−iv kj aj (v)e(αj +1+iv)s0 pe(Q −G(αj ,v))T1 ej dv,
(3.17)
0
where
−1
aj (v) = (αj2 + αj − v 2 ) + i(2αj v + v)
G(αj , v) = diag 8(v − i(αj + 1), −i) − (αj + 1 + iv) 8(−i, 0) + rf − 8(0, −i) + rd .
(j)
(j)
and where Qadj is defined in (3.16), kj = ln(K /Qadj ), ∆ is given in (3.14) and s0 = ln(S (0)). Proof. By analogy with Eq. (2.12), the jth expectation in (3.15) can be rewritten as e−αj kj
π
∞
e−iv kj aj (v)e(αj +1+iv)s0 Ed e−Ud (T1 )+(αj +1+iv)(Λ1 (T1 )+X1 (T1 )) 1M (T1 )=j dv.
0
The last expectation is easily calculated by applying the change of measure (3.11) and by using Lemma A.1, with the vector 1 in (A.2) replaced by ej . Example 5. We now model the process of the index S (t ) and the exchange rate Xd|f (t ) in (3.9) by choosing X1 and X2 as in Example 4 (and therefore similar as in [14]): X1 (t ) = Z1 (t ) + a1 (t )Z3 (t ), X2 (t ) = Z2 (t ) + a2 (t )Z3 (t ), where Z1 , Z2 and Z3 are three independent Markov-switching Variance Gamma processes with N = 3 and the parameters given in Table 1. The Markov-switching parameters and the interest rates processes are chosen to be equal to:
−3 Q =
2 2
3 −3 3
0 1 , −5
0.02 0.01 , 0.01
p= 1
0
0 ,
rf =
0.05 0.05 . 0.025
rd =
First, we assume T1 = T2 = T which implies the case of a call option on the index S. Fig. 5 compares the option prices related to the Markov-switching model (MMLP) with the prices when there is no regime-switching and the parameters are the ones of the first phase (LP 1), the second phase (LP 2) or the third phase (LP 3) for different maturity dates T , when S (0) = 100 and K = 90 days. We note that the prices are a convex function for short maturities but then turn to be a concave function of the maturity. Fig. 6 compares the option prices upon a quanto future on the index, and this for the four models for different maturity dates T2 when T1 is fixed at 30 days and p1 = 1. The prices seem then approximately to behave in a linear way.
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183
Fig. 5. Prices for the four models.
Fig. 6. Prices for the four models, T1 = 30 days.
4. From real world to risk neutral measure In this section, we start from a model of the dynamics of the underlying assets in the options under the real world measure. This can be useful for the estimation of the Markov chain parameters, or for ALM studies. We consider a historical probability space (Ω , F , P) on which we define a d-dimensional risky asset S (t ) = (S1 (t ), S2 (t ), . . . , Sd (t )) which has the following dynamics: Sl (t ) = Sl (0) eXl (t ) ,
l = 1, 2, . . . , d,
with X (t ) = (X1 (t ), X2 (t ), . . . , Xd (t )) being a d-dimensional MMLP controlled by a Markov process {M (t ) | t ∈ R+ } which is defined by a generator Q and an initial vector p. We assume that X (0) = 0 and, when M = j, X has the same behaviour as the d-dimensional Lévy process Yj determined by the characteristic exponent Ψj :
EP ei⟨u,Yj (t )⟩ = e−Ψj (u)t ,
for any d-dimensional vector u. The risk-free interest rate r (t ) and its integral U (t ) are defined as in Section 2. Following arbitrage theory, our aim is to find an equivalent martingale measure Q ∼ P under which the process {e−U (t ) S (t )} is a martingale. For this, we use the Markov switching Esscher transform, which is introduced in [5] in the context of geometric Markov-modulated Brownian motions and which can be generalized in a straightforward way to more general regime-switching models. Let Ft be the σ -algebra generated by both the Markov process and the MMLP X up to time t, and FtM the σ -algebra generated by only the Markov process up to time t. For any d × N matrix Θ = [θ 1 | θ 2 | . . . | θ N ] we define the change of measure QΘ ∼ P by
eZΘ (t ) = = L(t ), dP Ft EP eZΘ (t ) | FtM
dQΘ
(4.1)
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where {ZΘ (t )} is a one-dimensional MMLP such that, when M = j, ZΘ has the same behaviour as the Lévy process ⟨θ j , Yj ⟩. Eq. (4.1) can be rewritten in terms of the occupation times of the Markov process M. Indeed, let Jk (t ) be the total amount of time spent by the Markov process M in phase k up to time t. As the increments of the processes Yk are stationary, independent and not affected by the past of M, X (t ) equals in distribution: X (t ) =
N
Yk (Jk (t )) ,
(4.2)
k=1
and it follows then that
EP
eZΘ (t )
| Ft
M
N θ k ,Yk (Jk (t ))
= EP ek=1
N M θ ,Y (J (t )) F = EP e k k k Jk (t ) t
k=1
−
=e
N
Ψk (−iθ k )Jk (t )
k=1
,
and therefore the Radon–Nikodym derivative L(t ) is distributed as L( t ) = e
ZΘ (t )+
N
N θ k ,Yk (Jk (t )) +Ψk (−iθ k )Jk (t )
Ψk (−iθ k )Jk (t )
= ek=1
k=1
.
(4.3)
In order to prove that L(t ) is a martingale, we need the following lemma: Lemma 4.1. If a1 , a2 , . . . , aN are some vectors in Rd and b1 , b2 , . . . , bN some real numbers, then
N
EP ek=1
ak ,Yk (Jk (t )) +bk Jk (t )
= pe(Q −A)T 1,
where A is a diagonal matrix such that Akk = Ψk (−iak ) − bk . The proof of this lemma follows from the same arguments as in the proof of Lemma A.1, by using the strong Markov property applied to the Markov process M and the Lévy processes Yj , and is therefore omitted. Lemma 4.1 has already been proved in different forms: see e.g. Proposition 2 of Elliott and Osakwe [6] in the case where ak = 0 ∀k, and Lemma A.1 in [2]. This lemma allows to conclude that the process {L(t )} is a martingale: Lemma 4.2. The process L(t ) defined in (4.1) is a P-martingale with respect to Ft . Proof. Let T ≥ t. By Eq. (4.3):
N
EP [L(T ) | Ft ] = EP ek=1
θ k ,Yk (Jk (T )) +Ψk (−iθ k )Jk (T )
N
= L(t )EP ek=1
| Ft
θ k ,Yk (Jk (T ))−Yk (Jk (t )) +Ψk (−iθ k )(Jk (T )−Jk (t ))
| Ft .
Conditionally on the state j occupied by the Markov process at time t, the processes Yk (Jk (T )) − Yk (Jk (t )) and Jk (T ) − Jk (t ) are independent of Ft and have the same law as the processes Yk (Jk (T − t )) and Jk (T − t ) given that M (0) = j. We have so, if M (t ) = j:
N θ k ,Yk (Jk (T −t )) +Ψk (−iθ k )Jk (T −t )
EP [L(T ) | Ft ] = L(t )EP ek=1
| M (0) = j .
Calculating this last expectation with the help of Lemma 4.1 with ak = θ k and bk = Ψk (−iθ k ), yields to
EP [L(T ) | Ft ] = L(t )ej eQ (T −t ) 1 = L(t ), so that {L(t )} is a martingale.
The next proposition states a condition on the matrix Θ in order to ensure that the associated measure QΘ is a risk neutral measure. Similar constraints have already been derived in some one-dimensional cases, for example in the MMBM case (see e.g. Equation (2.16) of Elliott et al. [5]) and in the case of pure-jump processes with switching compensator (see e.g. Proposition 3 of Elliott and Osakwe [6]).
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185
Proposition 4.3. If the matrix Θ satisfies the following equations:
Ψk (−i (θ k + el )) − Ψk (−iθ k ) + rk = 0 ∀k = 1, 2, . . . , N ; ∀l = 1, 2, . . . , d, then the associated QΘ is a risk neutral measure. Proof. In order to check the QΘ -martingality of the processes e−U (t ) Sl (t )∀l = 1, 2, . . . , d, we concentrate upon the system of equations
EQΘ e−U (t )+Xl (t ) = 1 ∀l = 1, 2, . . . , d.
(4.4)
By noticing that the following equalities hold in distribution: U (t ) =
N
rk Jk (t ),
k=1
Xl (t ) =
N
el , Yk (Jk (t )) ,
k=1
and by using the change of measure (4.1), we see that (4.4) is equivalent with:
−U (t )+X (t ) l
EQΘ e
N
= EP ek=1
θ k +el ,Yk (Jk (t )) +(Ψk (−iθ k )−rk )Jk (t )
.
Using Lemma 4.1, one easily obtains
EQΘ e−U (t )+Xl (t ) = pe(Q −B)t 1,
where B is the diagonal matrix with Bkk = Ψk (−i (θ k + el )) − Ψk (−iθ k ) + rk . Since peQt 1 = 1 ∀t, the martingality condition can be reformulated as pe(Q −B)t 1 = peQt 1. This last equation is satisfied by taking B = 0, which gives the announced conditions.
Therefore, if a matrix Θ satisfies the conditions in Proposition 4.3, the pricing formulae found in the previous sections under a risk neutral measure can be easily rewritten in function of the parameters of the model under the historical probability. Indeed, let us determine the characteristic exponent of X under the measure QΘ by applying the change of measure (4.1) and Lemma 4.1:
EQΘ e
i u,X (t )
N
= EP ek=1
θ k +iu,Yk (Jk (t )) +Ψk (−iθ k )Jk (t )
= pe(Q −C )t 1,
where C is the diagonal matrix with Cjj = Ψj u − iθ j − Ψj (−iθ j ). The diagonal of that matrix C gives the function 8(.) to be used in the pricing formula when the parameters are known under P:
Proposition 4.4. Starting from the real world measure P under which the characteristic exponent of the Lévy process Yj is Ψ (.), the pricing formulae (2.9), (2.13), (3.7) and (3.17) hold with 8(u) the diagonal matrix such that
(8(u))jj = Ψj u − iθ j − Ψj −iθ j ,
(4.5)
where the matrix Θ = [θ 1 | θ 2 | . . . | θ N ] satisfies the conditions given in Proposition 4.3. Acknowledgements Griselda Deelstra acknowledges support of the ARC grant IAPAS ‘‘Interaction between Analysis, Probability and Actuarial Sciences’’ (Convention no AUWB - 2012 - 12/17 - ULB1). The research of Matthieu Simon was supported by the Belgian FNRS through a FRIA research grant. Appendix Lemma A.1. Let M be a Markov process determined by its generator Q and its initial vector p. Let X be a d-dimensional MMLP defined under a measure Q and modulated by M so that when M = j, X evolves as the d-dimensional Lévy process Yj determined
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by the characteristic exponent Φj (u):
EQ ei⟨u,Yj (t )⟩ = e−Φj (u)t ,
(A.1)
for any d-dimensional vector u. Consider the Markov-modulated drift process C : C (t ) =
t
c (s) ds
with c (t ) =
N
0
cj 1M (t )=j .
j =0
Then, for all a ∈ Cd
EQ eC (t )+⟨a,X (t )⟩ = pe(Q −A)t 1 ∀t ≥ 0,
(A.2)
where A is a diagonal matrix such that Ajj = Φj (−ia) − cj . Proof. We rewrite the left hand side of Eq. (A.2) by the following arguments, which are standard in the context of Markov additive processes (see for example Proposition 2.2 in [19]). Let us define for all states l and j the function Flj (t ) by Flj (t ) = EQ eC (t )+⟨a,X (t )⟩ 1M (t )=j | M (0) = l .
In order to determine Flj (t + h) up to o(h) for small h > 0, we condition upon the state occupied at time t and we note that the probability that there is more than one state change between t and t + h is equal to o(h) (see e.g. [19]): Flj (t + h) = EQ e(C (t )+cj h)+⟨a,X (t )⟩+⟨a,(Yj (t +h)−Yj (t ))⟩ 1M (t )=j | M (0) = l · P (M (t + h) = j | M (t ) = j)
+
EQ e(C (t )+ck h)+⟨a,X (t )⟩+⟨a,(Yk (t +h)−Yk (t ))⟩ 1M (t )=k | M (0) = l · P (M (t + h) = j | M (t ) = k) .
k̸=j
Since the increments (Yk (t + h) − Yk (t )) are independent of M and X , and have the same law as Yk (h), it follows that Flj (t + h) = Flj (t )ecj h EQ e⟨a,Yj (h)⟩
1 + Qjj h + o(h) +
Flk (t )eck h EQ e⟨a,Yk (h)⟩
Qkj h + o(h) .
k̸=j
Moreover, by applying (A.1) and the Taylor expansion e by Flj (t + h) = Flj (t ) − Flj (t ) Φj (−ia) − cj h +
(ck −Φk (−ia))h
N
= 1 − (Φk (−ia) − ck ) h + o(h), Flj (t + h) is determined
Flk (t )Qkj h + o(h),
k=1
and therefore the following equality holds: 1 h
(F (t + h) − F (t )) = F (t ) (Q − A) +
o(h) h
where F (t ) is the matrix with components Flj (t ) and A the diagonal matrix such that Ajj = Φj (−ia) − cj . When h tends to zero, this leads to the equation d dt
F (t ) = F (t ) (Q − A) ,
and therefore to the solution F (t ) = e(Q −A)t by noticing that F (0) = I. As a conclusion, the left-hand side of (A.2) becomes:
EQ eC (t )+⟨a,X (t )⟩ =
N N
EQ eC (t )+⟨a,X (t )⟩ 1M (t )=j | M (0) = l P (M (0) = l)
l =1 j =1
= pe(Q −A)t 1. References [1] [2] [3] [4] [5] [6] [7] [8]
M. Konikov, D. Madan, Option pricing using variance gamma markov chains, Rev. Derivatives Res. 5 (1) (2002) 81–115. J. Buffington, R. Elliot, American options with regime switching, Int. J. Theor. Appl. Finance 5 (5) (2002) 497–514. P. Boyle, T. Draviam, Pricing exotic options under regime switching, Insurance Math. Econom. 40 (2) (2007) 267–282. S. Zhu, A. Badran, X. Lu, A new exact solution for pricing European options in a two-state regime-switching economy, Comput. Math. Appl. 64 (8) (2012) 2744–2755. R. Elliott, L. Chan, T. Siu, Option pricing and Esscher transform under regime switching, Ann. Finance 1 (4) (2005) 423–432. R. Elliott, C. Osakwe, Option pricing for pure jump processes with markov switching compensators, Finance Stoch. 10 (2) (2006) 250–275. K. Chourdakis, Switching Lévy models in continuous time: finite distributions and option pricing. Working paper, 2005. P. Carr, D. Madan, Option valuation using the Fast Fourier Transform, J. Comput. Finance 2 (1999) 61–73.
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[9] T. Siu, H. Yang, J. Lau, Pricing currency options under two-factor markov-modulated stochastic volatility models, Insurance Math. Econom. 43 (3) (2008) 295–302. [10] L. Bo, Y. Wang, X. Yang, Markov-modulated jump diffusions for currency option pricing, Insurance Math. Econom. 46 (3) (2010) 461–469. [11] J. Yoon, B.-G. Jang, K.-H. Roh, An analytic valuation method for multivariate contingent claims with regime-switching volatilities, Oper. Res. Lett. 39 (3) (2011) 180–187. [12] S.-N. Chen, M.-H. Chiang, P.-P. Hsu, C.-Y. Li, Valuation of quanto options in a Markovian regime-switching market: A Markov-modulated Gaussian HJM model, Finance Res. Lett. 11 (2) (2014) 161–172. [13] L. Ballotta, G. Deelstra, G. Rayée, Multivariate FX models with jumps: triangles, Quantos and implied correlation, 2016, working paper. A previous version has been circulated with the title ‘Quanto Implied Correlation in a Multi-Lévy Framework’, 2015. Available at SSRN: http://ssrn.com/abstract=2569015 or http://dx.doi.org/10.2139/ssrn.2569015. [14] L. Ballotta, E. Bonfiglioli, Multivariate asset models using Lévy processes and applications, Eur. J. Financ. 22 (13) (2016) 1320–1350. [15] J. Bishwal, Parameter Estimation in Stochastic Differential Equations, in: Lecture Notes in Mathematics, Springer-Verlag, 2008. [16] H. Gerber, E. Shiu, Option pricing by Esscher transform, Trans. Soc. Actuar. 46 (1994) 99–140. [17] H. Geman, N.E. Karoui, J. Rochet, Changes of numeraire, changes of probability measure and option pricing, J. Appl. Probab. 32 (2) (1995) 443–458. [18] A. Giese, Quanto adjustments in the presence of stochastic volatility, Risk 25 (2012) 67–71. [19] S. Asmussen, Markov additive models, in: Applied Probability and Queues, in: Stochastic Modelling and Applied Probability, vol. 51, Springer, New York, 2003, pp. 302–339.
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Multivariate European option pricing in a Markov-modulated Lévy framework Griselda Deelstra, Matthieu Simon ∗ Université libre de Bruxelles (ULB), Département de Mathématique, Campus de la Plaine C.P. 210, B-1050 Bruxelles, Belgium
article
info
Article history: Received 10 February 2016 JEL classification: C02 G13 C63 D52 F31 Keywords: Regime-switching Esscher transform Exchange options Quanto options
abstract This paper studies the pricing of some multivariate European options, namely Exchange options and Quanto options, when the risky assets involved are modelled by MarkovModulated Lévy Processes (MMLPs). Pricing formulae are based upon the characteristic exponents by using the well known FFT methodology. We study these pricing issues both under a risk neutral martingale measure and the historical measure. The dependence between the asset’s components is incorporated in the joint characteristic function of the MMLPs. As an example, we concentrate upon a regime-switching version of the model of Ballotta et al. (2015) in which the dependence structure is introduced in a flexible way. Several numerical examples are provided to illustrate our results. © 2016 Elsevier B.V. All rights reserved.
1. Introduction It has already been recognized for a long time that the classical Black–Scholes model has limitations when pricing options. However, models based on the exponential Lévy processes are not able as well to capture all empirical features of the market because of the independence and stationarity of the Lévy process increments, as pointed out e.g. in [1]. A natural way to relax the stationarity and independence of the increments is to allow the process to change its parameters at certain times to reflect e.g. the fact that there exist business cycles in the market. This leads to the so called regime-switching models where a Markov process serves to describe parameter changes. The regime-switching framework has been largely developed in the literature, although most of the articles on the subject are dedicated to univariate options. The prices of the risky asset involved in univariate options are often modelled as the exponential of a Markov-modulated Brownian motion (MMBM). Buffington and Elliot [2] found the Black–Scholes equations in the two states regime-switching setting and used it to price vanilla and American options. In [3], a partial differential equations approach is developed to provide an algorithm allowing to price numerically Asian options and lookback options. More recently, Zhu et al. [4] deal with the Fourier Transform method to derive a closed-form formula to price vanilla options in the MMBM setting. We refer to Elliott et al. [5] for a good overview of the literature and financial applications of regimeswitching models. In this important paper, the authors introduce moreover a regime-switching Esscher transform in order to determine an equivalent martingale measure, since the market described by a MMBM is incomplete in general. Markov-modulated Lévy processes (MMLPs) have also been used to price univariate options. In [1], the authors consider the two state case and suppose that the risky asset is a Variance Gamma process in each phase, under a risk neutral
∗
Corresponding author. E-mail addresses: [email protected] (G. Deelstra), [email protected] (M. Simon).
http://dx.doi.org/10.1016/j.cam.2016.11.040 0377-0427/© 2016 Elsevier B.V. All rights reserved.
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G. Deelstra, M. Simon / Journal of Computational and Applied Mathematics 317 (2017) 171–187
measure. They calculate the corresponding characteristic function to estimate the parameters and price the options. Elliott and Osakwe [6] generalize this paper by considering a pure jump switching process with an arbitrary number of states. A more general approach is followed in [7], where the author prices vanilla and exotic options when the price of the risky asset is the exponential of a one-dimensional MMLP. A transition in the Markov process can also induce a jump in the asset price. The author starts under the risk neutral measure and employs the Carr–Madan formula (see [8]) to price vanilla options and gives numerical procedures to deal with exotic options. Regime-switching models further have shown to be useful in the study of currency options. In [9], the authors perform the valuation of European and American currency options in the context of MMBMs for the FX rate. They start from the real world measure and then employ the Esscher change of measure introduced in [5] to determine a risk neutral measure and explicit pricing formulae under that measure. For European option pricing, Bo et al. [10] generalize this model by adding jumps with lognormal distributions. The main focus of this paper is to concentrate on multivariate options in a regime-switching framework. In this context, Yoon et al. [11] start from a risk neutral measure and propose an analytic valuation method based on the occupation times to price European-type multivariate contingent claims. In particular, they price quanto options and exchange options when the risky asset is modelled by a two-dimensional geometric MMBM with two phases. Chen et al. [12] concentrate upon European quanto options when the forward interest rates are supposed to follow a regime-switching HJM model, the stock prices by a regime-switching jump–diffusion and the spot FX rate is supposed to be into the Black–Scholes setting. They adapt the approach in [5] in order to obtain a risk neutral measure. In these papers, dependence between the different assets is easily described by the correlation between the Brownian motions showing up in the MMBM parts. In this paper, we study three types of European options under the hypothesis that the risky assets involved are general Markov-Modulated Lévy Processes (MMLPs) modulated by a Markov chain with an arbitrary number of states. We begin by revisiting the pricing of European call options, for which we give some original examples in order to illustrate the large variety of Markov models that can be used to describe a given regime-switching situation, and to show how these models can lead to different option prices. Next, we apply the introduced methods into the pricing of European exchange and quanto options, which is the main contribution of this paper. As these options deal with two-dimensional MMLPs, the dependence between the two components has to be taken into account. We first provide pricing formulae that are valid whatever the correlation structure between the components. Then, we generalize the model introduced in [13] to the regime-switching case. As in [14], the dependence structure is then incorporated in a flexible way, and the pricing formulae can be easily implemented. We choose the Variance Gamma case to provide numerical illustrations. We derive the option prices by using the Fast Fourier Transform as presented in [8], and therefore calculate the necessary characteristic exponents. As usual, the pricing analysis is performed under a risk neutral measure. However it appears sometimes more interesting to start from the historical probability, e.g. for asset–liability management reasons or in order to estimate the Markov chain parameters (see e.g. [15]). The last section of this paper is therefore devoted to the derivation of the pricing formulae starting from a real world model. Since the financial market is incomplete, the risk-neutral measure is not uniquely determined. We follow the Esscher transform approach, introduced in [16] and adapted to the regime-switching case in [5], to define a risk neutral measure. We state conditions that ensure the existence of such a measure in our framework and then we give the link with the pricing formulae obtained in the preceding sections. The paper is organized as follows: Section 2 is devoted to vanilla options. Exchange options and quanto options are considered in Section 3. In Section 4, we concentrate upon the model under the real world measure and use a regimeswitching Esscher transform to obtain the dynamics in the risk-neutral world. 2. Markov models and vanilla call options In this section, we consider the pricing of a European call option when the price of the risky asset {S (t )} is modelled under a risk-neutral measure Q by the exponential of a one-dimensional Markov-Modulated Lévy Process. Although the pricing formulae presented here are either not new or represent straightforward generalizations of some known results (see for example [1,6,7]), we present the derivation methods since we will use them later when dealing with multivariate options. Moreover, we will give some original regime-switching examples that can be treated in this Markovian framework. To formulate the model we define the following processes:
• A Markov process {M (t ) | t ∈ R+ } on the state space {1, 2, . . . , N } and determined by the generator Q ∈ RN ×N and the initial vector p ∈ RN , • N independent Lévy processes {Yj (t ) | t ∈ R+ } j = 1, 2, . . . , N determined under the measure Q by the characteristic functions:
EQ eiuYj (t ) = e−Φj (u)t ,
u ∈ C.
(2.1)
Based on these processes, we define the MMLP {X (t ) | t ∈ R+ } as follows: X (0) = 0 and ∀t > 0: dX (t ) =
N
dYj (t ) 1M (t )=j .
j =1
Intuitively, when the Markov process M is in state j, the process X behaves like the Lévy process Yj .
(2.2)
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173
We start by recalling a well known result. It is proved in the Appendix in a d-dimensional setting, under the assumption that the expectations in Eqs. (2.3) and (A.2) below are finite. Lemma 2.1. Consider the MMLP X defined in (2.2) and a Markov-modulated drift process C (t ) =
N
t 0
c (s)ds where c (t ) =
j=0 cj 1M (t )=j . Then, ∀a ∈ C and ∀t ≥ 0:
EQ eC (t )+aX (t ) = pe(Q −A)t 1,
(2.3)
where 1 is a vector with each component equal to 1 and A is the diagonal matrix such that Ajj = Φj (−ia) − cj with Φj defined in (2.1). Under the measure Q, we assume that the price of the risky asset S has the form S (t ) = S (0) eΛ(t )+X (t ) ,
(2.4)
where the drift process Λ(t ) is defined as
Λ(t ) =
t
µ(s) ds with µ(t ) = 0
N
µj 1M (t )=j ,
j =0
where the coefficients µj are constant. We further denote the risk-free interest rate by r (t ) and assume that it also depends on the state occupied by the Markov process: r (t ) =
N
rj 1M (t )=j ,
j =0
where the coefficients rj are constant. In the following, we also need the integrated interest rate process, which we denote by U (t ): U (t ) =
t
r (s) ds. 0
We further introduce the vectorial notations: r = (r1 , r2 , . . . , rN ) , µ = (µ1 , µ2 , . . . , µN ) , 8(u) = (Φ1 (u), Φ2 (u), . . . , ΦN (u)) .
(2.5)
Since Q is a risk-neutral probability, the vector µ should be chosen in a particular way: Lemma 2.2. In order to ensure that the process e−U (t ) S (t )
t
is a Q-martingale, the vector µ should be determined by
µ = r + 8(−i).
(2.6)
Proof. In order to check the Q-martingality of e−U (t ) S (t ), we verify whether
EQ e−U (t )+Λ(t )+X (t ) = 1 ∀t .
(2.7)
The left hand side of this last equality follows easily from Lemma 2.1 with a = 1 and C (t ) = Λ(t ) − U (t ):
EQ e−U (t )+Λ(t )+X (t ) = pe(Q −A)t 1
with A = diag (r − µ + 8(−i)). Since peQt 1 = 1 ∀t, Eq. (2.7) can be rewritten as pe(Q −A)t 1 = peQt 1. The only solution of this equation is A = 0, so that we get the announced relation. Our first aim is to determine the price V (S , K , T ) of a European call option with maturity T and strike K : V (S , K , T ) = EQ e−U (T ) (S (T ) − K )+ .
(2.8)
Theorem 2.3. The price of the European call option at time t = 0 is given by V (S , K , T ) =
e−α k
π
∞
e−iv k a(v) e(α+1+iv)s0 pe(Q −G(α,v))T 1 dv,
(2.9)
0
where α is a damping factor,
−1
a(v) = (α 2 + α − v 2 ) + i(2αv + v)
,
G(α, v) = diag (8(v − i(α + 1)) − (α + 1 + iv)8(−i) − (α + iv)r ) , and s0 = ln(S (0)), k = ln(K ).
(2.10) (2.11)
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Proof. To calculate the right hand side of (2.8), we use the well known Carr–Madan formula (see [8]). We put s0 = ln(S (0)) and k = ln(K ) and write V (S , K , T ) =
∞
−∞
∞
e−u es − ek f (s, u) ds du,
k
where f (s, u) is the density function of the vector (s0 + Λ(T ) + X (T ), U (T )). Therefore, V (S , K , T ) =
e−α k
π
∞
e−iv k a(v) e(α+1+iv)s0 EQ e−U (T )+(α+1+iv)(X (T )+Λ(T )) dv
(2.12)
0
−1
where a(v) = (α 2 + α − v 2 ) + i(2αv + v) and α > 0 is chosen in such a way that the last integral exists. The expectation in (2.12) can be calculated by using Lemma 2.1 with a = α + 1 + iv and C (t ) = (α + 1 + iv)Λ(t ) − U (t ). The price of the European call option then easily follows.
Formula (2.9) allows the numerical computation of the prices in an efficient way thanks to e.g. the Fast Fourier Transform method (see [8]). For completeness, we also determine the price of the European call option at an arbitrary time t ≤ T , that is V (S , K , T , t ) = EQ e−U (T ) (S (T ) − K )+ | Ft
K = e−U (t ) S (t )EQ e−(U (T )−U (t )) e(X (T )−X (t ))+(Λ(T )−Λ(t )) − | Ft , S (t ) + where Ft is the σ -algebra generated by the history of the Markov process and by the MMLP X up to time t. Now, conditionally on the value of S (t ), U (t ), Λ(t ) and the state j occupied by the Markov process at time t, the processes X (T )−X (t ), U (T )−U (t ) and Λ(T ) − Λ(t ) are independent of Ft and have the same law as the processes X (T − t ), U (T − t ) and Λ(T − t ) given that M (0) = j. So, when denoting S = S (t ), U = U (t ) and M (t ) = j,
V (S , K , T , t ) = e−U EQ e−U (T −t ) S˜ (T − t ) − K
+
| M (0) = j ,
X (t )+Λ(t )
with S˜ (t ) = Se . The last formula corresponds to the price of a European call option with risky asset S˜ (t ), strike K and maturity T − t. Theorem 2.3 therefore immediately leads to the following corollary: Corollary 2.4. If at time t ≤ T , S (t ) = S, U (t ) = U and M (t ) = j, then e−α k V (S , K , T , t ) = e−U
∞
π
e−iv k a(v) e(α+1+iv)s ej e(Q −G(α,v))(T −t ) 1 dv,
(2.13)
0
with ej the jth vector of the usual basis of RN , with s = ln(S ) and k = ln(K ), and where a(v) and G(α, v) are given by (2.10) and (2.11). Examples The examples presented here aim to illustrate how the behaviour of the Markov process M influences the option prices in different ways, and how we can build some Markovian regime-switching models that differ from the classical Markovian setting. We consider the particular case when S (t ) is modelled as in (2.4) and X (t ) is assumed to be a Markov-modulated Merton jump–diffusion process. In particular, we suppose that there are N different phases and that the vector of characteristic exponents 8(u) is given by (2.5) with
8j (u) = u2 σj2 + λj 1 − e
− 21 u2 τj2
j = 1, . . . , N ,
so that the jumps in each phase have zero mean. In all examples, we will set S (0) = 100 and K = 90. We denote by λ, σ and τ the vectors containing the parameter values of the model in each phase. Example 1. We first consider a model with three phases A, B and C . The phase A corresponds to a normal behaviour of the market, when the volatilities are low. The phase C corresponds to a bad state of the market (e.g. with many jumps and high volatilities). Phase B corresponds to an intermediate state when there is a risk to fall in phase C but there also is a possibility that the market comes back into phase A. A first way to define a Markov process {M (t )} that controls the transitions between the three phases is by choosing the following classical rules:
• When M1 is in phase A, it stays in A during an exponential period of time with parameter α . Then the process goes to phase B with probability 1 − q1 or to phase C with probability q1 .
G. Deelstra, M. Simon / Journal of Computational and Applied Mathematics 317 (2017) 171–187
175
Fig. 1. Prices for different initial vectors.
• When M1 is in phase B, it stays in B during an exponential period of time with parameter β . Then the process goes to phase C with probability q2 or to phase A with probability 1 − q2 . • When M1 is in phase C , it stays in C during an exponential period of time with parameter γ before going to phase A. This leads to a first model (Model 1), where we choose the following parameters:
−α (1 − q1 )α q1 α 0.01 0 0.02 0 −β q2 β , Q = (1 − q2 )β r = 0.01 , λ= 2 , σ = 0.1 , τ = 0.1 , γ 0 −γ 0.01 10 0.25 0.25 and p = p1 p2 p3 . In this example, we focus on the effect of the initial distribution on the prices. We will consider the
basis vectors of R3 as initial vectors, as well as the stationary vector of Q . The latter is a reasonable choice when the initial vector is unknown. Fig. 1 shows the prices as a function of the maturity date T when the parameter values are α = 2.5, β = 5, γ = 4, q1 = 0.8 and q2 = 0.4. We note that the initial state vector influences the prices for short maturities but becomes less and less important as T increases. This is natural because the Markov process M converges to its stationary distribution for large maturities. Example 2. We now consider again the model defined in Example 1, but this time the Markov process {M ′ (t )} that controls the transitions between the three phases is chosen so that the time spent during a sojourn in phase A or B depends on the next phase to be visited. More precisely:
• When M ′ is in phase A, two situations are possible: with probability 1 − q1 the process stays in A during a period exponentially distributed with parameter α1 and then goes to B. With probability q1 the process stays in A during a period exponentially distributed with parameter α2 and then goes to C . • When M ′ is in phase B, two situations are possible: with probability 1 − q2 the process stays in B during a period exponentially distributed with parameter β1 and then goes to A. With probability q2 the process stays in B during a period exponentially distributed with parameter β2 and then goes to C . • When M ′ is in phase C , it stays in C during a period exponentially distributed with parameter γ before going to phase A. The Markov process {M ′ (t )} has then five states because we need to duplicate the phases A and B in order to include the different possible sojourn times. This yields the following generator:
−α1
0 Q ′ = (1 − q1 )β1 0 (1 − q1 )γ and p′ = p1 (1 − q1 )
r1 r1 r ′ = r2 , r 2 r3
(1 − q2 )α1
0
−α2
q1 β1 0 q1 γ
p1 q1
0
−β1 0 0
p2 (1 − q2 )
λ1 λ1 λ′ = λ2 , λ 2 λ3
q2 α1 0 0
0
α2 0 , β2 −γ
−β2 0 p2 q2
p3 . The other parameters are given by
σ1 σ1 σ ′ = σ2 , σ 2 σ3
τ1 τ1 τ ′ = τ2 . τ 2 τ3
176
G. Deelstra, M. Simon / Journal of Computational and Applied Mathematics 317 (2017) 171–187
Fig. 2. Prices for the different models.
In Fig. 2, we fix α = 0.5, β = 2, γ = 1, q1 = 0.3, q2 = 0.2 and p1 = 1 and we compare the prices obtained when the modulating process is M as in Example 1 (Model 1) with the ones obtained when the modulating process is M ′ (Model 2) in three situations. First, for Model 2.1, we set α1 = α2 = α , β1 = 5β and we choose β2 equal to
β2 = q2 /((1/β) − (1/(β1 )) ∗ (1 − q2 )),
(2.14)
so that the average duration of a visit in state B is 1/β as in Model 1. For Model 2.2, we take β1 = β2 = β , α1 = 2α and
α2 = 1/((1/α) − (1/(α 1)) ∗ (1 − q1 )).
(2.15)
In this way, the average duration of a visit in state A is 1/α as in Model 1. Finally, in Model 2.3 we take α1 = 2α , β1 = 5β and we choose β2 and α2 as in (2.14) and (2.15). Even if the four models presented in Fig. 2 have, by construction, the same mean sojourn time in each phase and the same transition probabilities, we observe that the prices can be different, especially when the maturity date is large enough. Example 3. As a last example in this section, we start from a model with two states A and B only and with the parameters
−α α , β −β
Q =
r =
0.05 , 0.02
λ=
0 , 10
σ=
0.05 , 0.2
τ=
0 , 0.2
and p = 1 0 . This is a classical Markov-switching situation in which the sojourn times in phase A (respectively B) are exponentially distributed with parameter α (respectively β ). In particular, the duration of a visit to phase A (B) has a mean 1/α (1/β ) and a variance 1/α 2 (1/β 2 ). We want to construct an analogous model where the sojourn times in A and B have the same mean durations but where they are now distributed as Erlang random variables. This leads to a model where a sojourn time in phase A (phase B) has an Erlang distribution with parameters n and nα (nβ ) for some n ∈ N0 . In this new situation the mean sojourn times in A and B remain unchanged but the variances become respectively 1/(nα 2 ) and 1/(nβ 2 ). As n increases, the variances of the sojourn times therefore decrease to zero. We will compare the option prices obtained in the two models. The model with Erlang sojourn times can be built by decomposing the two phases A and B into n successive states. A visit in state A (respectively B) is then translated by the successive visits to n states in which the sojourn time is exponentially distributed with parameter nα (respectively nβ ). The generator of the model with Erlang sojourn time is given by the 2n × 2n matrix
−n α 0 . .. 0 Q′ = 0 0 .. . nβ
nα −n α .. .
0 nα
··· ···
0 0 0
0 0 0
··· ··· ···
0
0
···
.. .
.. .
0 0
.. .
.. .
−nα 0 0
.. .
0
0 0
.. . nα −n β
0 0
.. .
0 0
··· ···
.. .
0
0 0 nβ
··· ··· ···
0
0
0
···
.. .
0 0 0
,
.. .
0 nβ −nβ
.. .
0 0
.. .
.. .
−nβ
G. Deelstra, M. Simon / Journal of Computational and Applied Mathematics 317 (2017) 171–187
(a) Prices for different values of n.
177
(b) Prices with or without regime switching.
Fig. 3. Option prices for different Erlang models.
and the other parameters are given by the following vectors (with n + n components): r′ =
r1 1 , r2 1
λ′ =
λ1 1 , λ2 1
σ′ =
σ1 1 , σ2 1
τ′ =
τ1 1 , τ2 1
p2 0 · · · 0 . and p′ = p1 0 · · · 0 Fig. 3(a) compares the option prices obtained for different values of n when α = 1, β = 1.5 and p1 = 1. We observe that the prices obtained when n > 1 oscillate around the curve associated with n = 1, and that the amplitude of the oscillations decreases with the number of transitions in the Markov Process. The intersections between the curves seem to correspond approximately to the transition instants of the Markov process. Fig. 3(b) compares the option prices obtained for n = 1 and n = 30 with the prices obtained where there is no regime switching and the parameters are the ones associated with the first phase or with the second one. It provides an interpretation of the oscillations: when n is large, the transition times in the Markov process are nearly deterministic. The prices associated with n show then a behaviour similar to the case where the parameters of the model change at deterministic instants. 3. Multivariate option pricing 3.1. Exchange options In this section, we concentrate upon exchange options in a MMLP framework and we derive prices by using the methods presented in the previous section. We recall that an analytic valuation method for exchange options was already proposed in [11] in the case of MMBMs, but as far as we know, no results are known in the MMLP setting. Whereas the dependence between the price of the two assets involved in an exchange option is easily modelled in the case of Brownian motions, specifying a dependence structure for general Lévy processes is a more challenging problem. In this section, we start by providing an option pricing formula which applies for arbitrary dependence structures as it is based upon the joined characteristic exponent of the two-dimensional MMLP X . We are thus interested in the evaluation of the price V (S1 , S2 , T ) at time 0 of an exchange option with maturity T : V (S1 , S2 , T ) = EQ e−U (T ) (S2 (T ) − S1 (T ))+ ,
(3.1)
where the two-dimensional process of asset prices S (t ) = (S1 (t ), S2 (t )) has the dynamics S1 (t ) = S1 (0)eΛ1 (t )+X1 (t )
and S2 (t ) = S2 (0)eΛ2 (t )+X2 (t ) ,
with X (t ) = (X1 (t ), X2 (t )) a two-dimensional MMLP controlled by a Markov process {M (t ) | t ∈ R+ } defined as before by a generator Q and an initial vector p. We assume that X (0) = 0 and, when M = j, X has the same behaviour as the two-dimensional Lévy process Yj determined by the characteristic exponent Φj (u):
EQ ei⟨u,Yj (t )⟩ = e−Φj (u)t ,
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with u = (u1 , u2 ). The processes Λ1 (t ) and Λ2 (t ) denote some drift processes:
Λ1 (t ) =
t
µ(s) ds,
µ(t ) =
0
Λ2 (t ) =
µj 1M (t )=j ,
(3.2)
j =0 t
N
ν(s) ds,
ν(t ) =
0
N
νj 1M (t )=j
(3.3)
j=0
where the coefficients µj and νj are constant. As in Section 1, U (t ) is the integral up to time t of the interest rate process: r (t ) =
N
rj 1M (t )=j ,
U (t ) =
t
r (s) ds. 0
j =0
Using the notations r = (r1 , r2 , . . . , rN ) , µ = (µ1 , µ2 , . . . , µN ) , ν = (ν1 , ν2 , . . . , νN ) , 8(u) = (Φ1 (u), Φ2 (u), . . . , ΦN (u)) , an analogous result as in Lemma 2.2 shows that the vectors µ and ν need to be defined in the following way:
µ = r + 8(−i, 0), ν = r + 8(0, −i).
(3.4) (3.5)
In order to evaluate (3.1), the following change of measure turns out to be useful (see e.g. [17]):
˜ dQ
e −U ( T ) S 1 ( T )
=
dQ
EQ
1
= e−U (T ) S1 (T ). S1 (0) 1 (T )
(3.6)
e−U (T ) S
Using this change of measure in (3.1) leads to:
EQ e−U (t ) (S2 (T ) − S1 (T ))+ = EQ˜
S1 (0) S1 (T )
(S2 (T ) − S1 (T ))+ = S1 (0) EQ˜ (H (T ) − 1)+ ,
S (0)
˜ , the problem boils down to the evaluation of where H (t ) = S2 (0) e(Λ2 (t )+X2 (t ))−(Λ1 (t )+X1 (t )) . Therefore, under the measure Q 1 the European Call Option price when H (t ) is the risky asset, T is the maturity, the strike is equal to 1 and the risk-free interest rate equals 0. We can therefore use the results of Section 2 to calculate V (S1 , S2 , T ): Theorem 3.1. The price of an exchange option as in (3.1) is given by V (S1 , S2 , T ) =
e−α k
∞
π
(1) (2) e−iv k a(v) e−(α+iv)s0 +(α+1+iv)s0 pe(Q −G(α,v))T 1 dv
,
(3.7)
k=0+
0
where G(α, v) = diag (8 (−v + iα, v − i(α + 1)) + (α + iv)8(−i, 0) − (α + 1 + iv)8(0, −i)) , (1)
(2)
and a(v) is given in (2.10), s0 = ln(S1 (0)), s0 = ln(S2 (0)). Proof. As we deal with European options, Eq. (2.12) holds modulo the fact that the process S is replaced by the process H. Adapting (2.12) to H yields the following expression for V (S1 , S2 , T ): S1 (0)
e−α k
π
∞
e−iv k a(v)e(α+1+iv)(ln(S2 (0))−ln(S1 (0))) EQ˜ e(α+1+iv)(−X1 (T )−Λ1 (T )+X2 (T )+Λ2 (T )) dv.
0
The above expectation can be calculated by the change of measure (3.6) under the original measure Q:
EQ˜ e(α+1+iv)(−X1 (T )−Λ1 (T )+X2 (T )+Λ2 (T )) = EQ e−U (T )−(α+iv)(X1 (T )+Λ1 (T ))+(α+1+iv)(X2 (T )+Λ2 (T )) ,
which can be rewritten by applying Lemma A.1 with d = 2, C (t ) = (α + 1 + iv)Λ2 (t ) − (α + iv)Λ1 (t ) − U (t ) and a = [−(α + iv)α + 1 + iv]. Using Eqs. (3.4) and (3.5) leads then to the announced result.
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179
Table 1 Parameter values.
a1 a2
Phase 1
Phase 2
Phase 3
−1.2
1 0
−1 −1
1
Z1 Z2 Z3
k
θ
σ
k
θ
σ
k
θ
σ
0.005 0.08 0.07
−0.02 0.15 −0.1
0.11 0.1 0.1
0.01 0.05 0.05
0.6 0.05 −0.5
0.03 0.15 0.1
0.12 0.1 0.05
0.2 0.15 0.1
0.11 0.1 0.1
Example 4. This example is inspired by Ballotta and Bonfiglioli [14]. Indeed, we concentrate upon a two-dimensional MMLP in which the dependence between the two components in each phase follows from the assumption of a common systematic component, besides the idiosyncratic parts Z1 (t ) and Z2 (t ): X1 (t ) = Z1 (t ) + a1 (t )Z3 (t ), X2 (t ) = Z2 (t ) + a2 (t )Z3 (t ),
(3.8)
with al ( t ) =
N
alj (t )1M (t )=j
(l = 1, 2)
j =1
some loading factors influencing the dependence between the two components of X . In this model, Z1 , Z2 and Z3 are assumed to be three independent one-dimensional MMLPs. As explained in [14], this factor construction implies a model with a flexible correlation structure which can be easily implemented since the characteristic function can be determined in a straightforward way: Lemma 3.2. The jth component Φj (u1 , u2 ) of the vector 8(u1 , u2 ) giving the characteristic exponents of X in phase j is determined by:
Φj (u1 , u2 ) = ϕ1j (u1 ) + ϕ2j (u2 ) + ϕ3j (a1j u1 + a2j u2 ), where ϕlj is the characteristic exponent of the process Zl when M = j. For a numerical experiment we suppose that when the Markov process M is in phase j, the process Zl behaves as a Variance Gamma process Ylj with parameters klj , θlj and σlj . The characteristic exponent ϕlj (u) of Ylj is well known:
ϕlj (u) =
1 klj
ln 1 − iuklj θlj +
1 2
u2 klj σlj2
(l = 1, 2, . . . , N , j = 1, 2, . . . , N ).
The goal of this example is to describe a regime-switching model which can switch between N = 3 phases with different correlation structures for X . The parameters for the Markov process are chosen to be equal to
−3 Q =
2 2
3 −3 3
0 1 , −5
p = p1
p2
p3 .
Table 1 gives the parameter values of the processes in (3.8) in the different phases, whereas the corresponding means, variances and correlations are stated in Table 2. Remark that phase 1 and phase 3 correspond respectively to a negative and positive correlation between X1 and X2 . Phase 2 corresponds to an uncorrelated situation. Fig. 4 compares the option prices related to this model (MMLP) with the prices when there is no regime-switching and the parameters are the ones of the first phase (LP 1), the second phase (LP 2) or the third phase (LP 3). The initial values of the two risky assets are S1 (0) = 90 and S2 (0) = 100, and the initial vector is given by p2 = 1. We observe that the last three models are relatively close to each other while the model LP 1 yields much higher prices, which can be explained by the assumption of a negative correlation between the two assets. As expected, the prices related to the model with regime-switching are always situated between the ones related to the models without switching. 3.2. Quanto options This section is concerned with the evaluation of quanto future options. Quanto options have already been explored in the setting of MMBM. Yoon et al. [11] provide some analytic valuation method under a risk neutral measure, Chen et al. [12] consider a more general MMBM model since they include a regime-switching HJM interest rate model. In this section, we generalize the model of Ballotta et al. [13] to a MMLP framework.
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G. Deelstra, M. Simon / Journal of Computational and Applied Mathematics 317 (2017) 171–187 Table 2 Means, variances and correlations.
E [X1 ] Var [X1 ] E [X2 ] Var [X2 ] Corr [X1 , X2 ]
Phase 1
Phase 2
Phase 3
0.1 0.027 0.05 0.023 −0.516
0.1 0.027 0.05 0.023 0
0.1 0.027 0.05 0.023 0.421
Fig. 4. Prices for the four models.
As before, we suppose that the market can switch between N phases 1, 2, . . . , N and that the phase transitions are controlled by a Markov process {M (t ) | t ∈ R+ } defined by a generator Q and an initial vector p. We consider that the spot FX rate between the foreign currency (f ) and the domestic currency (d) at time t, denoted by Xd|f (t ), is quoted as the amount of currency f per unit of currency d. We also consider an index S (t ) traded in the foreign currency. We denote by rf (t ) and rd (t ) resp. the interest rates in the foreign and the domestic market. We assume that these rates are constant in each phase of the market: rf ( t ) =
N
rf ,j 1M (t )=j ,
rd ( t ) =
N
j =0
rd,j 1M (t )=j ,
j =0
with rf ,j , rd,j ∈ R+ , and we introduce the notations: Uf (t ) =
t
rf (s) ds and Ud (t ) = 0
t
rd (s) ds. 0
We suppose that the processes S (t ) and Xd|f (t ) have the following form under the risk-neutral measure Qf defined in the foreign market: S (t ) = S (0)eΛ1 (t )+X1 (t )
and
Xd|f (t ) = Xd|f (0)eΛ2 (t )+X2 (t ) ,
(3.9)
where X (t ) = (X1 (t ), X2 (t )) is a two-dimensional MMLP controlled by {M (t )} such that X (0) = 0 and, when M = j, X has (1)
the same behaviour as the Lévy process Yj = Yj
, Yj(2) determined by the characteristic exponent Φj (u):
EQf ei⟨u,Yj (t )⟩ = e−Φj (u)t ,
and where the drift processes Λ1 (t ) and Λ2 (t ) are defined as in Eqs. (3.2) and (3.3). As before, we will use some vectorial notations: rf = rf ,1 , rf ,2 , . . . , rf ,N , rd = rd,1 , rd,2 , . . . , rd,N , µ = (µ1 , µ2 , . . . , µN ), ν = (ν1 , ν2 , . . . , νN ), 8(u) = (Φ1 (u), Φ2 (u), . . . , ΦN (u)). We also put Rf = diag(rf ) and Rd = diag(rd ). As in Lemma 2.2, the condition that the processes S (t ) and Xd|f (t ) are properly defined under a risk-neutral measure Qf , that is e−Uf (t ) S (t ) and e−Uf (t )+Ud (t ) Xd|f (t ) are Qf -martingales, implies the choice of µ and ν:
µ = rf + 8(−i, 0) and ν = rf − rd + 8(0, −i).
(3.10)
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181
It is well known that it is convenient to switch from the risk neutral foreign world to the risk neutral domestic market by defining the following change of measure from Qf to Qd :
Ud (t ) Xd|f (t ) = e , Uf (t ) dQf Ft e Xd|f (0)
dQd
(3.11)
where Ft is the σ -algebra generated by the history of the Markov process and by the MMLP X up to time t. The aim of this section is to derive the arbitrage free price of a European quanto call option with maturity date T1 on the quanto future of the index S, expressed in units of domestic currency, with maturity T2 and strike K , that is:
V FTd1 (T2 ), K , T1 = Ed e−Ud (T1 ) FTd1 (T2 ) − K
+
,
(3.12)
where Ed stands under the measure Qd , T2 is the maturity of the option contract and, for T1 < T2 , for the expectation FTd1 (T2 ) = Ed S (T2 ) | FT1 is the quanto future with underlying index S, under the assumption that the applied FX rate between the two currencies is set to 1 (d/f) (see e.g. [18]). This price can be reformulated in a simpler way by applying the change of measure (3.11): Lemma 3.3. FTd1 (T2 ) = S (T1 ) eM (T1 ) e(Q +Rf +∆)(T2 −T1 ) 1,
(3.13)
where eM (T1 ) is a random unitary vector in RN with the jth component equal to 1 if and only if M (T1 ) = j, and where
∆ = diag (8(−i, 0) + 8(0, −i) − 8(−i, −i)) .
(3.14)
Proof. In order to determine FTd1 (T2 ), we apply the change of measure (3.11): FTd1 (T2 ) = S (0)Ed eΛ1 (T2 )+X1 (T2 ) | FT1
Ef eA(T2 )+X1 (T2 )+X2 (T2 ) | FT1
= S (0)
Ef eB(T2 )+X2 (T2 ) | FT1
= S (0)eA(T1 )−B(T1 )+X1 (T1 )
Ef e(A(T2 )−A(T1 ))+(X1 (T2 )−X1 (T1 ))+(X2 (T2 )−X2 (T1 )) | FT1
Ef e(B(T2 )−B(T1 ))+(X2 (T2 )−X2 (T1 )) | FT1
,
where A(t ) = Λ1 (t ) + Ud (t ) − Uf (t ) + Λ2 (t ) and B(t ) = Ud (t ) − Uf (t ) + Λ2 (t ). Conditionally on the state j occupied by the Markov process at time T1 , the processes X (T2 ) − X (T1 ), A(T2 ) − A(T1 ) and B(T2 ) − B(T1 ) are independent of FT1 and have the same law as the processes X (T2 − T1 ), A(T2 − T1 ) and B(T2 − T1 ) given that M (0) = j. Therefore, if M (T1 ) = j:
Ef eA(T2 −T1 )+X1 (T2 −T1 )+X2 (T2 −T1 ) | M (0) = j
FTd1
(T2 ) = S (T1 )
Ef eB(T2 −T1 )+X2 (T2 −T1 ) | M (0) = j
.
By the use of Lemma A.1, assuming that M (T1 ) = j and using (3.10), it is easily found that
Ef eA(T2 −T1 )+X1 (T2 −T1 )+X2 (T2 −T1 ) | M (0) = j = ej e(Q +Rf +∆)(T2 −T1 ) 1
with ∆ as in (3.14), and
Ef eB(T2 −T1 )+X2 (T2 −T1 ) | M (0) = j = ej eQ (T2 −T1 ) 1 = 1.
This implies the announced result.
Remark. In a similar way, it can be shown that the quanto future with underlying index S under the measure Qf , that is f
FT1 (T2 ) = Ef S (T2 ) | FT1 , can be expressed as
FT1 (T2 ) = S (T1 ) eM (T1 ) e(Q +Rf )(T2 −T1 ) 1, f
which is the regime-switching generalization of Equation (10) in [13]. Note that the expression for FTd1 (T2 ) is the same as the f
one for FT1 (T2 ), except that the diagonal matrix ∆ is added in the exponential. This justifies the naming of quanto adjustment for ∆. When N = 1, and therefore no regime-switching, the matrix ∆ becomes the quanto adjustment q obtained in [13].
Lemma 3.3 is particularly useful for reformulating the price of V FTd1 (T2 ), K , T1 following lemma.
in (3.12), as can be observed in the
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Lemma 3.4. V FTd1 (T2 ), K , T1 =
N
(j)
Qadj Ed e−Ud (T1 ) S (T1 ) − Kj + 1M (T1 )=j ,
(3.15)
j =1
(j)
Kj = K /Qadj
where
Qadj = ej e(Q +Rf +∆)(T2 −T1 ) 1. (j)
and
(3.16)
Proof. Using Lemma 3.3 leads immediately to the result:
Ed e−Ud (T1 ) FTd1 (T2 ) − K
=
N
+
Ed e−Ud (T1 ) S (T1 )eM (T1 ) e(Q +Rf +∆)(T2 −T1 ) 1 − K
j =1
=
N
+
ej e(Q +Rf +∆)(T2 −T1 ) 1 Ed e−Ud (T1 ) S (T1 ) −
j =1
1M (T1 )=j
K
ej e(Q +Rf +∆)(T2 −T1 ) 1 +
1M (T1 )=j
.
As a result, the price of a quanto option can be seen as the sum of European call options upon the index S and strikes Kj and therefore, the results of Section 2 imply the following theorem: Theorem 3.5. The price of a quanto call option upon the quanto future on the index S with strike K and maturity date T1 is determined by V
FTd1
(T2 ), K , T1 =
N
(j)
Qadj
j =1
e−αj kj
∞
π
e−iv kj aj (v)e(αj +1+iv)s0 pe(Q −G(αj ,v))T1 ej dv,
(3.17)
0
where
−1
aj (v) = (αj2 + αj − v 2 ) + i(2αj v + v)
G(αj , v) = diag 8(v − i(αj + 1), −i) − (αj + 1 + iv) 8(−i, 0) + rf − 8(0, −i) + rd .
(j)
(j)
and where Qadj is defined in (3.16), kj = ln(K /Qadj ), ∆ is given in (3.14) and s0 = ln(S (0)). Proof. By analogy with Eq. (2.12), the jth expectation in (3.15) can be rewritten as e−αj kj
π
∞
e−iv kj aj (v)e(αj +1+iv)s0 Ed e−Ud (T1 )+(αj +1+iv)(Λ1 (T1 )+X1 (T1 )) 1M (T1 )=j dv.
0
The last expectation is easily calculated by applying the change of measure (3.11) and by using Lemma A.1, with the vector 1 in (A.2) replaced by ej . Example 5. We now model the process of the index S (t ) and the exchange rate Xd|f (t ) in (3.9) by choosing X1 and X2 as in Example 4 (and therefore similar as in [14]): X1 (t ) = Z1 (t ) + a1 (t )Z3 (t ), X2 (t ) = Z2 (t ) + a2 (t )Z3 (t ), where Z1 , Z2 and Z3 are three independent Markov-switching Variance Gamma processes with N = 3 and the parameters given in Table 1. The Markov-switching parameters and the interest rates processes are chosen to be equal to:
−3 Q =
2 2
3 −3 3
0 1 , −5
0.02 0.01 , 0.01
p= 1
0
0 ,
rf =
0.05 0.05 . 0.025
rd =
First, we assume T1 = T2 = T which implies the case of a call option on the index S. Fig. 5 compares the option prices related to the Markov-switching model (MMLP) with the prices when there is no regime-switching and the parameters are the ones of the first phase (LP 1), the second phase (LP 2) or the third phase (LP 3) for different maturity dates T , when S (0) = 100 and K = 90 days. We note that the prices are a convex function for short maturities but then turn to be a concave function of the maturity. Fig. 6 compares the option prices upon a quanto future on the index, and this for the four models for different maturity dates T2 when T1 is fixed at 30 days and p1 = 1. The prices seem then approximately to behave in a linear way.
G. Deelstra, M. Simon / Journal of Computational and Applied Mathematics 317 (2017) 171–187
183
Fig. 5. Prices for the four models.
Fig. 6. Prices for the four models, T1 = 30 days.
4. From real world to risk neutral measure In this section, we start from a model of the dynamics of the underlying assets in the options under the real world measure. This can be useful for the estimation of the Markov chain parameters, or for ALM studies. We consider a historical probability space (Ω , F , P) on which we define a d-dimensional risky asset S (t ) = (S1 (t ), S2 (t ), . . . , Sd (t )) which has the following dynamics: Sl (t ) = Sl (0) eXl (t ) ,
l = 1, 2, . . . , d,
with X (t ) = (X1 (t ), X2 (t ), . . . , Xd (t )) being a d-dimensional MMLP controlled by a Markov process {M (t ) | t ∈ R+ } which is defined by a generator Q and an initial vector p. We assume that X (0) = 0 and, when M = j, X has the same behaviour as the d-dimensional Lévy process Yj determined by the characteristic exponent Ψj :
EP ei⟨u,Yj (t )⟩ = e−Ψj (u)t ,
for any d-dimensional vector u. The risk-free interest rate r (t ) and its integral U (t ) are defined as in Section 2. Following arbitrage theory, our aim is to find an equivalent martingale measure Q ∼ P under which the process {e−U (t ) S (t )} is a martingale. For this, we use the Markov switching Esscher transform, which is introduced in [5] in the context of geometric Markov-modulated Brownian motions and which can be generalized in a straightforward way to more general regime-switching models. Let Ft be the σ -algebra generated by both the Markov process and the MMLP X up to time t, and FtM the σ -algebra generated by only the Markov process up to time t. For any d × N matrix Θ = [θ 1 | θ 2 | . . . | θ N ] we define the change of measure QΘ ∼ P by
eZΘ (t ) = = L(t ), dP Ft EP eZΘ (t ) | FtM
dQΘ
(4.1)
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where {ZΘ (t )} is a one-dimensional MMLP such that, when M = j, ZΘ has the same behaviour as the Lévy process ⟨θ j , Yj ⟩. Eq. (4.1) can be rewritten in terms of the occupation times of the Markov process M. Indeed, let Jk (t ) be the total amount of time spent by the Markov process M in phase k up to time t. As the increments of the processes Yk are stationary, independent and not affected by the past of M, X (t ) equals in distribution: X (t ) =
N
Yk (Jk (t )) ,
(4.2)
k=1
and it follows then that
EP
eZΘ (t )
| Ft
M
N θ k ,Yk (Jk (t ))
= EP ek=1
N M θ ,Y (J (t )) F = EP e k k k Jk (t ) t
k=1
−
=e
N
Ψk (−iθ k )Jk (t )
k=1
,
and therefore the Radon–Nikodym derivative L(t ) is distributed as L( t ) = e
ZΘ (t )+
N
N θ k ,Yk (Jk (t )) +Ψk (−iθ k )Jk (t )
Ψk (−iθ k )Jk (t )
= ek=1
k=1
.
(4.3)
In order to prove that L(t ) is a martingale, we need the following lemma: Lemma 4.1. If a1 , a2 , . . . , aN are some vectors in Rd and b1 , b2 , . . . , bN some real numbers, then
N
EP ek=1
ak ,Yk (Jk (t )) +bk Jk (t )
= pe(Q −A)T 1,
where A is a diagonal matrix such that Akk = Ψk (−iak ) − bk . The proof of this lemma follows from the same arguments as in the proof of Lemma A.1, by using the strong Markov property applied to the Markov process M and the Lévy processes Yj , and is therefore omitted. Lemma 4.1 has already been proved in different forms: see e.g. Proposition 2 of Elliott and Osakwe [6] in the case where ak = 0 ∀k, and Lemma A.1 in [2]. This lemma allows to conclude that the process {L(t )} is a martingale: Lemma 4.2. The process L(t ) defined in (4.1) is a P-martingale with respect to Ft . Proof. Let T ≥ t. By Eq. (4.3):
N
EP [L(T ) | Ft ] = EP ek=1
θ k ,Yk (Jk (T )) +Ψk (−iθ k )Jk (T )
N
= L(t )EP ek=1
| Ft
θ k ,Yk (Jk (T ))−Yk (Jk (t )) +Ψk (−iθ k )(Jk (T )−Jk (t ))
| Ft .
Conditionally on the state j occupied by the Markov process at time t, the processes Yk (Jk (T )) − Yk (Jk (t )) and Jk (T ) − Jk (t ) are independent of Ft and have the same law as the processes Yk (Jk (T − t )) and Jk (T − t ) given that M (0) = j. We have so, if M (t ) = j:
N θ k ,Yk (Jk (T −t )) +Ψk (−iθ k )Jk (T −t )
EP [L(T ) | Ft ] = L(t )EP ek=1
| M (0) = j .
Calculating this last expectation with the help of Lemma 4.1 with ak = θ k and bk = Ψk (−iθ k ), yields to
EP [L(T ) | Ft ] = L(t )ej eQ (T −t ) 1 = L(t ), so that {L(t )} is a martingale.
The next proposition states a condition on the matrix Θ in order to ensure that the associated measure QΘ is a risk neutral measure. Similar constraints have already been derived in some one-dimensional cases, for example in the MMBM case (see e.g. Equation (2.16) of Elliott et al. [5]) and in the case of pure-jump processes with switching compensator (see e.g. Proposition 3 of Elliott and Osakwe [6]).
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Proposition 4.3. If the matrix Θ satisfies the following equations:
Ψk (−i (θ k + el )) − Ψk (−iθ k ) + rk = 0 ∀k = 1, 2, . . . , N ; ∀l = 1, 2, . . . , d, then the associated QΘ is a risk neutral measure. Proof. In order to check the QΘ -martingality of the processes e−U (t ) Sl (t )∀l = 1, 2, . . . , d, we concentrate upon the system of equations
EQΘ e−U (t )+Xl (t ) = 1 ∀l = 1, 2, . . . , d.
(4.4)
By noticing that the following equalities hold in distribution: U (t ) =
N
rk Jk (t ),
k=1
Xl (t ) =
N
el , Yk (Jk (t )) ,
k=1
and by using the change of measure (4.1), we see that (4.4) is equivalent with:
−U (t )+X (t ) l
EQΘ e
N
= EP ek=1
θ k +el ,Yk (Jk (t )) +(Ψk (−iθ k )−rk )Jk (t )
.
Using Lemma 4.1, one easily obtains
EQΘ e−U (t )+Xl (t ) = pe(Q −B)t 1,
where B is the diagonal matrix with Bkk = Ψk (−i (θ k + el )) − Ψk (−iθ k ) + rk . Since peQt 1 = 1 ∀t, the martingality condition can be reformulated as pe(Q −B)t 1 = peQt 1. This last equation is satisfied by taking B = 0, which gives the announced conditions.
Therefore, if a matrix Θ satisfies the conditions in Proposition 4.3, the pricing formulae found in the previous sections under a risk neutral measure can be easily rewritten in function of the parameters of the model under the historical probability. Indeed, let us determine the characteristic exponent of X under the measure QΘ by applying the change of measure (4.1) and Lemma 4.1:
EQΘ e
i u,X (t )
N
= EP ek=1
θ k +iu,Yk (Jk (t )) +Ψk (−iθ k )Jk (t )
= pe(Q −C )t 1,
where C is the diagonal matrix with Cjj = Ψj u − iθ j − Ψj (−iθ j ). The diagonal of that matrix C gives the function 8(.) to be used in the pricing formula when the parameters are known under P:
Proposition 4.4. Starting from the real world measure P under which the characteristic exponent of the Lévy process Yj is Ψ (.), the pricing formulae (2.9), (2.13), (3.7) and (3.17) hold with 8(u) the diagonal matrix such that
(8(u))jj = Ψj u − iθ j − Ψj −iθ j ,
(4.5)
where the matrix Θ = [θ 1 | θ 2 | . . . | θ N ] satisfies the conditions given in Proposition 4.3. Acknowledgements Griselda Deelstra acknowledges support of the ARC grant IAPAS ‘‘Interaction between Analysis, Probability and Actuarial Sciences’’ (Convention no AUWB - 2012 - 12/17 - ULB1). The research of Matthieu Simon was supported by the Belgian FNRS through a FRIA research grant. Appendix Lemma A.1. Let M be a Markov process determined by its generator Q and its initial vector p. Let X be a d-dimensional MMLP defined under a measure Q and modulated by M so that when M = j, X evolves as the d-dimensional Lévy process Yj determined
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by the characteristic exponent Φj (u):
EQ ei⟨u,Yj (t )⟩ = e−Φj (u)t ,
(A.1)
for any d-dimensional vector u. Consider the Markov-modulated drift process C : C (t ) =
t
c (s) ds
with c (t ) =
N
0
cj 1M (t )=j .
j =0
Then, for all a ∈ Cd
EQ eC (t )+⟨a,X (t )⟩ = pe(Q −A)t 1 ∀t ≥ 0,
(A.2)
where A is a diagonal matrix such that Ajj = Φj (−ia) − cj . Proof. We rewrite the left hand side of Eq. (A.2) by the following arguments, which are standard in the context of Markov additive processes (see for example Proposition 2.2 in [19]). Let us define for all states l and j the function Flj (t ) by Flj (t ) = EQ eC (t )+⟨a,X (t )⟩ 1M (t )=j | M (0) = l .
In order to determine Flj (t + h) up to o(h) for small h > 0, we condition upon the state occupied at time t and we note that the probability that there is more than one state change between t and t + h is equal to o(h) (see e.g. [19]): Flj (t + h) = EQ e(C (t )+cj h)+⟨a,X (t )⟩+⟨a,(Yj (t +h)−Yj (t ))⟩ 1M (t )=j | M (0) = l · P (M (t + h) = j | M (t ) = j)
+
EQ e(C (t )+ck h)+⟨a,X (t )⟩+⟨a,(Yk (t +h)−Yk (t ))⟩ 1M (t )=k | M (0) = l · P (M (t + h) = j | M (t ) = k) .
k̸=j
Since the increments (Yk (t + h) − Yk (t )) are independent of M and X , and have the same law as Yk (h), it follows that Flj (t + h) = Flj (t )ecj h EQ e⟨a,Yj (h)⟩
1 + Qjj h + o(h) +
Flk (t )eck h EQ e⟨a,Yk (h)⟩
Qkj h + o(h) .
k̸=j
Moreover, by applying (A.1) and the Taylor expansion e by Flj (t + h) = Flj (t ) − Flj (t ) Φj (−ia) − cj h +
(ck −Φk (−ia))h
N
= 1 − (Φk (−ia) − ck ) h + o(h), Flj (t + h) is determined
Flk (t )Qkj h + o(h),
k=1
and therefore the following equality holds: 1 h
(F (t + h) − F (t )) = F (t ) (Q − A) +
o(h) h
where F (t ) is the matrix with components Flj (t ) and A the diagonal matrix such that Ajj = Φj (−ia) − cj . When h tends to zero, this leads to the equation d dt
F (t ) = F (t ) (Q − A) ,
and therefore to the solution F (t ) = e(Q −A)t by noticing that F (0) = I. As a conclusion, the left-hand side of (A.2) becomes:
EQ eC (t )+⟨a,X (t )⟩ =
N N
EQ eC (t )+⟨a,X (t )⟩ 1M (t )=j | M (0) = l P (M (0) = l)
l =1 j =1
= pe(Q −A)t 1. References [1] [2] [3] [4] [5] [6] [7] [8]
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