Pricing and Hedging in Stochastic Volatility Regime Switching Models ()
1. Introduction
In this paper, we consider general regime switching stochastic volatility models where both the asset and the volatility dynamics depend on the values of a Markov jump process. We are interested in finding formulae to solve the problem of pricing and hedging contingent claims in this framework. However, due to the stochastic volatility and the Markov regime switching, the market is incomplete and thus perfect pricing and hedging are not possible. Hence, to hedge derivatives, we adopt firstly the local-risk minimization approach studied by Föllmer and Schweizer in [1]. This approach consists of controlling the hedging errors at each instant
by minimizing the conditional variances of the instantaneous cost increments. Health et al. in [2], provided comparative theoretical and numerical results on risks, values and hedging strategies for this approach versus the meanvariance (see [3] or [4] for more details on this approach), in the specific case of stochastic volatility models. We propose in this paper to apply the local risk minimization approach to a more global class of stochastic volatility models since we will assume in the sequel that all parameters of the model depend on the values of a Markov jump process. Hence, we will work on a global class of regime switching stochastic volatility models. We will obtain formulae to give the price of contingent claims by solving a system of partial differential equations. We will also obtain the optimal strategy which solves the local risk minimization hedging problem.
There has been considerable interest in applications of regime switching models driven by a Markov process to various financial problems. Elliott et al. in [5] provided an overview of hidden Markov chain models. Indeed, the use of Markov switching in diffusions allows us to have different levels of drift or volatility during time. Moreover these regime switching activities are a better fit for economic times series data than non regime switching models and also allow us to better capture some market features or economics behaviors such as recession and financial crisis periods (see for example Goutte and Zou in [6]). Thus, Di Masi et al. in [7] considered the problem of hedging an European call option for a (nonstochastic volatility) diffusion models where the drift and volatility are functions of a Markov jump process.
Furthermore, we are also interested in pricing volatility derivatives, such as variance and volatility swaps. Broadie et al. in [8] studied the pricing and hedging of variance swaps and other volatility derivatives in the classical Heston stochastic volatility model introduced by Heston in [9]. Elliott et al. in [10] then developed a model for pricing the same class of derivatives but under a Markov-modulated version of this stochastic volatility model. In their paper, they only considered the case where not all the parameters of the model depend on the state of the Markov process. Moreover, they studied only the Heston stochastic volatility model. Hence, based on these two preceding works, we will extend this methodology to our global class of regime switching stochastic volatility models. Thus, we will obtain formulae to price options on the stochastic volatility process such as variance and volatility swaps.
This paper is arranged as follows: Section 1 gives the notion of the regime switching framework and presents our regime switching stochastic volatility models. Section 2 solves the problem of pricing and hedging using the local risk minimization approach. Section 3 then gives the formulas to price options on the volatility process.
2. The Stochastic Regime Switching Volatility Model
Let
be a filtered probability space with filtration
satisfying the usual conditions for some fixed but arbitrary time horizon
. We consider a general stochastic volatility model defined by the stochastic differential equations:
(1.1)
where
and
are two correlated Brownian motions. In fact, the process
represents the discounted price process of a tradable asset price process ![](https://www.scirp.org/html/6-1490111\4da2e1f3-3e6b-4e2b-bb23-2a8a868bd764.jpg)
divided by a Bond price process
. The process
represents the savings account in a riskless asset.
is a real stochastic process which is
adapted and
is a Markov jump process on finite state space
. It can be viewed as an observable exogenous quantity. We assume that the time invariant matrix
denotes the infinitesimal generator
of
, where
is an infinitesimal intensity of
. This generator is defined as
, for all
and
for all
.
Then, the semi-martingale decomposition for
is given by
![](https://www.scirp.org/html/6-1490111\a091ab79-91e1-4391-aa08-b0bce9e6e814.jpg)
where
is an
-valued martingale with respect to
, which is the natural filtration generated by the Markov process
under
.
Hence, in our model, there are three sources of randomness:
,
and
. We will denote by
the filtration generated by the two Brownian motions. Finally, we will denote by
the global filtration
.
Let
be the sequential jump times of
(i.e.
) and
a jump measure of
, i.e. the integer-valued random measure given by
(1.2)
Then the
-compensator of
is given by
(1.3)
Assumption 1.1
1) We assume all the hypothesis to ensure the existences and the regularities of the solutions of our model (1.1) (see for example [11] for more details). Hence, the system of stochastic differential Equations (1.1) admits a unique strong continuous solution for the vector process
with a strictly positive price process
and a volatility process
.
2) We assume moreover that the Markov process
is independent to both processes
and
.
Remark 1.1 This independence implies that the Markov process
is an exogenous factor of the market information. Thus, it can be viewed as an exogenous factor such as an economic impact factor. An economic interpretation of this is that this Markov process can represent a credit rating of a firm
. Indeed, assume that our stochastic volatility model describes the price of a commodity produced by the firm
, then the Markov process
can represent the credit rating of this firm given by an exogenous rating company as “Standard and Poors”. Thus, it is natural to think that the dynamic of the commodity’s price, produced by the firm
, depends on the value of this notation
.
To exclude arbitrage opportunities, we assume that the process S admits an equivalent local martingale measure (ELMM)
. In the sense that it is a probability measure with the same null sets as
and such that the process S is a local martingale under
. We will denote, in the sequel, by
the set of all ELMMs
.
Remark 1.2
1) We can rewrite the Brownian motion
as
, where the process
is another Brownian motion such that processes
and
are now independent.
2) The condition that S should be a
-local martingale fixes the effect of the Girsanov transformation of
but allows us for different transformations on the independent Brownian motion
defined in point 1. Consequently, if the correlation factor
satisfies
, then the set
contains more than one element and so the financial market is then incomplete. Moreover, there is an other source of incompleteness of the market which is the dependence of processes
and
with respect to the Markov process
.
We give now some examples of classical stochastic volatility models which are contained in our model (1.1).
Example 1.1 Hull-White:
![](https://www.scirp.org/html/6-1490111\f7b70672-feb3-46cd-8663-7392454acce1.jpg)
![](https://www.scirp.org/html/6-1490111\16cf1325-3345-4bed-b14c-daaa5c95b9c0.jpg)
![](https://www.scirp.org/html/6-1490111\ddde691d-08d1-4668-b456-44b9ecd9aa3c.jpg)
Stein-Stein:
![](https://www.scirp.org/html/6-1490111\117c0207-a4aa-492e-b6fb-beb6449e78e3.jpg)
![](https://www.scirp.org/html/6-1490111\94f30c91-0401-460a-85d2-4bbd20952669.jpg)
![](https://www.scirp.org/html/6-1490111\dfa69f5b-9489-4630-bb37-f015f3d9b643.jpg)
Heston:
![](https://www.scirp.org/html/6-1490111\2a066524-3f77-40b4-9e9d-8ff81ac07327.jpg)
![](https://www.scirp.org/html/6-1490111\beb9fc3a-b493-4c05-af6a-352f75f78d70.jpg)
![](https://www.scirp.org/html/6-1490111\be5845bc-c275-4bdd-9039-4d50fac1d985.jpg)
![](https://www.scirp.org/html/6-1490111\9030940a-bf68-461d-8914-4ef29a20fc4f.jpg)
3. Pricing and Hedging via Local Risk Minimization Approach
We are interested, in this section, in the hedging of an European contingent claims with an
-measurable squared integrable random variable
based on the dynamics given by (1.1). As an example of this payoff, we can take an European call option:
with maturity T and strike K > 0.
We consider here the local risk-minimization approach to hedge in this incomplete market. We recall some definitions of this approach.
Definition 2.1 An hedging strategy is a pair
such that
is a predictable process which satisfies
(2.4)
and
is an adapted process such that
(2.5)
We will denote by
the set of strategies which satisfy (2.4) and (2.5).
The hedging strategy
defines a portfolio where
denotes the number of shares of the risky asset
held by the investor at time
and
denotes the amount invested at time
in the bond.
Definition 2.2 Given a hedging strategy
, we call the Value process
of this corresponding portfolio the right continuous process given by
(2.6)
Definition 2.3 Given a hedging strategy
, we call the Cost process
of this corresponding portfolio the process given by
(2.7)
We can see that the quantity
represents the hedging gains or losses up to time
following the hedging strategy
. A hedging strategy
is called self-financing if its cost process is
-a.s. constant over the time
and mean self-financing if
is a
-martingale. If
is square integrable, then the risk process of
is defined by
(2.8)
Remark 2.3 Since the contingent claim
is
- measurable and
is adapted, there always exists a hedging strategy such that
. Indeed, we can take
and
for all
.
3.1. Local Risk Minimization Approach
We only consider hedging strategies which replicate contingent claim
at time
. This means that we only allow hedging strategies
such that
(2.9)
Thus, the hedging problem is so to find the strategy
which minimizes at time
the quadratic risk:
(2.10)
The idea is so to control the hedging errors at each instant
by minimizing the conditional variances of the instantaneous cost increments sequentially over time.
Remark 2.4 An alternative approach to hedge in incomplete market is the mean-variance approach (see [3]). In fact, in this approach, the aim is to minimize the global risk over the entire time
. Hence, it is a different approach than the local risk minimization which focuses on the minimization of the second moments of the infinitesimal cost increments (8).
Therefore, the study of this minimization problem in a general semimartingale case is due to Schweizer [3] and it requires more assumptions on the asset dynamic
. We assume firstly that
can be decomposed as
![](https://www.scirp.org/html/6-1490111\1e65f466-0991-437a-8d82-76b67bfb19b5.jpg)
where
is a real valued locally squared integrable local
-martingale null at zero and A is a real valued adapted continuous process of finite variation also null at zero.
We recall now the Definition of the Structure Condition (SC). We say that the process
satisfies the (SC) if there exists a predictable process
such that the process
is absolutely continuous with respect to
(i.e. the oblique bracket). In the sense that
![](https://www.scirp.org/html/6-1490111\5084a7b6-b9e0-44ee-b43e-d21a77b4c14c.jpg)
and such that the so called mean variance tradeoff process (MVT)
satisfies
![](https://www.scirp.org/html/6-1490111\9b777cee-17eb-4734-b3da-266df8a75f06.jpg)
Lemma 2.1 Since
, we have that if the process
is continuous then (SC) is satisfied.
Proof. See Theorem 1 of Schweizer [4].
Proposition 2.24 of Föllmer and Schweizer in [1] shows that finding a locally risk minimizing strategy for a given contingent claim
is equivalent to finding a decomposition of
of the form:
(2.11)
where
is a constant,
is a predictable process satisfying Condition (2.4) and
is a square integrable
-martingale null at 0 and strongly orthogonal to
(i.e.
is a P-martingale). The representation (2.11) is usually referred to as the Föllmer-Schweizer (FS) decomposition of the random variable
. Once we have (2.11), then the desired hedging strategy
, which is locally risk minimizing, is then given, for all
, by
(2.12)
and
(2.13)
where
(2.14)
with
(2.15)
In view of these results, finding the Föllmer-Schweizer decomposition (2.11) of a given contingent claim
is important because it allows us to obtain the locally risk minimizing strategy. Monat and Sticker in [12] and Pham et al. in [13] give sufficient conditions to prove the existence of this decomposition. We therefore explain how one can often obtain this decomposition by switching to a suitably chosen martingale measure for S. Indeed, as it is shown in [1] and [4], there exists a measure
, which is the so called minimal equivalent local martingale measure (minimal ELMM), such that
(2.16)
where
denotes the conditional expectation under
.
Remark 2.5 If there exists a locally risk minimizing strategy
, then we can use the expression of
appearing in (2.16) as a price of the contingent claim
at time
.
In the case where the process
is continuous (which is our case), Theorem 1 of [1] allows us to construct uniquely
. Indeed, we have the following result:
Proposition 2.1
exists if and only if for all ![](https://www.scirp.org/html/6-1490111\25d756f8-5d1c-4781-b107-5f2452e6a04b.jpg)
(2.17)
is a square integrable martingale under
.
Moreover,
defines a probability measure
equivalent to
which is in
since one easily verifies that
is a local
-martingale.
3.2. Markovian Regime Switching Case
Let
,
and
given by the model (1.1), then the local risk minimizing hedging strategy can be obtained in two steps:
1) Determine
and deduce the dynamic of
under
.
2) Find the Galtchouk-Kunita-Watanabe decomposition of
with respect to
under
.
Then, the optimal local risk minimizing hedging strategy is given by (2.12) and (2.13).
3.2.1. Finding ![](https://www.scirp.org/html/6-1490111\ac90eb6e-8495-442c-bcb5-cd97d8c9df40.jpg)
According to the previous subsection, the density process of the minimal ELMM
with respect to
is given by the Equation (2.17). We can firstly remark that
![](https://www.scirp.org/html/6-1490111\6f49bd3d-d9a1-4a45-8cc4-16173186b475.jpg)
Since
is continuous, we have first of all to determine the canonical decomposition
![](https://www.scirp.org/html/6-1490111\501e869e-40c4-42b3-82f8-a858995d120b.jpg)
of the asset process
under
.
Proposition 2.2 Assume that the regime stochastic volatility model follows (1.1) then we have, for all
, that
![](https://www.scirp.org/html/6-1490111\408e397d-08b1-4fe3-9fef-6d092917a7cb.jpg)
![](https://www.scirp.org/html/6-1490111\356a5337-92b8-4ac4-9ad1-82ba6292e916.jpg)
therefore we obtain
![](https://www.scirp.org/html/6-1490111\7dfbd083-5ad1-4bc1-a08e-7091d5804240.jpg)
Proof. It comes immediately from the definition of the dynamic of our model (1.1).
We are now able to determine the dynamic of our model under
.
Proposition 2.3 Assume that
is a true
-martingale, then the dynamic of the model (1.1) under
is given for all
by
(2.18)
![](https://www.scirp.org/html/6-1490111\174c883a-5383-4759-8bfd-fd6ba76ded4f.jpg)
with
(2.19)
Proof. Since
is a
-martingale, Girsanov’s theorem implies that
and
are independent
-Brownian motions. Hence
![](https://www.scirp.org/html/6-1490111\27141b5b-4401-4b22-b4df-fed993f1158d.jpg)
and
![](https://www.scirp.org/html/6-1490111\e8f87fd8-8f5d-4a69-88c3-228357fc6ea6.jpg)
3.2.2. Decomposition of the Contingent Claim
with Respect to
under ![](https://www.scirp.org/html/6-1490111\a354f888-be6d-4f51-8ba4-e8775013e8c6.jpg)
Let
be a contingent claim of the form
, then finding the Galtchouk-KunitaWatanabe decomposition of
under
reduces to solve a system of partial differential equations if one exploits the Markovian structure. Indeed, using the Markov property, we can rewrite (2.16):
![](https://www.scirp.org/html/6-1490111\ceb88767-3b78-4575-8150-7ee443c9d133.jpg)
for some function
defined on
.
Proposition 2.4 For all
, if
, then
is the solution to the system of partial differential equations given by
(2.20)
with terminal condition for all
given by
and where
,
,
,
and
.
Proof. Since
![](https://www.scirp.org/html/6-1490111\3e27e5de-f334-452c-a103-488797c019ac.jpg)
(2.21)
and notice that for any function
on
, which is right continuous and with left limits in
, we have
![](https://www.scirp.org/html/6-1490111\0a07f545-9d97-47e7-bd87-cbde54f842dc.jpg)
Hence, replacing the last equality in (2.21) gives
![](https://www.scirp.org/html/6-1490111\87703f47-3ecf-45ac-aa02-e3db814e5bc8.jpg)
The function
is a
-martingale so all boundary terms are null. Hence we obtain that
need to satisfy
![](https://www.scirp.org/html/6-1490111\efb39297-66a3-49ca-8d05-26ef3f87d7bf.jpg)
with terminal condition
. Moreover,
![](https://www.scirp.org/html/6-1490111\c7d8a74a-d4b3-49ce-b4e4-1e267d476d46.jpg)
where
means that at time
the Markov process is in state
(i.e.
). Hence with
, we obtain the expected result.□
Example 2.2 (Pricing European call options on the underlying process S) The value of an European call option on the stock price
with maturity T and strike K is given by
. Hence we can apply Proposition 2.4 with
![](https://www.scirp.org/html/6-1490111\840992ba-3a3f-4e83-85b0-4e044546b741.jpg)
where the terminal conditions for all
are given by
.
According to (2.12) and (2.15), we are now able to find the decomposition of
with respect to
under
and so the locally risk minimizing
-admissible strategy
.
Theorem 2.1 For all
, we have that the locally risk-minimizing hedging strategy of
,
is given by
(2.22)
(2.23)
where
![](https://www.scirp.org/html/6-1490111\2daed4fb-a682-4d71-b9b5-528fe7a2ea0a.jpg)
and
![](https://www.scirp.org/html/6-1490111\e816a275-7fd6-40e3-bb42-7c888416ed43.jpg)
Remark 2.6 Moreover, taking the problem at time
, we get
![](https://www.scirp.org/html/6-1490111\9d7b69af-dec0-498e-88f9-a8f24c1dc845.jpg)
which is the so-called Föllmer-Schweizer decomposition of the random variable
.
Proof. Let
, apply Ito’s formula to
, then by (2.21), we obtain:
![](https://www.scirp.org/html/6-1490111\dcbad303-f1ea-44a8-b2b9-ce39be92300e.jpg)
We apply now the result of Proposition 2.4 to obtain
![](https://www.scirp.org/html/6-1490111\0d91d291-ea8c-45d6-83d0-585040885909.jpg)
Combining this result with
gives the expected result.□
We can also obtain a formulation of the conditional expected squared cost on the time interval
for the locally risk-minimizing strategy
.
Proposition 2.5 We have, for all
, that the conditional expected squared cost on the interval
for the locally risk-minimizing strategy
is given by
![](https://www.scirp.org/html/6-1490111\cfb9f6c2-ed77-4a11-8fa8-f360e2b22d71.jpg)
Proof. Applying the result of Theorem 2.1 in (2.8) we obtain
![](https://www.scirp.org/html/6-1490111\293a9ca8-3a47-4677-8582-a38d0a214960.jpg)
We can also simplify the second expectation
![](https://www.scirp.org/html/6-1490111\0f284eee-54f5-4e8c-b822-7e5ed8c7f0f1.jpg)
To apply all the results about local risk minimizing hedging strategy , it remains to prove that
is a true P-martingale and square integrable under P. A wellknown sufficient condition for both is the boundedness of the mean variance tradeoff process
, stated in Proposition 0.2, uniformly in
and
(see [12-14]).
Proposition 2.6 If the mean variance tradeoff process
, defined in Proposition 0.2, is uniformly bounded in
and
then we have that:
1)
is a true
-martingale and square integrable under
.
2) H admits a Föllmer-Schweizer decomposition given by (2.11).
3)
, defined in (2.12) and (2.13), is the local risk minimizing hedging strategy.
Example 2.3 (Heston model) We can take for the model (1.1) a Heston model case. Hence, by Example 1.1, we take
![](https://www.scirp.org/html/6-1490111\aade245d-f111-4cc1-be5d-f55470977c98.jpg)
![](https://www.scirp.org/html/6-1490111\a222f92f-7588-4b65-9ed7-eef938834925.jpg)
![](https://www.scirp.org/html/6-1490111\3207ccac-5d64-443f-8fb2-c23eb4b17bfd.jpg)
![](https://www.scirp.org/html/6-1490111\ff24e8d5-e848-40dd-b4fd-04e5a1256d85.jpg)
The constants
,
and
are all nonnegative for all
. And we assume for the existence and positivity of the solution
that for all
,
.
The model is then given by
![](https://www.scirp.org/html/6-1490111\7ee56018-5cf3-4f8a-addf-234500ec5046.jpg)
![](https://www.scirp.org/html/6-1490111\bc090a82-0fb6-41bc-8ade-eca5e3bd5557.jpg)
and the corresponding mean variance tradeoff process is then given by
![](https://www.scirp.org/html/6-1490111\bd70f5eb-b9e6-4f6e-8c8a-fdfa9659d5bd.jpg)
Hence the MVT process K is deterministic so bounded uniformly in
and
. This implies that
is a P-martingale and so that we can apply all the results mentioned before.
4. Pricing Option on the Volatility: ![](https://www.scirp.org/html/6-1490111\5a70865f-00ea-4492-98e8-ad05add952cf.jpg)
We are now interested in establishing some formulae to price options based on the stochastic volatility process
.
4.1. Variance Swap
A variance swap is a forward contract on the annualized variance, which is the square of the realized annual volatility. Thus, let
denote the realized annual stock variance over the life of the contract. Then it is given by:
(3.24)
Let
and
denote the delivery price for variance and the notional amount of the swap in dollars per annualized volatility point squared. Then, the payoff
of the variance swap at the maturity time
is given by
. Intuitively, the buyer will receive
dollars for each point by which the realized annual variance
has exceeded the variance delivery price
. The results provided, in the sequel, are an extension of the results obtained in [10]. Indeed, firstly we study more general class of stochastic volatility models and secondly we apply our results, in the particular case of Heston model, but where not only the long-term volatility level depends on the regime but also the speed of mean reversion on the volatility of the volatility.
Hence, we start by considering the evaluation of the conditional price of a derivative H given the information about the sample path of the Markov process from time 0 to time T (i.e.
). This means that we assume to know all the historical path of the Markov process
. Assume, also, that we are under the minimal ELMM
. Thus, we recall that in this case our regime switching model is given by:
![](https://www.scirp.org/html/6-1490111\10493339-ed75-403f-b00e-a4608281d830.jpg)
![](https://www.scirp.org/html/6-1490111\325435ca-a152-475a-a67f-e2149f2af470.jpg)
with
. In particular, given
, the conditional price of the variance swap
is given by
(3.25)
Hence, if we denote as previous
, we obtain by Itô formula that for all
,
(3.26)
Thus, given
, we get
(3.27)
So
(3.28)
Assumption 3.2 Assume that we know the solution of Equation (3.28) which we will denote by
, for all
.
Proposition 3.7 Under Assumption 3.2, we have, for all
, that the conditional variance swap price P(X) is given by
(3.29)
Example 3.4 (Heston Model) Assume that we are in the Heston model case. Hence as mentioned in Example 1.1 we take
![](https://www.scirp.org/html/6-1490111\722c1732-945f-4fee-909c-f058c732373a.jpg)
and
![](https://www.scirp.org/html/6-1490111\60be8c81-3a45-4ab3-9828-a78a417e1f3c.jpg)
Then the dynamic of
is given by
.
Moreover, (3.28) becomes
![](https://www.scirp.org/html/6-1490111\fe1c55a1-5a6a-443c-b169-7920500049b3.jpg)
Let
, then we have to solve the differential equation
. The solution of this differential equation is given for all
by
![](https://www.scirp.org/html/6-1490111\84fc5c1e-3c3b-42e5-b908-98070d8f7f3c.jpg)
Thus, we get
(3.30)
We can now obtain the conditional variance swap price by applying Proposition 3.7:
![](https://www.scirp.org/html/6-1490111\6b875036-52a4-461f-9f6f-094dc488a1ca.jpg)
We can also obtain the value of the conditional variance given the full history of
.
Lemma 3.2 For all
, the conditional variance of
is given by
![](https://www.scirp.org/html/6-1490111\9a33069a-946c-4488-a2a4-9ff663868cfa.jpg)
Proof. By Ito’s lemma we get
![](https://www.scirp.org/html/6-1490111\53c8c20a-b78f-4482-b382-d535e8d082e3.jpg)
Hence, given
, we find that
(3.31)
Using (3.27) and the definition of the variance give the result.
Example 3.5 (Heston Model) We continue the study of the Heston model case (see Example 3.4). Thus Equation (3.31) gives
![](https://www.scirp.org/html/6-1490111\55677b21-5a98-499c-b1df-ba1b04a6e8e7.jpg)
Given (3.30), we know that
![](https://www.scirp.org/html/6-1490111\320e6f41-5845-409a-852a-bc9be812909b.jpg)
which is a function of time
and the Markov process
. Let
, then we have to solve the differential equation given by
![](https://www.scirp.org/html/6-1490111\0473f0b9-8760-4a4b-8501-cabc28345fcb.jpg)
The solution of this differential equation is given for all
by
![](https://www.scirp.org/html/6-1490111\fae605a9-c895-4050-92f1-5d10c3304254.jpg)
where
![](https://www.scirp.org/html/6-1490111\4fecb472-3c5a-42fb-871a-ae460e0415eb.jpg)
![](https://www.scirp.org/html/6-1490111\5ce976ef-e257-4702-9048-552ef99e7909.jpg)
We finally obtain in the Heston model case that
![](https://www.scirp.org/html/6-1490111\2f4732f0-6b6f-41c1-866e-3becc24e3378.jpg)
and that the conditional variance is equal to
![](https://www.scirp.org/html/6-1490111\68875440-226a-45da-9b64-c97e745a64e5.jpg)
4.2. Pricing Volatility Swaps
In this section, we follow the same methodology studied by Broadie and Jain in [8] where there is no regime switching component. We recall that the realized annual stock variance over the life of the contract is given by (3.24) and depends on the values of the Markov process
. Denote by
the accumulated variance between time 0 to
. We recall that the process
is the solution of the stochastic differential equation given by
![](https://www.scirp.org/html/6-1490111\539236fb-1c6f-432d-b297-5c5408804649.jpg)
Hence
is the solution of the stochastic differential equation given by
. Let define by
the expectation at time
with respect to ![](https://www.scirp.org/html/6-1490111\fb02448b-98ee-416c-a446-75b08f5d4864.jpg)
(3.32)
Hence
depends on the variance process
of the underlying asset and on the Markov process
. We call by fair conditional variance strike price the quantity
which is defined such that Equation 3.25 vanishes:
![](https://www.scirp.org/html/6-1490111\925f2d9b-d131-43d5-8c3c-0d1e3239d4c0.jpg)
Then, we have that
and for time
, we obtain that
. We define now the forward price process
as
(3.33)
Proposition 3.8 The forward price process
can be expressed as a function
and it is the solution of the system of stochastic differential equations given by
(3.34)
with boundary condition given by
.
Proof. Rewrite
as
.
Applying Itô Lemma to
, using (3.26) and the fact that the forward price measure
is a martingale give the expected result.
5. Conclusions
In this paper, we studied the problem of pricing and hedging options based on an asset which is modeled by a regime switching stochastic volatility model. We presented firstly this class of regime switching stochastic volatility models. We shown that this class of regime switching models encompasses a large panel of classical financial models. We also explained the interest in applications of this kind of models. Secondly, we used the local risk minimization approach to solve the problem of pricing and hedging contingent claims in this incomplete market. Thus, we obtained pricing and hedging formulae and also the formula of the optimal hedging strategy. Finally, we fund formulae to price volatility and variance swap options by solving a system of stochastic differential equations.
Some possible future directions in this field of research are the following:
- Develop a method to estimate all the parameters of this class of regime switching models, including the hidden Markov chain (its transition matrix);
- Apply these pricing and hedging formulae to economic and financial data. Indeed, regarding the existing literature, a good candidate could be the electricity spot price;
- Construct a method to evaluate quickly the solutions of the system of stochastic differential Equations (2.20) and (3.34). A system of spatial and time discretization grids could be considered and investigated.