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We look at the price of the European call option in a quanto market defined on a filtered probability space
when the exchange rate is being modeled by the process
where
*H _{t}* is a semimartingale. Precisely we look at an investor in a Sterling market who intends to buy a European call option in a Dollar market. The market consists of a Dollar bond, Sterling bond and and Sterling risky asset. We first of all convert the Sterling assets by using the exchange rate

*E*and later on derive an integro-differential equation that can be used to calculate the price on the option.

_{t}This paper considers the European call option in the Black-Scholes market when the exchange-rate is a semimar- tingale. Specifically, we consider a problem of a Dollar investor seeking to invest in a Sterling market. Theory of exchange rates has been widely discussed (see [_{t} is a Wiener process. This model assumes the logarithmic exchange which follows Brownian motion with drift. Using this as a benchmark model,

other models were developed, for example, a model given by the equation

where

_{t} is the exchange rate, λ_{t} is the instantaneous expected return, σ^{ }is the instantaneous volatility of the asset’s return subject to the Poisson jump not occurring, W_{t} is the Gauss- Wiener process, _{t} which makes it possible to capture the mean-reversion feature of the underlying process in addition to conditional heteroscedasticity and asymmetric jumps. The aim of our work is to provide further evidence for appropriateness of jump models. We do that by formulating an exchange rate model in terms of the general semimartingale on filtered probability space _{T}, which in our case is the European call option, in this market when the dynamics of the exchange rate is being modelled by the general semimartingale as described above.

We consider the quanto market model consisting of Dollar bond

space

tion generated by the stock price process while

Since our asset is in Sterling, we need first of all to find the Dollar equivalent of this asset. For convenience sake we let

Define

where

Then, using Ito’s formula for semimartingales [

where

(see Appendix). Let

It is important to take note that superscript

Since

where

Now, using the bilinear property of sharp bracket process [

We further note that

This is true because μt, and σW_{t} are continuous processes. It then follows that

Substituting X_{t} and Equations (3), (10), (11) into Equation (5) we have

Similarly, let Z_{t} be the dollar value of the Sterling bond given by

Let

It clearly follows that

Noting that

Note that Equation (15) follows because us is a continuous function so that

Hence

Z_{t} can now be written as

Its differential form, the dynamics of the dollar value of the Sterling bond can be written as

Which can be written as

where

And

For our analysis, we need to express the decompositions

In our case, we have

And from Equation (10) we have

Hence, process of bounded variation

where A is as defined in Equation (6) and D is the process of bounded variation for the semimartingale H_{t}. We can express these results in canonical form by using the random measure of jumps (see [

Now, if we assume that

where

Now if we substitute Equations (9) (10) and (19) into Equation (21) we obtain

where

i.e.

where

If we set

where

This means the dynamics in our market model are modeled by the equations can also be presented by the eququations

where

A question we must ask before we proceed is whether the market (27) allows arbitrage opportunities or not. In this market, an investment strategy or portfolio is a predictable process

Such that

inequality (29) ensures that the integrals

Let

be the worth the worth process. We need to know if our portfolio

or in differential form, if

Equations (31) and (32) imply that the portfolio is self-financing if changes in the value of the portfolio on an infinitesimal interval are due entirely to the changes in value of assets and not to an injection (or removal) of wealth from outside.

To show that our portfolio is self-financing, we use Lemma (5.1.3) in [

where

This means

Satisfying Equation (33). Hence

as defined in Equation (34) above is lower bounded, then

If

Similarly, the dynamic

From Equation (30)

Since the portfolio

Then

This means the differential form of of the dynamics of the discounted wealth process is

Hence the discounted wealth process will be

From the above equations,

Definition 1 A portfolio

Since the portfolio

Our stock price process as described in Equation (27) is a semimartingale. To use the martingale approach, we need to convert the price process into a martingale by finding another probability space _{t} becomes a martingale. Now we consider our price process

(see Equation (27)).

This means our price process is a local martingale iff _{t} becomes a martingale. To achieve this, we consider the following:

Suppose we have the triplet

Using the Gisanov’s Theorem for semimartingales,

where

And

where

where

definition, the

For all non-negative

where

And

We start by finding the values β and _{t}, as described by Equation (2). Using Itó’s formula for semimartingales, we have

It is easy to see that the continuous part of the semimartingale

Hint: in our calculations, we have made use of Equation (2) and the canonical decomposition of the semi- martingale hence the differential form of

From Equation (47) and using properties of conditional quadratic variation process for stochastic integrals with respect to semimartingales

Now

We can deduce from equating Equations (49) and (50), that Equation (40) can only hold if

And the Equation (42) which simplifies below

We arrive at a choice of

Hence from Equation (46)

Now under

(see [

From Equation (44),

But in our case, to really achieve the case

It is also important to take note that from the assumption we have made and lemma (2.13) in [

(see [

From Equation (38)

This means that under

Which is a martingale process.

This means that since our market has an equivalent local martingale measure

In our previous section (section 3.1), we have proved the existence of ELMM

Let

Definition 2

where

(see [

The relative entropy measures the minmal departure from a given measure

The minimal martingale measure

Let

where

Now solving (61) above gives

where

Now

And

(see Equation (10)). Hence

Hence

The process

which is the continuous and most familiar case, while if

We now come to the question fundamental of this study.

How much should the investor be willing to pay for a European call option at t = 0 in the case where Y_{t} is a semimartingale process as defined in Equation (65)?. We extend the theorem which was given in [

Theorem 1 Let

Proof. Before we proceed, we take note of the following:

This means that

The theory of pricing of the European call option (see [

where

□

But how do we evaluate the value

Now using the fact that a predictable local martingale with finite variation starting at zero is zero (theorem leads us to the equation i.e.

Suppose

From Equation (3),

where

Note that U, as it is defined in Equation (69), is an element of Borel sets which do not have a 0 element. Since

It follows that

Similarly

From Equations (39) and (51)), under measure

And

Hence

And hence (from the same theorem)

Suppose in our model, the exchange rate is not continuous and is modeled by the stochastic differential equation

where _{t} is the compensated _{t} (with intensity_{t}. We choose _{t}. We let _{t} where W_{t} is the Brownian motion as defined in Equation (3).

To find out what our X_{t} and

Using Itó’s formula for processes with jumps, we let

From which we obtain

And

Hence from Equation (77), we obtain

Clearly,

From (78) we obtain the process of the form

Equation (79) yields the sharp bracket process for

And hence under measure

Clearly

And

From Equation (80),

And

Hence from theorem (1),

And

We consider a situation where the exchange rate is

In this case

This means that

Hence Equation (80) becomes

This means

With

Equation (73) compares well with Equation (82) in the sense that (82) without the term

Gives Equation (73). This means that Equation (85) is the contribution of the jump to the price of the option. The effects of the jumps on the price of the the option can be easily observed from this Equation (82) through the role

Hence Equation (82) is reduced to Equation (73) which is a continuous case. This can be further justified from the definition of our

In Equation (82), the increase in

We also take note that expression (85) is also equal to zero if either

A case which can be handled numerically.

The method gives the general method of calculating the price of the option in the sense that it accommodates both continuous and processes with jumps. When

E. R.Offen,E. M.Lungu, (2015) Pricing a European Option in a Black-Scholes Quanto Market When Stock Price is a Semimartingale. Journal of Mathematical Finance,05,286-303. doi: 10.4236/jmf.2015.53025

Then, using Ito’s formula for semimartingales (Protter [

And in differential form, this can be expressed as