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In this paper, we present a stock model with Markov switching in the uncertainty markets, where the parameters of drift and volatility change according to the states of a Markov process. To price the option, we firstly establish a risk-neutral probability based on the uncertain measure given by Liu. Then a closed form of the European option pricing formula is obtained by applying the Laplace transforms and the inverse Laplace transforms.

The problem of option pricing is one of the most foundational problems in financial world. In 1900, Brownian motion was first introduced to finance by Bachelier [

In order to study uncertain phenomena in human systems, Liu [

In reality, some important information may greatly impact the volatility of stock returns, such as a change from “bull market” to “bear market”. Hamilton [

Inspired by the empirical phenomena of stock fluctuations related to the business cycle and the evidence from literatures that validated Markov switching model in the investigation of option pricing, in this paper, we deal with the pricing of options with Markov switching model in uncertainty markets. Specifically, we assume that stock prices are generated by a geometric uncertain process, and that the drift and volatility parameters take different values depending on the state of a Markov process. We finally provide an explicit formula for pricing when the Markov process has two states.

Uncertainty theory is a branch of axiomatic mathematics. It is an evolving system founded by Liu to deal with human uncertainty. Now the book Uncertainty theory has been updated to edition five [

Definition 1. An uncertain process

a)

b)

c) every increment

If

Definition 2. Let

provided that the limit exists almost surely and is an uncertain variable.

Definition 3. Suppose

is called an uncertain differential equation.

Definition 4. Let

where

Note that the stock price is

whose uncertainty distribution is

Definition 5. Assume a European call option has a strike price K and an expiration time s. Then the European call option price is

Definition 6. Assume a European put option has a strike price K and an expiration time s. Then the European put option price is

Consider the following uncertain stock model which incorporates different states of stock market quotation

where

Assume that

price change is not abnormal, in this state,

Suppose, further, that each piece of information flow is a random process

Then the volatility of stock price is driven by the canonical process

We aim to value the European call option based on the risk-neutral pricing theory, but it is easy to verify that the model is not accord with no-arbitrage hypothesis.

As we know, in a risk-neutral framework, the Option Put-Call Parity Relation is as follow:

where C is the European call option price, P is the European put option price, K is the same strike price and

But from the Definition 5 and Definition 6, we can learn:

Then

in which,

And according to Kai Yao (2010) [

so,

Therefore, no option put-call parity Relation was created between

Lemma 1. Consider the uncertain stock model (2.4), when

Proof: As we know, in the risk-neutral measure, the expected stock return

Following from (4.5), we have:

In the risk-neutral measure, the expected value of

Let

If:

Then:

Then we have:

So when

Thus, the risk-neutral uncertainty distribution function is verified.

Then we will present the following theorem for a two-state Markov switching model. Let

which

Theorem 1. Under the Markov switching uncertain stock model (3.1) and the risk-neutral uncertainty distribution (4.7), the arbitrage free price of European call option with expiration date T and strike price K is given by

where

where

Proof: Since the arbitrage price of the European option is the discounted expected value of

Consider

where

Next we will deduce the form of

Let

Assume that the time interval of state changing obeys the exponential distribution, as shown in (3.2), then by considering the total probability of

Then taking Laplace transforms with respect to T on both sides. By using the convolve formula, we get:

Specifically,

We obtain:

Then taking the inverse Laplace transforms on both

where

By using the delay and translation property of Laplace transform, and considering the following facts about the Laplace transform of Bessel functions:

We can take the inverse Laplace transforms on

where

and

So, when

Similarly,

Substitute

In this paper, a stock model with Markov switching in the uncertainty markets is proposed to capture the fluctuations related to the business cycle. Then the risk-neutral probability based on the uncertain measure is established for European call option pricing. Finally, an analytical formula of the option price is given by virtue of risk-neutral pricing theory. The model presented in this paper is applicable not only to two states Markov switching but also to general model with finite states Markov process.