THE ABSENCE OF ARBITRAGE ON THE COMPLETE BLACK-SCHOLES-MERTON REGIME-SWITCHING LÉVY MARKET

: The main aim of the paper was to prove that the complete Black-Scholes-Merton regime--switching Lévy market is characterized by an absence of arbitrage. In the considered model, the prices of financial assets are described by the Lévy process in which the coefficients depend on the states of the Markov chain. Such a market is incomplete; in order to complete this market, jump financial instruments and power-jump assets were added. Then, an equivalent martingale measure was indicated and the conditions were determined so that the above model is characterized by the absence of arbitrage. Arbitrage is a trade that profits by exploiting the price differences of identical or similar financial instruments in different markets or in different forms. Thus arbitrage can be understood as risk-free profit for the trader.


Introduction
The main purpose of the paper was to prove that the complete Black-Scholes-Merton regime-switching Lévy market is characterised by an absence of arbitrage. On this financial market, one conducts transactions of purchase and sale of risk-free and risky securities. The dynamics of the prices of risky financial instruments were described by a stochastic differential equation called the Black-Scholes-Merton model. The original ertain conditions. The Markov additive processes combine Levy processes with those modelled by the Markov chain, but other types of jumps also appear.
This study checked whether the market is free from arbitrage. The absence of arbitrage means that the investor cannot make a profit without taking risks. The possibility of arbitrage proves that there is a series of major errors in the valuation of instruments in the market. Such errors are very quickly noticed by arbitrators and as a result the market quickly returns to equilibrium. Thus, the market model should be one without the possibility of arbitrage (see Harrison and Pilska, 1981;Musiela and Rutkowski, 1997). The first theorem of financial mathematics is that the market is without arbitrage if and only if there is an equivalent martingale measure. Therefore, it was necessary to find an equivalent probability measure under which discounted price processes are martingales (see Jakubowski, Palczewski, Rutkowski, and Stettner, 2003), hence the author checked what conditions must be satisfied for the market to be arbitrage-free.

Market model
Let us assume that is complete filtered probability space. Let the period of time ] [0; := T T , where , be established and denote the maturity time for all types of securities. Let us consider an observable Markov chain J with a finite, canonical state space (Elliott, Aggoun, and Moore, 1994). Markov chains are stochastic processes, in which the probability of each event depends only on the previous one. They are a mathematical concept for describing random changes in an object. The states of the Markov chain can be identified with the states of the economy. The transition intensity matrix of the Markov chain is defined as follows: the elements of this matrix are constant transition intensities of the Markov chain J from state to stat within the period of time t  T, assuming that for .
The financial market under consideration consists of two basic instruments: 1. A bank account. Let us assume that the interest rate is modelled by the Markov chain J in the following way: where and is the scalar product in . Coefficient ri (i = 1, ..., N) is the value of the interest rate on the bank account when the Markov The absence of arbitrage on the complete Black-Scholes-Merton regime-switching Lévy market 75 chain is in i-th state. Thus, the dynamics of the monetary unit value process is given by the equation: And it is assumed that B(0) = 1. A bank account can be equated with a risk-free financial instrument.
2. A share. Let us start with defining the Itô-Lévy process, used to describe share price dynamics. Let be the appreciation rate of shares and be the share price volatility modelled by the Markov chain J in the following way: where , and for each i = 1,2, ..., N. The coefficients of vectors and represent the appreciation rate and volatility of the share price when the Markov chain is in the state of ei. Let us assume that for each i = 1,2, ..., N. This condition makes it possible to avoid arbitrage in the market. In addition, the following was defined: where is the vector of function value. The model in which the coefficients are modelled by the Markov chain is called a regime-switching model.

Process
is called the Itô-Lévy process if it has the following decomposition ( Oksendal and Sulem, 2004): where W stands for the standard Brownian motion independent of J, whereas is a compensated Poisson measure, independent of J and W.
With these notations, it was assumed that the dynamics of share prices S0 is given by the Markov-modelled Itô-Lévy process as follows: The market described above is called the Black-Scholes-Merton regime-switching Lévy market and it is a special case of the Markov-additive market (see Palmowski et al., 2019). The difference between these two market models is that in the Black--Scholes-Merton regime-switching Lévy model, one can explicitly determine optimal investment strategies (cf. Sulima, 2019), and in the other, one cannot.
The market consisting of a bank account and shares described above, is incomplete, meaning that some payoffs cannot be replicated by trading in marketed securities. The author completed this market by adding jump assets and power-jump assets.
Completeness is an important property of the market from a theoretical and practical point of view. In the complete market, one can value any financial instrument. The principle of martingale valuation says that the value of a financial instrument F at the initial moment equals , where B is a risk-free asset, and Q is a martingale measure. It is an expected value relative to the martingale measure Q of the discounted values of the instrument F until maturity T. On such a market, one can not only value the instruments, but also self-protect against risk. The complete market is one market where all risk factors can be perfectly prevented and investors can buy insurance contracts to protect all derivatives and themselves against the future situation in the world.
To complete the Black-Scholes-Merton regime-switching Lévy market, the study defined jump assets and power-jump assets.
Let us define the point process j. Let Tn (n = 1,2, …) denote the jump epochs of the Markov chain J, where . Then, the process is called a marked point process (cf. Zhang, Elliott, Siu, and Guo, 2012, p. 290 and ij  comes from matrix Λ defined before.
The absence of arbitrage on the complete Black-Scholes-Merton regime-switching Lévy market 77 Then, the process is the martingale and is called j-th jump martingale (cf. Elliott, 1976;Palmowski, Stettner, and Sulima, 2018).
Next, the author defined jump price processes of financial instruments, the dynamics of which is described by jump martingales. Let us assume that these coefficients are modulated by the Markov chain in the following way:

Coefficients
and represent the appreciation rate and volatility of the financial instrument when the economy is in i-th state of the Markov chain, also assuming that . It is required to avoid arbitrage in the market.
Let Sj be the price process of j-th jump asset, the dynamics of which is described as follows: One should note that when . Therefore, has no influence on the value on the right side (4). Otherwise, regardless of the value which has, it does not affect the dynamics of the price process . As a result, without the loss of generality, one can assume that , so is a vector with j-th coordinate that equals 0. In addition, the author extend the market with securities that were first introduced by Corcuera et al. (2005), namely the so-called power-jump assets. First, let us introduce the process defined as: . For k = 1 let us assume . The process is called the k-th power-jump process. One should note that is the Lévy-Itô process and these processes jump at the same time points as the original Lévy-Itô process, but the jump sizes are k-th powers of the jump size of the original Lévy-Itô process. For , from Protter (2005), it follows that: , and therefore processes are martingales, called Teugels martingales of order k (see Schoutens and Teugels, 1998). Now, the author completed the market by adding a set of k-th power-jump assets. The price process coefficients were determined in a similar way as for Sj, namely, The new prices processes were defined in the following way (for k ≥ 2): The interpretation of the additional financial instruments discussed above is described by Corcuera et al. (2005). The second degree power-jump process is identified with contracts implementing variance (see Barndorff-Nielsen andShephard, 2003, 2004). These contracts are the subject of regular transactions and they are quoted on OTC markets. The third degree power-jump process is related to the asymmetry of the underlying instrument, whereas the fourth degreeto kurtosis. What is more, assets of a power-jump of a degree greater than four are identified with insurance in the case of extreme events.
The complete Black-Scholes-Merton regime-switching Lévy market has the following form (the proof of completeness can be found in Sulima (2018) The absence of arbitrage on the complete Black-Scholes-Merton regime-switching Lévy market 79

Equivalent martingale measure and absence of arbitrage
A well-defined financial market model should meet the above assumptions, for example, the model would be unacceptable from an economic point of view if the investor were able to earn additional profit without the risk. Such a situation may take place when there is a possibility of arbitrage in the model. This study proved that arbitrage is not possible on the market under consideration.
The basic theorem about the valuation of financial assets is that the existence of an equivalent martingale measure is equivalent to the absence of arbitrage on the market: Theorem 1. (Björk and Näslund, 1998, Prop. 6.1). If there exists a martingale measure Q equivalent to P then the market is arbitrage-free. Now, let us define this type of measure. Theorem 2. (Boel and Kohlmann, 1980, p. 515). Let and be integrable processes for all j = 1,2,…, N. Therefore is a local martingale.
Let Q be a probability measure on (Ω, F, Ft, P) for t  T in such that Then , defined by the formula (6), is the density process of the new martingale measure Q. From the generalised version of the Girsanov theorem for the jumpdiffusion processes one has the following theorem: Theorem 3. (Boel and Kohlmann, 1980). On defined ), , is the standard Brownian motion with respect to Q. Moreover, for j = 1,2, ..., N, are martingales with respect to Q.
Let us note that for and do not change their form in relation to the new probability measure Q.
The complete Black-Scholes-Merton regime-switching Lévy market in relation to the new measure Q is described by the following equations: for j = 1, ... N and k = 2,3, ... . To find the conditions under which the probability measure Q is the martingale measure, thus it was necessary to investigate whether the discounted price processes are martingales. Using stochastic integration by parts of equations (7), discounted price processes take the following form: where and . Let us note that are martingales for k = 2,3, ... . In order for to be martingales, the following equations must be satisfied: if , the martingale condition will never be satisfied. Therefore, one cannot find an equivalent martingale measure so that the discounted price processes of all financial instruments in an extended market are martingales in relation to measure Q. This means that there are arbitrage possibilities in an extended market. In order to eliminate them, one must assume that for all j = 0,1, …, N. Thus, processes , j = 0,1, …, N. from (8), have the following form: ( ) = 0 otherwise.
Note that processes , (j = 0,1, …, N) are bounded therefore the density process L (defined in (6)) is martingale and discounted price processes (7) are martingales (see Palmowski et al. (2018, Remark 2)) . The above analysis leads to the following theorem.
Theorem 4. Let us assume that for all j = 0,1, …, N and ( j = 0,1, …, N) are expressed by equations (9). Then the discounted price processes in the complete market (7) are martingales with respect to the new martingale measure Q and the complete Black-Scholes-Merton regime-switching Lévy market is arbitrage--free. On a finite market, the absence of arbitrage is characterized by the existence of an equivalent martingale measure, while completeness holds if and only if the equivalent martingale measure is unique. Note that the complete Black-Scholes-Merton regime--switching Lévy market has infinitely many financial instruments. On an infinite market the property of uniqueness does not hold, i.e., if the market is complete one can have more than one equivalent martingale measure. Artzner and Heath (1995) constructed a market with many countable securities for which there are two equivalent martingale measures under which the market is complete.
On the complete arbitrage-free Black-Scholes-Merton regime-switching Lévy market the study determined the optimal investment strategy (see Sulima, 2019). The optimal investment strategy is one that maximizes the expected utility of the terminal wealth. Denote by π(t) = ( 0 ( ), 1 ( ), . . . , ( ), ( ) ( ), . . . ) a portfolio strategy, where 0 is the proportion of wealth invested in stock, , ( = 1,2, … , ) is the proportion of wealth invested in the j-th jump assets, and ( ) , ≥ 1 is the proportion of wealth invested in the k-th power-jump assets. Then the optimal investment strategies for the log utility function are as follows (see Sulima, 2019 These investment strategies are unique for each utility function.

Conclusions
The absence of arbitrage is an important property for financial markets, which means that an investor cannot make a profit without taking a risk. This research addressed determining the conditions so that the complete Black-Scholes-Merton regime--switching Lévy market model is characterised by the absence of arbitrage. An equally important characteristic of the financial market regarding the lack of arbitrage is completeness, because in a complete market one can value all financial instruments. In a complete market without arbitrage, the arbitral price (see Jakubowski, 2006) of a financial instrument is calculated as the expected value of discounted