A Martingale option pricing

Option pricing in exponential Lévy models with transaction costs

Abstract

We present an approach for pricing a European call option in presence of proportional transaction costs, when the stock price follows a general exponential Lévy process. The model is a generalization of the celebrated work of Davis, Panas and Zariphopoulou (1993), where the value of the option is defined as the utility indifference price. This approach requires the solution of two stochastic singular control problems in finite time, satisfying the same Hamilton-Jacobi-Bellman equation, with different terminal conditions. We solve numerically the continuous time optimization problem using the Markov chain approximation method and consider several Lévy processes such as diffusion, Merton and Variance Gamma processes to model the underlying stock log-price. This model takes into account the possibility of portfolio bankruptcy. We show numerical results for the simpler case of a big investor, whose probability of default can be ignored. Option prices are obtained for both the writer and the buyer.

Keywords: option pricing, transaction costs, Lévy processes, indifference price, singular stochastic control, variational inequality, Markov chain approximation.

1 Introduction

The problem of pricing a European call option was first solved mathematically in the paper of Black and Scholes (1973). Even if it is quite evident that this model is too simplistic to represent the real features of the market, it is still nowadays one of the most used model to price and hedge options. The reason for its success is that it gives a closed form solution for the option price, and that the hedging strategy is easily implementable. The Black-Scholes model considers a complete market, i.e. a market where it is possible to create a portfolio containing cash and shares of the underlying stocks, such that following a particular trading strategy it is always possible to replicate the payoff of the option. In this framework, this particular portfolio is called replicating portfolio and the trading strategy to hedge the option is called delta hedge. However, this model does not consider many features that characterize the real market.

In the Black-Scholes model the stock price follows a geometric Brownian motion. This is equivalent to assume that the log-returns are normally distributed. However, a rigorous statistical analysis of financial data reveals that the normality assumption is not a very good approximation of reality (see Cont (2001)). Indeed, it is easy to see that empirical log-return distributions have more mass around the origin and along the tails (heavy tails). This means that normal distribution underestimates the probability of large log-returns, and considers them just as rare events. In the real market instead, log-returns manifest frequently high peaks, that come more and more evident when looking at short time scales. The log-returns peaks correspond to sudden large changes in the price, which are called jumps.

There is a huge literature of option pricing models that consider an underlying process with a discontinuous path. Most of these models consider the log-prices dynamics following a Lévy process. These are stochastic processes with independent and stationary increments, that satisfy the additional property of stochastic continuity. Good references on the theory of Lévy processes are the books of Sato (1999) and Applebaum (2009). Financial applications are discussed in the book of Cont and Tankov (2003).

A second issue of the Black-Scholes model is that it does not consider the presence of market frictions such as bid/ask spread, transaction fees and budget constraints. The securities in the market are traded with a bid-ask spread, and this means that there are two prices for the same security. But the Black-Scholes formula just gives one price. Moreover, the replicating portfolio cannot be perfectly implemented, since the delta-hedging strategy involves continuous time trading. This is impractical because the presence of transaction costs makes it infinitely costly. Another kind of market friction that needs to be considered are the budget constraints. A bound in the budget or a restriction in the possibility of selling short, clearly restricts the set of possible trading strategies.

Many authors attempted to include the presence of proportional transaction costs in option pricing models. In Leland (1985), in order to avoid continuous trading, the author specifies a finite number of trading dates. He obtains a Black-Scholes-like nonlinear partial differential equation (PDE) with an adjusted volatility term, that takes into account the transaction costs. However, trading at fixed dates is not optimal, and the option price goes to infinity as the number of dates grows. Further work in this direction has been done by Boyle and Vorst (1992), which consider a multi-period binomial model (see Cox et al. (1979)) with transaction costs. Here again, the cost of the replicating portfolio depends on the number of time periods. Recent developments in this direction are for instance Mocioalca (2007), Florescu et al. (2014) and Sengupta (2014) who consider different features of the market such as jumps, stochastic volatility and stochastic interest rate respectively.

A different approach has been introduced by Hodges and Neuberger (1989). The authors use an alternative definition of the option price called indifference price, based on the concepts of expected utility and certainty equivalent. An overview of these concepts applied to several incomplete market model can be found in Carmona (2009). As long as the perfect replicating portfolio is no longer implementable in presence of transaction costs, the hedging strategy cannot be anymore riskless. The model has to take into account the risk profile of the writer/buyer to describe his trading preferences. Hodges and Neuberger (1989) define the option price as the value that makes an investor indifferent between holding a portfolio with an option and without, in terms of expected utility of the final wealth. They show that it is impossible to hedge perfectly the option. The optimal strategy is to keep the portfolio’s value within a band called no transaction region. Using numerical experiments, they verify that this strategy outperforms the one proposed in Leland (1985). This approach has been further developed in Davis et al. (1993), where the problem is formulated rigorously as a singular stochastic optimal control problem. The authors prove that the value functions of the two optimization problems can be interpreted as the solutions of the associated Hamilton-Jacobi-Bellman (HJB) equation in the viscosity sense. They prove also that the numerical scheme, based on the Markov chain approximation, converges to the viscosity solution. Numerical methods for this model are presented in Davis and Panas (1994), Clewlow and Hodges (1997) and Monoyios (2003), Monoyios (2004). In Whalley and Wilmott (1997) and Barles and Soner (1998) the problem is simplified by using asymptotic analysis methods for small levels of transaction costs. The authors, starting from the general HJB variational inequality, derive a simpler non-linear PDE for the option price. Further developments are presented in the thesis work of Damgaard (1998), where the author studies the robustness of the model from a theoretical and numerical point of view. He found that under particular conditions the model is quite robust with respect to the choice of the utility function.

In the present work, we want to develop a model for pricing options using the concept of indifference price, as proposed in Davis et al. (1993). We consider proportional transaction costs, and a stock dynamics following an exponential Lévy process. In contrast with Black-Scholes, with our model we obtain two different prices: the price for the buyer and the price for the writer of the option, and this feature makes this model more realistic. The presence of jumps implies to take in account the possibility of insolvency2. A sudden jump in the price can have dangerous consequences and cause the bankruptcy of the investor. It turns out that it is very difficult to solve the general maximization problem numerically. In order to simplify the problem, we consider the special case of an investor with infinite wealth (always solvent) and an exponential utility function to describe his risk profile. Under these assumptions it is possible to reduce by one the number of variables of the HJB equation.

In Section 2, after a short review of Lévy processes theory, we introduce the model equations and definitions. We derive the general HJB equation associated with the singular stochastic control problem. After that, we consider a simplification of the problem and reduce the number of variables choosing the exponential utility function. In Section 3 we present the properties of the Markov chain approximation of the continuous time problem and describe the algorithm for the solution of the singular control problem. We then introduce the Merton jump-diffusion and the Variance Gamma processes and refer to Appendix B for the construction of their specific Markov chain approximation. The numerical results are presented in Section 4 and a complete summary of all the outcomes is presented in the conclusive Section 5.

2 The model

2.1 Exponential Lévy models

Let be a Lévy process defined on a filtered probability space , where is the natural filtration and . We assume that has the characteristic Lévy triplet , where , and is a positive measure on , called Lévy measure which satisfies:

(1)

We model the log-prices dynamics with the Lévy process . The price of a stock follows the exponential Lévy process:

(2)

Motivated by practical reasons, we only consider processes with finite mean and variance. The condition for a finite second moment , is directly related to the integrability conditions of the Lévy measure:

(3)

Considering condition (3), the process has the following Lévy-Itô decomposition:

(4)

where is a standard Brownian motion and the term is the compensated Poisson martingale measure, defined by:

(5)

where is the Poisson random measure with intensity . By applying the Itô lemma to (2) we obtain the SDE describing the evolution of the price:

(6)

In the following we indicate the drift term as

(7)

Lévy processes and their exponentials are Markov processes. The infinitesimal generator associated with the price process (6) is given by:

(8)

with , where is the space of twice differentiable functions and is the space of continuous functions with polynomial growth of second degree at infinity. We can define the transition probabilities associated with a Lévy process. For any Borel set and for every :

(9)

Transition probabilities are connected with the conditional expectation by the simple relation:

(10)

2.2 Portfolio dynamics with transaction costs

In this section we introduce the market model with proportional transaction costs that generalizes the model of Davis et al. (1993). Let us consider a portfolio composed by one risk-free asset (bank account) paying a fixed interest rate and a stock . We denote by the number of shares of the stock that the investor holds. The state of the portfolio at time is and evolves following the SDE:

(11)

The parameters , are the proportional transaction costs when buying and selling respectively. The process is the trading strategy (the control process) and represents the cumulative number of shares bought and sold respectively in . The strategy is a left-continuous, -progressively measurable, nondecreasing process with bounded variation in every finite time interval and such that . We define the cash value function as the value in cash when the shares in the portfolio are liquidated: long positions are sold and short positions are covered.

(12)

For , we define the total wealth process:

(13)

We say that a portfolio is solvent if the portfolio’s wealth is greater than a fixed constant , with for all . We define the solvency region:

(14)

As long as we describe the underlying stock as a process with jumps, we cannot guarantee that the portfolio stays solvent for all . Holding short positions, it is possible that a sudden increase in the value of the stock can cause the total wealth to jump instantaneously out of the solvency region. The same can happen with a downward jump when the investor is long in stocks and negative in cash. The sudden decrease of the stock’s price makes him unable to pay his debts. If the investor goes bankrupt, there are no trading strategies to save him. We define the first exit time from the solvency region as

(15)

We define the set of admissible trading strategies , as the set of all left-continuous, nondecreasing, -progressively measurable processes , such that is a solution of (11) for , and with initial values .

Remark 2.1.

The original model formulated by Hodges and Neuberger (1989) and Davis et al. (1993), considers a portfolio starting with zero total wealth. It does not consider the possibility of insolvency. The writer (or buyer) of the option creates a portfolio at time in order to hedge the option. Therefore it is reasonable to assume that it does not own any shares in the underlying stock at time . Following the literature, we consider a portfolio with zero initial shares and an initial amount in the cash account. This assumption can be easily relaxed to include an initial number of shares if needed.

2.3 Utility maximization

The objective of the investor is to maximize the expected utility of the wealth at over all the admissible strategies. This expectation is conditioned on the initial value of cash , number of shares and value of the stock . The value function of the maximization problem is:

(16)

where is a concave and increasing utility function, such that . Here we use the superscript “” to indicate that this is the case of a portfolio with zero options. The wealth and the exit time are defined as in (13) and (15) (with the solvency region as (14)).

Remark 2.2.

It is important to stress that the utility function has to be defined also for negative numbers, and this requirement excludes the logarithmic utility for instance. The indicator functions separate the two cases of solvency at maturity date , and bankruptcy at . This feature is not present in the work of Davis et al. (1993). In the case of default the value function assumes the lowest possible value attainable by , multiplied by the factor .

Assume an investor builds a portfolio with cash, shares of a stock and in addition he sells or purchases a European call option written on the same stock, with strike price and expiration date . This means that at time the initial amount in the cash account increases by the option’s value (in the writer case), or decreases by (in the buyer case). We define the wealth processes for the writer and buyer portfolios respectively:

  • Writer:

    (17)
  • Buyer:

    (18)

In the case the option is exercised, , the buyer pays the writer the strike in cash, and the writer delivers one share to the buyer. In a market with transaction costs the real value (in cash) of a share is given by the bilinear cash cost function (12). The buyer of the option does not exercise when , but when . The solvency regions for the writer/buyer are:

The investor wishes to maximize the expected utility of the wealth of his portfolio.

(19)

where is the initial cash amount in the writer or buyer portfolio and can assume the two values and . The exit time is associated with the writer/buyer solvency region , respectively for .

With this model we can compute two option prices: the price for the writer and the price for the buyer. These prices are defined, respectively, as the amount required to get the same maximal expected utility of the wealth of the option-free portfolio. To compute the option price, it is necessary to solve two portfolio optimization problems: the problem without the option and the problem with the option. We define the

  • Writer price:

    (20)
  • Buyer price:

    (21)

The prices and can be obtained implicitly by these conditions.

2.4 Hamilton-Jacobi-Bellman Equation

We present the HJB equation associated to the singular stochastic optimal control problems described before. These problems are called singular because the controls are allowed to be singular with respect to the Lebesgue measure . A rigorous derivation of the following equation can be found in Fleming and Soner (2005) and Øksendal and Sulem (2007). The HJB equation associated with the singular control problems (16) and (19) is a variational inequality:

(22)

for and . The boundary conditions are given by Eqs. (16) and (19). The main difference between this model and the previous models in the literature is that this HJB equation is a partial integro-differential equation (PIDE), which involves an additional integral operator. The presence of this non-local operator implies that we need to define the lateral conditions not only on the boundary of the solvency region, but also beyond. This is given by the condition

(23)

The variational inequality (22) says that the maximum of three operators is equal to zero. This feature can be interpreted better if we consider the state space divided into three different regions: the Buy, the Sell and the No Transaction (NT) regions. The optimization problem is a free boundary problem, and its solution consists of finding the value function and the optimal boundaries that divide the three regions. The boundaries completely characterize the investor’s trading strategy. The optimal strategy consists in keeping the portfolio process inside the NT region. If the portfolio jumps outside the NT region, the optimal strategy is to trade in order to bring back the portfolio on the boundary with the NT region. We can argue that Buy and Sell regions are separated by the NT region, since it is clearly not optimal to buy and sell a stock at the same time. In the Buy and Sell regions the value functions remain constant along the directions of the trades. We have respectively:

  • Buy:       

  • Sell:       

where and are the number of shares respectively bought or sold in the trade. The second and third terms in the HJB equation (22) are the gradients of the value function along the optimal trading directions from the Buy and Sell regions to the NT boundaries. In the NT region the portfolio evolves according to the portfolio equation (11), with . Therefore the number of shares remains constant as long as the portfolio stays in the NT region. If we assume that the process does not leave the NT region in the (small) time interval , by the dynamic programming principle we can write the value function simply as:

(24)

where we indicate with and the change in the cash account and in the stock price after . The formula (24) is the integral representation of the first operator in Eq. (22). This can be proved easily by applying the Itô lemma, and sending .

2.5 Variable reduction

In this model we introduce the feature of portfolio insolvency, which is directly reflected in our definition of the set of admissible trading strategies.

In the literature, all the models considering diffusion processes use to define the set of admissible trading strategies as the set of all the measurable processes solution of (11) with initial values , such that:

(25)

This is a static set and is completely determined by the initial values of the portfolio. In the diffusion case the portfolio is solvent for every and therefore it is always possible to calculate the utility of the wealth at the terminal time .

In the current model, the stock process can jump, and the portfolio can go bankruptcy at any time before the maturity date . For this reason the set of admissible trading strategies has to be a dynamic set, and at any time it has to depend on the current state. In other words, the set of possible trading strategies cannot be evaluated at the beginning, but has to be updated continuously in time depending on the current wealth.

However, the HJB Eq. (22), associated with the maximization problems (16), (19) turns out to be very difficult to solve numerically. In order to simplify the problem, we have to restrict our attention only on the solvent strategies. We consider the subset of composed by only the portfolio’s paths that remain inside the solvency region, while all the paths that jump outside are ignored. We indicate this subset with:

(26)

The set of only depends on the initial state of the investor (like in the diffusion case). Under this assumption we have:

(27)

In the following we maximize the cost functions (16) and (19) over the new set of trading strategies .

We can interpret this simplification as the case of a large investor with a small probability of default, and thus the non-solvent paths can be ignored. Consequently, the solvency constraint (14) loses meaning, and we can set for convenience . The lateral boundary conditions (23) lose importance as well, and are ignored. This simplification is very important for the numerical computations.

Considering the solvent set , we can use the properties of the exponential utility function to reduce by one the number of variables. The exponential utility is defined as:

(28)

This is a common choice and has already been used in Hodges and Neuberger (1989) and Davis et al. (1993). The exponential utility has the property that the risk aversion coefficient is constant, and does not depend on the wealth . This means that the amount invested in the risky asset at time is independent of the total wealth at time . This choice of utility function permits to simplify the optimization problem by reducing the number of variables from four to three. As long as the amount in the risky asset is independent of the total wealth, the amount in the cash account at the maturity is irrelevant to the trading strategy. We can thus remove from the state dynamics. The integral representation of the evolution of the cash account in (11) is:

(29)

where . Using together (26), (27), (28) and (29), and the wealth processes (13),(17),(18), we can write the value functions (16) and (19) in a compact form for :

(30)

where assumes the values , and respectively for . The expectations taken on the domain are zero because . The new minimization problem is:

(31)

where the first term in the product inside the expectation can be considered as a discount factor, and the second term is the terminal payoff:

  • No option:

    (32)
  • Writer:

    (33)
  • Buyer:

    (34)

From (2.5) it is straightforward to see that for the value function can be written as . Since does not depend on we can define it as:

(35)

In order to simplify further the equation, it is convenient to pass to the log-variable . The derivative operators change as:

(36)

Using (35) and (36), the HJB Eq. (22) becomes:

(37)

with . The HJB Eq. (37) has the following integral representation:

(38)

Each term inside the “min” is the integral form of the corresponding term inside Eq. (37). This can be proved by sending .

Using conditions (20), (21) together with (2.5), we obtain the explicit formulas for the option prices:

(39)
(40)

3 Markov chain approximation

To solve the minimization problem (31) we use the Markov chain approximation method developed by Kushner and Dupuis (2001). The numerical technique for singular controls has been developed in the work of Kushner and Martins (1991). The portfolio dynamics (11) is approximated by a discrete state controlled Markov chain in discrete time. The method consists in creating a backward recursive dynamic programming algorithm, in order to compute the value function at time , given its value at time like in (38). Kushner and Dupuis (2001) prove that the value function obtained through the discrete dynamic programming algorithm converges to the value function of the original continuous time problem as . Their proof uses a weak convergence in probability argument. Another approach to prove convergence has been introduced by Barles and Souganidis (1991). It consider the convergence of the discrete value function to the viscosity solution of the original HJB equation. In the work of Davis et al. (1993) the authors prove existence and uniqueness of the viscosity solution of the HJB Eq. (22) for the diffusion case, and using the method developed by Barles and Souganidis (1991) prove that the value function obtained through the Markov chain approximation converges to it.

In this work we model the stock dynamics as a general exponential Lévy process. For practical computations we need to specify which Lévy process we are using, and this is equivalent to specify the Lévy triplet. Since every Lévy process satisfies the Markov property, we are allowed to use a Markov chain approximation approach. A possible technique to construct the Markov chain is to discretize the infinitesimal generator of the continuous process by using an explicit finite difference approximation (see for instance Kushner and Dupuis (2001) or Fleming and Soner (2005)). This method is straightforward for Lévy processes of jump-diffusion type with finite activity of jumps. For Lévy processes with infinite activity it is not possible to obtain the transition probabilities from the infinitesimal generator. A common procedure is to approximate the small jumps with a Brownian motion, as explained in Cont and Voltchkova (2005). This serves to remove the singularity of the Lévy measure near the origin, and permits to create a Markov chain approximation using the same framework used for the jump-diffusion processes. In this work we present examples of option prices computed by using Brownian motion, Merton jump-diffusion and Variance Gamma processes.

3.1 The discrete model

Thanks to the variable reduction introduced in the previous section, the optimization problem (31) only depends on two state variables. The portfolio dynamics (11) has the simpler form (using ):

(41)

where the SDE for the log-variable can be obtained by putting together (4) and (7). If the process has finite activity , thanks to the moment condition 3, we can define and such that the SDE of can be written as

(42)

If the process has infinite activity , we can approximate the “small jumps” martingale component with a Brownian motion with same variance. After fixing a truncation parameter , we can split the integrals in (41) in two domains and . The integrand on the domain , is approximated by Taylor expansion such that:

(43)

where we defined the new parameters:

(44)

The process is a compensated Poisson process with finite activity and variance .

Now we can discretize the time and space to create a Markov chain approximation of the portfolio process (41). For , define the discrete time step such that . Define the set , where we consider the discrete log-price step and . The values of and can be different to capture the possible asymmetry in the jump sizes. Define also the set , where is a discrete step and . The discretized version of the SDE (41) is:

(45)

where , and . The term assumes values in 3 which is a subset of . It is a random process with and . The term is the discrete version of the compensated Poisson jump term. It is a discrete random process with and , for . When the continuous time jump term is , the corresponding discrete process can assume all the values in . If the integral has a truncation term , as in , we define the subset , such that .

The two increments and , which describe the change in the number of shares bought or sold are non-negative multiples of , and cannot assume values different from at the same time . The process assumes values in at each time . The action of the controls and is supposed to happen instantaneously. We indicate with and the number of shares just before the transaction.

The desired Markov chain approximation of (from now on we assume and indicate with ) has to satisfy two conditions to be admissible:

  1. the transition probabilities (9) are represented as:

    (46)

    where is the jumps activity, and are the transition probabilities of the diffusion and jump components respectively. See Appendix B.1.

  2. The transition probabilities have to be locally consistent with the continuous SDE. This means that, at each time step, the first two moments of the Markov chain have to be equal to the first two moments of the continuous processes (42) or (3.1) at first order in . Respectively:

    (47)

    or

    (48)

Kushner and Dupuis (2001) prove that if the conditions above are satisfied, then the Markov chain approximation converges in probability to the continuous process when . In the Appendix B.2 we construct a discrete Markov chain and prove that it is locally consistent.

3.2 Discrete dynamic programming algorithm

The backward algorithm that uses the dynamic programming equation (38) considers two different steps: a jump-diffusion step and a control step. However, in order to implement a numerical algorithm we cannot use an equation written in that form, because the value function at time on the right hand side is still unknown. The solution is to represent the value functions at time on the right hand side as the expectation of their values at time . If at time it is optimal to trade (for example to buy shares), the portfolio state changes instantaneously from in the buy region, to in the NT region. It is reasonable to assume that after a trade, for a small time interval , the process remains in the NT region. Therefore, following the same argument used to obtain (24), we can use the dynamic programming principle and express the value function as