Dynamic option pricing with endogenous stochastic arbitrage

Abstract

Only few efforts have been made in order to relax one of the key assumptions of the Black–Scholes model: the no-arbitrage assumption. This is despite the fact that arbitrage processes usually exist in the real world, even though they tend to be short-lived. The purpose of this paper is to develop an option pricing model with endogenous stochastic arbitrage, capable of modelling in a general fashion any future and underlying asset that deviate itself from its market equilibrium. Thus, this investigation calibrates empirically the arbitrage on the futures on the S&P 500 index using transaction data from September 1997 to June 2009, from here a specific type of arbitrage called “arbitrage bubble”, based on a $t$-step function, is identified and hence used in our model. The theoretical results obtained for Binary and European call options, for this kind of arbitrage, show that an investment strategy that takes advantage of the identified arbitrage possibility can be defined, whenever it is possible to anticipate in relative terms the amplitude and timespan of the process. Finally, the new trajectory of the stock price is analytically estimated for a specific case of arbitrage and some numerical illustrations are developed. We find that the consequences of a finite and small endogenous arbitrage not only change the trajectory of the asset price during the period when it started, but also after the arbitrage bubble has already gone. In this context, our model will allow us to calibrate the B–S model to that new trajectory even when the arbitrage already started.

Introduction

For almost 35 years, since the seminal articles by Black and Scholes [1] and Merton [2], the Black–Scholes (B–S) model has been widely used in financial engineering to model the price of a derivative on equity.¹ Thus, the research agenda in the option price modelling has been concentrated almost exclusively on testing the empirical validity of the model and relaxing some of the most restrictive assumptions of the original B–S model.

Indeed, the B–S model for an equity makes several well-known assumptions, such as: (i) the price of the underlying instrument follows a geometric Brownian motion, with constant drift $\mu$ and volatility $\sigma$ (a lognormal random walk), (ii) it is possible to short sell the underlying stock, (iii) there are no dividends on the underlying, (iv) trading in the stock is continuous, in other words delta hedging is done continuously, (v) there are no transaction costs or taxes, (vi) all securities are perfectly divisible (it is possible to buy any fraction of a share), (vii) it is possible to borrow and lend at a constant risk-free interest rate $r$, (viii) there are no-arbitrage opportunities.

In analytic terms, if $B\left(t\right)$ and $S\left(t\right)$ are the risk-free asset and underlying stock prices, the price dynamics of the bond and the stock in this model are given by the following equations:

$$\begin{array}{c}dB\left(t\right)=rB\left(t\right)dt\hfill \\ dS\left(t\right)=\mu S\left(t\right)dt+\sigma S\left(t\right)dW\left(t\right)\hfill \end{array}$$

where $r$, $\mu$ and $\sigma$ are constants and $W\left(t\right)$ is a Wiener process.

In order to price the financial derivative, it is assumed that it can be traded, so we can form a portfolio based on the derivative and the underlying stock (no bonds are included). Considering only non-dividend paying assets and no consumption portfolios, the purchase of a new portfolio must be financed only by selling from the current portfolio.

Calling $\overrightarrow{h}\left(t\right)=(h_S,h_{\Pi })$ the portfolio, $\overrightarrow{P}\left(t\right)=(S,\Pi )$ the price vector of shares and $V\left(t\right)$ the value of the portfolio at time $t$; the dynamic of a self-financing portfolio with no consumption is given by

$$dV\left(t\right)=\overrightarrow{h}\left(t\right)d\overrightarrow{P}\left(t\right)\text{.}$$

In other words, in a model without exogenous incomes or withdrawals, any change of value is due to changes in asset prices.

Another important assumption for deriving B–S equation is that the market is efficient in the sense that is free of arbitrage possibilities. This is equivalent with the fact that there exists a self-financed portfolio with value process $V\left(t\right)$ satisfying the dynamic:

$$dV\left(t\right)=rV\left(t\right)dt$$

which means that any locally riskless portfolio has the same rate of return than the bond.

For the classical model presented above, there exists a well-known solution for the price process of the derivative $\Pi \left(t\right)$ (See, for example Ref. [4]). Given its simplicity and predictive power this formulation can be described as one of the most popular standards in the profession.

Today however, it is possible to find models that have addressed and relaxed almost all of the assumptions mentioned above, including models with transaction costs, different probability distribution functions, stochastic volatility, imperfect information, etc; all of which have improved the prediction capabilities of the B–S model. See [4], [5], [6], [7] for some complete reviews of these extensions.

Nevertheless, only few efforts have been made in order to address one of the key assumption of the model: the no-arbitrage assumption. Since the 80’s economists have realized that, in a real market, futures contracts are not always traded at the price predicted by the simple no-arbitrage relation. Strong empirical evidence have supported this point many times and in different settings, however, economists have tended to develop several alternative explanations for the variability of the arbitrage, such as: differential tax treatment for spots and futures [8], and marking-to-market requirements for futures, but it was argued that noise is the main source of the mispricing [9]. It was also noted that there are certain factors that influence the arbitrage strategies and slow down the market reaction on the arbitrage. The factors include constrained capital requirements [10], position limits, and transaction costs [11]. Nevertheless, there have not been any attempts to modified the B–S model in terms of arbitrage processes, all modifications have been looking for alternative ways of facing the problem, for recent developments in this direction, see Refs. [12], [13], [14]. Notoriously, most of the attempts to incorporated explicitly arbitrage have been made by physicists. Perhaps, the reasons are the importance that academic economists give to the notion of equilibrium, or the sophisticated mathematical treatment necessary for modelling the arbitrage. Despite of this, it is clear that in practice there are arbitrage opportunities in the market, where a lot of people can make (lose) considerable sums of money everyday, see for example Refs. [15], [16], [17] among others.

Most of the attempts to take into account arbitrage in option pricing assume that the return from the B–S portfolio is not equal to the constant risk-free interest rate, changing the no-arbitrage principle (3) to an equation

$$dV\left(t\right)=(r+x\left(t\right))V\left(t\right)dt\text{,}$$

where $x\left(t\right)$ is a random arbitrage return. This formulation gives great flexibility to the model, since $x\left(t\right)$ can be seen as any deviations of the traditional assumed equilibrium, and not just as an arbitrage return. For instance, Ilinski [18] and Ilinski and Stepanenko [19] assume that $x\left(t\right)$ follows an Ornstein–Uhlenbeck process.

Other distinct effort in this direction is Otto [20], who reformulated the original B–S model through a stochastic interest rate. However, as Panayides [21] pointed out, the main problem with this approach is that the random interest rate is not a tradable security, and therefore the classical hedging cannot be applied. This difficulty leads to the appearance of an unknown parameter, the market price of risk, which cannot be directly estimated from financial data.

Finally, Panayides [21] and Fedotov and Panayides [22] follow an approach suggested by Papanicolaou and Sircar [23], where option pricing with stochastic volatility is modelled. These studies instead of finding the exact equation for option price focus on the pricing bands for options that account for random arbitrage opportunities. The approach yields pricing bands that are independent of the detailed statistical characteristics of the random arbitrage return.

The purpose of this paper is to develop an option pricing model with endogenous stochastic arbitrage, capable of modelling in a general fashion any underlying asset that deviate itself from its market equilibrium. To our knowledge this is the first attempt in this direction in the literature, modelling endogenously arbitrage process. It is expected that the introduction of this stylized fact of the financial markets could improve the forecasting performance of the original B–S model. It is important to notice that in the light of the recent events occurred in the global financial markets, where economists as well as financial analysts have been hardly questioned for their results and naive models, modelling efforts like this one, that assume more realistic views, considering departures from the traditional equilibrium approach, must be properly examined. ² It is expected that given the flexibility of our approach, where arbitrage can vary in terms of shape, size and length, for example using “arbitrage bubbles”, a new theoretical and empirical research agenda could be developed.

Section 2 of this paper describes the general model of option pricing with endogenous stochastic arbitrage, and it discussed its economic interpretation. Section 3 presents empirical evidence that supports our theoretical model and allow us to choose in a justified manner our main assumptions. Section 4 develops the analytic results of a particular function of the arbitrage amplitude, and some numerical results. Finally, conclusions and future research are developed.