Modern Stochastics: Theory and Applications

Empirical studies of financial time series led to the conclusion that the log-return increments exhibit long-range dependence. This fact supports the idea of modelling the randomness of a risky asset using a fractional Brownian motion with Hurst parameter $H>1/2$. However, markets driven by a fractional Brownian motion have been extensively disputed, as this motion fails to be a semimartingale and, hence, they allow for a free lunch with vanishing risk (see [6]).

Many attempts were proposed to overcome this drawback of the fractional Brownian motion. In this work, we deal with the regularization method proposed by Cheridito in [2, 1] when $H>3/4$. This method consists in adding to the fractional Brownian motion a multiple of an independent Brownian motion, the resulting process, called mixed fractional Brownian motion, being Gaussian with the long-range dependence property. Moreover, as shown in [2, 1], when $H>3/4$ the mixed fractional Brownian motion is equivalent to a multiple of a Brownian motion. Therefore, a Black–Scholes type model in which the randomness of the risky asset is driven by a mixed fractional Brownian motion is arbitrage free and complete. We call such a model a mixed fractional Black–Scholes model.

On one hand the fractional Black–Scholes model admits arbitrage. On the other hand, when the Hurst parameter $H>3/4$, adding a Brownian component (in the above explained way) makes the arbitrage disappear. In this paper we aim to go a step further and study the sensitivity to arbitrage of the mixed fractional Black–Scholes model when the Brownian component asymptotically vanishes. In [3, 4] it was argued that a good way of seeing the sensitivity to arbitrage of a market when one of its parameters converges to zero (or infinity), is to consider the family of markets indexed by the corresponding parameter and to use methods from large financial markets. To be precise, we study the asymptotic arbitrage opportunities in the sequence of mixed fractional Black–Scholes models when the scaling factor in front of the Brownian motion converges to zero. We focus on the notion of strong asymptotic arbitrage (SAA) introduced by Kabanov and Kramkov in [9] as the possibility of getting arbitrarily rich with probability arbitrarily close to one by taking a vanishing risk. Our model fits the standard framework of large financial markets, as each mixed fractional Black–Scholes model is arbitrage free (and even complete). We point out that the existence of arbitrage in the limiting market does not directly imply the existence of any kind of asymptotic arbitrage in the approximating sequence of mixed markets. In [9] the existence of strong asymptotic arbitrage was shown to be equivalent to the entire asymptotic separation of the sequence of objective probability measures and the sequence of equivalent martingale measures.

In order to show the existence of strong asymptotic arbitrage in the sequence of mixed fractional Black–Scholes models we use the result of [9] that was mentioned above. That means we show that the sequence of objective probability measures is entirely asymptotically separable from the sequence of equivalent martingale measures. Our main contribution is the proof of this entire asymptotic separability in the given model. We use the notion of relative entropy and a dichotomy result for sequences of Gaussian measures. Indeed, inspired by the work of Cheridito [2, 1], we first show, for each fixed market, that the entropy of the objective probability measure relative to the equivalent martingale measure, both restricted to a discrete partition, converges to infinity. Our proof then follows using tightness arguments for the sequence of Radon–Nikodym derivatives of the objective probability measures with respect to equivalent martingale measures and the fact that two sequences of Gaussian measures are either mutually contiguous or entirely separable. The latter is known in the literature as the equivalence/singularity dichotomy for sequences of Gaussian processes, see [5].

The paper is structured as follows. In Section 2, we set the mixed fractional Black–Scholes model and recall the framework of the large financial market. At the end of this part, we state the main result (Theorem 1). Section 3 is dedicated to the proof of Theorem 1, whereas Section 4 provides a discussion about the existence of strong asymptotic arbitrage using self-financing strategies constrained to jump only in a finite set of times. We end our work with Appendix A in which we recall the definition of relative entropy and an equivalent characterization in terms of the Radon–Nikodym derivative.

In order to prove Theorem 1, we provide a series of lemmas from which the desired result is obtained as a direct consequence. Before proceeding, we introduce some notations.

Following the lines of [2, 1], we define, for all $n\in \mathbb{N}$, ${Y_{n}}:C[0,1]\to {\mathbb{R}^{n}}$ by: and denote by ${Q^{H,\alpha ,n}}$ and ${Q_{\frac{\mu \alpha }{\sigma }}^{n}}$ the restrictions of ${Q^{H,\alpha }}$ and ${Q_{\frac{\mu \alpha }{\sigma }}}$ to the σ-algebra ${\mathcal{F}_{n}}:=\sigma ({Y_{n}})$. We fix the Hurst parameter $H\in (3/4,1)$ and we avoid to mention the dependence on it by setting ${Q^{H,\alpha }}\equiv {Q^{\alpha }}$ and ${Q^{H,\alpha ,n}}\equiv {Q^{\alpha ,n}}$.

We denote by ${C_{n}}$ the covariance matrix of the increments of the fractional Brownian motion ${Z^{H}}$: and by ${\lambda _{1}^{n}},\dots ,{\lambda _{n}^{n}}$ its eigenvalues. Since the matrix ${C_{n}}$ is symmetric and positive semi-definite, all the ${\lambda _{i}^{n}}$, $1\le i\le n$, are real and nonnegative.

We moreover set where ${I_{n}}$ is the identity matrix and ${1_{n\times n}}$ is the $n\times n$ matrix with all coefficients equal to 1. Clearly, the matrices ${\varSigma _{0}}$ and ${\varSigma _{1}}$ are positive definite and therefore invertible.

The proof of Theorem 1 strongly relies on the concept of relative entropy (also called sometimes Kullback–Leibler divergence) of the probability measure ${Q^{\alpha ,n}}$ (respectively, ${Q^{\alpha }}$) relative to ${Q_{\frac{\mu \alpha }{\sigma }}^{n}}$ (respectively, ${Q_{\frac{\mu \alpha }{\sigma }}}$), denoted by $H({Q^{\alpha ,n}}|{Q_{\frac{\mu \alpha }{\sigma }}^{n}})$ (respectively, $H({Q^{\alpha }}|{Q_{\frac{\mu \alpha }{\sigma }}})$), see [7, Section 6]. We recall the definition of relative entropy and some relevant results in the Appendix A.

Using standard properties of the trace and the determinant, it is not difficult to see that We set ${a_{n}}:=\frac{1}{n}\frac{{\mu ^{2}}{\alpha ^{2}}}{{\sigma ^{2}}}$ and note that ${\varSigma _{1}}=\frac{1}{n}({I_{n}}+{a_{n}}{1_{n\times n}})$. The next lemma summarizes the properties of the matrix ${\varSigma _{1}}$.

For each $n\ge 1$, we denote the Radon–Nikodym derivative of ${Q^{\alpha ,n}}$ relative to ${Q_{\frac{\mu \alpha }{\sigma }}^{n}}$ by Using [7, Lemma 6.1] (see Lemma 7 in Appendix A), we see that

Moreover, let us recall the notion of ${({Q^{\alpha ,n}})_{\alpha >0}}$-tightness: ${({L_{\alpha }^{n}})_{\alpha >0}}$ is ${({Q^{\alpha ,n}})_{\alpha >0}}$-tight if the following holds:

Before we can state and prove the last lemma of this section, we recall now the definition of contiguity of sequences/families of probability measures.

Lemma 5 might suggest that, for each $n>1$ (or following Remark 5, for each $n\ge 1$ if $\mu =0$), there exists also some kind of asymptotic arbitrage in the large financial market consisting of the restrictions of the α-mixed fractional Black–Scholes models, $\alpha >0$, to the grid ${E_{n}}:=\{0,\frac{1}{n},\dots ,\frac{n-1}{n},1\}$. However, we will show that this is impossible.

For simplicity, we only consider the case $n=1$ and $\mu =0$. We also assume that ${S_{0}^{H,\alpha }}=1$. Thus, for each $\alpha >0$, the corresponding market is In this case all possible strategies are constants and hence the value process ${V_{1}^{\alpha }}$ takes the following form where ${c_{\alpha }}\in \mathbb{R}$. Obviously, we cannot hope for admissibility (boundedness from below), see the discussion in the introduction of [10]. But even if we do not require any admissibility here, there is no way to choose a sequence of ${\alpha _{\ell }}\to \infty $ and corresponding value processes ${V^{{\alpha _{\ell }}}}$ such that the following hold: there exists $\beta >0$ and ${\varepsilon _{\ell }}\to 0$ with This is not possible since ${Z_{1}^{H}}$ as well as ${B_{1}}$ are independent $N(0,1)$ and hence are strictly positive as well as strictly negative, with positive P-probability (and here neither letting $\alpha \to \infty $ nor multiplying ${S_{1}^{H,\alpha }}-1$ by some constants, either positive or negative, will be of any help: whenever there will be a strictly positive part in the limit there will also be a strictly negative part in the limit with a non-disappearing probability). Hence there is no such thing as (14) which, in our discrete time $t=0,1$ situation, is the appropriate version of an asymptotic arbitrage.

The reason behind this apparent contradiction is that in contrast to the continuous time large financial market its discrete counterpart is not complete. Under the original measure P (which induces ${Q^{\alpha }}$ on $\mathcal{C}[0,1]$) we have that ${Z_{1}^{H}}\sim N(0,1)$ and ${B_{1}}\sim N(0,1)$ and the two random variables are independent. We know that the Wiener measure ${Q_{W}}$ is a martingale measure for ${S^{\alpha }}$ (understood on $\mathcal{C}[0,1]$) for all α, hence ${Q_{W}}{|_{{\mathcal{F}_{1}}}}$ is an equivalent martingale measure for (13). We will now construct a different martingale measure for the process (13) which is equivalent to P on $(\varOmega ,\mathcal{F})$.

Indeed, define a measure $\tilde{P}$ on $(\varOmega ,\mathcal{F})$ as follows: $\frac{d\tilde{P}}{dP}=g(X)$ where we have $X:=\exp (\sigma {Z_{1}^{H}})$ and $g(x)={e^{-x}}\frac{1}{h(x)}$ where $h(x)=\frac{1}{\sqrt{2\pi }\sigma x}\exp (-\frac{1}{2}{(\frac{\ln (x)}{\sigma })^{2}})$ is the density of a lognormal distribution, i.e., the density of the law of X under P. Obviously this measure change has the purpose to make the distribution of X exponential with parameter 1. Recall that the measure ${Q^{\alpha ,1}}$ is considered as a measure on $\mathbb{R}$ (the measure induced by ${M_{1}^{H,\alpha }}:=\alpha {Z_{1}^{H}}+{B_{1}}$).

In conclusion, Lemma 6 shows that there exists a family of equivalent martingale measures for the model (13) with good properties, in this case with the property of mutual contiguity. And this fact is reflected by the impossibility to find asymptotic arbitrage opportunities for the family of models (13), $\alpha >0$.