Arbitrage Problems with Reflected Geometric Brownian Motion

Throughout this paper, we assume the existence of a filtered probability space $(\Omega,\mathscr{F},\mathfrak{F},\textsf{P})$, whose filtration $\mathfrak{F}=(\mathscr{F}_{t})_{t\geq 0}$ satisfies the usual conditions of completeness and right-continuity. It is understood that all random variables and stochastic processes are defined on this space and all stochastic processes are adapted to its filtration. Given a continuous semimartingale $X$, we shall write $\mathsf{L}(X)$ to denote the family of predictable processes $\varphi$, such that the stochastic integral $\int_{0}^{\cdot}\varphi_{s}\,\mathrm{d}X_{s}$ exists.

Consider a financial market comprising a risk-free security and a single risky security. For convenience, we shall refer to the former as a bank account and to the latter as a stock. The value $B$ of the bank account is given by $B_{t}\coloneqq\mathrm{e}^{rt}$, for all $t\geq 0$, where $r\geq 0$ is a continuously compounding risk-free interest rate. The stock price $S$ is determined by the SDE

for all $t\geq 0$, with initial value $S_{0}>0$, where $A$ is a continuous finite variation process and $M$ is a continuous local martingale. For convenience, we shall assume that $\langle M\rangle_{t}<\infty$, for all $t\geq 0$, which ensures that the stock price is strictly positive, by virtue of the law of large numbers for local martingales (see Revuz and Yor 1999, Exercise V.1.16).

Investors in the market described above are able to trade self-financing portfolios comprising the bank account and the stock. The following definition makes this concept precise.

Given a trading strategy $\xi\in\mathsf{L}(S)$ and an initial endowment $v\geq 0$, we interpret $V^{v,\xi}_{t}$ as the value at time $t\geq 0$ of a self-financing portfolio that holds $\xi_{s}$ shares of the stock at each time $s\in[0,t]$, with initial value $V^{v,\xi}_{0}=v$. The SDE (2.2) follows from the self-financing requirement that the portfolio must hold $\nicefrac{{(V^{v,\xi}_{t}-\xi_{t}S_{t})}}{{B_{t}}}$ units of the bank account at time $t\geq 0$, if it holds $\xi_{t}$ shares of the stock at that time.

Following Jacod and Shiryaev (2003, Proposition II.2.9), there exists a continuous (and hence predictable) increasing process $G$, such that

for all $t\geq 0$, where $\rho$, $b$ and $a$ are some predictable processes. One possible choice for $G$ is obtained by setting $G_{t}\coloneqq t+\textsf{Var}(A)_{t}+\langle M\rangle_{t}$, for all $t\geq 0$. The next definition formulates the so-called structure condition in terms of the processes $\rho$, $b$ and $a$.

The structure condition was first identified by Schweizer (1995), who formulated it somewhat differently. To demonstrate that the two formulations are equivalent, let $\hat{S}$ denote the discounted stock price, defined by $\hat{S}_{t}\coloneqq\nicefrac{{S_{t}}}{{B_{t}}}=\mathrm{e}^{-rt}S_{t}$, for all $t\geq 0$. It satisfies the SDE

for all $t\geq 0$, with $\hat{S}_{0}=S_{0}>0$, by an application of Itô’s formula. Next, define the continuous finite variation process $\hat{A}$ and the continuous local martingale $\hat{M}$, by setting

for all $t\geq 0$. If the structure condition holds, for some predictable process $\vartheta$ satisfying (2.4), then

for $\textsf{P}\otimes G\text{-a.a}\;(\omega,t)\in\Omega\times\mathbb{R}_{+}$. On the other hand, if there is a predictable process $\lambda$, such that

for $\textsf{P}\otimes G\text{-a.a}\;(\omega,t)\in\Omega\times\mathbb{R}_{+}$, then the same reasoning shows that the structure condition holds, with the predictable process $\vartheta\coloneqq\lambda\hat{S}$ satisfying (2.4). So, the structure condition is equivalent to the existence of a predictable process $\lambda$ that satisfies (2.6), which is how Schweizer (1995) originally formulated it.

The reformulation of the structure condition above offers a useful insight that will help us interpret the arbitrage properties of the reflected geometric Brownian motion model, studied in Section 3. First, note that the finite variation processes $\hat{A}$ and $\langle\hat{M}\rangle$ may be regarded as (possibly signed) random measures on $(\mathbb{R}_{+},\mathscr{B}(\mathbb{R}_{+}))$. With that interpretation in mind, (2.6) states that $\hat{A}\ll\langle\hat{M}\rangle$, with predictable density $\lambda$. This means that if $\int_{0}^{\infty}\mathbf{1}_{U}\,\mathrm{d}\langle\hat{M}\rangle_{t}(\omega)=0$ then $\int_{0}^{\infty}\mathbf{1}_{U}\,\mathrm{d}\hat{A}_{t}(\omega)=0$, for $\textsf{P}\text{-a.a.}\;\omega\in\Omega$ and any Borel-measurable set $U\in\mathscr{B}(\mathbb{R}_{+})$. In particular, if there are measurable subsets of $\mathbb{R}_{+}$ over which the sample paths of $\hat{A}$ increase or decrease but the sample paths of $\langle\hat{M}\rangle$ remain constant, then the structure condition cannot hold.

An increasing profit is the strongest form of arbitrage for continuous-time financial market models. This concept of arbitrage was introduced by Karatzas and Kardaras (2007) and thoroughly investigated by Fontana (2015). The following definition provides a formulation that is appropriate for our setting.

Putting the previous definition into words, a model admits an increasing profit if there is a trading strategy, for which the value of the associated portfolio with an initial endowment of zero is a non-decreasing process with a positive probability of ultimately becoming strictly positive. Such arbitrage opportunities are unequivocally pathological, since they offer investors a chance of making something from nothing, without incurring any risk. Equilibrium asset prices cannot exist in a model that offers such opportunities, since there would be insatiable demand for the portfolios that exploit them. Consequently, a viable financial market model must satisfy the NIP condition.

The next theorem reveals that the NIP condition is equivalent to the structure condition. Fontana (2015, Theorem 3.1) proved essentially the same result, but the formulation and proof in our setting are quite different and the proof is sufficiently instructive to merit inclusion. Karatzas and Shreve (1991, Lemma 1.4.6) and Delbaen and Schachermayer (1995, Theorem 3.5) proved related results in the “only iff” direction.

Inspection of (2.4) reveals that the structure condition fails if and only if there are periods during which the process $a$ is zero and processes $b$ and $\rho$ assume different values. Whenever that happens, the market effectively contains two risk-free securities with different rates of return. During such periods, the trading strategy constructed in the first part of the proof of Thoerem 2.4 generates an increasing profit by borrowing at the lower rate and investing the proceeds at the higher rate. In light of Theorem 2.4, the NIP condition may be interpreted economically as stating that the market never contains two risk-free securities with different rates of return.

In order for a model to be suitable for contingent claim valuation, it should admit either a numéraire portfolio or an equivalent risk-neutral probability measure. Informally, a numéraire portfolio in our modelling framework is a well-behaved trading strategy $\xi^{*}$, whose portfolio value $V^{1,\xi^{*}}$, with initial endowment $v=1$, provides a natural benchmark against which the value of every other well-behaved portfolio can be measured. More formally, the benchmarked processes $\nicefrac{{B}}{{V^{1,\xi^{*}}}}$ and $\nicefrac{{S}}{{V^{1,\xi^{*}}}}$ are required to be supermartingales. On the other hand, an equivalent risk-neutral probability measure in our setting is an equivalent probability measure $\textsf{{Q}}\sim\textsf{P}$, such that the discounted stock price $\hat{S}\coloneqq\nicefrac{{S}}{{B}}$ is a local martingale under Q. If a model admits a numéraire portfolio, then the benchmark approach to contingent claim valuation (see Platen and Heath 2006) can be used, while traditional risk-neutral valuation can be used if it admits an equivalent risk-neutral probability measure.

Both the existence of well-behaved numéraire portfolio and the existence of an equivalent risk-neutral probability measure can be characterised in terms of no-arbitrage conditions. In more general settings than ours, Karatzas and Kardaras (2007) established that the existence of a well-behaved numéraire portfolio is equivalent to the no unbounded profit with bounded risk (NUPBR) condition, while Delbaen and Schachermayer (1994) famously demonstrated that the no free lunch with vanishing risk (NFLVR) condition is necessary and sufficient for the existence of an equivalent risk-neutral probability measure. Based on those results, a model that does not satisfy NUPBR or NFLVR cannot be used for pricing options and other contingent claims.

Crucially, both NUPBR and NFLVR are stronger conditions that NIP.²²2The logical dependencies between the various no-arbitrage conditions for continuous financial market models are explained in the surveys by Hulley (2010, Chapter 1) and Fontana (2015). Consequently, a model that does not satisfy the NIP condition will fail to satisfy the NUPBR and NFLVR conditions as well, which means that it will not admit a well-behaved numéraire portfolio or an equivalent risk-neutral probability measure. Such a model is unsuitable for contingent claim valuation. This is unsurprising, since we have already argued that a model cannot sustain equilibrium asset prices if it does not satisfy the NIP condition.

Skorokhod (1961) showed that the SDE

for all $t\geq 0$, with a lower reflecting boundary $b>0$ and initial value $S_{0}>b$, admits a unique strong solution. This solution comprises a pair of processes $S$ and $L$, such that

for all $t\geq 0$, and where the integrals above are well-defined. The process $S$ is called a reflected geometric Brownian motion (RGBM) and the process $L$ is called the reflection term.

Condition (3.2c) ensures that the value of $L$ does not change while the value of $S$ exceeds $b$, in which case condition (3.2d) ensures that $S$ behaves like a vanilla geometric Brownian motion (GBM). However, as soon as $S$ reaches the boundary $b$, the value of $L$ increases, instantaneously reflecting $S$ back into the interval $(b,\infty)$, after which it behaves like a vanilla GBM once again. It follows that the points of increase of $L$ are limited to times when $S$ visits $b$. It is natural to interpret $L_{t}$ as the cumulative amount of reflection of $S$ from the boundary $b$, up time $t\geq 0$.

In addition to establishing the existence of a unique solution to (3.1), Skorokhod (1961) showed that the reflection term is given by

for all $t\geq 0$, where the integral in this expression can be defined rigorously as a limit of integral sums. There is, however, a more useful interpretation of the reflection term. Define the process $\ell^{b}$, by setting

for all $t\geq 0$. This process, which is known as the local time of $S$ at $b$, provides a non-trivial measure of the amount of time $S$ spends in the vicinity of $b$. Pilipenko (2014, Theorem 1.3.1) showed that $L_{t}=\nicefrac{{\ell^{b}_{t}}}{{2}}$, for all $t\geq 0$, which is to say that the reflection term is just the scaled local time at the reflecting boundary.

We shall now revisit the financial market described in Section 2, comprising a risk-free bank account and a non-dividend-paying stock. As before, the value $B$ of the bank account is given by $B_{t}\coloneqq\mathrm{e}^{rt}$, for all $t\geq 0$, where $r\geq 0$ is the risk-free interest rate. However, following Veestraeten (2008), we now assume that the stock price $S$ follows the RGBM (3.1).

We begin by noting that the RGBM model falls within the scope of the framework introduced in Section 2, which ensures that the analysis in that section is fully applicable to it. Indeed, (3.2c) allows us to rewrite (3.1) as

for all $t\geq 0$. This equation may in turn be expressed in the form (2.1), with the continuous finite variation process $A$ and the continuous local martingale $M$ given by

for all $t\geq 0$. As noted in Section 2, there exists a continuous increasing process $G$ satisfying (2.3), for some predictable processes $\rho$, $b$ and $a$. That is to say,

for all $t\geq 0$. Setting $G_{t}\coloneqq t+\ell^{b}_{t}$, for all $t\geq 0$, provides a convenient choice for the process $G$.

The behaviour of the stock price at the reflecting boundary raises concerns about the arbitrage properties of the RGBM model. Indeed, it seems plausible that an arbitrageur could exploit this behaviour by purchasing the stock when its price reaches the boundary and unwinding the position immediately afterwards, as the stock price is reflected off the boundary. The guiding intuition is that, upon reaching the boundary, the behaviour of the stock price is completely predictable over the next instant of time, and therefore arbitrageable. The following proposition confirms this intuition.³³3The trading strategy constructed in the proof of Proposition 3.1 is similar in spirit to the trading strategy in Fontana (2015, Example 7.1).

Figure 3.1 plots a simulated path for the stock price and the corresponding path for the reflection term, together with the path for the stock position in the arbitrage portfolio described in the proof of Proposition 3.1. Figure 3.11(a) illustrates the reflection of the stock price off the boundary, while Figure 3.11(b) illustrates the behaviour of the the reflection term (which is effectively the local time of the stock price at the boundary). Each time the stock price visits the boundary, we see an instantaneous increase in the refection term, which nudges the stock price away from the boundary. Figure 3.11(c) shows that the stock holding in the arbitrage portfolio switches from zero to one in that instant, before immediately switching back to zero as the stock price is instantaneously reflected off the boundary. In effect, the stock is purchased and immediately sold when its price reaches the boundary, realising a profit equal to the increment in the reflection term in that instant. This profit is deposited in the bank account, so that the value of the arbitrage portfolio at any time is the cumulative sum of the instantaneous profits realised up to that time, plus interest.

In the case when the risk-free interest rate is zero, it follows from (3.6) that the value of the arbitrage portfolio is simply the value of the reflection term (or the scaled local time process). In that case, Figure 3.11(b) also illustrates the behaviour of the value of the arbitrage portfolio. We see that the portfolio requires no initial investment of capital, since its initial value is zero. We also see that its value is non-decreasing, which reflects the riskless nature of the strategy as well as the fact that it requires no trading capital to fund margin calls. Moreover, the portfolio value increases instantaneously from zero at the first-passage time of the stock price to the reflecting boundary, and increases again each subsequent time the stock price visits the boundary. Effectively, the arbitrage strategy harvests the local time of the stock price at the boundary as a riskless profit.

Veestraeten (2008) claimed that the RGBM model does not admit arbitrage opportunities. He justified this assertion by citing two properties of the stock price in the model. First, since reflection off the boundary is instantaneous, the Lebesgue measure of the time spent there by the stock price is zero. Second, the stock price follows a continuous process. Based on those two observations (both of which are correct), he argued that the model is arbitrage-free and therefore admits an equivalent risk-neutral probability measure. Although Proposition 3.1 refutes this argument, we shall nevertheless dwell on it, because it has been repeated by several other authors.

The main problem is that Veestraeten’s (2008) argument focuses on the wrong measure of the time spent by the stock price at the reflecting boundary. Although the Lebesgue measure of the time spent there is indeed zero, that is irrelevant. Instead, we should focus on the local time of the stock price at the boundary. Being a continuous process whose paths are of unbounded total variation on compact intervals means that the sample paths of the stock price are extremely irregular. As a result of this irregularity, the stock price spends a non-zero amount of time infinitesimally close to the boundary, each time it visits, even though it spends no time actually on the boundary. The time spent arbitrarily close to the boundary is what the local time process measures. Each time the stock price reaches the boundary, the clock measuring its local time there ticks over and the local time increment is added to the stock price, nudging it away from the boundary. The arbitrage strategy constructed in the proof of Proposition 3.1 exploits this behaviour, by purchasing and immediately selling the stock each time it reaches the boundary, risklessly harvesting the local time increments in the process.

Another problem with Veestraeten’s (2008) argument is its apparent reliance on the idea that it is practically impossible for an arbitrageur to time their trades to coincide with the stock price reaching and leaving the reflecting boundary. Mathematically, an arbitrage is merely predictable process that satisfies certain technical conditions, and a model is arbitrage-free (in the appropriate sense) if no such process exists. The question of whether the strategy could be implemented in practice has no bearing on its existence as a well-defined mathematical object, and is thus irrelevant to the arbitrage properties of the model. Even though no trader could implement the strategy described in the proof of Proposition 3.1, it nevertheless exists as a well-defined predictable process satisfying the conditions in Definition 2.3.

Theorem 2.4 showed that a continuous financial market model satisfies the NIP condition if and only if it satisfies the structure condition. Proposition 3.1 proved that the RGBM model does not satisfy the NIP condition, by explicitly constructing an immediate profit strategy. Hence, the structure condition must fail as well. The next proposition proves this directly.

We can approach Proposition 3.2 from a slightly more intuitive angle. Using (3.4), we obtain the following expressions for the continuous finite variation process $\hat{A}$ and the continuous local martingale $\hat{M}$, defined by (2.5):

for all $t\geq 0$, where $\hat{S}\coloneqq\nicefrac{{S}}{{B}}$ is the discounted stock price. Since the local time of the stock price at the reflecting boundary increases instantaneously each time gets there, while the Lebesgue measure of the time spent by the stock price at the boundary is zero, we have

for $\textsf{P}\text{-a.a.}\;\omega\in\{\tau_{b}<\infty\}$. Consequently,

for $\textsf{P}\text{-a.a.}\;\omega\in\{\tau_{b}<\infty\}$. Since $\textsf{P}(\tau_{b}<\infty)>0$, it follows that $\hat{A}\centernot\ll\langle\hat{M}\rangle$, in violation of the structure condition.⁴⁴4See the interpetation of the structure condition following Definition 2.2. In other words, the structure condition fails because a non-trivial set of paths of $\hat{A}$ (those for which the first-passage time of the stock price to the reflecting boundary is finite) increase on sets of Lebesgue-measure zero (corresponding to the times when the stock price is at the reflecting boundary), while the paths of $\langle\hat{M}\rangle$ cannot change on sets of Lebesgue-measure zero.

Proposition 3.1 established that the RGBM model does not satisfy the NIP condition. As a result, the NFLVR condition also fails, which rules out the existence of an equivalent risk-neutral probability measure. Operating under the mistaken belief that risk-neutral valuation is possible for the RGBM model, Veestraeten (2008) derived pricing formulae for European options written on the stock. Subject to an amendment to the pricing formula for European put options by Hertrich and Veerstraeten (2013), these formulae have also been used by Hertrich (2015) and Hertrich and Zimmermann (2017). Since their derivations are predicated on an incorrect assumption, we expect them to exhibit some inconsistencies.

Suppose we make the counterfactual assumption that an equivalent risk-neutral probability measure $\textsf{{Q}}\sim\textsf{P}$ does in fact exist for the RGBM model. In that case, the discounted stock price process $\hat{S}$ is a Q-local martingale. Given a European contingent claim written on the stock, with maturity $T\geq 0$ and payoff function $h:[b,\infty)\rightarrow\mathbb{R}$, we may (by hypothesis) apply a standard risk-neutral valuation approach to obtain its pricing function $V^{h}:[0,T]\times[b,\infty)\rightarrow\mathbb{R}$, as follows:

for all $(t,S)\in[0,T]\times[b,\infty)$. Here $\textsf{E}^{\textsf{{Q}}}_{t,S}(\,\cdot\,)$ is the expected value operator with respect to the risk-neutral probability measure $\textsf{{Q}}_{t,S}$, under which $S_{t}=S\geq b$.

Consider European call and put options on the stock, with a common strike price of $K\geq b$ and a common maturity date $T\geq 0$. The pricing functions for these instruments are given by

for all $(t,S)\in[0,T]\times[b,\infty)$. Veestraeten (2008) derived a putative risk-neutral transition density for the stock price, which Veestraeten (2008), Hertrich and Veerstraeten (2013) and Hertrich (2015) used to evaluate (3.7) and (3.8). In so-doing, they obtained the following pricing functions for European calls and puts:⁵⁵5The option pricing formulae are formulated differently (but equivalently) in the cited papers. We have chosen the formulations in Hertrich (2015, Equations (11) and (12)), which produce compact expressions.

and

for all $(t,S)\in[0,T]\times[b,\infty)$, where

Here $\Phi(\,\cdot\,)$ denotes the cumulative distribution function for a standard normal random variable.

As an immediate consequence of (3.7), we deduce that

for all $(t,S)\in[0,T]\times[b,\infty)$, where the second inequality follows because $\hat{S}$ is a supermartingale under Q, by virtue of Fatou’s lemma.⁶⁶6Since $\hat{S}$ is (by assumption) a local martingale under Q, which is bounded from below, an application of Fatou’s lemma reveals that it is also a Q-supermartingale. Consequently, $$\textsf{E}^{\textsf{{Q}}}\bigl{(}\mathrm{e}^{-r(T-t)}S_{T}\,|\,\mathscr{F}_{t}\bigr{)}=\mathrm{e}^{rt}\textsf{E}^{\textsf{{Q}}}(\hat{S}_{T}\,|\,\mathscr{F}_{t})\leq\mathrm{e}^{rt}\hat{S}_{t}=S_{t},$$ for all $t\geq 0$. Note that (3.11) is a model-independent upper bound on call prices, whose violation contradicts the assumption that Q is an equivalent risk-neutral probability measure. The next lemma reveals that this bound can be violated if call prices are determined by (3.9).