Endogenous Stochastic Arbitrage Bubbles
and the Black–Scholes model.

Since its introduction by Fischer Black, Myron Scholes [1], and Robert C. Merton [2], the Black–Scholes (B–S) model has been widely used in financial engineering to price a derivative on equity. Several generalizations of the initial model premises have since been made. For example, some of these generalizations include stochastic volatility models [3]–[7]; the incorporation of jumps, which gives rise to integrodifferential equations for the option price [8]; and, the consideration of many assets which gives its multi-asset extension [9], [10], among others.
However, one of the last assumptions of the initial model to be changed was the no-arbitrage hypothesis. In effect, in the last decade, several efforts to overcome the no-arbitrage assumption have been made in the literature [11], [12], [13]. In addition, [14], [15], and [16] suggested that the arbitrage can be taken into account in option pricing model by changing the usual return rate of the B–S portfolio $P$ from

$$dP=rdt,$$

(1)

to

$$dP=(r+x(t))Pdt,$$

(2)

where $x(t)$ follows an Ornstein–Uhlenbeck process. Using these ideas, an endogenous arbitrage model is presented in [17]. Here, equation (2) is replaced by the stochastic differential equation

$$dP=rPdt+f(S,t)PdW,$$

(3)

and where the deterministic function $f(S,t)$ was called an arbitrage bubble, and $dW$ is the same Brownian motion that is present in the underlying asset dynamics given by

$$dS=S\mu dt+S\sigma dW.$$

(4)

In [17], using (3) and (4), the following Black–Scholes equation in the presence of an arbitrage bubble is obtained

$\displaystyle\frac{\partial V}{\partial t}+\frac{1}{2}\sigma^{2}S^{2}\frac{\partial^{2}V}{\partial S^{2}}+\left(r+v(S,t)\right)\left[S\frac{\partial V}{\partial S}-V\right]=0$

,

(5)

where $V=V(S,t)$ and

$$v(S,t)=\frac{\left(r-\mu\right)}{(\sigma-f(S,t))}f(S,t),$$

(6)

is a potential term that is equivalent to an electromagnetic potential that is induced by the arbitrage bubble $f(S,t)$. An approximate solution of this equation for an arbitrary bubble form $f(S,t)$ is given in [18] and a method to determine the bubble $f$ from the real financial data is proposed in [19]. The resonances that appear in the model are also discussed in [20].

The interacting B–S equation (5) has two limit behaviors. The first is the “weak bubble” limit $f/\sigma<<1$ or $f\approx 0$, in which case the potential is $v(S,t)\approx 0$ and (5) becomes the usual “free” Black–Scholes equation

$$\frac{\partial V}{\partial t}+\frac{1}{2}\sigma^{2}S^{2}\frac{\partial^{2}V}{\partial S^{2}}+r\left[S\frac{\partial V}{\partial S}-V\right]=0.$$

(7)

The second is the “strong bubble” limit $f/\sigma>>1$ or $f\rightarrow\infty$, in which case

$$v(S,t)=-(r-\mu)$$

(8)

and equation (5) again becomes a “free” Black–Scholes equation

$$\frac{\partial V}{\partial t}+\frac{1}{2}\sigma^{2}S^{2}\frac{\partial^{2}V}{\partial S^{2}}+\mu\left[S\frac{\partial V}{\partial S}-V\right]=0,$$

(9)

where the value of the interest rate has been changed to the mean of the underlying asset value $\mu$.

In this paper, I want to incorporate possible stochastic effects on the arbitrage bubble. Hence, instead of $f$ being a given deterministic function, it becomes a random variable. I will explore its consequences on the dynamic of the option price, and I will obtain the respective weak and strong bubble limits for this case.

Consider the usual underlying asset dynamics given in (4). Now, I generalize the deterministic bubble given in [17] to the stochastic case. To do that, one can assume that the arbitrage bubble satisfies the generic stochastic differential equation

$$df=\mu_{f}dt+\Gamma dW,$$

(10)

where $\ \mu=\mu(S,f,t)$, $\ \sigma=\sigma(S,f,t)$, $\ \mu_{f}=\mu_{f}(S,f,t)$ and $\ \Gamma=\Gamma(S,f,t)\$ are arbitrary functions of $S$, $f$ and $t$, which defines the stochastic model completely. Note that for both equations (4) and (10), there is a unique Brownian motion $dW$. Therefore, this model is endogenous in the same sense of [17].
In this case, the option price $V$ then also becomes a function of $f$, so $V=V(S,f,t)$ and by the Itô lemma one has that

$$dV=\frac{\partial V}{\partial t}dt+\frac{\partial V}{\partial S}dS+\frac{\partial V}{\partial f}df+\frac{1}{2}\frac{\partial^{2}V}{\partial S^{2}}dS^{2}+\frac{1}{2}\frac{\partial^{2}V}{\partial f^{2}}df^{2}+\frac{\partial^{2}V}{\partial S\partial f}dSdf,$$

(11)

and by replacing (4) and (10) in (11), one has that

$\displaystyle dV=$

$\displaystyle\ L(V)\ dt+\left[\sigma S\frac{\partial V}{\partial S}+\Gamma\frac{\partial V}{\partial f}\right]dW,$

(12)

where $L(V)$ denotes the differential operator action

$$L(V)=\frac{\partial V}{\partial t}+\frac{1}{2}\sigma^{2}S^{2}\frac{\partial V}{\partial S^{2}}+\frac{1}{2}\Gamma^{2}\frac{\partial^{2}V}{\partial f^{2}}+S\sigma\Gamma\frac{\partial^{2}V}{\partial S\partial f}+S\mu_{S}\frac{\partial V}{\partial S}+\mu_{f}\frac{\partial V}{\partial f}.$$

(13)

To derive the corresponding Black–Scholes equation, one must consider a portfolio $P$ that is constructed by a number of $N_{V}$ options and $N_{S}$ underlying assets according to

$$P=N_{S}S+N_{V}V,$$

(14)

so one has that (see [10], [9])

$$dP=N_{S}dS+N_{V}dV.$$

(15)

According to equation (3) [17], the portfolio return in the presence of a arbitrage bubble $f$ has the form

$$dP=Prdt+PfdW,$$

(16)

so

$$N_{S}dS+N_{V}dV=Prdt+PfdW.$$

(17)

By replacing (4), (12) in (17), one obtains

$$N_{S}\left(\mu Sdt+\sigma SdW\right)+N_{V}\left(Ldt+\sigma S\frac{\partial V}{\partial S}dW+\Gamma\frac{\partial V}{\partial f}dW\right)=\left(N_{s}S+N_{V}V\right)rdt+\left(N_{s}S+N_{V}V\right)fdt.$$

(18)

By equalling terms in $dt$ and $dW$ in this equation, one finds the system

$$\begin{array}[]{l}\left(\mu_{S}S-Sr\right)N_{S}+(L-rV)N_{V}=0\\ (\sigma S-Sf)N_{S}+\left(\sigma S\frac{\partial V}{\partial S}+\Gamma\frac{\partial V}{\partial f}-Vf\right)N_{V}=0,\end{array}$$

(19)

To obtain a solution with $N_{S}\neq 0$ and $N_{V}\neq 0$, the determinant associated to the matrix form of this system (19) must be equal to zero; that is,

$$\left(\mu S-Sr\right)\left(\sigma S\frac{\partial V}{\partial S}+\Gamma\frac{\partial V}{\partial f}-Vf\right)-(\sigma S-Sf)(L-rV)=0,$$

(20)

that is,

$$(L-rV)=\frac{\left(\mu S-Sr\right)\left(\sigma S\frac{\partial V}{\partial S}+\Gamma\frac{\partial V}{\partial f}-Vf\right)}{(\sigma S-Sf)}.$$

(21)

Now, by replacing $L(V)$ in (13) and simplifying terms, one finally ends with the following explicit Black–Scholes equation for the option price

$\displaystyle\frac{\partial V}{\partial t}+\frac{1}{2}\sigma^{2}S^{2}\frac{\partial^{2}V}{\partial S^{2}}+\frac{1}{2}\Gamma^{2}\frac{\partial^{2}V}{\partial f^{2}}+S\sigma\Gamma\frac{\partial^{2}V}{\partial S\partial f}+\left(r+v(f)\right)\left[S\frac{\partial V}{\partial S}-V\right]+\left(\mu_{f}-\frac{\left(\mu-r\right)}{(\sigma-f)}\Gamma\right)\frac{\partial V}{\partial f}=0.$

(22)

where

$$v(f)=\frac{\left(r-\mu\right)}{(\sigma-f)}f,$$

(23)

is the “electromagnetic” potential mentioned in [17]. Note that the (22) is the same form of the Blacks–Scholes equation for a stochastic volatility model, but without external undetermined functions as the market price of risk [9], [7].

For the $\Gamma=0$ case, equation (22) reduces to

$\displaystyle\frac{\partial V(S,f,t)}{\partial t}+\frac{1}{2}\sigma^{2}S^{2}\frac{\partial^{2}V(S,f,t)}{\partial S^{2}}+\left(r+v(f)\right)\left[S\frac{\partial V(S,f,t)}{\partial S}-V(S,f,t)\right]+\mu_{f}\frac{\partial V(S,f,t)}{\partial f}=0.$

(24)

Here, $f$ is, due to (10), the deterministic function

$$\frac{df}{dt}=\mu_{f}(S,f,t),$$

(25)

so $f=f(S,t)$ and the option price becomes a function of $S$ and $t$ only, which is defined by

$$V(S,t)=V(S,f(S,t),t),$$

(26)

This means that

$$\frac{\partial V(S,t)}{\partial t}=\frac{\partial V(S,f(S,t),t)}{\partial f}\frac{df(S,t)}{dt}+\frac{\partial V(S,f(S,t),t)}{\partial t},$$

(27)

that is,

$$\frac{\partial V(S,t)}{\partial t}=\frac{\partial V(S,f(S,t),t)}{\partial f}\ \mu_{f}+\frac{\partial V(S,f(S,t),t)}{\partial t},$$

(28)

Consequently, in terms of $V(S,t)$, equation (24) is finally

$\displaystyle\frac{\partial V(S,t)}{\partial t}+\frac{1}{2}\sigma^{2}S^{2}\frac{\partial^{2}V(S,t)}{\partial S^{2}}+\left(r+v(f)\right)\left[S\frac{\partial V(S,t)}{\partial S}-V(S,t)\right]=0.$

(29)

which is the same equation (5). Thus, the case $\Gamma=0$ recovers the deterministic arbitrage bubble case.

In the rest of this paper, I will test the effect of the stochastic bubble on the Black–Scholes solution on analytical grounds. I will analyze two special cases: one is the Gaussian bubble, and the other is the lognormal bubble. For these two models, one can find an analytical solution valid for some asymptotic regions in the $(S,f,t)$ space.
Of course, for more general models, to find solutions of equation (22) one must use numerical methods [9]. Nevertheless, the analytical solutions obtained in this work can be used to test the grade of exactitude of the numerical solutions. In a further incoming paper, I will tackle the numerical analysis in a detailed manner and I will then compare it with the analytical solutions obtained in the following sections.

For the Gaussian bubble, one can consider that the asset’s dynamics (4) is given by the usual Black–Scholes case; that is, $\mu$ and $\sigma$ are constants. In addition, for the Gaussian bubble, the $f$–dynamic is given by (10) with $\mu_{f}$ and $\Gamma$ constants. In fact, these parameters represent the mean height and the variance of the bubble.
Thus, one needs to find solutions of (22), with all parameters being constant. An analytical solution can be obtained that is valid in the following regions of the $(S,f,t)$ space:

(a) the region $f\approx 0$, in which case $v(f)=\frac{\left(r-\mu\right)}{(\sigma-f)}f\approx 0$ and (22) takes the form

$\displaystyle\frac{\partial V}{\partial t}+\frac{1}{2}\sigma^{2}S^{2}\frac{\partial^{2}V}{\partial S^{2}}+\frac{1}{2}\Gamma^{2}\frac{\partial^{2}V}{\partial f^{2}}+S\sigma\Gamma\frac{\partial^{2}V}{\partial S\partial f}+r\left[S\frac{\partial V}{\partial S}-V\right]+\left(\mu_{f}-\frac{\left(\mu-r\right)}{\sigma}\Gamma\right)\frac{\partial V}{\partial f}=0,$

(30)

and

(b) the asymptotic limit $\ f>>\sigma$  or  $f\rightarrow\infty$, in which case $v(f)=\frac{\left(r-\mu\right)}{(\sigma-f)}f\rightarrow-(r-\mu)$ so the asymptotic Black–Scholes equation becomes

$\displaystyle\frac{\partial V}{\partial t}+\frac{1}{2}\sigma^{2}S^{2}\frac{\partial^{2}V}{\partial S^{2}}+\frac{1}{2}\Gamma^{2}\frac{\partial^{2}V}{\partial f^{2}}+S\sigma\Gamma\frac{\partial^{2}V}{\partial S\partial f}+\mu\left[S\frac{\partial V}{\partial S}-V\right]+\mu_{f}\frac{\partial V}{\partial f}=0.$

(31)

One can consider equation (30) as the “weak bubble limit” of (22), whereas (31) can be considered as the “strong bubble limit” of (22).

Instead of working directly on equations (30) and (31) to obtain the analytical solutions, one can again consider the “full” equation (22) for the Gaussian bubble, and take the following transformation

$$\left\{\begin{array}[]{l}\bar{u}=\ln S-\left(r-\frac{1}{2}\sigma^{2}\right)t\\ f=f\\ t=t.\end{array}\right.$$

(32)

This maps (22) to the following equation

	$\displaystyle\frac{\partial V}{\partial t}+\frac{1}{2}\sigma^{2}\frac{\partial V}{\partial\bar{u}^{2}}+\frac{1}{2}\Gamma^{2}\frac{\partial^{2}V}{\partial f^{2}}+\sigma\Gamma\frac{\partial^{2}V}{\partial\bar{u}\partial f}+v(f)\frac{\partial V}{\partial\bar{u}}$		(33)
	$\displaystyle+\left(\mu_{f}-\frac{(\mu-r)\Gamma}{(\sigma-f)}\right)\frac{\partial V}{\partial f}-(r+v(f))V=0.$

By defining

$$V(\bar{u},f,t)=e^{-r(T-t)}\psi(\bar{u},f,t)$$

(34)

one has that

$$\frac{\partial\psi}{\partial t}+\left(\frac{1}{2}\sigma^{2}\frac{\partial^{2}\psi}{\partial\bar{u}^{2}}+\frac{1}{2}\Gamma^{2}\frac{\partial^{2}\psi}{\partial f^{2}}+\sigma\Gamma\frac{\partial^{2}\psi}{\partial\bar{u}\partial f}\right)+\\ v(f)\left(\frac{\partial\psi}{\partial\bar{u}}-\psi\right)+\left(\mu_{f}-\frac{(\mu-r)\Gamma}{(\sigma-f)}\right)\frac{\partial\psi}{\partial f}=0.$$

(35)

Now, by performing the following transformation

$$\left\{\begin{array}[]{l}\bar{x}=\frac{1}{2}\left(\frac{\bar{u}}{\sigma}+\frac{f}{\Gamma}\right)-\frac{\mu_{f}}{2\Gamma}t\\ \bar{y}=\frac{1}{2}\left(\frac{\bar{u}}{\sigma}-\frac{f}{\Gamma}\right)+\frac{\mu_{f}}{2\Gamma}t\\ \tau=T-t,\end{array}\right.$$

(36)

one arrives to

$$-\frac{\partial\psi}{\partial\tau}+\frac{1}{2}\frac{\partial^{2}\psi}{\partial\bar{x}^{2}}+\left[\frac{1}{2\sigma}v(f)-\frac{1}{2}\frac{(\mu-r)}{(\sigma-f)}\right]\frac{\partial\psi}{\partial\bar{x}}+\left[\frac{1}{2\sigma}v(f)+\frac{1}{2}\frac{(\mu-r)}{(\sigma-f)}\right]\frac{\partial\psi}{\partial\bar{y}}-v(f)\psi=0,$$

(37)

where $f$ denotes the function

$$f=f(\bar{x},\bar{y},\tau)=\Gamma\left(\bar{x}-\bar{y}-\frac{\mu_{f}}{\Gamma}(T-\tau)\right).$$

(38)

Now, by replacing $v(f)$ explicitly one has that

$$-\frac{\partial\psi}{\partial\tau}+\frac{1}{2}\frac{\partial^{2}\psi}{\partial\bar{x}^{2}}+\left[\frac{(r-\mu)}{2\sigma}\frac{(1+f/\sigma)}{(1-f/\sigma)}\right]\frac{\partial\psi}{\partial\bar{x}}+\left[-\frac{(r-\mu)}{2\sigma}\right]\frac{\partial\psi}{\partial\bar{y}}-\frac{\left(r-\mu\right)}{(1-f/\sigma)}(f/\sigma)\ \psi=0,$$

(39)

One can now consider the “weak” and “strong” limits of (39).

For the lognormal bubble, the underlying $S$–dynamics are the same as the Gaussian but for $f$ one takes instead

$$\begin{array}[]{l}\mu_{f}=f\ \bar{\mu}_{f}\\ \Gamma=f\ \bar{\Gamma},\end{array}$$

(56)

where $\bar{\mu}_{f}$ and $\bar{\Gamma}$ are constants. Consequently, (10) becomes

$$df=\bar{\mu}_{f}fdt+f\bar{\Gamma}dW.$$

(57)

In this case, both the underlying asset and the stochastic bubble have lognormal dynamics. For this case, equation (22) becomes

$\displaystyle\frac{\partial V}{\partial t}+\frac{1}{2}\sigma^{2}S^{2}\frac{\partial^{2}V}{\partial S^{2}}+\frac{1}{2}\bar{\Gamma}^{2}f^{2}\frac{\partial^{2}V}{\partial f^{2}}+\sigma\bar{\Gamma}Sf\frac{\partial^{2}V}{\partial S\partial f}+\left(r+v(f)\right)\left[S\frac{\partial V}{\partial S}-V\right]+\left(\bar{\mu}_{f}-\frac{\left(\mu-r\right)}{(\sigma-f)}\bar{\Gamma}\right)f\frac{\partial V}{\partial f}=0,$

(58)

Now by taking the coordinate transformation

$$\left\{\begin{array}[]{ll}\bar{u}&=\ln S-\left(r-\frac{1}{2}\sigma^{2}\right)t\\ \bar{v}&=\ln f-\left(\bar{\mu}_{f}-\frac{1}{2}\bar{\Gamma}^{2}\right)t\\ t&=t,\end{array}\right.$$

(59)

and defining

$$V(\bar{u},\bar{v},t)=e^{-r(T-t)}\psi(\bar{u},\bar{v},t),$$

(60)

equation (58) maps to

	$\displaystyle\frac{\partial\psi}{\partial t}+\frac{1}{2}\sigma\frac{\partial^{2}\psi}{\partial\bar{u}^{2}}+\frac{1}{2}\bar{\Gamma}^{2}\frac{\partial^{2}\psi}{\partial\bar{v}^{2}}+\sigma\Gamma\frac{\partial^{2}\psi}{\partial\bar{u}\partial\bar{v}}$		(61)
	$\displaystyle+v(f)\left(\frac{\partial\psi}{\partial\bar{u}}-\psi\right)-\frac{(\mu-r)}{(\sigma-f)}\bar{\Gamma}\frac{\partial\psi}{\partial\bar{v}}=0.$

Now, by doing the following transformation

$$\left\{\begin{array}[]{ll}x=\frac{1}{2}\left(\frac{\bar{u}}{\sigma}+\frac{\bar{v}}{\bar{\Gamma}}\right)&\\ \\ y=\frac{1}{2}\left(\frac{\bar{u}}{\sigma}-\frac{\bar{v}}{\bar{\Gamma}}\right)&\\ \\ \tau=T-t,\end{array}\right.$$

(62)

the equation (61) gets

$$-\frac{\partial\psi}{\partial\tau}+\frac{1}{2}\frac{\partial^{2}\psi}{\partial x^{2}}+\left(\frac{1}{2\sigma}v(f)-\frac{1}{2}\frac{(\mu-r)}{(\sigma-f)}\right)\frac{\partial\psi}{\partial x}\\ +\left(\frac{1}{2\sigma}v(f)+\frac{1}{2}\frac{(\mu-r)}{(\sigma-f)}\right)\frac{\partial\psi}{\partial y}=0.$$

(63)

where $f$ denotes the function

$$f=f(x,y,\tau)=e^{\bar{\Gamma}(x-y)+\left(\bar{\mu}_{f}-\frac{1}{2}\bar{\Gamma}^{2}\right)(T-\tau)}.$$

(64)

By replacing $v(f)$, one finally obtains

$$-\frac{\partial\psi}{\partial\tau}+\frac{1}{2}\frac{\partial^{2}\psi}{\partial x^{2}}+\left[\frac{(r-\mu)}{2\sigma}\frac{(1+f/\sigma)}{(1-f/\sigma)}\right]\frac{\partial\psi}{\partial x}-\frac{(r-\mu)}{2\sigma}\frac{\partial\psi}{\partial y}=0.$$

(65)

The asymptotic equations (42), (49), (54), (68) and (71) are particular cases of the generic equation

$$-\frac{\partial\psi}{\partial\tau}+\frac{1}{2}\frac{\partial^{2}\psi}{\partial x^{2}}+\alpha_{x}\frac{\partial\psi}{\partial x}+\alpha_{y}\frac{\partial\psi}{\partial y}=0,$$

(72)

where $\alpha_{x}$ and $\alpha_{y}$ are constants. In fact, the propagator of (72) is

$$P(x,y,\tau)=\frac{1}{\sqrt{2\pi\tau}}\ e^{-\frac{(x+\alpha_{x}\tau)^{2}}{2\tau}}\delta\left(y+\alpha_{y}\tau\right),$$

(73)

where $\delta(x)$ is the Dirac’s delta function. So, if $\Phi(x,y)$ is some initial condition for equation (72), then its solution is

$$\psi(x,y,\tau)=\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}P(x-x^{\prime},y-y^{\prime},\tau)\ \Phi(x^{\prime},y^{\prime})\ dx^{\prime}dy^{\prime},$$

(74)

that is

$$\psi(x,y,\tau)=\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}\frac{1}{\sqrt{2\pi\tau}}\ e^{-\frac{(x-x^{\prime}+\alpha_{x}\tau)^{2}}{2\tau}}\delta\left(y-y^{\prime}+\alpha_{y}\tau\right)\ \Phi(x^{\prime},y^{\prime})\ dx^{\prime}dy^{\prime}.$$

(75)