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Markets with random lifetimes and private values: mean reversion and option to trade

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Abstract

We consider a market in which traders arrive at random times, with random private values for the single-traded asset. A trader’s optimal trading decision is formulated in terms of exercising the option to trade one unit of the asset at the optimal stopping time. We solve the optimal stopping problem under the assumption that the market price follows a mean-reverting diffusion process. The model is calibrated to experimental data taken from Alton and Plott (Principles of continuous price determination in an experimental environment with flows of random arrivals and departures. Working paper, Caltech, 2010), resulting in a very good fit. In particular, the estimated long-term mean of the traded prices is close to the theoretical long-term mean at which the expected number of buys is equal to the expected number of sells. We call that value long-term competitive equilibrium, extending the concept of flow competitive equilibrium of Alton and Plott (Principles of continuous price determination in an experimental environment with flows of random arrivals and departures. Working paper, Caltech, 2010).

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Fig. 1

Notes

  1. 1.

    It should be mentioned that we use the Ornstein–Uhlenbeck process for the purpose of illustrating our approach, but one could also use other tractable mean-reverting processes. In practice, a statistical data analysis should be used to decide which model to use.

  2. 2.

    It should be noted that the reason why we opted for fitting the model to experimental data rather than real-market data is that in the experiments the private values are known, in fact, chosen by the experimenter, while it would be hard or impossible to estimate what they are in real markets. However, see the Sect. 5 for a possible future research on reverse engineering the distributions of private values from real-market data.

  3. 3.

    The iid assumptions and exponential distribution are assumed for tractability. Our aim is not to have a realistic model of real-life markets, but to test, in a simple model, whether the prices that are formed by trading are consistent with risk-neutral traders maximizing their expected profit/loss.

  4. 4.

    See, e.g., http://mathworld.wolfram.com/ParabolicCylinderFunction.html.

  5. 5.

    Functions \(\phi \) and \(\psi \) are the general solutions of the ODE (3.2), for \(X>\log K\).

  6. 6.

    The phenomenon that individually participants in experiments do not behave optimally and nevertheless in aggregate the price formation is not far away to what it would be if they did, has been found before in experimental asset pricing, for the CAPM model; see, e.g., Bossaerts et al. (2007).

  7. 7.

    In doing this, we discard initial data points which are far away from “equilibrium price,” as this is a period in which the participants are basically learning. Moreover, we smooth out the price values grouped in narrow time intervals, because our diffusion process would not be a good fit for the big jumps in price that often occur during those intervals.

  8. 8.

    See, e.g., http://mathworld.wolfram.com/ParabolicCylinderFunction.html.

References

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Author information

Correspondence to Jakša Cvitanić.

Additional information

The research of J. Cvitanić is supported in part by NSF Grant DMS 10-08219.

Appendix

Appendix

Proof of Proposition 3.1

As in Carr (1998), \(P(X)\) is the Laplace transform of the standard American put price, and thus satisfies the ordinary differential equation (ODE)

$$\begin{aligned} \frac{\sigma ^2}{2}P''(X)+\kappa (\theta -X)P'(X)-rP(X)=\lambda \left[ P(X)-\left( K-e^X\right) ^+\right] ,\quad X>\underline{X}\nonumber \\ \end{aligned}$$
(6.1)

subject to the boundary conditions

$$\begin{aligned} \lim _{X\rightarrow \infty }P(X)=0,\quad \lim _{X\rightarrow \underline{X}}P(X)=K-e^{\underline{X}},\quad \lim _{X\rightarrow \underline{X}}P'(X)=-e^{\underline{X}} \end{aligned}$$
(6.2)

In the region \(X>\log K\equiv \underline{X_0}\), the ODE is reduced to homogenous ODE

$$\begin{aligned} \frac{\sigma ^2}{2}P''(X)+\kappa (\theta -X)P'(X)-(r+\lambda )P(X)=0,\quad X>\underline{X_0} \end{aligned}$$
(6.3)

Introducing the change in variables

$$\begin{aligned} z=\dfrac{\sqrt{2\kappa }}{\sigma }(X-\theta ) \end{aligned}$$

and letting \(P(X)=e^{z^2/4}\omega (z)\), Eq. (6.3) becomes

$$\begin{aligned} \omega ''(z)+\left( \frac{1}{2}-\nu -\frac{z^2}{4}\right) \omega (z)=0 \end{aligned}$$
(6.4)

with

$$\begin{aligned} \nu =(r+\lambda )/\kappa . \end{aligned}$$

The general solution of (6.4) can be represented as the linear combination of the so-called parabolic cylinder functions:Footnote 8

$$\begin{aligned} \omega (z)=CD_{-\nu }(z)+ED_{-\nu }(-z). \end{aligned}$$
(6.5)

From \(\lim _{X\rightarrow \infty }P(X)=0\), we get \(E=0\). Therefore,

$$\begin{aligned} P(X)=Ce^{z^2/4}D_{-\nu }(z),\quad X>\underline{X_0}\equiv \log K \end{aligned}$$
(6.6)

In the region \(\underline{X}<X<\underline{X_0}\), the solution can be written as the general solution plus a particular solution,

$$\begin{aligned} P(X)=A\phi (X)+B\psi (X)+Q(X), \end{aligned}$$
(6.7)

where \(Q(X)\) is a particular solution that can be taken as in (3.8) (see, e.g., Johnson (2006)). From the boundary conditions Eq. (6.2) at \(X=\underline{X}\) and using the continuity of \(P(X)\) and \(P'(X)\) at \(X=\underline{X_0}\), it is not difficult to obtain \(B=0\), and \(A,\,C\), and \(\underline{X}\) as in the statement of the proposition.

Proof of Proposition 3.2

Because the maturity date is exponential and independent of process \(X\), we have

$$\begin{aligned} P_{min}(x~|~{\underline{X}},\lambda )&= \lambda \int \limits _0^\infty e^{-\lambda t} F_{\underline{X}}(x~|~t)~\hbox {d}t\end{aligned}$$
(6.8)
$$\begin{aligned}&= -\left[ F_{\underline{X}}(x~|~t) e^{-\lambda t}|_0^\infty -\int \limits _0^\infty e^{-\lambda t}f_{\underline{X}}(x~|~t)~\hbox {d}t\right] \end{aligned}$$
(6.9)
$$\begin{aligned}&= \int \limits _0^\infty e^{-\lambda t}f_{\underline{X}}(x~|~t)~dt\end{aligned}$$
(6.10)
$$\begin{aligned}&= \hat{f}_{\underline{X}}(x~|~\lambda ) \end{aligned}$$
(6.11)

Similarly,

$$\begin{aligned} P_{\mathrm{max}}(x~|~{\overline{X}},\lambda )=\hat{f}_{\overline{X}}(x~|~\lambda ). \end{aligned}$$
(6.12)

For our Ornstein–Uhlenbeck process \(X\), function \(p(x~|~y,t)\) satisfies the PDE

$$\begin{aligned} \frac{\partial p}{\partial t}=\kappa (\theta -x)\frac{\partial p}{\partial x}+\frac{\sigma ^2}{2}\frac{\partial ^2 p}{\partial x^2} \end{aligned}$$
(6.13)

with initial and boundary conditions \(p(\infty ~|~y,t)=p(-\infty ~|~y,t)=0,\,p(x~|~y,0)=\delta (x-y)\). Taking the Laplace transform of Eq. (6.13), we get

$$\begin{aligned} \lambda \hat{p} -\delta (x-y)=\kappa (\theta -x)\frac{\mathrm{d}\hat{p}}{\mathrm{d} x}+\frac{\sigma ^2}{2}\frac{\mathrm{d}^2\hat{p}}{\mathrm{d}x^2} \end{aligned}$$
(6.14)

Therefore, we have

$$\begin{aligned} \hat{p}(x~|~y,\lambda )=\left\{ \begin{array}{l@{\quad }l} \psi (x)\phi (y), &{} y\ge x\\ \phi (x)\psi (y), &{} y\le x \end{array}\right. \end{aligned}$$
(6.15)

up to a constant factor. The result follows now from Theorem 3.1 in Darling and Siegert (1953).

Proof of Proposition 3.3

Note that we can write our Ornstein–Uhlenbeck process \(X\) in the form

$$\begin{aligned} X(t)=xe^{-\kappa t}+\theta (1-e^{-\kappa t})+\sigma \int \limits _0^t e^{-\kappa (t-u)}~\mathrm{d}W(u) \end{aligned}$$
(6.16)

and that there is a Brownian motion \(B(t)\) such that

$$\begin{aligned} \int \limits _0^t e^{\kappa u}\mathrm{d}W(u) = \frac{1}{\sqrt{2\kappa }}B(e^{2\kappa t}-1). \end{aligned}$$
(6.17)

Therefore, we have

$$\begin{aligned} X(t)=xe^{-\kappa t}+\theta (1-e^{-\kappa t})+\frac{\sigma }{\sqrt{2\kappa }}e^{-\kappa t}B(e^{2\kappa t}-1),~~X(0)=x \end{aligned}$$
(6.18)

It is then not difficult to show that the transition density is given by

$$\begin{aligned} p(x~|~y,t)=\sqrt{\frac{\kappa }{\pi \sigma ^2}\frac{1}{(1-e^{-2\kappa t)}}}\exp {\left\{ -\frac{\kappa }{\sigma ^2}\frac{\left[ y-xe^{-\kappa t}-\theta (1-e^{-\kappa t})\right] ^2}{1-e^{-2\kappa t}}\right\} }\nonumber \\ \end{aligned}$$
(6.19)

Buyer \(i\) lives during random interval \([\tau ^B_i,\tau ^B_i+\Delta \tau ^B_i]\) with

$$\begin{aligned} Pr\{\tau _i^B\in \mathrm{d}t\}=\lambda _B e^{-\lambda _B t}\mathrm{d}t,\quad \quad Pr\{\Delta \tau ^B_i\in \mathrm{d}t\}=\rho _B e^{-\rho _B t}\mathrm{d}t \end{aligned}$$
(6.20)

Then, the probability that the minimum of \(X(t)\) is less than \(\underline{X}\) during a buyer’s lifetime is

$$\begin{aligned} \mathbf {P}_{\mathrm{min}}(x~|~\underline{X},\lambda _B,\rho _B)&= \int \limits _0^\infty \lambda _B e^{-\lambda _B\tau }~\hbox {d}\tau \nonumber \\&\times \left[ \int \limits _{\underline{X}}^\infty P_{\mathrm{min}}(y~|~\underline{X},\rho _B) p(x~|~y,\tau )~\mathrm{d}y+\int \limits _{-\infty }^{\underline{X}} p(x~|~y,\tau )~\mathrm{d}y\right] \nonumber \\&= \int \limits _{\underline{X}}^\infty \lambda _BP_{\mathrm{min}}(y~|~\underline{X},\rho _B)\hat{p}(x~|~y,\lambda _B)~\mathrm{d}y\nonumber \\&+\int \limits _{-\infty }^{\underline{X}}\lambda _B\hat{p}(x~|~y,\lambda _B)~\mathrm{d}y, \end{aligned}$$
(6.21)

where we use the fact that if \(X(\tau _i^B)\le \underline{X}\), the buyer will make a transaction immediately after she enters the market, and if \(X(\tau _i^B)\ge \underline{X}\), there is probability \(P_{\mathrm{min}}(y~|~{\underline{X}},\rho _B)\) that \(X(t)\) will hit \(\underline{X}\) during the random period. The expression for \( P_{\mathrm{min}}(y~|~{\underline{X}},\rho _B)\) follows from the previous section. The corresponding expression for the seller follows using the same method.

Next, we calculate the Laplace Transform of \(p(x~|~y,t)\),

$$\begin{aligned} \hat{p}(x~|~y,\lambda )=\int \limits _0^\infty e^{-\lambda t}p(x~|~y,t)~\mathrm{d}t \end{aligned}$$
(6.22)

We know that \(p(x~|~y,t)\) satisfies Kolmogorov equation

$$\begin{aligned} \frac{\partial f}{\partial t}=\kappa (\theta -x)\frac{\partial f}{\partial x}+\frac{\sigma ^2}{2}\frac{\partial ^2 f}{\partial x^2} \end{aligned}$$
(6.23)

subject to \(f(\infty ~|~y,t)=f(-\infty ~|~y,t)=0\) and \(f(x~|~y,0)=\delta (x-y)\). Taking Laplace transform on both sides of Eq. (6.23), we get

$$\begin{aligned} \lambda \hat{p}-\delta (x-y)=\kappa (\theta -x)\frac{\mathrm{d}\hat{p}}{\mathrm{d}x}+\frac{\sigma ^2}{2}\frac{\mathrm{d}^2\hat{p}}{\mathrm{d}x^2} \end{aligned}$$
(6.24)

Letting \(z\equiv \dfrac{\sqrt{2\kappa }(x-\theta )}{\sigma }\) and \(z_y\equiv \dfrac{\sqrt{2\kappa }(y-\theta )}{\sigma }\), Eq. (6.24) becomes

$$\begin{aligned} \frac{\mathrm{d}^2\hat{p}}{\mathrm{d}z^2}-z\frac{\mathrm{d}\hat{p}}{\mathrm{d}z}-\frac{\lambda }{k}\hat{p}=-\sqrt{\frac{2}{\kappa \sigma ^2}}~\delta (z-z_y) \end{aligned}$$
(6.25)

Imposing the boundary conditions, we get

$$\begin{aligned} \hat{p}=\left\{ \begin{array}{ll} Ae^{z^2/4}D_{-\nu }(z),\quad z\ge z_y\\ Be^{z^2/4}D_{-\nu }(-z),\quad z\le z_y \end{array}\right. \end{aligned}$$
(6.26)

From Eq. (6.25), we know that \(\dfrac{d\hat{p}}{dz}\) cannot be continuous. Integrating both sides of Eq. (6.25) from \(z_y^-\) to \(z_y^+\), and because \(\hat{p}\) is continuous, it is straightforward to get

$$\begin{aligned} \frac{\mathrm{d}\hat{p}}{\mathrm{d}z}({z_y^+})-\frac{\mathrm{d}\hat{p}}{\mathrm{d}z}({z_y^-})=-\sqrt{\frac{2}{\kappa \sigma ^2}} \end{aligned}$$
(6.27)

With Eq. (6.27) and \(\hat{p}\) continuous, we get

$$\begin{aligned} \hat{p}(x~|~y,\lambda )=\left\{ \begin{array}{ll} \sqrt{\frac{2}{\kappa \sigma ^2}}e^{-z_y^2/4}\frac{D_{-\nu }(-z_y)}{D_{1-\nu }(z_y)D_{-\nu }(-z_y)+D_{1-\nu }(-z_y)D_{-\nu }(z_y)}e^{z^2/4}D_{-\nu }(z), \quad z\ge z_y\\ \sqrt{\frac{2}{\kappa \sigma ^2}}e^{-z_y^2/4}\frac{D_{-\nu }(z_y)}{D_{1-\nu }(z_y)D_{-\nu }(-z_y)+D_{1-\nu }(-z_y)D_{-\nu }(z_y)}e^{z^2/4}D_{-\nu }(-z), \quad z\le z_y \end{array}\right. \nonumber \\ \end{aligned}$$
(6.28)

with \(\nu =\lambda /\kappa \). We can further simplify the answer by calculating \(T(\nu ,z)\equiv D_{1-\nu }(z)D_{-\nu }(-z)+D_{1-\nu }(-z)D_{-\nu }(z)\). First, we prove \(T(\nu ,z)\) is independent of \(z\):

$$\begin{aligned} \frac{\mathrm{d} T(\nu ,z)}{\mathrm{d}z}&= D'_{1-\nu }(z)D_{-\nu }(-z)-D_{1-\nu }(z)D'_{-\nu }(-z)\nonumber \\&-D'_{1-\nu }(-z)D_{-\nu }(z)+D_{1-\nu }(-z)D'_{-\nu }(z)\nonumber \\&= D_{1-\nu }(z)\left[ -zD_{-\nu }(-z)+\nu D_{-\nu -1}(-z)\right] \nonumber \\&-D_{1-\nu }(-z)\left[ zD_{-\nu }(z)+\nu D_{-\nu -1}(z)\right] \nonumber \\&= D_{1-\nu }(z)D_{1-\nu }(-z)-D_{1-\nu }(-z)D_{1-\nu }(z)=0 \end{aligned}$$
(6.29)

Here, we use the recursion relation for parabolic cylinder functions,

$$\begin{aligned} D_{\nu +1}(z)-zD_{\nu }(z)+\nu D_{\nu -1}(z)&= 0\end{aligned}$$
(6.30)
$$\begin{aligned} D'_{\nu }(z)+\frac{1}{2} zD_{\nu }(z)-\nu D_{\nu -1}(z)&= 0 \end{aligned}$$
(6.31)

From these, it is also not difficult to get

$$\begin{aligned} T(\nu )=\nu T(\nu +1) \end{aligned}$$
(6.32)

(Note we dropped dependence on \(z\) here.) Then, we have

$$\begin{aligned} T(\nu )=\frac{T(1)}{\Gamma (\nu )},\quad \nu >0, \end{aligned}$$
(6.33)

where \(T(1)=\sqrt{2\pi }\). Plugging this result into Eq. (6.28), we get the stated expression for \(\hat{p}\).

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Cvitanić, J., Plott, C. & Tseng, C. Markets with random lifetimes and private values: mean reversion and option to trade. Decisions Econ Finan 38, 1–19 (2015). https://doi.org/10.1007/s10203-014-0155-4

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Keywords

  • Trading with private values
  • Equilibrium price
  • Optimal exercise of options
  • Experimental markets
  • Tick-by-tick trading

JEl Classification

  • G11
  • G12