On Optimal Pricing Model for Multiple Dealers in a Competitive Market

Abstract

In this paper, the optimal pricing strategy in Avellande and Stoikov (Quant. Finance 8:217–224, 2008) for a monopolistic dealer is extended to a general situation where multiple dealers are present in a competitive market. The dealers’ trading intensities, their optimal bid and ask prices and therefore their spreads are derived when the dealers are informed the severity of the competition. The effects of various parameters on the bid-ask quotes and profits of the dealers in the competitive market are also discussed. This study gives some insights on the average spread, profits of the dealers in the competitive trading environment.

Appendix

1.1 Appendix A

Proof

Consider first the arrival rate of market order. When $N=1$,

$$\begin{aligned} \lambda ^a = \lambda ^a(\delta ^a)=A e^{-k\delta ^a}. \end{aligned}$$

By condition (i), when $N=2$

$$\begin{aligned} \lambda ^a= \lambda ^a(\delta _1^a,\delta _2^a)=f(\delta _1^a)^{\beta _1}f(\delta _2^a)^{\beta _2} \end{aligned}$$

where $(\beta _1+\beta _2=1)$; when $\delta _1^a=\delta _2^a=\delta ^a$, this is equivalent to the case when $N=1$, that is

$$\begin{aligned} f(\delta ^a)^{\beta _1}f(\delta ^a)^{\beta _2}=\lambda ^a(\delta ^a,\delta ^a)=\lambda ^a(\delta ^a)=A e^{-k \delta ^a} \end{aligned}$$

Consequently

$$\begin{aligned} f(\delta ^a)=A e^{-k \delta ^a} \quad \mathrm{and} \quad \lambda ^a=\lambda ^a(\delta _1^a,\delta _2^a)=A e^{-k(\beta _1 \delta _1^a+\beta _2 \delta _2^a)}. \end{aligned}$$

By Condition (i), for any N,

$$\begin{aligned} \lambda ^a=\lambda ^a(\delta _1^a,\ldots ,\delta _n^a)=f(\delta _1^a)^{\beta _1} \ldots f(\delta _N^a)^{\beta _N} \end{aligned}$$

where $\beta _1+\cdots +\beta _N=1$. When $\delta _1^a=\cdots =\delta _N^a=\delta ^a$ this situation is equivalent to the case when $N=1$, i.e.,

$$\begin{aligned} \lambda ^a(\delta ^a,\ldots , \delta ^a)=f(\delta ^a)^{\beta _1} \ldots f(\delta ^a)^{\beta _n}=\lambda ^a(\delta ^a)=A e^{-k \delta ^a} \end{aligned}$$

and $f(\delta ^a)=A e^{-k \delta ^a}$. Consequently,

$$\begin{aligned} \lambda ^a=\lambda ^a(\delta _1^a,\ldots ,\delta _N^a) =A e^{-k(\beta _1 \delta _1^a +\cdots +\beta _N\delta _N^a)}. \end{aligned}$$

To calculate the arrival rate of buy and sell orders that will reach the dealer, we need the following information:

• (i) the overall frequency of market orders;

• (ii) the distribution of market orders’ size;

• (iii) the temporary impact of a large market order.

From above, we are given one of the estimation of market orders’ frequency. For the other conditions, from a lot of studies, see, for instance (Gabaix et al. 2006; Gopikrishnan et al. 2000; Maslow and Mills 2001, we have some statistical properties of the limit order book, such as, the distribution of the size of market orders Q obeys a power law:

$$\begin{aligned} f^Q(x)\varpropto x^{-1-\alpha } \end{aligned}$$

and the market impact follows a “$\log$ law” Bouchaud et al. (2002):

$$\begin{aligned} \Delta p \varpropto \frac{1}{K(N)} \ln (Q), \end{aligned}$$

where K(N) is an increasing function in N which satisfies

$$\begin{aligned} K(1)= 0 \quad \mathrm{and} \quad \lim _{N\rightarrow \infty }K(N)=c \end{aligned}$$

with c being a positive constant. This means if the market consists of a single dealer, its order will be executed immediately whenever there is a counterpart arriving at the system; and if there are infinite many participants in the market, then the market will achieve an equilibrium. Here we recall that $\Delta p = p^Q -s$ where $p^Q$ is the price of the highest limit order executed in the trade and s is stock mid-price. Aggregating the information of limit order book’s statistical properties, we have

$$\begin{aligned} \begin{array}{lll} \lambda _i^a(\delta _i^a)&{}=&{}\lambda ^a \cdot P(\Delta p>\delta _i^a)\\ &{} \varpropto &{}\lambda ^a \cdot P(\ln (Q)>K(N)\delta _i^a)\\ &{}=&{}\lambda ^a \cdot p(Q>\exp (K(N)\delta _i^a))\\ \end{array} \end{aligned}$$

and

$$\begin{aligned} \begin{array}{lll} \displaystyle p(Q>\exp (K(N)\delta _i^a))&{} \varpropto &{} \displaystyle \int _{\exp {(K(N)\delta _i^a)}}^{\infty } x^{-1-\alpha }dx\\ &{}=&{}\displaystyle \frac{1}{\alpha }\exp (-\alpha K(N)\delta _i^a). \end{array} \end{aligned}$$

Therefore, for some constant ${\hat{A}}$, we have

$$\begin{aligned} \lambda _i^a(\delta _i^a)= {\hat{A}} \exp (-k(\beta _1\delta _1^a+\cdots +\beta _N\delta _N^a)) \cdot \exp (-\alpha K(N)\delta _i^a). \end{aligned}$$

1.2 Appendix B

Proof

We note that there is no trading in the period $(t_n,T)$,

$$\begin{aligned} E\left[ \exp (-\gamma _i\int _{t_n}^T (\delta _u^{i,a}dN_u^{i,a}+\delta _u^{i,b}dN_u^{i,b}))\Big |\mathcal {F}_{n-1}\right] =1. \end{aligned}$$

Thus the expected exponential utility of Dealer i’s ultimate wealth equals

$$\begin{aligned}&E\left[ -\exp (-\gamma _i(X_T^i+q_T^iS_T))| \mathcal {F}_{n-1}\right] \\&\quad =-\left\{ \lambda _{n-1}^{i,a}\Delta t_{n-1}\exp (-\gamma _i\left( x_{n-1}^i+q_{n-1}^is_{n-1}+\delta _{n-1}^{i,a}\right) \right. \\&\qquad \left. \times \exp \left( \frac{\gamma _i^2\sigma ^2(q_{n-1}^i-1)^2(T-t_{n-1})}{2}\right) \right. \\&\qquad \left. \lambda _{n-1}^{i,b}\Delta t_{n-1}\exp (-\gamma _i(x_{n-1}^i+q_{n-1}^is_{n-1}+\delta _{n-1}^{i,b}))\right. \\&\qquad \left. \times \exp \left( \frac{\gamma _i^2\sigma ^2(q_{n-1}^i+1)^2(T-t_{n-1})}{2}\right) \right. \\&\qquad \left. \times \left[ 1-\lambda _{n-1}^{i,a}\Delta t_{n-1} -\lambda _{n-1}^{i,b}\Delta t_{n-1}\right] \exp (-\gamma _i(x_{n-1}^i+q_{n-1}^is_{n-1}))\right. \\&\qquad \left. \times \exp \left( \frac{\gamma _i^2\sigma ^2(q_{n-1}^i)^2(T-t_{n-1})}{2}\right) \right\} . \end{aligned}$$

By considering the first-order optimality conditions, we can obtain the optimal bid and ask quotes as follows:

$$\begin{aligned} \left\{ \begin{array}{lll} \delta _{n-1}^{i,b}&{}=&{}\displaystyle \frac{1}{\gamma _i}\ln \left( 1+\frac{\gamma _i}{k\beta _i+{\hat{c}}(1-\frac{1}{N})}\right) + \frac{\gamma _i \sigma ^2 (T-t_{n-1})}{2}(2q_{n-1}^i+1)\\ \delta _{n-1}^{i,a}&{}=&{}\displaystyle \frac{1}{\gamma _i}\ln \left( 1+\frac{\gamma _i}{k\beta _i+{\hat{c}}(1-\frac{1}{N})}\right) + \frac{\gamma _i \sigma ^2 (T-t_{n-1})}{2}(-2q_{n-1}^i+1). \end{array} \right. \end{aligned}$$

(71)

Substituting the optimal bid and ask quotes into the expected utility function, one can get the utility for Dealer i. $\square$

1.3 Appendix C

Proof

We apply the principle of Dynamic Programming (DP) to Dealer i’s utility function, and obtain

$$\begin{aligned}&V^i\left( s_{n-2},\gamma _i,x_{n-2}^i, q_{n-2}^i,\gamma _j,q_{n-2}^j(j\ne i), t_{n-2}\right) \nonumber \\&\quad =\displaystyle \max _{\delta _{n-2}^{i,a},\delta _{n-2}^{i,b},\delta _{n-1}^{i,a},\delta _{n-1}^{i,b}} E\left[ -\exp (-\gamma _i(X_T^i+q_T^iS_T))\Big |\mathcal {F}_{n-2}\right] \nonumber \\&\quad =\displaystyle \max _{\delta _{n-2}^{i,a},\delta _{n-2}^{i,b},\delta _{n-1}^{i,a},\delta _{n-1}^{i,b}} E\left[ E[-\exp (-\gamma _i(X_T^i+q_T^iS_T))\big |\mathcal {F}_{n-1}]\Big |\mathcal {F}_{n-2}\right] \nonumber \\&\quad =\displaystyle \max _{\delta _{n-2}^{i,a},\delta _{n-2}^{i,b}}-\exp (-\gamma _i(x_{n-2}^i+q_{n-2}^is_{n-2})) \nonumber \\&\qquad \times \Big \{\lambda _{n-2}^{i,a}\Delta t_{n-2}\Big (\exp (-\gamma _i\delta _{n-2}^{i,a})\exp \left( \frac{\gamma _i^2\sigma ^2 (q_{n-2}^i-1)^2\Delta t_{n-2}}{2}\right) g_{n-1}^i(q_{n-2}^i-1,t_{n-1})\Big ) \nonumber \\&\qquad +\, \lambda _{n-2}^{i,b}\Delta t_{n-2}\Big (\exp (-\gamma _i\delta _{n-2}^{i,b})\exp \left( \frac{\gamma _i^2\sigma ^2 (q_{n-2}^i+1)^2\Delta t_{n-2}}{2}\right) g_{n-1}^i(q_{n-2}^i+1,t_{n-1})\Big ) \nonumber \\&\qquad + \left[ 1-\lambda _{n-2}^{i,a}\Delta t_{n-2}-\lambda _{n-2}^{i,b}\Delta t_{n-2}\right] \Big (\exp \left( \frac{\gamma _i^2\sigma ^2 (q_{n-2}^i)^2\Delta t_{n-2}}{2}\right) g_{n-1}^i(q_{n-2}^i,t_{n-1})\Big )\Big \}\nonumber \\&\quad =\displaystyle \max _{\delta _{n-2}^{i,a},\delta _{n-2}^{i,b}}-\exp (-\gamma _i(x_{n-2}^i+q_{n-2}^is_{n-2})) \exp \left( \frac{\gamma _i^2\sigma ^2 (q_{n-2}^i)^2(T-t_{n-2})}{2}\right) h_{n-1}^i \nonumber \\&\qquad \times \Big \{\lambda _{n-2}^{i,a}\Delta t_{n-2}\exp (-\gamma _i\delta _{n-2}^{i,a})\exp \left( \frac{\gamma _i^2\sigma ^2 (-2q_{n-2}^i+1)(T-t_{n-2})}{2}\right) \nonumber \\&\qquad +\,\lambda _{n-2}^{i,b}\Delta t_{n-2}\exp (-\gamma _i\delta _{n-2}^{i,b})\exp \left( \frac{\gamma _i^2\sigma ^2 (-2q_{n-2}^i+1)(T-t_{n-2})}{2}\right) \nonumber \\&\qquad +\,1-\lambda _{n-2}^{i,a}\Delta t_{n-2}-\lambda _{n-2}^{i,b}\Delta t_{n-2}\Big \}. \end{aligned}$$

(72)

By considering the first order optimality conditions, we obtain Dealer i’s optimal ask quote as follows:

$$\begin{aligned} \begin{array}{lll} \delta _{n-2}^{i,a}= & {} \displaystyle \frac{1}{\gamma _i}\ln \left( 1+\frac{\gamma _i}{k\beta _i+{\hat{c}}(1-\frac{1}{N})}\right) + \frac{\gamma _i \sigma ^2 (T-t_{n-2})}{2}(-2q_{n-2}^i+1). \end{array} \end{aligned}$$

(73)

Similarly, we can give his bid quote

$$\begin{aligned} \begin{array}{lll} \delta _{n-2}^{i,b}= & {} \displaystyle \frac{1}{\gamma _i}\ln \left( 1+\frac{\gamma _i}{k\beta _i+{\hat{c}}(1-\frac{1}{N})}\right) + \frac{\gamma _i \sigma ^2 (T-t_{n-2})}{2}(2q_{n-2}^i+1). \end{array} \end{aligned}$$

(74)

Substituting the optimal bid and ask quotes into the utility function, one can obtain Dealer i’s utility function:

$$\begin{aligned} \begin{array}{lll} &{}&{}V^i\left( s_{n-2},\gamma _i, x_{n-2}^i q_{n-2}^i,\gamma _j,q_{n-2}^j(j\ne i), t_{n-2}\right) \\ &{}&{}\quad =\displaystyle -\exp (-\gamma _i(x_{n-2}^i+q_{n-2}^is_{n-2})) \exp \left( \frac{\gamma _i^2\sigma ^2(q_{n-2}^i)^2(T-t_{n-2})}{2}\right) \\ &{}&{}\qquad \times \left[ 1-\frac{\gamma _i\Delta t_{n-2}}{k\beta _i+{\hat{c}}(1-\frac{1}{N})}\left( \lambda _{n-2}^{i,a}+ \lambda _{n-2}^{i,b}\right) \right] h_{n-1}^i. \end{array} \end{aligned}$$

(75)

$\square$