Insider Trading in Sequential Auction Markets with Risk-aversion and Time-discounting

Transcription

1 Insider Trading in Sequential Auction Markets with Risk-aversion and Time-discounting Paolo Vitale University of Pescara September 2015 ABSTRACT We extend Kyle s (Kyle, 1985) analysis of sequential auction markets to the case in which the insider is risk-averse and discounts her trading profits as her private information is long-lived. We see that time-discounting exacerbates the impact of risk-aversion on the optimal trading strategy of the insider. Ceteris paribus, a larger degree of riskaversion or a smaller time-discount factor induces the informed agent to consume more rapidly her informational advantage increasing the liquidity and efficiency of the securities market. JEL Classification Numbers: C61, G14. Keywords: Risk-aversion, Sequential Auction Markets, Long-lived Private Information. I wish to thank the Editor and two anonymous referees for valuable comments and suggestions. I am alone responsible for the views expressed in the paper and for any errors that may remain. Department of Economics, Università d Annunzio, Viale Pindaro 42, Pescara (Italy); telephone ; webpage: p.vitale@unich.it

2 1 Introduction Kyle (1985) investigates the behavior of an informed trader (insider) in the market for a risky asset. Such an agent possesses an incentive to act strategically and exploit her informational advantage to gain speculative profits from her trading activity. She acts strategically because in choosing the timing and size of her transactions she takes into account the impact that her trades will have on the equilibrium price of the risky asset and hence on her profits. Kyle s seminal model has stimulated an endless list of extensions and contributions. Thus, Holden and Subrahmanyam (1994) introduce multiple insiders and risk-aversion, analyzing the effects of risk-aversion and imperfect competition on insider trading, while Foster and Viswanathan (1996) consider the case in which insiders have different bits of information, investigating the impact of strategic complementarities on the diffusion of private information. Huddart et al. (2001) analyze the effect of trade disclosure on the trading activity of the insiders, while Daher et al. (2014) study the interplay between public and private signals. Chau and Vayanos investigate the impact of an infinite sequence of private signals, while Caldentey and Stacchetti (2010) allow for the random publication of the fundamental value of the risky asset. Finally, Vitale (2012) considers the scenario in which the insider possesses private information on several risky assets. We consider a particular extension of Kyle s model, which combines two different departures from his original formulation: we assume that the insider is risk-averse and that her private information is so long-lived as to require the discounting of her future profits. We deem this extension of Kyle s model important for two reasons. On the one hand, Kyle s assumption that future profits needn t be discounted can be justified either by assuming that, even if she can trade over several rounds of trading, her informational advantage is not very long-lived and it dissipates in a relatively short period of time, such as a day or a week, or by normalizing the interest rate to zero. The latter assumption, however, is not neutral with respect to the insider s attitude towards risk. In fact, the interplay between 1

3 the inter-temporal rate of substitution and risk-aversion is a crucial facet of portfolio theory and has potentially important implications for insider trading and market quality. On the other hand, combining risk-aversion and time-discounting is also a complex analytical endeavor which is worth undertaking. In this respect, we rely on some recent advances in optimal control theory and employ a recursive optimization criterion recently proposed by Vitale (2013). We are then able to investigate what happens to the insider s optimal trading strategy and the characteristics of the market for the risky asset, such as its efficiency and liquidity. An important conclusion of our analysis is that both time-discounting and risk-aversion induce the insider to trade more aggressively, revealing her informational advantage at a greater pace than that observed in Kyle s seminal contribution, and make the market for the risky asset more efficient. In addition, we show that the simultaneous presence of time-discounting and risk-aversion reinforces each other s impact on market quality. Our work is related to that of Holden and Subrahmanyam (1994), as we add timediscounting to their formulation with only one insider. Moreover, our model is also related to the contribution of Caldentey and Stacchetti (2010). In fact, in their extension of Kyle s model the insider receives in any round of trading a private signal on the fundamental value of the risky asset, while there is a positive probability that in any period the fundamental value is publicly announced. Caldentey and Stacchetti then show that this implies that in choosing her optimal trading strategy a risk-neutral insider maximizes the expected value of her discounted future profits. Hence, our contribution can be considered analogous to an extension of their formulation with a risk-averse insider. 2

4 2 Strategic Informed Trading According to Kyle s formulation, in a securities market a market maker trades with a group of customers a numeraire, which pays no return, for a risky asset with an uncertain liquidation value. Clients include an insider, who knows the liquidation value of the risky asset, and a group of liquidity (noise) traders, who trade purely for liquidity reasons. The liquidation value, v, of the risky asset is determined at time 0, before trading in the market starts, and is publicly announced at time 1, when no more trading is possible. Apart from the insider, no one knows the actual realization of v at time 0. However, unconditionally it is normally distributed, v N (µ v, Σ 0 ). This information is common knowledge. Between the instant the liquidation value is realized and that in which it is announced N rounds of trading, in the form of call auctions, are conducted by the market maker. Any call auction is identified by the subscript n and takes place at time t n, with 0 < t 1 <... < t N 1. When auction n is called, the market maker s clients, the liquidity traders and the insider, select their market orders, i.e. the amount of the risky asset they desire to trade. Their orders, x l n and x i n, are batched together and the overall market order, x n = x i n + x l n, is passed to the market maker. He then fixes a transaction price, p n, at which he executes all orders. The market maker cannot observe either the individual orders or the identity of his clients. This permits the insider to exploit over time her informational advantage. As the market maker is risk-neutral, Bertrand competition in the market making industry forces him to set the risky asset s transaction price according to a semi-strong form efficiency condition. Since he just observes the flow of total market orders he receives along the sequence of auctions and since these market orders may contain some information, impounded in the insider s orders, in any auction n the transaction price, p n, is equal to the 3

5 conditional expectation of the risky asset s liquidation value, p n = E[v x 1,..., x n 1, x n ]. (1) The liquidity traders place unpredictable market orders. Thus, in any auction, n, their market order is normally distributed, x l n N (0, σ 2 l t n) with t n = t n t n 1. The random values { x l n} N n=1 are independent of each other and of the liquidation value v. This means that the market orders of the liquidity traders follow a white noise process. In Kyle s original formulation the insider is risk-neutral. As she does not have any initial endowment of the risky asset, risk-neutrality implies that in any auction, n, the insider chooses her market order to maximize the expected value of her aggregate future profits. As no time-discounting is considered in Kyle s formulation these profits are equal to Π n = N k=n π k, where π k denotes the per-auction profits (v p k ) x k. Since the prices charged by the market maker are function of the unpredictable orders of the liquidity traders, these profits are uncertain. Thus, when auction n is called, the insider solves the problem x i n = argmax E[Π n p 1, p 2,..., p n 1, v]. (2) Private information may last longer than a day or a week, that is it may last longer than the span of time usually associated with the interval [0, 1] within which trading takes place in Kyle s formulation. When private information is very long-lived the insider needs to discount her future profits when her performance is assessed on a per-period basis. 1 Under risk-neutrality, this is dealt with straightforwardly, by introducing a time-discount factor δ (with 0 < δ < 1) and by assuming that the insider chooses her optimal market order in auction n by maximizing the expectation of the discounted value of her future profits. Thus, assume the N auctions are equally spaced in the interval [0,1]. This implies that t n = 1/N. We then conclude that in auction n the discounted value of the insider s 4

6 future profits whose expectation she maximizes is Π n = N k=n δ(k n)/n π k. In our extension of Kyle s formulation we are interested in combining time-discounting and risk-aversion. This is analytically more involving than introducing individually either risk-aversion or time-discounting in his model. A possibility is to consider a CARA utility function of the insider s discounted future profits. This would imply that in any period n the insider would maximize the expected value of u(π n ) exp( ρπ n ). However, according to this formulation the impact of risk-aversion on the insider s trading strategy would dissipate over time. In addition, when the number of auctions approaches infinity such strategy would be non-stationary and would rapidly converge to the risk-neutral counter-part. 2 To avoid such unpleasant features of the CARA utility function we rely on the optimization criterion presented by Vitale (2013) which discriminates between timediscounting and risk-aversion in the formulation of individual preferences. Specifically, because profits are time-separable and the dynamics of the transaction price is Markovian, we assume that in any auction n the insider chooses her optimal market order, x i n, solving the following recursive optimization 3 V n = min x i n { 2 ρ ln ( E n [exp ( ρ 2 (c n + δ n V n+1 ))]) }, (3) where ρ (with ρ > 0) is the coefficient of risk-aversion, δ n (with δ n = δ t n+1 ) is the perperiod discount factor, c n is a scalar-valued cost function equal to the opposite of her per-auction profits, π n = (p n v) x i n, and V n is the optimization criterion in n (with terminal condition V N+1 = 0). The optimization criterion in (3) accommodates risk-aversion through the curvature of the exponential function, 4 while the coefficient δ n, which pre-multiplies next period optimization criterion V n+1, captures discounting from time t n+1 to time t n. The optimization criterion in (3) represents a particular formulation of Epstein-Zin preferences and hence it inherits most of their properties. 5 In particular, differently from standard time-separable preferences, in the optimization criterion in (3) the inter-temporal rate of substitution and 5

7 the coefficient of relative risk-aversion are separated. Importantly, this property implies that this optimization criterion allows to discriminate between the effects of risk-aversion and time-discounting on the insider s trading activity and market quality. For ρ 0, the recursive optimization in (3) converges to V n = min x i n E n [c n + δ n V n+1 ]. 6 This is analogous to the Bellman equation which solves the insider s optimization exercise within Caldentey and Stacchetti s model with the random publication of the fundamental value of the risky asset. 7 One should however note that in their formulation δ n indicates the probability that the fundamental value of the risky asset is announced at the end of auction n given that it has not been previously disclosed, whereas in our formulation it represents a time-discount factor in the insider s preferences. In other words, δ n has a different role within our and their formulations. For δ 1 the argmin of the recursive optimization in (3) corresponds to the argmax of E n [ exp( ρ N 2 k=n π k)], i.e. the expected value of the CARA utility function of the risk-averse insider investigated by Holden and Subrahmanyam (1994). Given the optimization criterion in (3), the insider needs to solve an optimal control problem characterized by a clear trade-off. In fact, a larger market order today generates larger profits now at the expense of future ones, since a more informative order is passed to the market maker reducing his uncertainty on the liquidation value. On the other hand, the market maker needs to solve a filtering problem. He uses the signal contained in the flow of orders to up-date his expectation of the liquidation value. This will induce a process of convergence of the transaction price to the actual liquidation value. To solve simultaneously and consistently these two problems Kyle introduces a special notion of sequential equilibrium. We adapt it to the scenario in which the insider recursively solves (3). First, we need to define the strategies that characterize an equilibrium. These are two collections of functions, X and P, that indicate the trading strategy of the 6

8 insider and the pricing rule of the market maker for any auction n, X = X 1, X 2,..., X n,..., X N, P = P 1, P 2,..., P n,..., P N, where (4) x i n = X n (p 1,..., p n 1, v), p n = P n ( x 1,..., x n ). (5) We can now define a sequential auction equilibrium. Definition 1 A sequential auction equilibrium is a couple (X, P ) such that: (1) n, the insider chooses her market order by solving the recursive optimization in (3); (2) n, the market maker sets the transaction price according to the efficiency condition (1). We can then define a Markovian linear equilibrium as follows. Definition 2 A sequential auction equilibrium is linear if the component functions of strategies X and P are linear. A linear sequential auction equilibrium is Markovian if there exist constants λ 1, λ 2,..., λ N, such that for any n = 1,..., N p n = p n 1 + λ n x n. (6) 3 A Markovian Linear Equilibrium To find a Markovian linear equilibrium for the model with risk-aversion and time-discounting, let us concentrate on the insider s trading strategy, assuming that the market maker sets the transaction price of the risky asset according to equation (6). Then, we introduce the discounted stress proposed by Vitale (2013) to solve the insider s optimization exercise. Definition 3 The (discounted) stress function in n is S n c n + δ n V n+1 1 ρ ( xl n) 2 /(σ 2 l t n). The following Lemma presents some important properties for this function. 7

9 Lemma 1 If: i) the market maker sets the transaction price according to equation (6); ii) the optimization criterion in n + 1, V n+1, is a quadratic form in v p n ; and iii) the stress S n satisfies a saddle point condition with respect to x l n and x i n, so that min x i n max x l n S n exists, then: 1) the saddle point condition identifies the optimal insider s market order; and 2) the optimization criterion in n is a quadratic form in v p n 1 equal to the extremized stress plus a constant, ϑ n, independent of v p n 1, V n = ϑ n + min x i n max S n. x l n Proof. See Appendix. As corollary of Lemma 1 we establish a particularly useful result. Proposition 1 If the stress respects the saddle point condition in the periods N, N 1,, n + 2, n + 1, (i.e. if min x i n+j max x l n+j S n+j exists for j = N n,..., 1), the insider s optimal market order in auction n is determined by extremizing the stress, that is by simultaneously maximizing S n with respect to x l n and minimizing it with respect to x i n. Proof. See Appendix. Proposition 1 indicates that the extremization of the stress can be undertaken recursively. In particular, starting from N one proceeds backward imposing the saddle point condition for the stress in periods N, N 1,..., 1. We are now ready to establish our main result. Proposition 2 A linear Markovian sequential Nash equilibrium with N auctions is identified 8

10 by constants α n, β n, λ n, Σ n and θ n such that for any n p n = p n 1 + λ n x n, (6) x i n = β n t n (v p n 1 ), (7) V n = θ n 1 α n 1 (v p n 1 ) 2, (8) Σ n = Var(v x 1,..., x n ). (9) Given the initial value of Σ 0, the constants α n, β n, λ n, Σ n and θ n, with n = 1, 2,..., N, are a solution of the following recursive system β n t n = 2(1 2α n δ n λ n ) 4λ n (1 α n δ n λ n ) + λ 2 nρσl 2 t, (10) n Σ n = σ 2 l Σ n 1 β 2 n t n Σ n 1 + σ 2 l, (11) λ n = β nσ n σ 2 l α n 1 =, (12) 1 4λ n (1 α n δ n λ n ) + λ 2 nρσl 2 t, (13) n θ n 1 = 1 ρ ln(1 + α nδ n λ 2 nρσ 2 l t n ) + δ n θ n, (14) subject to the terminal conditions α N = 0, θ N = 0 and the second order conditions 4λ n (1 α n δ n λ n ) + ρσ 2 l t n λ 2 n > 0 n. (15) Proof. See Appendix. 9

11 4 Equilibrium Properties and Comparative Statics We now discuss the properties of the equilibrium described in Proposition 2. In particular we intend to unveil the impact of risk-aversion and time-discounting on the trading strategy of the insider and the characteristics of the market for the risky asset, by investigating the the dynamics of the coefficients β n, λ n and Σ n. β n represents the insider s trading intensity and determines how aggressively her trading strategy is; λ n reflects the liquidity of the market, as its inverse corresponds to the market s depth, a standard measure of liquidity for securities markets; Σ n is the residual uncertainty of the market maker on the liquidation value of the risky asset and hence it represents an indicator of market efficiency. Inspection of the expression for β n in equation (10) immediately reveals that both the coefficients of risk-aversion, ρ, and time-discounting, δ, affect the optimal trading strategy of the insider and the characteristics of the market. Indeed, risk-aversion makes the insider care about the variance of her profits. The uncertainty she faces results from the randomness of the liquidity traders orders. Given her information, the insider s expectation of the transaction price in auction n is E[p n In] i = p n 1 + λ n x i n and consequently the corresponding conditional variance is Var [p n In] i = λ 2 nσl 2 t n. This conditional variance and the insider s risk-aversion enter into the specification of β n and hence affect the sequential auction equilibrium. Similarly, time-discounting conditions the relevance that future payoffs have in shaping the insider s trading strategy and hence affects the sequential auction equilibrium. To determine the exact impact of time-discounting and risk-aversion on the trading strategy of the insider, the pricing process and the characteristics of the market for the risky asset we need to solve the system of recursive equations (10) to (15) which characterizes the sequential auction equilibrium in Proposition 2. To find the solution to this system we rely on a numerical algorithm based on a backward routine. To define this routine 10

12 consider that for any n, given α n and Σ n, there is a unique positive value of λ n satisfying the condition λ n (1 α n δ n λ n )+ 1 4 λ2 nρσ 2 l t n > 0 that the optimization problem of the insider must satisfy. This value is given by the appropriate root of the following equation 4 (1 α n δ n λ n )(Σ n σ 2 l t n λ 2 n) = 2 Σ n + ρσ 4 l t 2 nλ 3 n, which is obtained by substituting out the expression for β n in equation (10) into that for λ n in equation (12). Then, given λ n, alongside α n and Σ n, equation (10) yields β n. Using the projection theorem for normal distributions, one can write equation (12) as λ n = β n Σ n 1 β 2 n t n Σ n 1 + σ 2 l. Combining this expression with equation (11) it is found that Σ n 1 = (1 β n λ n ) 1 Σ n, so that Σ n 1 is derived from β n, Σ n and λ n. Equations (13) and (14) then provide α n 1 and θ n 1, completing the backward routine, (α n 1, β n, λ n, Σ n 1, θ n 1 ) = R(α n, β n+1, λ n+1, Σ n, θ n ). Since we have the terminal values α N = 0 and θ N = 0, while β N+1 = λ N+1 = 0, we can define a function of Σ N, G, that gives the initial variance of the liquidation value in 0, Σ (0) = G(Σ N ). Since G(Σ N ) is increasing in Σ N it is easy to find via the Newton-Raphson method the root of the equation Σ 0 = G(Σ N ) that gives the unique value of Σ N consistent with the boundary value Σ 0. This completes the algorithm which finds the solution to the system (10) to (15). 8 [Figure 1 about here] Using the algorithm above we can derive the dynamics of the equilibrium coefficients for any specific choice of the model s parameters. In Figure 1 we represent the dynamics of the conditional variance Σ n, top panel, and the liquidity coefficient λ n, bottom panel, across the N actions. The values of these coefficients are derived for N, the total number of auctions, equal to 100, σ l, the volatility of liquidity trading over the entire trading interval 11

13 [0, 1], and Σ 0, the unconditional variance of the liquidation value of the risky asset, equal to 1. In Figure 1 four different combinations of the coefficient of risk-aversion and the time-discount factor are considered. For ρ = 0 and δ = 1 we have the equilibrium derived in Kyle s original formulation with risk-neutrality and no time-discounting, while for ρ = 5 and δ = 1 we find the equilibrium described by Holden and Subrahmanyam (1994) under risk-aversion with only one insider. For ρ = 0 and δ = 0.5 we obtain an equilibrium which is analogous to that derived by Caldentey and Stacchetti (2010) with the random publication of the fundamental value of the risky asset. Finally, for ρ = 5 and δ = 0.5 we have the more general formulation with both risk-aversion and time-discounting. Figure 1 shows that in all the scenarios we consider the conditional variance of the liquidation value given the information the market maker possesses at the end of auction n, Σ n, is monotonically decreasing with n, indicating that private information is gradually incorporated into the asset price as it is disclosed through time by order flow. As eventually Σ n vanishes, the market maker learns the liquidation value of the risky asset by the time trading halts. How quickly the transaction price converges to the liquidation value depends on the insider s trading strategy. For ρ = 0 and δ = 1, i.e. in Kyle s original formulation, private information is disclosed at a constant pace (the derivative of Σ n with respect to n is constant). This is because the insider finds it optimal to trade with constant intensity and maintain overtime a stable news-to-noise ratio in order flow. Consequently the price sensitiveness, or liquidity coefficient, λ n is constant throughout most of the auctions. Figure 1 shows that in the risk-averse case considered by Holden and Subrahmanyam (ρ = 5 and δ = 1), instead, the insider places larger market orders in the initial auctions, as β n is larger than in the risk-neutral case (ρ = 0 and δ = 1) and so is the liquidity coefficient, λ n. This is because the inter-temporal substitution between present and future profits is reduced by risk-aversion and, therefore, the insider prefers exploiting sooner her information advantage. Consequently, for ρ > 0 she trades more aggressively, order flow is more informative and the market maker learns at a higher speed the liquidation value 12

14 of the risky asset. This implies that the conditional variance, Σ n, declines more rapidly. As the market maker progressively learns the liquidation value, the volatility of the price p n and the insider s uncertainty over future profits fall and consequently the inter-temporal substitution between present and future profits dissipates. Hence, as the last auction approaches the impact of risk-aversion on the trading activity of the insider resembles that of the static version of Kyle s model studied by Subrahmanyam (1991): risk-aversion induces the insider to be more cautious and trade less aggressively. This means that as time elapses the informational content of order flow decreases (λ n declines through time) and hence the reduction in the value of Σ n is smaller. In the end, in the risk-averse case the information gain from order flow becomes smaller than that of the risk-neutral one while market liquidity is larger (λ n is now smaller for ρ > 0). Anyway, despite the reduction in the information gain, the informativeness of prices is always larger in the risk-averse case as Σ n is always smaller for ρ larger than 0. The impact of time-discounting on market efficiency is similar to that of risk-aversion (ρ = 0 and δ = 0.5), in that a smaller δ reduces the inter-temporal substitution between present and future profits, inducing the insider to act more aggressively and reveal more rapidly her informational advantage. Figure 1 also shows how time-discounting and riskaversion interact and exacerbate each other s effect on the insider s trading strategy. In fact, for ρ = 5 and δ = 0.5 we see that the market maker learns at an even faster pace the liquidation value, as the insider trades even more aggressively. A similar result is proved by Caldentey and Stacchetti in their formulation with the random publication of the fundamental value of the risky asset. For a larger probability of the publication of the fundamental value, in steady state, the insider s trading intensity is larger, while the market maker s conditional variance of the fundamental value is smaller. This is not surprising, since an increase in the probability of the publication of the fundamental value has an impact on the insider s payoffs equivalent to that of a reduction in the time-discount factor δ. 13

15 Whereas Figure 1 applies to specific parametric constellations, the properties of the equilibrium it illustrates are actually general. In particular, the reduced inter-temporal trade-off between present and future profits that a smaller δ entails holds whatever the value of δ. Similarly, the impact of a larger degree of risk-aversion would be to make the insider more aggressive and the market more efficient whatever the value of ρ. This is because as shown by Tallarini 2000 for any choice of ρ > 0 in the optimization criterion (3) the coefficient of relative risk-aversion is larger than the inverse of the inter-temporal elasticity of substitution. 9 Exploiting results from Kreps and Porteus (1978), Epstein and Zin (1989) show that under such condition their recursive preferences induce earlier resolution of uncertainty vis-a-vis the case of expected utility. Given that our optimization criterion is a special version of their recursive preferences, this property extends to our formulation. Therefore, when ρ is positive the insider is willing to trade-off her expected future profits in order to reduce their uncertainty. This means that she will be willing to reveal more information to the market maker than it would be optimal under risk-neutrality by trading more aggressively in the earlier auctions. Appendix Proof of Lemma 1. To prove this Lemma we first need to establish a preliminary result. Lemma 2 If Q(u, ɛ) is a quadratic form in the vectors u and ɛ which admits the saddle point max u min ɛ Q(u, ɛ), then the following holds min u [ exp 12 ] [ Q(u, ɛ) dɛ exp 1 ] 2 max min Q(u, ɛ). u ɛ 14

16 Proof. Consider the quadratic form Q(u, ɛ) in the vectors u and ɛ, where Q(u, ɛ) = u ɛ Q u u Q ɛ u Q u ɛ Q ɛ ɛ u. ɛ Assume Q admits a minimum in ɛ in that Q ɛ ɛ is positive definite. Then, the following holds exp [ 12 ] [ Q(u, ɛ) d ɛ exp 1 ] 2 min Q(u, ɛ). (16) ɛ In fact, for ˆɛ the vector ɛ minimizing Q, we can write Q(u, ɛ) = Q(u, ˆɛ)+(ɛ ˆɛ) Q ɛ ɛ (ɛ ˆɛ). Consider that as Q ɛ ɛ is positive definite and invertible, the minimum of Q with respect to ɛ is obtained for ˆɛ = Q 1 ɛ ɛ Q ɛ u u and is equal to Q(u, ˆɛ) = u [Q u u Q u ɛ Q 1 ɛ ɛ Q ɛ u ]u. Thus, Q(u, ɛ) Q(u, ˆɛ) = ɛ Q ɛ ɛ ɛ + ɛ Q ɛ u u + u Q u ɛ ɛ + u Q u ɛ Q 1 ɛ ɛ Q ɛ u u = ɛ Q ɛ ɛ ɛ ɛ Q ɛ ɛ ˆɛ ˆɛ Q ɛ ɛ ɛ + ˆɛ Q ɛ ɛ ˆɛ = (ɛ ˆɛ) Q ɛ ɛ (ɛ ˆɛ). As Q(u, ˆɛ) = min ɛ Q(u, ɛ) is a constant in the integral in equation (16), we find that exp [ 12 ] [ Q(u, ɛ) d ɛ = exp 1 ] 2 min Q(u, ɛ) ɛ exp[ 1 2 (ɛ ˆɛ) Q ɛ ɛ (ɛ ˆɛ)] d ɛ. Therefore, the constant of proportionality in equation (16) is exp( 1 2 Q ɛ ɛ ) d = (2π) m/2 det(q ɛ ɛ ) 1/2, where m is the dimension of ɛ, and hence it is independent of u. Then, suppose that we solve the program min u exp [ 1 2 Q(u, ɛ)]. Assume that Q admits a saddle point with respect to ɛ and u, so that max u min ɛ Q(u, ɛ) exists. From equa- 15

17 tion (16) min u exp [ 12 ] [ Q(u, ɛ) d ɛ min exp 1 ] u 2 min Q(u, ɛ) ɛ [ = exp 1 ] 2 max min Q(u, ɛ). u ɛ It is worth noting this result applies also when Q is a non-homogeneous quadratic form, which depends on x and ɛ, alongside a third vector z, insofar it admits a saddle point max x min ɛ Q(x, ɛ, z). We are now ready to prove Lemma 1. Proof. First, we notice that because the exponential function is monotonic exp( ρ V 2 n) = [ ( min x i n E n exp ρ (c 2 n + δ n V n+1 ) )]. Second, we consider that v p n is linearly dependent on x l n via equation (6). Third, since c n = (p n 1 v + λ n x i n) x i n + λ n x i n x l n, the per-period cost depends on x l n. Then, the distribution of c n + δ n V n+1 depends on that of x l n and hence, given that x l n N(0, σl 2 t n), min x i n ( ρ )] E n [exp 2 (c n + δ n V n+1 ) = ( 2πσ 2 l t n ) 1/2 min x i n ( exp ρ S ) n d x l n, where 2 S n = c n + δ n V n+1 1 ρ ( xl n) 2 /(σ 2 l t n ). Now, since V n+1 is assumed to be a quadratic form in v p n and this is linear in x l n, x i n and v p n 1, V n+1 can be expressed as a quadratic form in x l n, x i n and v p n 1. Similarly, c n is a quadratic form in x l n, x i n and v p n 1 and so is S n. Thus, if the stress in n admits the saddle point min x i n statement of Lemma 2. Exploiting this Lemma min x i n ( exp ρ S ) n d x l n = min 2 x i n max x l n S n, then S n admits the saddle point in the exp ( ( = K n exp 1 2 max x i n 1 2 ( ρs n) }{{} Q( x i n, xl n ) ) min ( ρs n ) x l n d x l n ) ( ρ = K n exp 2 min x i n ) max S n, x l n 16

18 where, using the result outlined in the proof of Lemma 2, we establish that K n = (2π/Q x l n x l )1/2 n with Q x l n x equal to the second derivative of ρs l n n with respect to x l n. This implies that [ ( min x i n E n exp ρ (c 2 n + δ n V n+1 ) )] = (σl 2 t nq x l n x l ) 1/2 exp( ρ min n 2 x i max n x S l n n). Extremizing the stress S n, i.e. maximizing it with respect to x l n and minimizing the resulting function with respect to x i n, we find that min x i n max x l n S n is a quadratic form in v p n 1. Because exp( ρ V [ ( 2 n) = min x i n E n exp ρ (c 2 n + δ n V n+1 ) )] we conclude that the saddle point condition pins down the optimal market order for the insider and that the optimization criterion in n is a quadratic form in v p n 1 equal to the extremized stress plus a constant independent of v p n 1, V n = ϑ n + min max S n, where ϑ n = 1 x i n x l n ρ ln(σ2 l t n Q x l n x ). l n Proof of Proposition 1. Notice that in N c N is a positive definite quadratic form in x i N, while V N+1 = d N+1 = 0. This implies that S N is a quadratic form in x i N and xl n and hence that the conditions to apply Lemma 1 are met, so that the saddle point conditions for S N yields the optimal control x i N, with the extremized stress, min x i max N x S N, and l N the optimization criterion, V N, both quadratic forms in v p N 1. By backward induction the statement is proved. Proof of Proposition 2. Assume n the market maker sets the transaction price according to (6). From Lemma 1 we know that for V n+1 a quadratic form in v p n, V n = ϑ n +min x i n max x l n S n, where ϑ n = 1 ρ ln(σ2 l t nq x l n x ) and Q l n x is the second l n x l n derivative of ρs n with respect to x l n. Therefore, assume that V n+1 = θ n α n (v p n ) 2, where θ n is independent of v p n. Clearly, as V N+1 = 0, it must be that α N = 0 and θ N = 0. 17

19 Considering that S n = c n + δ n V n+1 1 ρ ( xl n) 2 /(σ 2 l t n), it follows that min x i n max x l n { [ S n = min max c n + δ n θ n δ n α n (v p n ) 2 1 ] } x i n x l n ρ ( xl n) 2 /(σl 2 t n ) = δ n θ n + min x i n { [ max c n δ n α n (v p n ) 2 1 ] } x l n ρ ( xl n) 2 /(σl 2 t n ) (17). Maximizing the argument in the square brackets with respect to x l n, we have x l n = γ n {λ n (1 2α n δ n λ n ) x i n 2α n δ n λ n (v p n 1 )}, with γ n = ρσ 2 l t n 2(1 + α n δ n λ 2 nρσ 2 l t n). Notice that we have maximum as the second order condition holds. In fact, the second derivative is 2(α n δ n λ 2 n + 1/(ρσl 2 t n)), which is negative for α n 0. Plugging our expression for x l n in the argument inside the square brackets in the right hand side of (17), we find that minimizing the resulting expression in curly brackets with respect to x i n, under the second order condition (15), gives the optimal market order of the insider at time n, x i n = β n t n (v p n 1 ), where β n t n = 2(1 2α n δ n λ n ) 4λ n (1 α n δ n λ n ) + λ 2 nρσl 2 t. n Condition (15) guarantees that the saddle point condition of Proposition 1 is met. Plugging the optimal value of x i n in the right hand side of (17) we find that min x i n max x l n S n = δ n θ n α n 1 (v p n 1 ) 2 where α n 1 = 1 4λ n (1 α n δ n λ n ) + λ 2 nρσl 2 t. n Coherently with the original assumption on V n+1, we see that V n = θ n 1 α n 1 (v p n 1 ) 2, with θ n 1 = ϑ n + δ n θ n. Since Q x l n x = l 1/(σ2 n l t n) + α n δ n λ 2 nρ, ϑ n = 1 ln(1 + ρ α n δ n λ 2 nρσn t 2 n ). Finally, the projection theorem for normal distributions shows that if equation (7) holds n, the conditional expectation of the liquidation value in n, p n, is a linear function of the total market order, p n = p n 1 + λ n x n, where λ n is given by equation (12), while its conditional variance, Σ n, respects equation (11). 18

20 References Boukaiz, M., and M. Sobel (1984): Non Stationarity Policies Are Optimal for Risk-sensitive Markov Decision Processes, Discussion paper, California Institute of Technology. Caldentey, R., and E. Stacchetti (2010): Insider Trading with a Random Deadline, Econometrica, 78, Chau, M. and D. Vayanos (2008): Strong-Form Efficiency with Monopolistic Insiders, Review of Financial Studies, 21, Daher, W., L.J. Mirman and E.G. Saleeby (2014): Two Period Model of Insider Trading with Correlated Signals, Journal of Mathematical Economics, 52, Epstein, L.G., and S.E. Zin (1989): Substitution, Risk Aversion, and the Temporal Behavior of Consumption and Asset Returns: A Theoretical Framework, Econometrica, 57, Foster, F.D. and S. Viswanathan (1996): Strategic Trading When Agents Forecast the Forecasts of Others, Journal of Finance, 51, Hansen, L., and T. J. Sargent (1994): Discounted Linear Exponential Quadratic Gaussian Control, Discussion paper, University of Chicago mimeo. (1995): Discounted Linear Exponential Quadratic Gaussian Control, IEEE Transactions on Automatic Control, 40, Holden, C. W., and A. Subrahmanyam (1994): Risk-aversion, Imperfect Competition, and Long-lived Information, Economic Letters, 44, Huddart, S., J.S. Hughes and C. Levine (2001): Public Disclosure and Dissimulation of Insider Trading, Econometrica, 69, Kreps, D. M., and E.L. Porteus (1978): Temporal Resolution of Uncertainty and Dy- namic Choice Theory, Econometrica, 46,

21 Kyle, A. S. (1985): Continuous Auction and Insider Trading, Econometrica, 53, Subrahmanyam, A. (1991): Risk Aversion, Market Liquidity, and Price Efficiency, Review of Financial Studies, 4, Tallarini, T. D. (2000): Risk-Sensitive Real Business Cycles, Journal of Monetary Economics, 45, Vitale, P. (2012): Risk-averse Insider Trading in Multi-asset Sequential Auction Markets, Economic Letters, 117, (2013): Pessimistic Optimal Choice for Risk-averse Agents, Discussion paper, CASMEF , vitale/pessimistic-optimal-choice-for-risk-averse- Agents-Quater.pdf. Whittle, P. (1990): Risk-sensitive Optimal Control. John Wiley & Sons, New York. 20

22 Notes 1 The market maker keeps setting the transaction price according to equation (1), as he calculates his profits/losses only when the liquidation value of the risky asset is announced. On the contrary, the insider can calculate the profits associated with any round of trading, π n, at the end of any auction, n, and assess her performance on a per-period basis. 2 See Bouakiz and Sobel (1984), Whittle (1990) and Hansen and Sargent (1994). 3 Hansen and Sargent (1994,1995) have proposed a similar optimization criterion. 4 The functional form ln(e[exp( ρ X )]) is monotonic increasing and convex in X. In 2 the optimization criterion in (3) X c n + δ n V n+1. We will show that c n and V n+1 are positive definite quadratic forms in x i n and v p n 1. This implies that X is convex in x i n and v p n 1. Given the convexity of the functional form ln(e[exp( ρ X )]), we see 2 that the optimization criterion in (3) is convex in x i n and v p n 1. This means that our optimization criterion is well-defined, in that the insider s market order, x i n, is her choice variable in auction n, whereas the risky asset mis-pricing, v p n 1, is the fundamental information she carries into such an auction. As the convexity of ln(e[exp( ρ X )]) increases 2 with ρ, this coefficients determines the insider s degree of risk-aversion. 5 See Tallarini (2000) and Vitale (2013) for extensive discussions of the properties of the optimization criterion in (3) and of its relation to Epstein-Zin preferences. 6 For δ = 1, this is the Bellman equation solved by Kyle in his original formulation. 7 Indeed, in their formulation Caldentey and Stacchetti assume that the fundamental value is subject to unpredictable shocks. In addition they do not impose a terminal date for the announcement of the fundamental value. 21

23 8 The corresponding MatLab code is available on request. 9 See also Vitale (2013). 22

24 1 Conditional Variance of Liquidation Value, n =5, =0.5 =5, =1 =0, =0.5 =0, = Auction, n Market Liquidity, n =5, =0.5 =5, =1 =0, =0.5 =0, = Auction, n Figure 1: Market efficiency (Σ n, top panel) and liquidity (λ n, bottom panel) for N = 100, σ 2 l = 1, Σ 0 = 1. 23