A Multiperiod Equilibrium Pricing Model

Abstract

We propose an equilibrium pricing model in a dynamic multiperiod stochastic framework with uncertain income. There are one tradable risky asset (stock/commodity), one nontradable underlying (temperature), and also a contingent claim (weather derivative) written on the tradable risky asset and the nontradable underlying in the market. The price of the contingent claim is priced in equilibrium by optimal strategies of representative agent and market clearing condition. The risk preferences are of exponential type with a stochastic coefficient of risk aversion. Both subgame perfect strategy and naive strategy are considered and the corresponding equilibrium prices are derived. From the numerical result we examine how the equilibrium prices vary in response to changes in model parameters and highlight the importance of our equilibrium pricing principle.

1. Introduction

Hitherto, there has been an increasing literature on pricing contingent claims written on nontradable underlyings in a dynamic multiperiod equilibrium framework. One example of such contingent claim is a weather derivative, in which case the underlying is the temperature process. One approach in pricing this financial instrument is to use a multiperiod stochastic equilibrium model. In financial economics there is a huge amount of literature on this issue. Rubinstein [1] considers a multiperiod state-preference equilibrium model without explicit modeling of production/investment. Brennan [2] looks at a multiperiod equilibrium problem in which the representative agent exhibits constant risk aversion. Bhattacharya [3] extends the model of Rubinstein [1] to show that risk/return tradeoffs are linear relations linking instantaneous expected returns of assets to the instantaneous covariance of returns with aggregate consumption. Bizid and Jouini [4] derive restrictions on the equilibrium state-price deflator independent of the choices of utility function in an incomplete market. Câmara [5] obtains preference-free option prices in a discrete equilibrium model where representative agent has exponential utility and aggregate wealth together with the underlying asset price has transformed normal distributions.

Our paper presents a partial equilibrium model with two exogenous assets, one tradable and one nontradable. The goal is to find the price of a derivative security written on tradable and nontradable underlyings in equilibrium. We assume that the tradable asset is priced in a different market so its price is exogenously given. Moreover the interest rate risk is not considered because we take the money market account as numeraire. The derivative security, written on the tradable asset and nontradable underlying, is priced in equilibrium by a representative agent who receives (unspanned) random income within an incomplete multiperiod market. The optimal strategies are obtained by backward induction. First order conditions together with the market clearing give the equilibrium prices.

Cao and Wei [6], Lee and Oren [7, 8], and Cheridito et al. [9] are related to our work. Cao and Wei [6] generalize the model of Lucas [10] to provide an equilibrium framework for valuing weather derivatives in a multiperiod setting. Lee and Oren [7] explore a single-period equilibrium pricing model in a multicommodity setting and mean-variance preferences. Lee and Oren [8] made a followup in a multiperiod framework. Cheridito et al. [9] establish results on the existence and uniqueness of equilibrium in dynamically incomplete financial markets with preferences of monetary type and heterogeneous agents. Utility indifference pricing is also a way for pricing derivatives written on nontradable underlying. The utility indifference price was introduced by Hodges and Neuberger [11]. By now, there are several papers on this topic; we recall only a few (on discrete time Musiela et al. [12]; on continuous time Henderson [13]; Musiela and Zariphopoulou [14]; for an overview see Henderson and Hobson [15]). Pirvu and Zhang [16] derive utility indifference prices in a model with time changing risk aversion.

In our model the representative agent has risk preferences of exponential type with time and state-dependent coefficient of risk aversion. Inspired by Gordon and St-Amour [17], we assume that the representative agent's risk aversion coefficient is a stochastic process. Gordon and St-Amour [17] motivate this change by the fact that stochastic risk aversion can explain asset-price movements which cannot be explained by constant risk aversion. Lately, the issue of time changing risk aversion received some attention in the financial literature. For instance, Barberis and Huang [18] consider a model in which the loss aversion depends on prior gains and losses, so it may change through time. Danthine et al. [19] allow the representative agent's coefficient of relative risk aversion to vary with the underlying economy's growth rate. Gordon and St-Amour [20] explain equity premium puzzle by state-dependent risk preferences. Yuan and Chen [21] show that dynamic risk aversion plays a critical role in the dynamics of asset price fluctuations. Recently, Berrada et al. [22] consider growth regime-dependent risk aversion of representative agent and examine the effect of regime-dependent preference on the structure of equilibrium.

A time changing risk preference leads to time inconsistent investment strategies. It means that an investor may have an incentive to deviate from the classical optimal strategy which he/she computed at some past time. Let us define the naive strategies as the classical optimal strategies given that the agent does not update his/her risk preferences. The naive strategies cannot be implemented in the absence of a precommitment. In order to overcome the time inconsistency issue, Pirvu and Zhang [16] and Björk and Murgoci [23] develop a theory for stochastic control problems which are time inconsistent in the sense that they do not admit the Bellman optimality principle; they consider the subgame perfect equilibrium as a substitute for the Bellman optimality principle. In Wu [24], subgame perfect equilibrium strategies are studied for a multiperiod mean-variance portfolio selection problem with wealth-dependent risk aversion. In our paper, we follow Pirvu and Zhang [16] in introducing subgame perfect strategies. Subgame perfect strategies denote the strategies obtained from the subgame perfect equilibrium. The corresponding equilibrium price obtained by imposing the market clearing condition with subgame perfect strategies of representative agent is called as subgame perfect equilibrium price. On the other hand, market clearing condition with naive strategies gives naive equilibrium price. The naive equilibrium price is considered as a benchmark price. By comparing the subgame perfect equilibrium price and the naive equilibrium price, the difference and the importance of subgame perfect equilibrium price are highlighted.

The exogenous assets in our model may have stochastic drifts and volatilities and the assets may be correlated. A derivative security is priced in equilibrium within this model. Our main result is an iterative algorithm which yields the subgame perfect equilibrium prices. At each stage the equilibrium prices depend on the current risk aversion level and all previous wealth and risk aversion levels. The algorithm constructs recursively one period pricing kernels. Moreover, the naive equilibrium pricing kernel equals the marginal utility. We prove that the equilibrium pricing measures are martingale measures so the equilibrium prices are arbitrage free.

Numerical experiments shed light into the importance of the pricing principle to find subgame perfect equilibriums price. For the numerical result, we consider a two-state regime switching model (bull and bear markets) and assume that the representative agent becomes more risk averse in the bear market. The result shows that the equilibrium prices (both subgame perfect equilibrium price and naive equilibrium price) are decreasing in risk aversion. It's because the derivative can be regarded as a risky asset so agent's demand for the derivative is reduced if he/she is more risk averse. Since the risk aversions of the agent in two states are different, we may try to approximate the subgame perfect equilibrium price with a new risk aversion, say average risk aversion which is a weighted average of two risk aversion (risk aversions in bull market and bear market) with the steady state probabilities of two states as the weights. However, we find that there is significant gap between the exact subgame perfect equilibrium price and the naive equilibrium price with average risk aversion if the risk aversion in the bear market is much higher than the risk aversion in the bull market. This means that the naive equilibrium price with average risk aversion cannot be a useful estimate of subgame perfect equilibrium price especially when the agent's preference changes a lot depending on the market environment.

We also examine the effect of the number of units of derivatives in the market on the equilibrium prices and we find out that the equilibrium prices are decreasing in number of units of derivatives. This result is consistent with Gârleanu et al. [25] which examine the effects of demand pressure on option prices. Moreover, it is observed that the gap between the naive equilibrium price with average risk aversion and the subgame perfect equilibrium price gets larger if the number of units of derivatives is large enough. Hence the naive equilibrium price is not a good estimate of the subgame perfect equilibrium price in this case.

The remainder of this paper is organized as follows. Section 2 presents the model. Section 3 provides the equilibrium pricing valuation. Numerical results are presented in Section 4. Proofs of the results are delegated to an Appendices A, B, and C.

2. The Model

We consider a multi-period stochastic model of investment. The trading horizon is [0,𝑇], with 𝑇 an exogenous finite horizon. There are 𝑁+1 trading dates: 𝑡_𝑛=𝑛ℎ, for 𝑛=0,1,…,𝑁, and ℎ=𝑇/𝑁. Let (𝑏¹,𝑏²,…,𝑏^𝑑)∶=(𝑏¹_𝑡𝑛,𝑏²_𝑡𝑛,…,𝑏^𝑑_𝑡𝑛)_{𝑛=0,1,…,∞}, be a 𝑑-dimensional binomial random walk on a complete probability space (Ω,ℱ,{ℱ_𝑡𝑛},ℙ). The random walk is assumed to be symmetric under ℙ in the sense that ℙΔ𝑏^𝑖_𝑡𝑛=1=±12,𝑖=1,2,…,𝑑.(1) There are three securities available for trading in our model; a money market account, a primary asset (e.g., stock or commodity) and a derivative security. We take the money market account as numeraire, thus it can be assumed to offer zero interest rate. The primary asset price process 𝐶∶={𝐶_𝑡𝑛;𝑛=0,1,…,𝑁}, follows the difference equation: Δ𝐶_𝑡𝑛=𝐶_𝑡𝑛𝜇^𝑐_𝑡𝑛ℎ+𝜎^𝑐_𝑡𝑛√ℎΔ𝑏¹_𝑡𝑛𝐶,𝑛=0,1,…,𝑁−1,₀=𝑐>0,(2) for some adapted drift process 𝜇^𝑐∶={𝜇^𝑐_𝑡;𝑡=0,ℎ,…,(𝑁−1)ℎ,𝑁ℎ} and volatility process 𝜎^𝑐∶={𝜎^𝑐_𝑡;𝑡=0,ℎ,…,(𝑁−1)ℎ,𝑁ℎ} which are chosen so that the commodity price remains positive. The derivative security 𝐷∶={𝐷_𝑡𝑛;𝑛=0,1,…,𝑁} is written on the primary asset 𝐶 and a nontradable underlying 𝑆∶={𝑆_𝑡𝑛;𝑛=0,1,…,𝑁} which follows the difference equation: Δ𝑆_𝑡𝑛=𝑆_𝑡𝑛𝜇^𝑠_𝑡𝑛ℎ+𝜎^𝑠_𝑡𝑛√ℎ𝜌Δ𝑏¹_𝑡𝑛+1−𝜌²Δ𝑏²_𝑡𝑛,𝑆𝑛=0,1,…,𝑁−1,₀=𝑠>0,(3) for some adapted drift process 𝜇^𝑠∶={𝜇^𝑠_𝑡;𝑡=0,ℎ,…,(𝑁−1)ℎ,𝑁ℎ}, volatility process 𝜎^𝑠∶={𝜎^𝑠_𝑡;𝑡=0,ℎ,…,(𝑁−1)ℎ,𝑁ℎ}, and a correlation coefficient 𝜌, with |𝜌|<1. The two-dimensional process 𝑃=(𝐶,𝑆), exogenously given, is referred to as the forward process.

Since the derivative security 𝐷 is to be priced in equilibrium, we have a partial equilibrium model. It is motivated by a situation in which the primary asset and derivative are priced in different markets. As an example, consider energy market and weather derivatives. Although the energy price and weather derivatives are correlated (in California, a high correlation between energy prices and temperature process was observed), energy prices and weather derivative prices are determined in different markets.

2.1. Trading Strategies

Let 𝛼_𝑡𝑛 be the amount invested in the primary asset at time 𝑡_𝑛, and let𝛽_𝑡𝑛 be the number of shares of derivative held at time 𝑡_𝑛; denote 𝜋_𝑡𝑛∶={𝛼_𝑡𝑛,𝛽_𝑡𝑛}∈ℱ_𝑡𝑛,  𝑛=0,1,…,𝑁−1. The value of a self-financed portfolio satisfies the following stochastic difference equation: Δ𝑋^𝜋_𝑡𝑛∶=𝛼_𝑡𝑛𝜇^𝑐_𝑡𝑛ℎ+𝜎^𝑐_𝑡𝑛√ℎΔ𝑏¹_𝑡𝑛+𝛽_𝑡𝑛Δ𝐷_𝑡𝑛.(4) At maturity, 𝑡_𝑁∶=𝑇=𝑁ℎ, the representative agent in this economy receives random income 𝐼_𝑡𝑁, which is ℱ_𝑇 adapted. Thus, his/her final wealth is 𝑊^𝜋_𝑡𝑁=𝑋^𝜋_𝑡𝑁+𝐼_𝑡𝑁.(5) The random income may depend on all the random walks {𝑏¹,𝑏²,…,𝑏^𝑑}, so it may not be spanned by the existing assets.

2.2. Risk Preferences

The representative agent utility is assumed to be of exponential type, time, and state dependent. The coefficient of absolute risk aversion is a stochastic process 𝛾_𝑡𝑛, ℱ_𝑡𝑛 adapted, 𝑛=0,1,…,𝑁−1. More precisely, 𝑈𝑥,𝑡_𝑛,𝜔=−exp−𝛾_𝑡𝑛(𝜔)𝑥,𝜔∈Ω.(6) As we pointed out in Section 1, this modeling approach is not new and consistent with a number of papers that consider time changing, state-dependent risk aversion. The performance of an investment strategy 𝜋 is measured by the expected utility criterion applied to the final wealth; that is, the optimization criterion at time 𝑡_𝑛 is given by sup_{𝜋∈Π𝑡𝑛}𝔼−𝑒^{−𝛾𝑡𝑛𝑊𝜋𝑡𝑁}∣ℱ_𝑡𝑛.(7) Here 𝑛=0,1,…,𝑁−1, 𝑊^𝜋_𝑡𝑁 is given by (5), and Π_𝑡𝑛 denotes the set of admissible trading strategies: Π_𝑡𝑛∶=𝜋_𝑡𝑛,𝜋_𝑡𝑛+1,…,𝜋_𝑡𝑁−1∶𝜋_𝑡𝑘∈ℱ_𝑡𝑘,suchthat𝔼𝑋^𝜋_𝑡𝑘<∞,𝑘=𝑛,𝑛+1,…,𝑁−1.(8)

2.3. Naive Strategies

The naive strategies are the classical optimal strategies given that the risk preferences are not updated. More precisely 𝜋∈Π_𝑡0 is a naive strategy if it satisfies 𝜋=argsup_{𝜋∈Π𝑡0}𝔼−exp−𝛾_𝑡0𝑊^𝜋_𝑡𝑁∣ℱ_𝑡0.(9) They fail to remain optimal at later times 𝑡_𝑛, in the sense that 𝜋≠argsup_{𝜋∈Π𝑡𝑛}𝔼−exp−𝛾_𝑡𝑛𝑊^𝜋_𝑡𝑁∣ℱ_𝑡𝑛.(10)

2.4. Subgame Perfect Strategies

In this section, we introduce the subgame perfect strategies by following Pirvu and Zhang [16]. First, let us consider the time period [(𝑁−1)ℎ,𝑁ℎ] (recall that 𝑡_𝑁=𝑇=𝑁ℎ). At time (𝑁−1)ℎ consider the optimization problem: sup_{𝜋∈Π𝑡𝑁−1}𝔼−exp−𝛾_𝑡𝑁−1𝑊^𝜋_𝑡𝑁∣ℱ_𝑡𝑁−1.P₁ In our model sup in P₁ is attained and we denote 𝜋^∗_𝑡𝑁−1=argmax_{𝜋∈Π𝑡𝑁−1}𝔼−exp−𝛾_𝑡𝑁−1𝑊^𝜋_𝑡𝑁∣ℱ_𝑡𝑁−1.(11) On the time period [(𝑁−2)ℎ,𝑁ℎ], consider trading strategies 𝜋 be of the form: 𝜋𝜋=^{∗𝑡𝑁−1})),𝜋,on[(𝑁−1ℎ,𝑁ℎ_𝑡𝑁−2,on[(𝑁−2)ℎ,(𝑁−1)ℎ),(12) for an arbitrary ℱ_𝑡𝑁−2 adapted control 𝜋_𝑡𝑁−2 such that (𝜋_𝑡𝑁−2,𝜋^∗_𝑡𝑁−1)∈Π_𝑡𝑁−2; consider the optimization problem sup_{𝜋∈Π𝑡𝑁−2}𝔼−exp−𝛾_𝑡𝑁−2𝑊^𝜋_𝑡𝑁∣ℱ_𝑡𝑁−2.P₂ In our model sup in P₂ is attained and we denote 𝜋^∗_𝑡𝑁−2,𝜋^∗_𝑡𝑁−1=argmax_{𝜋∈Π𝑡𝑁−2}𝔼−exp−𝛾_𝑡𝑁−2𝑊^𝜋_𝑡𝑁∣ℱ_𝑡𝑁−2.(13) Further we proceed iteratively. On the time period [(𝑁−𝑛)ℎ,𝑁ℎ) one restricts to trading strategies 𝜋 of the form: 𝜋𝜋=^∗𝑡𝑘()()𝜋,for𝑘=𝑁−𝑛−1,𝑁−𝑛−2,…,𝑁−1,_𝑡𝑘,for𝑘=𝑁−𝑛,(14) for an arbitrary ℱ_{𝑡𝑁−𝑛} adapted control 𝜋_{𝑡𝑁−𝑛} such that (𝜋_{𝑡𝑁−𝑛},𝜋^∗_𝑡𝑘)_{{𝑘=𝑁−𝑛+1,…,𝑁−1}}∈Π_{𝑡𝑁−𝑛}. Consider the optimization problem max_{𝜋∈Π𝑡𝑁−𝑛}𝔼−exp−𝛾_{𝑡𝑁−𝑛}𝑊^𝜋_𝑡𝑁∣ℱ_{𝑡𝑁−𝑛}.P_𝑛 The sup in P_𝑛 is attained and we denote 𝜋^∗_{𝑡𝑁−𝑛},𝜋^∗_{𝑡𝑁−𝑛+1},…,𝜋^∗_𝑡𝑁−1=argmax_{𝜋∈Π𝑡𝑁−𝑛}𝔼−exp−𝛾_{𝑡𝑁−𝑛}𝑊^𝜋_𝑡𝑁∣ℱ_{𝑡𝑁−𝑛}.(15) The subgame perfect strategy is 𝜋^∗=(𝜋^∗_𝑡0,𝜋^∗_𝑡1,…,𝜋^∗_𝑡𝑁).

2.5. Subgame Perfect Strategies versus Naive Strategies

Let us recall that time inconsistency in this model is due to time changing, state-dependent risk aversion. Indeed in the case of constant risk aversion, the subgame perfect strategies and the naive strategies coincide; that is, 𝜋^∗=𝜋. In general, it is hard to show that the subgame perfect strategy outperforms the naive strategy. We show that this is the case if the risk preferences are updated in the following two period models. Let us assume that 𝐼_𝑡2=0, and only one asset is available for trading (the primary asset with constant drift and volatility). It is claimed that 𝔼−exp−𝛾_𝑡1𝑊^𝜋∗_𝑡2∣ℱ_𝑡1≥𝔼−exp−𝛾_𝑡1𝑊𝜋_𝑡2∣ℱ_𝑡1.(16) Indeed in this model it can be shown that 𝜋^∗_𝑡1=𝜋_𝑡1, and hence 𝑊^𝜋∗_𝑡1=𝑊𝜋_𝑡1. By the definition of the subgame perfect strategies, we have (16).

3. Equilibrium Valuation

We assume that there exists a representative agent with risk preferences given by (6). The representative agent trades 𝐶 and 𝐷 in order to maximize the expected utility of his/her final wealth. This can be achieved by the naive strategy 𝜋=(𝛼,𝛽) if the representative agent does not update his/her risk preferences. Otherwise, the subgame perfect strategy 𝜋^∗=(𝛼^∗,𝛽^∗) will be used. Thus, depending on whether or not the representative agent updates his/her risk preferences, we introduce two notions of equilibrium prices: subgame perfect equilibrium price and naive equilibrium price. They are given by the market clearing condition in the formal definition below.

Definition 1. Given the terminal payoff 𝐷_𝑡𝑁 and 𝑚_𝐷>0 units of derivative, 𝐷_𝑡𝑛 is the subgame perfect equilibrium price if and only if 𝛽^∗_𝑡𝑛=𝑚_𝐷,(17) for every 𝑛=0,1,…,𝑁−1. Likewise, 𝐷_𝑡𝑛 is the naive equilibrium price if and only if 𝛽_𝑡𝑛=𝑚_𝐷,(18) for every 𝑛=0,1,…,𝑁−1.

This simply says that there is 𝑚_𝐷 units of derivative in the market and it is priced such that “it is optimal” for the representative agent to acquire it. The interest of the representative agent in holding the derivative comes from the fact that the primary asset is correlated with the nontradable asset which is an underlying of the derivative. Moreover, agent's income is assumed to be related to the nontradable asset 𝑆; thus, the risk of income fluctuations due to 𝑆 can be hedged by trading 𝐷. Let 𝑟^𝑐_𝑡𝑛, given by 𝑟^𝑐_𝑡𝑛𝜇∶=^𝑐𝑡𝑛𝜎^𝑐_𝑡𝑛,(19) be the market price of risk (MPR) for the primary asset which is assumed positive. We choose time length ℎ small enough such that 1≥𝑟^𝑐_𝑡𝑛√ℎ.

Remark 2. For the naive equilibrium price to be implemented the representative agent has to exhibit the same risk aversion throughout time. This may not be realistic since different factors (e.g., economic and financial) may change the risk preference of the representative agent. In the light of this, the naive equilibrium price can be seen as a benchmark price. A more realistic price is the subgame perfect equilibrium price; this, unlike the naive equilibrium price, takes into account the changes of the representative agent's risk aversion.

3.1. Single Period

Define 𝐴_𝑡𝑁−1∶={𝜔∈Ω∶Δ𝑏¹_𝑡𝑁−1=1} and 𝐴^𝑐_𝑡𝑁−1∶={𝜔∈Ω∶Δ𝑏¹_𝑡𝑁−1=−1}.

Theorem 3. The subgame perfect equilibrium price at time 𝑡_𝑁−1 is given by 𝐷_𝑡𝑁−1=𝔼^ℚ∗𝐷_𝑡𝑁∣ℱ_𝑡𝑁−1,(20) where the probability measure ℚ^∗ is defined by 𝑑ℚ^∗𝑑ℙ=Λ^∗_𝑡𝑁𝔼𝑑ℚ^∗𝑑ℙ∣ℱ_𝑡𝑁−1.(21) The pricing kernel Λ^∗_𝑡𝑁 is Λ^∗_𝑡𝑁𝜆∶=^𝑡𝑁−1𝑒^{−𝛾𝑡𝑁−1(𝑚𝐷𝐷𝑡𝑁+𝐼𝑡𝑁)}𝔼𝑒^{−𝛾𝑡𝑁−1(𝑚𝐷𝐷𝑡𝑁+𝐼𝑡𝑁)}∣𝐴_𝑡𝑁−1∨ℱ_𝑡𝑁−1,𝑖𝑓𝜔∈𝐴_𝑡𝑁−1𝜆_𝑡𝑁−1𝑒^{−𝛾𝑡𝑁−1(𝑚𝐷𝐷𝑡𝑁+𝐼𝑡𝑁)}𝔼𝑒^{−𝛾𝑡𝑁−1(𝑚𝐷𝐷𝑡𝑁+𝐼𝑡𝑁)}∣𝐴^𝑐_𝑡𝑁−1∨ℱ_𝑡𝑁−1,𝑖𝑓𝜔∈𝐴^𝑐_𝑡𝑁−1,(22) with 𝜆_𝑡𝑁−1=1−𝑟^𝑐_𝑡𝑁−1√ℎ,𝑖𝑓𝜔∈𝐴_𝑡𝑁−11+𝑟^𝑐_𝑡𝑁−1√ℎ,𝑖𝑓𝜔∈𝐴^𝑐_𝑡𝑁−1.(23) The optimal trading strategy is given by 𝛼^∗_𝑡𝑁−1=12𝛾_𝑡𝑁−1𝜎^𝑐_𝑡𝑁−1√ℎlog1+𝑟^𝑐_𝑡𝑁−1√ℎ1−𝑟^𝑐_𝑡𝑁−1√ℎ+12𝛾_𝑡𝑁−1𝜎^𝑐_𝑡𝑁−1√ℎ𝔼𝑒×log^{−𝛾𝑡𝑁−1(𝑚𝐷𝐷𝑡𝑁+𝐼𝑡𝑁)}∣𝐴_𝑡𝑁−1∨ℱ_𝑡𝑁−1𝔼𝑒^{−𝛾𝑡𝑁−1(𝑚𝐷𝐷𝑡𝑁+𝐼𝑡𝑁)}∣𝐴^𝑐_𝑡𝑁−1∨ℱ_𝑡𝑁−1.(24)

Proof. Proof of this theorem is done in Appendix A.

Remark 4. For the naive equilibrium price just replace 𝛾_𝑡𝑁−1 with 𝛾_𝑡0 in the above formulas.

Next we prove that the probability measure ℚ^∗ is a martingale measure so the subgame perfect equilibrium price is arbitrage-free (similarly it can be shown that the naive equilibrium price is arbitrage free).

Lemma 5. The tradable assets {𝐶_𝑡𝑛}_{𝑛=𝑁−1,𝑁} and {𝐷_𝑡𝑛}_{𝑛=𝑁−1,𝑁} are martingales under ℚ^∗.

Proof. {𝐷_𝑡𝑛}_{𝑛=𝑁−1,𝑁} is martingale under ℚ^∗ by definition. Next we show that {𝐶_𝑡𝑛}_{𝑛=𝑁−1,𝑁} is martingale under ℚ^∗. It suffices to prove that with 𝑛=𝑁−1𝔼^ℚ∗𝐶_𝑡𝑛+1𝐶_𝑡𝑛∣ℱ_𝑡𝑛=𝔼^ℚ∗1+𝜎^𝑐_𝑡𝑛√ℎ𝑟^𝑐_𝑡𝑛√ℎ+Δ𝑏¹_𝑡𝑛∣ℱ_𝑡𝑛=1+𝜎^𝑐_𝑡𝑛√ℎ𝔼^ℚ∗𝑟^𝑐_𝑡𝑛√ℎ+Δ𝑏¹_𝑡𝑛∣ℱ_𝑡𝑛=1.(25) This is the case if 𝔼^ℚ∗𝑟^𝑐_𝑡𝑛√ℎ+Δ𝑏¹_𝑡𝑛∣ℱ_𝑡𝑛=0.(26) When 𝑛=𝑁−1, (26) is equivalent to 𝑟^𝑐_𝑡𝑁−1√ℎ+12𝔼Λ^∗_𝑡𝑁𝔼𝑑ℚ^∗𝑑ℙ∣ℱ_𝑡𝑁−1∣𝐴_𝑡𝑁−1∨ℱ_𝑡𝑁−1+𝑟^𝑐_𝑡𝑁−1√ℎ−12𝔼Λ^∗_𝑡𝑁𝔼𝑑ℚ^∗𝑑ℙ∣ℱ_𝑡𝑁−1∣𝐴^𝑐_𝑡𝑁−1∨ℱ_𝑡𝑁−1=1−𝑟^𝑐_𝑡𝑁−1²ℎ2𝔼𝑑ℚ^∗𝑑ℙ∣ℱ_𝑡𝑁−1+𝑟^𝑐_𝑡𝑁−1²ℎ−12𝔼𝑑ℚ^∗𝑑ℙ∣ℱ_𝑡𝑁−1=0,(27) and this completes the proof.

3.2. Multiple Periods

Let us define the sets 𝐴_{𝑡𝑁−𝑛}∶={𝜔∈Ω∶Δ𝑏¹_{𝑡𝑁−𝑛}=1} and 𝐴^𝑐_{𝑡𝑁−𝑛}∶={𝜔∈Ω∶Δ𝑏¹_{𝑡𝑁−𝑛}=−1}; to ease notations, let us denote 𝔼_𝑡𝑛[⋅]∶=𝔼⋅∣ℱ_𝑡𝑛,𝔼_𝑡𝑛[⋅∣𝒢]∶=𝔼⋅∣𝒢∨ℱ_𝑡𝑛,(28) for every 𝒢⊂ℱ. The equilibrium prices are computed by a recursive algorithm. Imagine that the equilibrium prices after 𝑡_𝑁−𝑛 were found and now we want to find the equilibrium price at 𝑡_𝑁−𝑛. At time 𝑡_𝑁−𝑛, define the random variables 𝑌^∗_{𝑡𝑁−𝑛+1} and 𝑌_{𝑡𝑁−𝑛+1} by 𝑒^{−𝛾𝑡𝑁−𝑛𝑌∗𝑡𝑁−𝑛+1}∶=𝔼_{𝑡𝑁−𝑛+1}×exp−𝛾_{𝑡𝑁−𝑛}^𝑁−1_{𝑘=𝑁−𝑛+1}Δ𝑋^∗_𝑡𝑘+𝐼_𝑡𝑁,𝑒(29)_−𝛾𝑡0𝑌_{𝑡𝑁−𝑛+1}∶=𝔼_{𝑡𝑁−𝑛+1}×exp−𝛾_𝑡0^𝑁−1_{𝑘=𝑁−𝑛+1}Δ𝑋_𝑡𝑘+𝐼_𝑡𝑁.(30) Here Δ𝑋^∗_𝑡𝑘=𝛼^∗_𝑡𝑘𝜇^𝑐_𝑡𝑘ℎ+𝜎^𝑐_𝑡𝑘√ℎΔ𝑏¹_𝑡𝑘+𝑚_𝐷Δ𝐷_𝑡𝑘,(31) for any 𝑘=𝑁−𝑛+1,…,𝑁−2,𝑁−1; the subgame perfect strategy 𝛼^∗_𝑡𝑘 is given by 𝛼^∗_𝑡𝑘=12𝛾_𝑡𝑘𝜎^𝑐_𝑡𝑘√ℎlog1+𝑟^𝑐_𝑡𝑘√ℎ1−𝑟^𝑐_𝑡𝑘√ℎ+12𝛾_𝑡𝑘𝜎^𝑐_𝑡𝑘√ℎ𝔼log^𝑡𝑘𝑒^{−𝛾𝑡𝑘(𝑚𝐷𝐷𝑡𝑘+1+𝑌∗𝑡𝑘+1)}∣𝐴_𝑡𝑘𝔼_𝑡𝑘𝑒^{−𝛾𝑡𝑘(𝑚𝐷𝐷𝑡𝑘+1+𝑌∗𝑡𝑘+1)}∣𝐴^𝑐_𝑡𝑘.(32) Moreover Δ𝑋_𝑡𝑘=𝛼_𝑡𝑘𝜇^𝑐_𝑡𝑘ℎ+𝜎^𝑐_𝑡𝑘√ℎΔ𝑏¹_𝑡𝑘+𝑚_𝐷Δ𝐷_𝑡𝑘,(33) for any 𝑘=𝑁−𝑛+1,…,𝑁−2,𝑁−1; the naive strategy 𝛼_𝑡𝑘 is given by 𝛼_𝑡𝑘=12𝛾_𝑡0𝜎^𝑐_𝑡𝑘√ℎlog1+𝑟^𝑐_𝑡𝑘√ℎ1−𝑟^𝑐_𝑡𝑘√ℎ+12𝛾_𝑡0𝜎^𝑐_𝑡𝑘√ℎ𝔼log^𝑡𝑘𝑒^{−𝛾𝑡0(𝑚𝐷𝐷𝑡𝑘+1+}𝑌_𝑡𝑘+1)∣𝐴_𝑡𝑘𝔼_𝑡𝑘𝑒^{−𝛾𝑡0(𝑚𝐷𝐷𝑡𝑘+1+}𝑌_𝑡𝑘+1)∣𝐴^𝑐_𝑡𝑘.(34) Next, define the one-step period pricing kernels Λ^∗_{𝑡𝑁−𝑛+1} and Λ_{𝑡𝑁−𝑛+1} by Λ^∗_{𝑡𝑁−𝑛+1}𝜆∶=^{𝑡𝑁−𝑛}𝑒^{−𝛾𝑡𝑁−𝑛(𝑚𝐷𝐷𝑡𝑁−𝑛+1+𝑌∗𝑡𝑁−𝑛+1)}𝔼_{𝑡𝑁−𝑛}𝑒^{−𝛾𝑡𝑁−𝑛(𝑚𝐷𝐷𝑡𝑁−𝑛+1+𝑌∗𝑡𝑁−𝑛+1)}∣𝐴_{𝑡𝑁−𝑛},if𝜔∈𝐴_{𝑡𝑁−𝑛},𝜆_{𝑡𝑁−𝑛}𝑒^{−𝛾𝑡𝑁−𝑛(𝑚𝐷𝐷𝑡𝑁−𝑛+1+𝑌∗𝑡𝑁−𝑛+1)}𝔼_{𝑡𝑁−𝑛}𝑒^{−𝛾𝑡𝑁−𝑛(𝑚𝐷𝐷𝑡𝑁−𝑛+1+𝑌∗𝑡𝑁−𝑛+1)}∣𝐴^𝑐_{𝑡𝑁−𝑛},if𝜔∈𝐴^𝑐_{𝑡𝑁−𝑛},Λ(35)_{𝑡𝑁−𝑛+1}𝜆∶=^{𝑡𝑁−𝑛}𝑒^{−𝛾𝑡0(𝑚𝐷𝐷𝑡𝑁−𝑛+1+}𝑌_{𝑡𝑁−𝑛+1)}𝔼_{𝑡𝑁−𝑛}𝑒^{−𝛾𝑡0(𝑚𝐷𝐷𝑡𝑁−𝑛+1+}𝑌_{𝑡𝑁−𝑛+1)}∣𝐴_{𝑡𝑁−𝑛},if𝜔∈𝐴_{𝑡𝑁−𝑛},𝜆_{𝑡𝑁−𝑛}𝑒^{−𝛾𝑡0(𝑚𝐷𝐷𝑡𝑁−𝑛+1+}𝑌_{𝑡𝑁−𝑛+1}⁾𝔼_{𝑡𝑁−𝑛}𝑒^{−𝛾𝑡0(𝑚𝐷𝐷𝑡𝑁−𝑛+1+}𝑌_{𝑡𝑁−𝑛+1)}∣𝐴^𝑐_{𝑡𝑁−𝑛},if𝜔∈𝐴^𝑐_{𝑡𝑁−𝑛}.(36) Here 𝜆_{𝑡𝑁−𝑛}=1−𝑟^𝑐_{𝑡𝑁−𝑛}√ℎ,if𝜔∈𝐴_{𝑡𝑁−𝑛},1+𝑟^𝑐_{𝑡𝑁−𝑛}√ℎ,if𝜔∈𝐴^𝑐_{𝑡𝑁−𝑛}.(37)

Remark 6. Notice that 𝑌_{𝑡𝑁−𝑛+1}, defined by (30), is the certainty equivalent at time 𝑡_{𝑁−𝑛+1} of the naive strategy which does not update the risk aversion and assumes constant risk aversion 𝛾_𝑡0. Similarly, 𝑌^∗_{𝑡𝑁−𝑛+1} can be seen as the certainty equivalent of the subgame perfect strategy with updating risk preference over time. In the special case of constant coefficient of absolute risk aversion, 𝑌_{𝑡𝑁−𝑛+1} and 𝑌^∗_{𝑡𝑁−𝑛+1} are equal. Furthermore, it can be easily seen that 𝑒^−𝛾𝑡0𝑌_{𝑡𝑁−𝑛+1}=𝔼_{𝑡𝑁−𝑛+1}×exp−𝛾_𝑡0Δ𝑋_{𝑡𝑁−𝑛+1}+𝑌_{𝑁−𝑛+2},(38) where 𝑋_{𝑡𝑁−𝑛+1} is the change of self-financed portfolio value during the time period [(𝑁−𝑛+1)ℎ,(𝑁−𝑛+2)ℎ] so that 𝑌_{𝑡𝑁−𝑛+2} can be used to find out the values of 𝑌_{𝑡𝑁−𝑛+1}. However, we can observe that 𝑒^{−𝛾𝑡𝑁−𝑛𝑌∗𝑡𝑁−𝑛+1}≠𝔼_{𝑡𝑁−𝑛+1}×exp−𝛾_{𝑡𝑁−𝑛}Δ𝑋^∗_{𝑡𝑁−𝑛+1}+𝑌^∗_{𝑁−𝑛+2}(39) because the risk aversion changes over time in the definition of 𝑌^∗ in (29). More precisely, 𝑌^∗_{𝑡𝑁−𝑛+1} is defined with 𝛾_{𝑡𝑁−𝑛}, while 𝑌^∗_{𝑡𝑁−𝑛+2} is defined with 𝛾_{𝑡𝑁−𝑛+1} which can be different from 𝛾_{𝑡𝑁−𝑛}. Therefore, all possible future changes of risk aversion until the time horizon 𝑇 should be considered to compute the value of 𝑌^∗_{𝑁−𝑛+1}. Moreover, the pricing kernel Λ^∗_{𝑡𝑁−𝑛+1} in (35) for subgame perfect equilibrium price contains 𝑌^∗_{𝑡𝑁−𝑛+1} as well as 𝛾_{𝑡𝑁−𝑛}. As a consequence, all possible future changes of risk aversion should be examined to find the pricing kernel Λ^∗_{𝑡𝑁−𝑛+1}.

Remark 7. The myopic term in 𝛼^∗_𝑡𝑘 in (32) is (1/2𝛾_𝑡𝑘𝜎^𝑐_𝑡𝑘√ℎ)log((1+𝑟^𝑐_𝑡𝑘√ℎ)/(1−𝑟^𝑐_𝑡𝑘√ℎ)) and in 𝛼_𝑡𝑘 is (1/2𝛾_𝑡0𝜎^𝑐_𝑡𝑘√ℎ)log((1+𝑟^𝑐_𝑡𝑘√ℎ)/(1−𝑟^𝑐_𝑡𝑘√ℎ)). Moreover, it can be verified that they satisfy lim_ℎ↓012𝛾_𝑡𝑘𝜎^𝑐_𝑡𝑘√ℎlog1+𝑟^𝑐_𝑡𝑘√ℎ1−𝑟^𝑐_𝑡𝑘√ℎ=𝑟^𝑐_𝑡𝑘𝛾_𝑡𝑘𝜎^𝑐_𝑡𝑘,(40)lim_ℎ↓012𝛾_𝑡0𝜎^𝑐_𝑡𝑘√ℎlog1+𝑟^𝑐_𝑡𝑘√ℎ1−𝑟^𝑐_𝑡𝑘√ℎ=𝑟^𝑐_𝑡𝑘𝛾_𝑡0𝜎^𝑐_𝑡𝑘,(41) respectively. Notice that the right-hand sides of (41) are the well-known myopic term of the optimal investment strategy with constant absolute risk aversion 𝛾_𝑡0 in continuous time model. The only difference in (40) with subgame perfect strategy is that the risk aversion is updated; that is, 𝛾_𝑡𝑘 is used, while it is not in the naive strategy (41). The second terms of (32) and (34) are hedging terms which take into account the other factors, such as investment in derivative, stochastic parameters of underlying processes, income, and stochastic risk aversion (in case of subgame perfect strategy).

The following theorem is the main result of the paper.

Theorem 8. The subgame perfect equilibrium price at time 𝑡_𝑁−𝑛 is given by 𝐷_{𝑡𝑁−𝑛}=𝔼^ℚ∗_{𝑡𝑁−𝑛}𝐷_{𝑡𝑁−𝑛+1},(42) where the probability measure ℚ^∗ is defined by 𝑑ℚ^∗𝑑ℙ=Λ^∗_𝑡𝑁Λ^∗_𝑡𝑁−1⋯Λ^∗_𝑡1.(43) The subgame perfect strategy (in the primary asset) is 𝛼^∗=(𝛼^∗_𝑡0,𝛼^∗_𝑡1,…,𝛼^∗_𝑡𝑁), with 𝛼^∗_𝑡𝑘 defined by (32). The naive equilibrium price at time 𝑡_𝑁−𝑛 is given by 𝐷_{𝑡𝑁−𝑛}ℚ=𝔼_{𝑡𝑁−𝑛}𝐷_{𝑡𝑁−𝑛+1},(44) where the probability measure ℚ is defined by 𝑑ℚ=Λ𝑑ℙ^𝑡𝑁Λ_𝑡𝑁−1⋯Λ_𝑡1.(45) The naive strategy (in the primary asset) is 𝛼=(𝛼_𝑡0,𝛼_𝑡1,…,𝛼_𝑡𝑁), with 𝛼_𝑡𝑘 defined by (34).

Proof. Proof of this theorem is done in Appendix B.

For the naive equilibrium price, we recover the following classical result.

Corollary 9. The pricing kernel for naive equilibrium price equals the marginal utility; that is, Λ_{𝑡𝑁−𝑛+1}=𝔼_{𝑡𝑁−𝑛+1}𝑈^′𝑊_𝑡𝑁𝔼_{𝑡𝑁−𝑛}𝑈^′𝑊_𝑡𝑁,(46) where 𝑈(𝑥)=−𝑒^−𝛾𝑥, and 𝑊_𝑡𝑁 (see (5)) is the wealth obtained by naive strategies.

Proof. Proof of this corollary is done in Appendix C.

Lemma 10. The tradable assets {𝐶_𝑡𝑛}_{𝑛=0,1,…,𝑁} and {𝐷_𝑡𝑛}_{𝑛=0,1,…,𝑁} are martingales under 𝑄^∗.

Proof. It can be proved by following similar argument to the proof of Lemma 5.

4. Numerical Results

Throughout this section, we specialize to a regime switching model. A discrete time finite state homogeneous Markov chain (MC) 𝐽∶=(𝐽_𝑡𝑛)_{𝑛=0,1,…,∞} is defined on (Ω,ℱ,{ℱ_𝑡𝑛},ℙ) and it takes values in the state space 𝒮={𝟎,𝟏} (which represents two states of the market: bull and bear). The 𝑛-step transition matrix 𝑃^(𝑛)=(𝑃^𝑛_𝑖𝑗), is defined by 𝑃^(𝑛)_𝑖𝑗∶=ℙ𝐽_𝑡𝑛=𝑗∣𝐽_𝑡0=𝑖,𝑖,𝑗=𝟎,𝟏,𝑛=0,1,…,∞,(47) where 𝑃⁽⁰⁾_𝑖𝑗=1 when 𝑖=𝑗; otherwise 𝑃⁽⁰⁾_𝑖𝑗=0. We assume that the distribution of 𝐽₀ is known, and ℙ𝐽₀=𝑖∣ℱ₀=ℙ𝐽₀=𝑖,𝑖=𝟎,𝟏.(48) The risk aversion is changed depending on the state and denoted by 𝛾_𝑡𝑛=𝛾(𝐽_𝑡𝑛⋅)

In this section we give a concrete example. Consider the electricity industry one of the most weather-sensitive businesses in the economy. Energy price and temperature are denoted by 𝐶 and 𝑆, respectively. When the temperature increases, there is a higher demand for electricity due to the usage of air conditioners. In turn this will lead to higher energy prices. Therefore, 𝐶 and 𝑆 are assumed to be positively correlated. In our model an energy provider hedges the weather exposure by selling 𝑚_𝐷 shares of weather derivative to the representative agent. This derivative is designed to have a higher payoff when temperature is high (e.g., call option). It is supposed that the representative agent's income depends on the weather, so he/she has an incentive to buy this product because of his/her income exposure to the weather. Moreover, weather derivative also allows the representative agent to construct a more diversified portfolio. The baseline parameters are 𝐶₀=1,𝜇^𝑐=0.1,𝜎^𝑐=0.2,𝑆₀𝜇=1,_𝑠=0.1,𝜎^𝑠1=0.2,𝜌=0.5,𝑇=1,𝑁=52,ℎ=(0)𝛾(1)52,𝛾=1,=2,𝑚_𝐷𝐼=1,_𝑡𝑁=4exp−0.1𝑆_𝑡𝑁−𝑆₀,𝐷_𝑡𝑁=𝑆_𝑡𝑁−𝐾⁺,𝐾=1.(49) One-step transition matrix is given by 𝑃⁽¹⁾=0.60.40.60.4.(50) Notice that 𝑆 is the normalized temperature. The income of the agent is assumed to be exponential function of 𝑆 to prevent negative income. However, our result in this section still can be obtained if we use different models for income (e.g., linear function of 𝑆). Since the income is decreasing in 𝑆, the agent has an incentive to have long position in 𝑆 which is impossible because 𝑆 is a nontradable asset. Instead, there is a demand for 𝐷 which is a call option on nontradable asset 𝑆. Letting 𝑁=52 with 𝑇=1 means weekly rebalancing of portfolio throughout a year. In fact, equilibrium price changes if we consider large 𝑁. However, the change of price is very small (less than 0.2% even if we consider 𝑁=200) so we may only focus on the results with 𝑁=52.

Let 𝐷₀^{𝛾(0),𝛾(1)} be the subgame perfect equilibrium price of 𝐷 at time 0 with 𝐽₀=0 and time changing risk aversion. Likewise, 𝐷₀^{𝛾(1),𝛾(0)} is the subgame perfect equilibrium price of 𝐷 at time 0 with 𝐽₀=1. It is natural to define 𝐷₀^𝛾,𝛾 as the naive prices of 𝐷 at time 0 with risk aversion 𝛾. Recall that this naive price ignores the time changing risk aversion of the agent.

4.1. Effect of Risk Aversion

First of all, let us consider the effect of risk aversion on the naive equilibrium price. As we can see in Figure 1, the naive equilibrium price 𝐷₀^𝛾,𝛾 decreases as risk aversion 𝛾 increases. This is intuitive because the derivative 𝐷 is also a risky asset and there is a less demand for risky asset if the agent is more risk averse. A lower demand for 𝐷 leads to a lower naive equilibrium price as a result. In sum, a higher risk aversion results in a lower naive equilibrium price.

Figure 1 

Effect of risk aversion on the equilibrium prices.

Therefore, if we consider the naive prices 𝐷₀^{𝛾(0),𝛾(0)} and 𝐷₀^{𝛾(1),𝛾(1)} for different risk aversions 𝛾(0) and 𝛾(1) which satisfy 𝛾(0)<𝛾(1), we have 𝐷₀^{𝛾(0),𝛾(0)}>𝐷₀^{𝛾(1),𝛾(1)}.(51) From (51), we can infer that if 𝛾(0)<𝛾(1), the subgame perfect equilibrium prices 𝐷₀^{𝛾(0),𝛾(1)} and 𝐷₀^{𝛾(1),𝛾(0)} are in between 𝐷₀^{𝛾(0),𝛾(0)} and 𝐷₀^{𝛾(1),𝛾(1)}; that is, 𝐷₀^{𝛾(1),𝛾(1)}<𝐷₀^{𝛾(0),𝛾(1)}<𝐷₀^{𝛾(0),𝛾(0)},𝐷₀^{𝛾(1),𝛾(1)}<𝐷₀^{𝛾(1),𝛾(0)}<𝐷₀^{𝛾(0),𝛾(0)}.(52) In Figure 1, we can verify the inequalities in (52).

Let 𝛾_𝑝 be the weighted average of 𝛾(0) and 𝛾(1) with steady-state probabilities 𝑃₀ and 𝑃₁ of states 𝟎 and 𝟏 as weights; that is, 𝛾_𝑝∶=𝑃₀𝛾(0)+𝑃₁𝛾(1).(53) Consider the naive equilibrium price 𝐷^{𝛾𝑝,𝛾𝑝}₀ with 𝛾_𝑝, and then a question arises. Is the naive equilibrium price 𝐷^{𝛾𝑝,𝛾𝑝}₀ a good estimate of the subgame perfect equilibrium prices 𝐷₀^{𝛾(0),𝛾(1)} and 𝐷₀^{𝛾(1),𝛾(0)}? If the gap between 𝐷^{𝛾𝑝,𝛾𝑝}₀ and the subgame perfect equilibrium prices is small enough, we may use the naive equilibrium price 𝐷^{𝛾𝑝,𝛾𝑝}₀ instead of the subgame perfect equilibrium prices. However, the answer is no because the gap is considerable for many cases.

In Figure 2, 𝛾(0) is fixed at 𝛾(0)=1 and we examine the changes of equilibrium prices as 𝛾(1) changes. The left panel shows the equilibrium prices; the solid line is 𝐷^{𝛾𝑝,𝛾𝑝}₀, the dashed line is 𝐷₀^{𝛾(0),𝛾(1)}, and the dotted line is 𝐷₀^{𝛾(1),𝛾(0)}. In the right panel, percentage changes from 𝐷^{𝛾𝑝,𝛾𝑝}₀ are illustrated; that is, 𝐷Dashedline:^{0𝛾(0),𝛾(1)}−𝐷^{𝛾𝑝,𝛾𝑝}₀𝐷^{𝛾𝑝,𝛾𝑝}₀𝐷×100,Dottedline:^{0𝛾(1),𝛾(0)}−𝐷^{𝛾𝑝,𝛾𝑝}₀𝐷^{𝛾𝑝,𝛾𝑝}₀×100.(54) We can observe that there are significant gaps between the naive equilibrium price 𝐷^{𝛾𝑝,𝛾𝑝}₀ and the subgame perfect equilibrium prices 𝐷₀^{𝛾(0),𝛾(1)} and 𝐷₀^{𝛾(1),𝛾(0)} for large enough 𝛾(1), and the gaps become wider as 𝛾(1) increases. This implies that if the agent becomes much more risk averse in the bear market than in the bull market, the naive equilibrium price with 𝛾_𝑝 cannot be a good estimate of the subgame perfect equilibrium prices and thus the subgame perfect equilibrium prices should be derived using our pricing principle.

(a)

(b)

Figure 2 

The gap between 𝐷𝛾𝑝,𝛾𝑝0 and the subgame perfect equilibrium prices. 𝛾(0) is fixed at 1 in both panels.

4.2. Effect of 𝑚_𝐷

Now we examine the effect of 𝑚_𝐷, the number of units of derivative 𝐷 in the market, on the equilibrium prices.

It is observed that the equilibrium prices (both subgame perfect and naive) are decreasing with respect to 𝑚_𝐷 in the left panel of Figure 3. This is a consistent result with Gârleanu et al. [25]. They considered a derivative market with two participants: representative dealer and end users. End users' aggregate demand for derivative 𝑑 is assumed to be a random variable. On the contrary, dealer's demand for derivative 𝑞 is optimally determined by utility maximization. The market clearing condition is 𝑞+𝑑=0. Their result is 𝜕𝑝𝜕𝑑≥0.(55) Notice that as 𝑑 decreases, there are more derivatives which should be purchased by representative dealer to achieve market clearing. So the decreasing 𝑑 in Gârleanu et al. [25] corresponds to increasing 𝑚_𝐷 in our model. Therefore, our result 𝜕𝐷₀𝜕𝑚_𝐷<0(56) is consistent with (55). Moreover, as we can observe in the right panel of Figure 3, the gaps between the subgame perfect equilibrium prices 𝐷₀^{𝛾(0),𝛾(1)} and 𝐷₀^{𝛾(1),𝛾(0)} and the naive equilibrium price 𝐷^{𝛾𝑝,𝛾𝑝}₀ is considerable for large enough 𝑚_𝐷.

(a)

(b)

Figure 3 

The gap between 𝐷𝛾𝑝,𝛾𝑝0 and the subgame perfect equilibrium prices.

5. Conclusion

We consider a multiperiod incomplete market model with tradable risky asset, nontradable underlying and contingent claim (derivative) written on both tradable risky asset, and nontradable underlying. We develop an equilibrium pricing principle for the derivative; the price of the derivative is priced by combining the optimal strategies of representative agent and market clearing condition for the derivative. Since the agent's risk aversion is assumed to be stochastic, classical approach with precommitment gives the naive strategies and the naive equilibrium price which do not update the change of risk preference. Alternatively, we consider the subgame perfect strategies and derive the corresponding subgame perfect equilibrium price. We consider two-state regime switching model for numerical experiments. Then the numerical result shows that naive equilibrium price cannot be a good estimate for subgame perfect equilibrium price in many cases. In particular, if the representative agent's risk aversion changes a lot depending on regime or the number of units of derivative we consider is large, then the gap between naive equilibrium price and subgame perfect equilibrium price is considerable. This shows the importance of our subgame perfect equilibrium pricing principle.

Appendices

A. Proof of Theorem 3

From (4) and (5), it follows that 𝔼_𝑡𝑁−1−exp−𝛾_𝑡𝑁−1𝑋^𝜋_𝑡𝑁+𝐼_𝑡𝑁=𝔼_𝑡𝑁−1−exp−𝛾_𝑡𝑁−1𝑋_𝑡𝑁−1+Δ𝑋^𝜋_𝑡𝑁−1+𝐼_𝑡𝑁=−𝑒^{−𝛾𝑡𝑁−1𝑋𝜋𝑡𝑁−1}𝔼_𝑡𝑁−1exp−𝛾_𝑡𝑁−1Δ𝑋^𝜋_𝑡𝑁−1+𝐼_𝑡𝑁=−𝑒^{−𝛾𝑡𝑁−1𝑋𝜋𝑡𝑁−1}𝑔_𝑁−1𝛼,𝛽,⋅,(A.1) where 𝑔_𝑁−1𝛼,𝛽,⋅∶=𝔼_𝑡𝑁−1exp−𝛾_𝑡𝑁−1Δ𝑋^𝜋_𝑡𝑁−1+𝐼_𝑡𝑁=12𝑒^{−𝛾𝑡𝑁−1𝛼(𝜇𝑐𝑡𝑁−1ℎ+𝜎𝑐𝑡𝑁−1√ℎ)}×𝔼_𝑡𝑁−1𝑒^{−𝛾𝑡𝑁−1𝛽Δ𝐷𝑡𝑁−1}𝑒^{−𝛾𝑡𝑁−1𝐼𝑡𝑁}∣𝐴_𝑡𝑁−1+12𝑒^{−𝛾𝑡𝑁−1𝛼(𝜇𝑐𝑡𝑁−1ℎ−𝜎𝑐𝑡𝑁−1√ℎ)}×𝔼_𝑡𝑁−1𝑒^{−𝛾𝑡𝑁−1𝛽Δ𝐷𝑡𝑁−1}𝑒^{−𝛾𝑡𝑁−1𝐼𝑡𝑁}∣𝐴^𝑐_𝑡𝑁−1.(A.2) Recall that 𝐴_𝑡𝑘∶={𝜔∈Ω∶Δ𝑏¹_𝑡𝑘=1} and 𝐴^𝑐_𝑡𝑘∶={𝜔∈Ω∶Δ𝑏¹_𝑡𝑘=−1}. The function 𝑔_𝑁−1(𝛼,𝛽,⋅) has the following properties: 𝑔_𝑁−1(0,0,⋅)=𝔼_𝑡𝑁−1𝑒^{−𝛾𝑡𝑁−1𝐼𝑡𝑁}≤1.(A.3) For a fixed 𝛽, it follows that for small ℎ𝑔_𝑁−1∞,𝛽,⋅=∞;𝑔_𝑁−1−∞,𝛽,⋅=∞.(A.4) By arbitrage argument it follows that 𝐷_𝑡𝑁−1 belongs to the interval 𝐷_𝑡𝑁−1(𝜔)∈inf_ℚ𝔼^ℚ𝐷_𝑡𝑁,sup_ℚ𝔼^ℚ𝐷_𝑡𝑁,𝜔∈Ω,(A.5) where ℚ ranges over the set of probability measures. Consequently, Δ𝐷_𝑡𝑁−1(𝜔)∈𝐷_𝑡𝑁−sup_ℚ𝔼^ℚ𝐷_𝑡𝑁,𝐷_𝑡𝑁−inf_ℚ𝔼^ℚ𝐷_𝑡𝑁.(A.6) Thus, the sets {𝜔∈Ω∶Δ𝐷_𝑡𝑁−1(𝜔)>0} and {𝜔∈Ω∶Δ𝐷_𝑡𝑁−1(𝜔)<0} have positive probability. This implies that 𝑔_𝑁−1(𝛼,∞,⋅)=∞,𝑔_𝑁−1(𝛼,−∞,⋅)=∞.(A.7) From the above analysis, it follows that the minimum of the function of 𝑔 is a critical point. First order conditions lead to 𝜕𝑔_𝑁−1=𝜕𝛼−𝛾_𝑡𝑁−1𝜇^𝑐_𝑡𝑁−1ℎ+𝜎^𝑐_𝑡𝑁−1√ℎ2⋅𝑒^{−𝛾𝑡𝑁−1𝛼(𝜇𝑐𝑡𝑁−1ℎ+𝜎𝑐𝑡𝑁−1√ℎ)}×𝔼_𝑡𝑁−1𝑒^{−𝛾𝑡𝑁−1𝛽Δ𝐷𝑡𝑁−1}𝑒^{−𝛾𝑡𝑁−1𝐼𝑡𝑁}∣𝐴_𝑡𝑁−1+−𝛾_𝑡𝑁−1𝜇^𝑐_𝑡𝑁−1ℎ−𝜎^𝑐_𝑡𝑁−1√ℎ2⋅𝑒^{−𝛾𝑡𝑁−1𝛼(𝜇𝑐𝑡𝑁−1ℎ−𝜎𝑐𝑡𝑁−1√ℎ)}×𝔼_𝑡𝑁−1𝑒^{−𝛾𝑡𝑁−1𝛽Δ𝐷𝑡𝑁−1}𝑒^{−𝛾𝑡𝑁−1𝐼𝑡𝑁}∣𝐴^𝑐_𝑡𝑁−1=0,𝜕𝑔_𝑁−1=1𝜕𝛽2𝑒^{−𝛾𝑡𝑁−1𝛼(𝜇𝑐𝑡𝑁−1ℎ+𝜎𝑐𝑡𝑁−1√ℎ)}×𝔼_𝑡𝑁−1−𝛾_𝑡𝑁−1Δ𝐷_𝑡𝑁−1𝑒^{−𝛾𝑡𝑁−1𝛽Δ𝐷𝑡𝑁−1}×𝑒^{−𝛾𝑡𝑁−1𝐼𝑡𝑁}∣𝐴_𝑡𝑁−1+12𝑒^{−𝛾𝑡𝑁−1𝛼(𝜇𝑐𝑡𝑁−1ℎ−𝜎𝑐𝑡𝑁−1√ℎ)}×𝔼_𝑡𝑁−1−𝛾_𝑡𝑁−1Δ𝐷_𝑡𝑁−1𝑒^{−𝛾𝑡𝑁−1𝛽Δ𝐷𝑡𝑁−1}×𝑒^{−𝛾𝑡𝑁−1𝐼𝑡𝑁}∣𝐴^𝑐_𝑡𝑁−1=0.(A.8) Recall that Δ𝐷_𝑡𝑁−1=𝐷_𝑡𝑁−𝔼^ℚ∗_𝑡𝑁−1𝐷_𝑡𝑁,(A.9) for an equilibrium pricing measure ℚ^∗ to be found. Since 𝔼^ℚ∗_𝑡𝑁−1[𝐷_𝑡𝑁] is ℱ_𝑡𝑁−1measurable, it follows that 𝛼^∗_𝑡𝑁−1=12𝛾_𝑡𝑁−1𝜎^𝑐_𝑡𝑁−1√ℎlog1+𝑟^𝑐_𝑡𝑁−1√ℎ1−𝑟^𝑐_𝑡𝑁−1√ℎ+12𝛾_𝑡𝑁−1𝜎^𝑐_𝑡𝑁−1√ℎ𝔼×log^𝑡𝑁−1𝑒^{−𝛾𝑡𝑁−1𝛽∗𝑡𝑁−1𝐷𝑡𝑁}𝑒^{−𝛾𝑡𝑁−1𝐼𝑡𝑁}∣𝐴_𝑡𝑁−1𝔼_𝑡𝑁−1𝑒^{−𝛾𝑡𝑁−1𝛽∗𝑡𝑁−1𝐷𝑁}𝑒^{−𝛾𝑡𝑁−1𝐼𝑡𝑁}∣𝐴^𝑐_𝑡𝑁−1.(A.10) By the equilibrium condition 𝛽^∗_𝑡𝑁−1=𝑚_𝐷. This together with 𝜕𝑔_𝑁−1/𝜕𝛽=0 leads to 𝔼_𝑡𝑁−1Δ𝐷_𝑡𝑁−1𝑒^{−𝛾𝑡𝑁−1𝑚𝐷Δ𝐷𝑡𝑁−1}𝑒^{−𝛾𝑡𝑁−1𝐼𝑡𝑁}∣𝐴_𝑡𝑁−1=−𝑒^{2𝛾𝑡𝑁−1𝛼∗𝑡𝑁−1𝜎𝑐𝑡𝑁−1√ℎ}𝔼_𝑡𝑁−1×Δ𝐷_𝑡𝑁−1𝑒^{−𝛾𝑡𝑁−1𝑚𝐷Δ𝐷𝑡𝑁−1}𝑒^{−𝛾𝑡𝑁−1𝐼𝑡𝑁}∣𝐴^𝑐_𝑡𝑁−1,𝑒(A.11)_{2𝛾𝑡𝑁−1𝛼∗𝑡𝑁−1𝜎𝑐𝑡𝑁−1√ℎ}=1+𝑟^𝑐_𝑡𝑁−1√ℎ𝔼_𝑡𝑁−1𝑒^{−𝛾𝑡𝑁−1𝑚𝐷𝐷𝑡𝑁}𝑒^{−𝛾𝑡𝑁−1𝐼𝑡𝑁}∣𝐴_𝑡𝑁−11−𝑟^𝑐_𝑡𝑁−1√ℎ𝔼_𝑡𝑁−1𝑒^{−𝛾𝑡𝑁−1𝑚𝐷𝐷𝑡𝑁}𝑒^{−𝛾𝑡𝑁−1𝐼𝑡𝑁}∣𝐴^𝑐_𝑡𝑁−1.(A.12) Combing the above equations leads to 21−𝑟^𝑐_𝑡𝑁−1√ℎ𝔼^ℚ∗_𝑡𝑁−1𝐷_𝑡𝑁=𝔼_𝑡𝑁−1𝐷_𝑡𝑁𝑒^{−𝛾𝑡𝑁−1𝑚𝐷𝐷𝑡𝑁}𝑒^{−𝛾𝑡𝑁−1𝐼𝑡𝑁}∣𝐴_𝑡𝑁−1𝔼_𝑡𝑁−1𝑒^{−𝛾𝑡𝑁−1𝑚𝐷𝐷𝑡𝑁}𝑒^{−𝛾𝑡𝑁−1𝐼𝑡𝑁}∣𝐴_𝑡𝑁−1+1+𝑟^𝑐_𝑡𝑁−1√ℎ1−𝑟^𝑐_𝑡𝑁−1√ℎ×𝔼_𝑡𝑁−1𝐷_𝑡𝑁𝑒^{−𝛾𝑡𝑁−1𝑚𝐷𝐷𝑡𝑁}𝑒^{−𝛾𝑡𝑁−1𝐼𝑡𝑁}∣𝐴^𝑐_𝑡𝑁−1𝔼_𝑡𝑁−1𝑒^{−𝛾𝑡𝑁−1𝑚𝐷𝐷𝑡𝑁}𝑒^{−𝛾𝑡𝑁−1𝐼𝑡𝑁}∣𝐴^𝑐_𝑡𝑁−1.(A.13) This together with (A.9) implies that 𝐷_𝑡𝑁−1=1−𝑟^𝑐_𝑡𝑁−1√ℎ2×𝔼_𝑡𝑁−1𝐷_𝑡𝑁𝑒^{−𝛾𝑡𝑁−1𝑚𝐷𝐷𝑡𝑁}𝑒^{−𝛾𝑡𝑁−1𝐼𝑡𝑁}∣𝐴_𝑡𝑁−1𝔼_𝑡𝑁−1𝑒^{−𝛾𝑡𝑁−1𝑚𝐷𝐷𝑡𝑁}𝑒^{−𝛾𝑡𝑁−1𝐼𝑡𝑁}∣𝐴_𝑡𝑁−1+1+𝑟^𝑐_𝑡𝑁−1√ℎ2×𝔼_𝑡𝑁−1𝐷_𝑡𝑁𝑒^{−𝛾𝑡𝑁−1𝑚𝐷𝐷𝑡𝑁}𝑒^{−𝛾𝑡𝑁−1𝐼𝑡𝑁}∣𝐴^𝑐_𝑡𝑁−1𝔼_𝑡𝑁−1𝑒^{−𝛾𝑡𝑁−1𝑚𝐷𝐷𝑡𝑁}𝑒^{−𝛾𝑡𝑁−1𝐼𝑡𝑁}∣𝐴^𝑐_𝑡𝑁−1.(A.14) Thus, the equilibrium price is 𝐷_𝑡𝑁−1=𝔼_𝑡𝑁−1𝐷_𝑡𝑁Λ^∗_𝑡𝑁,(A.15) where Λ^∗_𝑡𝑁 was defined in (22).

B. Proof of Theorem 8

We will prove the result for the subgame perfect equilibrium price; the proof for the naive equilibrium price is similar and hence omitted. First consider the time period [(𝑁−𝑛)ℎ,(𝑁−𝑛+1)ℎ) and choose an arbitrary control 𝜋=(𝛼,𝛽) as follows: 𝜋𝜋=^∗𝑡𝑛()()𝜋,for𝑛=𝑁−𝑛−1,𝑁−𝑛−2,…,𝑁−1,_𝑡𝑛,for𝑛=𝑁−𝑛.(B.1) From 𝑋^𝜋_𝑡𝑁=𝑋^𝜋_{𝑡𝑁−(𝑛−1)}+^𝑁−1_{𝑘=𝑁−(𝑛−1)}Δ𝑋^∗_𝑡𝑘,(B.2) it follows that 𝔼_{𝑡𝑁−𝑛}−exp−𝛾_{𝑡𝑁−𝑛}𝑋^𝜋_𝑡𝑁+𝐼_𝑡𝑁=−𝑒^{−𝛾𝑡𝑁−𝑛𝑋𝜋𝑡𝑁−𝑛}𝔼_{𝑡𝑁−𝑛}×𝑒^{−𝛾𝑡𝑁−𝑛Δ𝑋𝜋𝑡𝑁−𝑛}𝔼_{𝑡𝑁−𝑛+1}×exp−𝛾_{𝑡𝑁−𝑛}^𝑁−1_{()𝑘=𝑁−𝑛−1}Δ𝑋^∗_𝑡𝑘+𝐼_𝑡𝑁=−𝑒^{−𝛾𝑡𝑁−𝑛𝑋𝜋𝑡𝑁−𝑛}𝔼_{𝑡𝑁−𝑛}𝑒^{−𝛾𝑡𝑁−𝑛Δ𝑋𝜋𝑡𝑁−𝑛}𝑒^{−𝛾𝑡𝑁−𝑛𝑌∗𝑡𝑁−(𝑛−1)}=−𝑒^{−𝛾𝑡𝑁−𝑛𝑋𝜋𝑡𝑁−𝑛}𝑔_𝑁−𝑛𝛼,𝛽,⋅.(B.3) Here for 𝑘=𝑁−(𝑛−1),…,𝑁−2,𝑁−1, Δ𝑋^∗_𝑡𝑘=𝛼^∗_𝑡𝑘𝜇^𝑐_𝑡𝑘ℎ+𝜎^𝑐_𝑡𝑘√ℎΔ𝑏¹_𝑡𝑘+𝑚_𝐷Δ𝐷_𝑡𝑘,𝑔_𝑁−𝑛𝛼,𝛽,⋅∶=𝔼_{𝑡𝑁−𝑛}𝑒^{−𝛾𝑡𝑁−𝑛Δ𝑋𝜋𝑡𝑁−𝑛}𝑒^{−𝛾𝑡𝑁−𝑛𝑌∗𝑡𝑁−(𝑛−1)}=12𝑒^{−𝛾𝑡𝑁−𝑛𝛼(𝜇𝑐𝑡𝑁−𝑛ℎ+𝜎𝑐𝑡𝑁−𝑛√ℎ)}𝔼_{𝑡𝑁−𝑛}×𝑒^{−𝛾𝑡𝑁−𝑛𝛽Δ𝐷𝑡𝑁−𝑛}𝑒^{−𝛾𝑡𝑁−𝑛𝑌∗𝑡𝑁−𝑛+1}∣𝐴_{𝑡𝑁−𝑛}+12𝑒^{−𝛾𝑡𝑁−𝑛𝛼(𝜇𝑐𝑡𝑁−𝑛ℎ−𝜎𝑐𝑡𝑁−𝑛√ℎ)}𝔼_{𝑡𝑁−𝑛}×𝑒^{−𝛾𝑡𝑁−𝑛𝛽Δ𝐷𝑡𝑁−𝑛}𝑒^{−𝛾𝑡𝑁−𝑛𝑌∗𝑡𝑁−𝑛+1}∣𝐴^𝑐_{𝑡𝑁−𝑛}(B.4) with 𝐴_{𝑡𝑁−𝑛}∶={𝜔∈Ω∶Δ𝑏¹_{𝑡𝑁−𝑛}=1} and 𝐴^𝑐_{𝑡𝑁−𝑛}∶={𝜔∈Ω∶Δ𝑏¹_{𝑡𝑁−𝑛}=−1}. Arguing as in the one period case we get 𝑔_𝑁−𝑛(0,0,⋅)=𝔼_{𝑡𝑁−𝑛}𝑒^{−𝛾𝑡𝑁−𝑛𝑌∗𝑡𝑁−𝑛+1}𝑔≤∞;_𝑁−𝑛∞,𝛽,⋅=∞;𝑔_𝑁−𝑛−∞,𝛽,⋅=∞.(B.5) From arbitrage considerations it follows that 𝐷_{𝑡𝑁−𝑛}(𝜔)∈inf_ℚ𝔼^ℚ𝐷_{𝑡𝑁−𝑛+1},sup_ℚ𝔼^ℚ𝐷_{𝑡𝑁−𝑛+1},(B.6) where ℚ is the set of probability measures. Thus Δ𝐷_{𝑡𝑁−𝑛}(𝜔)∈𝐷_{𝑡𝑁−𝑛+1}−sup_ℚ𝔼^ℚ𝐷_{𝑡𝑁−𝑛+1},𝐷_{𝑡𝑁−𝑛+1}−inf_ℚ𝔼^ℚ𝐷_{𝑡𝑁−𝑛+1},(B.7) so the sets {𝜔∶Δ𝐷_{𝑡𝑁−𝑛}(𝜔)>0} and {𝜔∶Δ𝐷_{𝑡𝑁−𝑛}(𝜔)<0} have positive probability. Consequently, it follows that 𝑔_𝑁−𝑛(𝛼,∞,⋅)=∞;𝑔_𝑁−𝑛(𝛼,−∞,⋅)=∞.(B.8) Therefore the minimum of 𝑔_𝑁−𝑛(𝛼,𝛽,⋅) is a critical point. Hence 𝜕𝑔_𝑁−𝑛=𝜇𝜕𝛼^{𝑐𝑡𝑁−𝑛}ℎ+𝜎^𝑐_{𝑡𝑁−𝑛}√ℎ2𝑒^{−𝛾𝑡𝑁−𝑛𝛼(𝜇𝑐𝑡𝑁−𝑛ℎ+𝜎𝑐𝑡𝑁−𝑛√ℎ)}×𝔼_{𝑡𝑁−𝑛}𝑒^{−𝛾𝑡𝑁−𝑛𝛽∗Δ𝐷𝑡𝑁−𝑛}𝑒^{−𝛾𝑡𝑁−𝑛𝑌∗𝑡𝑁−𝑛+1}∣𝐴_{𝑡𝑁−𝑛}+𝜇^𝑐_{𝑡𝑁−𝑛}ℎ−𝜎^𝑐_{𝑡𝑁−𝑛}√ℎ2𝑒^{−𝛾𝑡𝑁−𝑛𝛼(𝜇𝑐𝑡𝑁−𝑛ℎ−𝜎𝑐𝑡𝑁−𝑛√ℎ)}×𝔼_{𝑡𝑁−𝑛}𝑒^{−𝛾𝑡𝑁−𝑛𝛽∗Δ𝐷𝑡𝑁−𝑛}𝑒^{−𝛾𝑡𝑁−𝑛𝑌∗𝑡𝑁−𝑛+1}∣𝐴^𝑐_{𝑡𝑁−𝑛}=0.(B.9) By direct calculation, we verify that the subgame perfect strategy is 𝛼^∗_{𝑡𝑁−𝑛}=12𝛾_{𝑡𝑁−𝑛}𝜎^𝑐_{𝑡𝑁−𝑛}√ℎlog1+𝑟^𝑐_{𝑡𝑁−𝑛}√ℎ1−𝑟^𝑐_{𝑡𝑁−𝑛}√ℎ+12𝛾_{𝑡𝑁−𝑛}𝜎^𝑐_{𝑡𝑁−𝑛}√ℎ𝔼×log^{𝑡𝑁−𝑛}𝑒^{−𝛾𝑡𝑁−𝑛𝛽∗𝑡𝑁−𝑛𝐷𝑡𝑁−𝑛+1}𝑒^{−𝛾𝑡𝑁−𝑛𝑌∗𝑡𝑁−𝑛+1}∣𝐴_{𝑡𝑁−𝑛}𝔼_{𝑡𝑁−𝑛}𝑒^{−𝛾𝑡𝑁−𝑛𝛽∗𝑡𝑁−𝑛𝐷𝑡𝑁−𝑛+1}𝑒^{−𝛾𝑡𝑁−𝑛𝑌∗𝑡𝑁−𝑛+1}∣𝐴^𝑐_{𝑡𝑁−𝑛}.(B.10) From the equilibrium conditions it follows that 𝛽^∗_{𝑡𝑁−𝑛}=1. This, combined with 𝜕𝑔_𝑁−𝑛/𝜕𝛽=0, yields the equilibrium price at 𝑇_𝑁−𝑛. First, from 𝜕𝑔_𝑁−𝑛/𝜕𝛽=0, one gets 𝔼_{𝑡𝑁−𝑛}Δ𝐷_{𝑡𝑁−𝑛}𝑒^{−𝛾𝑡𝑁−𝑛𝑚𝐷Δ𝐷𝑡𝑁−𝑛}𝑒^{−𝛾𝑡𝑁−𝑛𝑌∗𝑡𝑁−𝑛+1}∣𝐴_{𝑡𝑁−𝑛}=−𝑒^{2𝛾𝑡𝑁−𝑛𝛼∗𝑡𝑁−𝑛𝜎𝑐𝑡𝑁−𝑛√ℎ}𝔼_{𝑡𝑁−𝑛}×Δ𝐷_{𝑡𝑁−𝑛}𝑒^{−𝛾𝑡𝑁−𝑛𝑚𝐷Δ𝐷𝑡𝑁−𝑛}𝑒^{−𝛾𝑡𝑁−𝑛𝑌∗𝑡𝑁−𝑛+1}∣𝐴^𝑐_{𝑡𝑁−𝑛}.(B.11) From 𝜕𝑔_𝑁−𝑛/𝜕𝛼=0 it follows that 𝑒^{2𝛾𝑡𝑁−𝑛𝛼∗𝑡𝑁−𝑛𝜎𝑐𝑡𝑁−𝑛√ℎ}=1+𝑟^𝑐_{𝑡𝑁−𝑛}√ℎ𝔼_{𝑡𝑁−𝑛}×𝑒^{−𝛾𝑡𝑁−𝑛𝑚𝐷𝐷𝑡𝑁−𝑛+1}𝑒^{−𝛾𝑡𝑁−𝑛𝑌∗𝑡𝑁−𝑛+1}∣𝐴_{𝑡𝑁−𝑛}×1−𝑟^𝑐_{𝑡𝑁−𝑛}√ℎ𝔼_{𝑡𝑁−𝑛}×𝑒^{−𝛾𝑡𝑁−𝑛𝑚𝐷𝐷𝑡𝑁−𝑛+1}𝑒^{−𝛾𝑡𝑁−𝑛𝑌∗𝑡𝑁−𝑛+1}∣𝐴^𝑐_{𝑡𝑁−𝑛}⁻¹.(B.12) This, together with Δ𝐷_{𝑡𝑁−𝑛}=𝐷_{𝑡𝑁−𝑛+1}−𝔼^ℚ∗_{𝑡𝑁−𝑛}𝐷_{𝑡𝑁−𝑛+1},(B.13) (here ℚ^∗ is the equilibrium probability measure to be found) yields 21−𝑟^𝑐_{𝑡𝑁−𝑛}√ℎ𝔼^ℚ∗_{𝑡𝑁−𝑛}𝐷_{𝑡𝑁−𝑛+1}=𝔼_{𝑡𝑁−𝑛}𝐷_{𝑡𝑁−𝑛+1}𝑒^{−𝛾𝑡𝑁−𝑛𝑚𝐷𝐷𝑡𝑁−𝑛+1}×𝑒^{−𝛾𝑡𝑁−𝑛𝑌∗𝑡𝑁−𝑛+1}∣𝐴_{𝑡𝑁−𝑛}×𝔼_{𝑡𝑁−𝑛}𝑒^{−𝛾𝑡𝑁−𝑛𝑚𝐷𝐷𝑡𝑁−𝑛+1}×𝑒^{−𝛾𝑡𝑁−𝑛𝑌∗𝑡𝑁−𝑛+1}∣𝐴_{𝑡𝑁−𝑛}⁻¹+1+𝑟^𝑐_{𝑡𝑁−𝑛}√ℎ1−𝑟^𝑐_{𝑡𝑁−𝑛}√ℎ×𝔼_{𝑡𝑁−𝑛}𝐷_{𝑡𝑁−𝑛+1}𝑒^{−𝛾𝑡𝑁−𝑛𝑚𝐷𝐷𝑡𝑁−𝑛+1}×𝑒^{−𝛾𝑡𝑁−𝑛𝑌∗𝑡𝑁−𝑛+1}∣𝐴^𝑐_{𝑡𝑁−𝑛}×𝔼_{𝑡𝑁−𝑛}𝑒^{−𝛾𝑡𝑁−𝑛𝑚𝐷𝐷𝑡𝑁−𝑛+1}×𝑒^{−𝛾𝑡𝑁−𝑛𝑌∗𝑡𝑁−𝑛+1}∣𝐴^𝑐_{𝑡𝑁−𝑛}⁻¹.(B.14) Consequently 𝐷_{𝑡𝑁−𝑛}=1−𝑟^𝑐_𝑡𝑛√ℎ2×𝔼_{𝑡𝑁−𝑛}𝐷_{𝑡𝑁−𝑛+1}𝑒^{−𝛾𝑡𝑁−𝑛𝑚𝐷𝐷𝑡𝑁−𝑛+1}×𝑒^{−𝛾𝑡𝑁−𝑛𝑌∗𝑡𝑁−𝑛+1}∣𝐴_{𝑡𝑁−𝑛}×𝔼_{𝑡𝑁−𝑛}𝑒^{−𝛾𝑡𝑁−𝑛𝑚𝐷𝐷𝑡𝑁−𝑛+1}×𝑒^{−𝛾𝑡𝑁−𝑛𝑌∗𝑡𝑁−𝑛+1}∣𝐴_{𝑡𝑁−𝑛}⁻¹+1+𝑟^𝑐_𝑡𝑛√ℎ2×𝔼_{𝑡𝑁−𝑛}𝐷_{𝑡𝑁−𝑛+1}𝑒^{−𝛾𝑡𝑁−𝑛𝑚𝐷𝐷𝑡𝑁−𝑛+1}×𝑒^{−𝛾𝑡𝑁−𝑛𝑌∗𝑡𝑁−𝑛+1}∣𝐴^𝑐_{𝑡𝑁−𝑛}×𝔼_{𝑡𝑁−𝑛}𝑒^{−𝛾𝑡𝑁−𝑛𝑚𝐷𝐷𝑡𝑁−𝑛+1}×𝑒^{−𝛾𝑡𝑁−𝑛𝑌∗𝑡𝑁−𝑛+1}∣𝐴^𝑐_{𝑡𝑁−𝑛}⁻¹.(B.15) Thus, the equilibrium price is 𝐷_{𝑡𝑁−𝑛}=𝔼_{𝑡𝑁−𝑛}𝐷_{𝑡𝑁−𝑛+1}Λ^∗_{𝑡𝑁−𝑛+1},(B.16) with Λ^∗_{𝑡𝑁−𝑛+1} defined in (35).

C. Proof of Corollary 9

We consider the time period [(𝑁−𝑛)ℎ,(𝑁−𝑛+1)ℎ). Recall that 𝑋_𝑡𝑁=𝑋_{𝑡𝑁−𝑛}+Δ𝑋_{𝑡𝑁−𝑛}+^𝑁−1_{𝑘=𝑁−(𝑛−1)}Δ𝑋_𝑡𝑘,(C.1) where Δ𝑋_𝑡𝑘=𝛼_𝑡𝑘𝜇^𝑐_𝑡𝑘ℎ+𝜎^𝑐_𝑡𝑘√ℎΔ𝑏¹_𝑡𝑘+𝑚_𝐷Δ𝐷_𝑡𝑘,(C.2) for 𝑘=𝑁−(𝑛−1),…,𝑁−2,𝑁−1. By definition of 𝑌_𝑡𝑘, we have 𝔼_{𝑡𝑁−𝑛}𝑈^′𝑊_𝑡𝑁=𝛾_𝑡0𝑒^−𝛾𝑡0𝑋_{𝑡𝑁−𝑛}𝔼_{𝑡𝑁−𝑛}𝑒^{−𝛾𝑡0Δ}𝑋_{𝑡𝑁−𝑛}𝑒^−𝛾𝑡0𝑌_{𝑡𝑁−𝑛+1},𝔼_{𝑡𝑁−𝑛+1}𝑈^′𝑊_𝑡𝑁=𝛾_𝑡0𝑒^−𝛾𝑡0𝑋_{𝑡𝑁−𝑛}𝑒^{−𝛾𝑡0Δ}𝑋_{𝑡𝑁−𝑛}𝑒^−𝛾𝑡0𝑌_{𝑡𝑁−𝑛+1},𝔼_{𝑡𝑁−𝑛+1}𝑈^′𝑊_𝑡𝑁𝔼_{𝑡𝑁−𝑛}𝑈^′𝑊_𝑡𝑁=𝑒^{−𝛾𝑡0Δ}𝑋_{𝑡𝑁−𝑛}𝑒^−𝛾𝑡0𝑌_{𝑡𝑁−𝑛+1}𝔼_{𝑡𝑁−𝑛}𝑒^{−𝛾𝑡0Δ}𝑋_{𝑡𝑁−𝑛}𝑒^−𝛾𝑡0𝑌_{𝑡𝑁−𝑛+1}=𝑒^−𝛾𝑡0𝛼_{𝑡𝑁−𝑛𝜎}^𝑐_{𝑡𝑁−𝑛}^√_ℎΔ𝑏¹_{𝑡𝑁−𝑛}𝑒^{−𝛾𝑡0(𝑚𝐷𝐷𝑡𝑁−𝑛+1+}𝑌_{𝑡𝑁−𝑛+1)}𝔼_{𝑡𝑁−𝑛}𝑒^−𝛾𝑡0𝛼_{𝑡𝑁−𝑛}^𝜎𝑐_{𝑡𝑁−𝑛}^{√ℎΔ𝑏1}_{𝑡𝑁−𝑛}𝑒^{−𝛾𝑡0(𝑚𝐷𝐷𝑡𝑁−𝑛+1+}𝑌_{𝑡𝑁−𝑛+1}⁾.(C.3) Recall that 𝑒^2𝛾𝑡0𝛼_{𝑡𝑁−𝑛𝜎}^𝑐_{𝑡𝑁−𝑛}^√_ℎ=1+𝑟^𝑐_{𝑡𝑁−𝑛}𝔼_{𝑡𝑁−𝑛}𝑒^{−𝛾𝑡0(𝑚𝐷𝐷𝑡𝑁−𝑛+1+}𝑌_{𝑡𝑁−𝑛+1)}∣𝐴_{𝑡𝑁−𝑛}1−𝑟^𝑐_{𝑡𝑁−𝑛}𝔼_{𝑡𝑁−𝑛}𝑒^{−𝛾𝑡0(𝑚𝐷𝐷𝑡𝑁−𝑛+1+}𝑌_{𝑡𝑁−𝑛+1)}∣𝐴^𝑐_{𝑡𝑁−𝑛}.(C.4) Using (C.4), we have 𝔼_{𝑡𝑁−𝑛}𝑒^−𝛾𝑡0𝛼_{𝑡𝑁−𝑛𝜎}^𝑐_{𝑡𝑁−𝑛}^√_ℎΔ𝑏¹_{𝑡𝑁−𝑛}×𝑒^{−𝛾𝑡0(𝑚𝐷𝐷𝑡𝑁−𝑛+1+}𝑌_{𝑡𝑁−𝑛+1)}=12𝑒^−𝛾𝑡0𝛼_{𝑡𝑁−𝑛𝜎}^𝑐_{𝑡𝑁−𝑛}^√_ℎ𝔼_{𝑡𝑁−𝑛}×𝑒^{−𝛾𝑡0(𝑚𝐷𝐷𝑡𝑁−𝑛+1+}𝑌_{𝑡𝑁−𝑛+1)}∣𝐴_{𝑡𝑁−𝑛}+12𝑒^𝛾𝑡0𝛼_{𝑡𝑁−𝑛𝜎}^𝑐_{𝑡𝑁−𝑛}^√_ℎ𝔼_{𝑡𝑁−𝑛}×𝑒^{−𝛾𝑡0(𝑚𝐷𝐷𝑡𝑁−𝑛+1+}𝑌_{𝑡𝑁−𝑛+1)}∣𝐴^𝑐_{𝑡𝑁−𝑛}=11+𝑟^𝑐_{𝑡𝑁−𝑛}√ℎ𝑒^𝛾𝑡0𝛼_{𝑡𝑁−𝑛𝜎}^𝑐_{𝑡𝑁−𝑛}^√_ℎ𝔼_{𝑡𝑁−𝑛}×𝑒^{−𝛾𝑡0(𝑚𝐷𝐷𝑡𝑁−𝑛+1+}𝑌_{𝑡𝑁−𝑛+1)}∣𝐴^𝑐_{𝑡𝑁−𝑛}=11−𝑟^𝑐_{𝑡𝑁−𝑛}√ℎ𝑒^−𝛾𝑡0𝛼_{𝑡𝑁−𝑛𝜎}^𝑐_{𝑡𝑁−𝑛}^√_ℎ𝔼_{𝑡𝑁−𝑛}×𝑒^{−𝛾𝑡0(𝑚𝐷𝐷𝑡𝑁−𝑛+1+}𝑌_{𝑡𝑁−𝑛+1)}∣𝐴_{𝑡𝑁−𝑛}.(C.5) Therefore, it follows that on the set of {𝜔∶𝜔∈𝐴^𝑐_{𝑡𝑁−𝑛}}𝔼_{𝑡𝑁−𝑛+1}𝑈^′𝑊_𝑡𝑁𝔼_{𝑡𝑁−𝑛}𝑈^′𝑊_𝑡𝑁=1+𝑟^𝑐_{𝑡𝑁−𝑛}√ℎ𝑒^{−𝛾𝑡0(𝑚𝐷𝐷𝑡𝑁−𝑛+1+}𝑌_{𝑡𝑁−𝑛+1)}𝔼_{𝑡𝑁−𝑛}𝑒^{−𝛾𝑡0(𝑚𝐷𝐷𝑡𝑁−𝑛+1+}𝑌_{𝑡𝑁−𝑛+1}⁾∣𝐴^𝑐_{𝑡𝑁−𝑛}.(C.6) Moreover, on the set of {𝜔∶𝜔∈𝐴_{𝑡𝑁−𝑛}}, 𝔼_{𝑡𝑁−𝑛+1}𝑈^′𝑊_𝑡𝑁𝔼_{𝑡𝑁−𝑛}𝑈^′𝑊_𝑡𝑁=1−𝑟^𝑐_{𝑡𝑁−𝑛}√ℎ𝑒^{−𝛾𝑡0(𝑚𝐷𝐷𝑡𝑁−𝑛+1+}𝑌_{𝑡𝑁−𝑛+1)}𝔼_{𝑡𝑁−𝑛}𝑒^{−𝛾𝑡0(𝑚𝐷𝐷𝑡𝑁−𝑛+1+}𝑌_{𝑡𝑁−𝑛+1}⁾∣𝐴_{𝑡𝑁−𝑛}.(C.7) Therefore 𝔼_{𝑡𝑁−𝑛+1}𝑈^′𝑊_𝑡𝑁𝔼_𝑁−𝑛𝑈^′𝑊_𝑡𝑁=Λ_{𝑡𝑁−𝑛+1},(C.8) with Λ_{𝑡𝑁−𝑛+1} defined in (36).

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work is supported by NSERC Grants 371653-09 and 88051 and MITACS Grants 5-26761 and 30354 and by the Natural Science Foundation of China (10901086). The authors would like to thank Tom Hurd for helpful comments.