Liquidity and Valuation in an Uncertain Market with Multiple Risky Assets and Difference of Opinions

Transcription

1 Liquidity and Valuation in an Uncertain Market with Multiple Risky Assets and Difference of Opinions Qi Nan Zhai FDG School of Finance and Economics, UTS Business School University of Technology, Sydney PO Box 123, Broadway NSW, Australia, 2007 April 20, 2012 Abstract This paper explores the implication of asset correlation on illiquid risky assets arise from ambiguity in a market populated with traders who agree to disagree. An equilibrium model is proposed in a mean-variance framework. It focuses on how heterogeneity beliefs and ambiguity about a risky asset affects another through correlation; and in turn affects demand and market prices. After experiencing a common shock, traders become uncertain about future asset values respectively to their beliefs; asset prices deviate from equilibrium, forming bid and ask spreads. The findings seem to suggest (i) given enough ambiguity about expected payoffs, assets become illiquid in which trades cease to exist between bid and ask prices. The size of the spreads however depends on the correlation between risky assets. A spread reduction is generally observed in a more risky asset and an increase in a less risky asset with positive correlation. Thus, traders are more likely to trade an asset with greater expected payoff to reflect diminishing portfolio diversification advantage. (ii) Given trades are nonexistent for some level of ambiguity; there must be a condition that allows trades to exist. Threshold to trade depends on the level of common shock and the diversity of traders beliefs, but interestingly independent of correlation. Due to this disconnection between correlation and threshold to trade; in a market with many illiquid assets, a risky asset that is liquid cannot influence the liquidity of another related asset. Rather, liquidity is affected by the diversity of traders beliefs and the size of common shock. Trading prices are thus derived. (iii) Contrary to general thinking, heterogeneity does not always decrease bid and ask spread, it depend on the level of initial equilibrium determined by market belief; in particular, the variance-covariance matrix depends on correlation. By relaxing independence between risky assets, the model captures the interdependence and in some cases lack thereof between bid and ask spreads, and thus prices. 1

2 1 Introduction Easley and O Hara (2010) characterise the lack of trading activities observed in the asset-back securities during the recent Global Financial Crisis (GFC), a result of traders uncertainty about the future payoff of an asset. The absence of trade is described by a range of prices in which traders are not willing to trade. In comparison, this lack of trading is different to the well-known no-trade theorem in Milgrom and Stokey (1982) which is studied under the traditional asset pricing theory. This paper adds to the literature by extending the work of Easley and O Hara (2010) and studies price impact on illiquid assets arise from ambiguity through relaxation of independence between risky assets in a multi-risky assets market. Traders are also assume to be heterogeneous and beliefs are aggregated with method proposed in Chiarella et al. (2010) to obtain market belief. Traditional asset price models base on the assumption that traders are rational and homogeneous in beliefs. They maximise expected utility and use all available information in their decision making process. However, empirical works have emphasise on the importance of heterogeneity; and often than not, heterogeneous beliefs are associated with excessive market volatility studied in the seminal paper of Shiller (1981) 1. To account for this contradiction observed in the GFC, Bewley (2002) s incomplete preferences can provide an explanation. According to his paper, traders can no longer maximise expected utility and is unable to arrive at equilibrium due to uncertainty. Ambiguity models arise from the classic work of Knight (2002) where he defines two kinds of uncertainty, risk and Knightian uncertainty (also known as ambiguity). The latter uncertainty is associated with situation when events do not have an obvious probability assignment, and is closely related to Bewley (2002) s incomplete preferences. To study the implication of correlation on bid and ask spread, hence prices; market microstructure is often a useful platform as bid and ask spread is a distinctive feature of this strand of literature. However, different from most models where spread arises from traders ambiguity about the future value of assets rather than commission, inventory costs discussed or asymmetric information, see Hasbrouck (1997) for a good discussion on the models 2. To study price impact on a particular trading feature; many ambiguity models have focused on the implication of limited market participation on price in the context of market microstructure 3. Different to limited market participation in which traders are frequently assumed to exhibit Schmeidler and Gilboa (1989) s preferences, and are always tempted to move out of the market, therefore generate trades. Easley and O Hara (2010) steer in a different direction. The authors develop a model that specifically characterise the absence of trades 1 See Beber et al. (2010), Buraschi and Jiltsov (2006), Buraschi et al. (2008) and Ziegler (2007) for empirical evidence on heterogeneous beliefs. 2 The paper do not rule out asymmetric information; it recognises the difference of opinion traders have about future performance of risky assets. However, the model does not distinguish between informed and uninformed traders as in the seminal work of Kyle (1985). 3 Among these literature, some have considered heterogeneity in addition to ambiguity, works like Cao et al. (2005) and Easley and O Hara (2009); while others have focused on scenario where ambiguity arises from individual risky asset rather than a systematic one, works like Mukerji and Tallon (2001) and Guidolin and Rinaldi (2010). 2

3 observed in asset-backed securities during the GFC. The market contains a simple one-risky asset, onerisk-free asset and assume heterogeneous traders experience an unexpected adverse shock in a future period. Further, traders are ambiguous about the future value of the asset, where traders ambiguity is described by? s inertia assumption. That is, traders will always remain at status quo unless the alternative trade gives a greater expected utility in case of all events. This assumption is consistent with the empirical observation in Samuelson and Zeckhauser (1988) 4. Intuitively, there are range of prices that traders are not willing to trade, the authors define the range as the bid and ask spread arise from ambiguity, namely equilibrium no-trade spread. The trader who is most optimistic about the largest possible decline in the asset value sets the bid price; while the trader who is most pessimistic about the smallest possible decline in asset value sets the ask price. Since Easley and O Hara (2010) assume risky assets are i.i.d., only a single risky asset is studied. The model does not capture the implication of correlation on the market, more specifically, the interaction between bid and ask spreads between different but correlated risky assets. By relaxing independence between risky assets, our model captures the interdependence of spreads. Further, Easley and O Hara (2010) only allow traders to differ in their opinion about the expected payoff of some risky asset. Consider traders to differ not only about the first and second moments of risky assets, but their appetite for risk. This paper studies the implication of bid and ask spread arise from ambiguity in a multiple risky asset market and found, (i) given enough ambiguity about expected payoffs, assets become illiquid in which trades cease to exist between bid and ask prices. The size of the spreads however depends on the correlation between risky assets. A spread reduction is generally observed in a more risky asset and an increase in a less risky asset with positive correlation. Thus, traders are more likely to trade an asset with greater expected payoff to reflect diminishing portfolio diversification advantage. (ii) Given trades are nonexistent for some level of ambiguity; there must be a condition that allows trades to exist. Threshold to trade depends on the level of common shock and the diversity of traders beliefs, but interestingly independent of correlation. Due to this disconnection between correlation and threshold to trade; in a market with many illiquid assets, a risky asset that is liquid cannot influence the liquidity of another related asset. Rather, liquidity is affected by the diversity of traders beliefs and the size of common shock. Trading prices are thus derived. (iii) Contrary to general thinking, heterogeneity does not always decrease bid and ask spread, it depend on the level of initial equilibrium determined by market belief; in particular, the variance-covariance matrix depends on correlation. The remainder of the paper is organised as follows. Section 2 setups the general model for a two-period. Traders enter period 1 with optimal heterogeneous portfolios from period 0; market equilibrium is studied under two scenarios: I) traders experience an unambiguous or II) an ambiguous shock to the future value of assets. Section 3 analysis model under special cases. The implication of 4 Others that have adopted the same preference include Rigotti and Shannon (2005), and Illeditsch (2010). The other two common preferences in the ambiguity literature are the maximin expected utility studied in Schmeidler and Gilboa (1989) and Choquet expected utility studied in Schmeidler (1989); the above preferences relax one of the axioms made famous by Savage (1954) and is different to that of? 3

4 correlation on equilibrium bid and ask spreads are visually demonstrated and discussed. In particular, the discussion focuses on scenarios where market is pessimistic and under-confident; in line with the characteristics of the GFC. Section 4 concludes. The Appendix provides technical derivations and proofs. 2 Model Setup In a mean-variance framework, consider an economy consisting I (i=1,2,...i) number of traders; J (j=1,2,...j) number of risky assets, and one risk-free asset with its current value at $1 and increases at a rate of R f = 1 + r f. Unlike Easley and O Hara (2010) which set value of the risk-free asset at 1, the rate is endogenously derived. Intuitively, the value of risk-free asset should not be constant but varies inversely with demand for risky assets. Further, rather than to account for a single source of heterogeneity, traders may be heterogeneous about first and second moments of risky assets defined as market sentiment and confidence respectively; the difference of opinion are due to the way traders interpret information. Traders may also differ in appetite for risk. To study the implication of correlation, let there be some form of relationship ρ i;jk between risky assets j and k. Traders may also differ in belief about the correlation coefficient. Finally, future payoff vector of J risky assets is denoted ṽ = (ṽ 1, ṽ 2,..., ṽ J ). They are jointly normal with expected payoff vector v i and variancecovariance matrix Ω i, again they may differ for each trader. The model considers two periods, and trades may occur at both times. At time t=0, traders maximise the expected CARA utility respective to their beliefs about the future value of risky assets, and hold heterogeneous portfolios. At time t=1, trader may experience one of the two scenarios. Apply the same structure as in Easley and O Hara (2010), but account for correlation impact and the effect of multi-dimensional heterogeneity beliefs of a risky asset has on another related asset. In the former scenario, traders experience an unexpected systematic adverse shock to the expected payoffs with known magnitude, the expected payoff vector at the current period for trader i becomes v i1 = αv i, where 0 < α < 1. In the first scenario, traders are utility maximisers hence traders rebalance portfolios. However in the second scenario, traders are ambiguous about the magnitude of the adverse shock and can no longer arrive at a single belief for some risky asset. Each trader has a range of possible events ranging from worst to best [α v i, ᾱ v i ], where αɛ[α, ᾱ] is the range of possible falls in asset prices and 0 < α α ᾱ 1 5. Rather to trade at equilibrium, each trader s action to trade is determined by Bewley (2002) s incomplete preferences. Bewley s model shows under standard assumptions, incomplete preferences can be represented with a single utility function and a set of probabilities. He describes this with an inertia assumption. Given some level of ambiguity, each trader will arrive at a different spread where they cease to trade. The equilibrium bid and ask prices of risky asset j is therefore bounded by individual traders, such that the bid price is the 5 α is an adverse shock, consistent with the falling prices during the GFC. For an absolute positive shock such that α > 1, we derive a similar spread. 4

5 highest price a trader is willing to buy the asset given the largest possible drop in asset value, while the ask price is the lowest price a trader is willing to sell the asset given the smallest drop in value. Traders may also trade at period Market equilibrium at period 0 At the start of the period, trader i chooses a portfolio of assets to maximise the expected utility. Let x i = (x i1, x i2,..., x ij ) be the position vector of J risky assets and m i be the position of the risk-free asset. Then the future wealth of trader i s portfolio is w i = x i ṽ + m ir f, subjected to a budget constraint m i = w io x i p o, where w io is the initial wealth and p o = (p o1, p o2,..., p oj ) is a price vector of J risky assets. Assume trader i s expected utility is given by a constant absolute risk aversion (CARA) utility function U i ( w i ) = exp( 1 τ i w i ), and τ i is the risk tolerance coefficient. Trader i s belief is in the form of E i [ṽ] = v i = (v i1, v i2,..., v ij ) and Ω i = (ρ i;j,k σ ij σ ik ) JxJ, where ρ i;j,k and σij 2 are trader i s opinion of the correlation between risky assets j and k and the variance of risky asset j respectively. Under these assumptions, maximising trader i s expected utility is equivalent to maximising the certain equivalent of wealth max x i {x i v i + (w io x i p o)r f 1 2τ i x i Ω ix i }. (1) By applying First Order Condition (FOC), we find the optimal portfolio composition for trader i x i = τ iω 1 i (v i p o R f ). (2) If risky assets are i.i.d, the market belief will just be given as the average of all traders beliefs as in Easley and O Hara (2010). However, the model assumes risky assets are jointly normal. To find market equilibrium at t=0, we refer to Proposition 1 in Chiarella et al. (2010). We let τ a be the average risk tolerance coefficient across I number of traders, τ a = 1 I I i=1 τ i, the market variance-covariance matrix and expected payoff vector are then given Ω 1 a = 1 I I i=1 τ i τ a Ω 1 i, and v a = 1 I Ω a I i=1 τ i τ a Ω 1 i v i. (3) Applying market clearing condition 1 I ΣI i=1 x i satisfies = 1 I ΣI i=1 x i = x m, the market equilibrium price p o x m = τ a Ω 1 a (v a p o R f ); (4) 5

6 after re-arranging equation (4), we obtain the equilibrium price vector p o = 1 R f (v a Ω ax m τ a ). (5) A crucial difference between this models and that in Easley and O Hara (2010) is they assume a constant risk free rate increasing at zero rate. To generalise this assumption and derive an equilibrium risk-free rate Rf ; let risk-free asset be zero net supply I i=1 (w io x i p o) = 0, and substitute equation (5) into p o to re-express Rf Rf = I (v a Ω ax m ) x m, (6) w mo τ a where I i=1 w io = w mo is the total market wealth. Equation (5) shows the equilibrium prices for J risky assets. Price vector is determined by the difference between market expected payoff and a compensation factor for holding market risk where assets are correlated in one way or another, Ω ax m τ a. It is also affected by the average market tolerance and is inversely related with the risk-free rate, the higher the risk-free rate the lower the risky asset prices. Because the risk-free rate is endogenous derived, the rate is affected by market belief regardless of correlation, shown in Equation (6). Finally, substitute equation (5) back into equation (2) to obtain trader i s optimal portfolio composition x io = τ iω 1 i [(v i v a ) + Ω ax m ]. (7) τ a Equation (7) shows trader i s optimal portfolio composition. The composition of the portfolio depends on trader s relative belief to the market, the trader is said to be optimistic (pessimistic) if the expected payoff is higher (lower) than the market v a and holds positive (negative) positions. Further, let the status quo be the initial portfolio composition x io. 2.2 Market Equilibrium at period 1 After establishing the initial equilibrium, each trader enters period 1 with a different portfolio. Scenario I assumes heterogeneous traders experience an unanticipated adverse shock to the expected payoff of J risky assets, with known magnitude. Find equilibrium price and position vectors at t=1; traders trade to adjust risk depending on their relative beliefs to the market. In scenario II, traders experience a similar shock as above but the magnitude of the shock is unknown and establishes bid and ask spreads arise from ambiguity Scenario I - unexpected adverse shock with known magnitude At period 1, traders experience an unexpected adverse unambiguous shock α, and the expected payoff vector at the current period becomes v i1 = αv i. Derive equilibrium with shock the same way as in 6

7 the previous period, it then follows p 1 = 1 R f (αv a Ω ax m τ a ), (8) x i1 = τ iω 1 i (α(v i v a ) + Ω ax m ). (9) τ a This unanticipated shock disturbs traders optimal holdings from period 0. Because α is known with certainty, the expected payoff vector is just reduced by α, the market becomes pessimistic, and prices fall. Further, traders tend to hold less aggressive portfolios in comparison to portfolio before common shock. The trading rule for trader i is given by the difference between the status quo and the current optimal portfolio compositions t i = x i1 x io = τ i(1 α)ω 1 i (v a v i ). (10) According to the rule above, trader i s trading direction and aggressiveness do not only depend on their relative beliefs to the market but the type of relationship between risky assets. The further the trader s expected payoff vector is from the market expectation, the more aggressive is the trade to adjust for risk, holding all else constant. Trader s aggressiveness to trade is also directly linked to one s risk appetite; a smaller position is traded if the trader is more risk adverse; this is because the trader whom has a lower risk tolerance would have hold a less aggressive portfolio from the beginning. Further, trader tend to trade more aggressively for large positive and negative correlation. Finally, if trader i has the same expectation as the market, than we expect trade to remain at status quo and observe no trades. Equation (11) shows the aggregated trading volume t a = 1 2 I t i = 1 2 i=1 I i=1 τ i Ω 1 i (1 α)(v a v i ) > 0. (11) Scenario II - unexpected adverse shock with unknown magnitude In this scenario, traders experience an unexpected adverse shock to the expected payoff of risky assets at period 1; the magnitude of the shock is unknown. Traders are ambiguous in the sense that they do not have a single expected payoff for risky asset j but a range of expected payoffs for every possible outcome [αv i, ᾱv i ] 6. According to Bewley (2002), traders have incomplete preferences if they cannot rank choices. In this model, traders will only trade to rebalance their portfolio if there is a set of trades gives a larger expected utility than the status quo for every possible outcome [αv i, ᾱv i ]. This by trade basis is a feature of market microstructure and the terminal value is expressed in terms of set of trades 6 In this paper, the range of possible events [α, ᾱ] is assumed to be the same across all traders. In general, the range may vary from trader to trader, that is [α i, ᾱ i], for trader i, α i is the worse case scenario and ᾱ i is the best case scenario. 7

8 t i. The random future wealth at period 1 is thus given by w i1 = (x io + t i) ṽ + (m io + m i )R f = x io ṽ + t i (ṽ p 1R f ) + m ior f, (12) after substituting the budget constraint equation m i + t i p 1 = 0. Then, trader i s trading vector for J risky assets is t i = (t i1, t i2,..., t ij ) ; m i is the position for risk-free asset; and p 1 = (p 11, p 12,..., p 1J ) is some price vector. The expected utility at period 1 is again equivalent to the certain equivalent of wealth max{x t io v i1 + t i (v i1 p 1 Rf ) + m i iorf 1 (x io 2τ + t i) Ω i (x io + t i)}. (13) i Because traders exhibits Bewley s incomplete preferences, their decisions whether to remain at status quo or to trade depend on the following two conditions. Condition i) Condition ii) x io v i1 + t i (v i1 p 1 R f ) + m ior f 1 2τ i (x io + t i) Ω i (x io + t i) (14) x io v i1 + m ior f 1 2τ i x io Ω ix io ; x io v i1 + t (v i1 p 1 R f ) + m ior f 1 2τ i (x io + t ) Ω i (x io + t ) (15) x io v i1 + t i (v i1 p 1 R f ) + m ior f 1 2τ i (x io + t i) Ω i (x io + t i) for all v i1 ɛ[α v i, ᾱ v i ]. Condition (i) implies trader i will trade to rebalance portfolio if and only if this inequality satisfies the set of beliefs v i1 ɛ[α v i, ᾱ v i ]. Then, condition (ii) implies, if there are a set of trades that satisfy condition (i), intuitively, there will not be another set of trade t that will give a greater expected utility than t i. This incomplete preferences introduce spreads that is specific to an ambiguity model. Because traders may differ about the possible range of expected payoffs due to their heterogeneous beliefs at period 0, they may also derive different no-trade spreads. Before finding the equilibrium no-trade spread, p 1 is required 7. To maximise the certainty equivalent of wealth function with respect to t i, the equality becomes (v i1 p 1 R f ) + 1 2τ i Ω i (x io + t i) = 0. (16) Since it is to study the change in price vector when no trade occurs, let t i = 0 and rearrange equation 7 It is important to note p 1 is not an equilibrium price vector, but a price vector for trader i. 8

9 (16), and find p 1 = 1 R f (v i1 Ω ix io τ i ), (17) where p 1 is the no-trade price vector for trader i. However, the price vector is meaningless unless it is expressed in terms of no-trade spread. Because the trader only knows the range of the expected payoff vector v i1 ɛ[α v i, ᾱ v i ], trader i s no-trade spread becomes p 1 = 1 Rf (αv i 1 Ω i x io τ ) p 1 i 1 Rf (ᾱv i 1 Ω i x io τ ) = p 1. (18) i To establish equilibrium no-trade spread for risky assets, the same condition is followed as in Bewley (2002), and assume the interval between spreads is a non-empty intersection I i=1 [ 1 Rf (αv i 1 Ω i x io τ ), 1 i Rf (ᾱv i 1 ] Ω i x io τ ). (19) i Due to heterogeneity nature of traders, there will be I number of no-trade spread for any given risky asset. The upper price p 1 and lower price p 1 are the prices some trader is willing to sell and buy; any price that falls between the spread, no one will trade because both selling and buying will make them worse off than to do nothing. Equilibrium bid and ask spread of risky asset j is defined such that the bid price is the highest price a trader is willing to buy an asset given the largest possible drop on the expected payoff; and the ask price is the lowest price a trader is willing to sell given the smallest possible drop on the expected payoff. Intuitively, in a market where there is ambiguity, traders will always be inclined to consider the best case scenario when selling; that is, they are never prepared to sell at a lower price if the market only falls by a little. On the other hand, traders are most likely to consider the worst case scenario when buying, that is, they are not willing to buy at a higher price if the market falls considerably; we may call this the what-if attitude. To re-express equilibrium no-trade spreads in terms of p o, we make use of equations (5) and (7) (see appendix A.1 for proof). p bid = max i { 1 Rf (αv i 1 Ω i x io τ )} = p o 1 i Rf min i {(1 α)v i } (20) and the ask price is p ask = min i { 1 Rf (ᾱv i 1 Ω i x io τ } = p o 1 i Rf max i {(1 ᾱ)v i }. (21) Equations (20) and (21) show the bid and ask price vectors. Any prices above the ask price vector are prices at which traders are willing to sell but no one is willing to buy; and any prices below the bid price vector are prices at which traders are willing to buy but no one is willing to sell. It is apparent from above equations, equilibrium no-trade spreads are affected by market belief implicitly implied through p o and Rf. Market sentiment, confidence, risk tolerance as well as the correlation 9

10 coefficient determine the level of bid and ask prices and the magnitude of the equilibrium no-trade spread. Intuitively, when risky assets are correlated in some form, traders belief of a risky asset should have a spillover effect related assets. This relationship will in term affects traders demand for other risky and risk-free assets; prices will then reflect traders preferences for holding these type of assets. To spread size is the difference of bid and ask prices and is strictly positive for trades to cease to exist. Liquidity is restored if and only if the inequality is less or equal to zero. The necessary condition for equilibrium no-trade spread is thus in the form of p ask p bid = 1 Rf ((1 α)min i {v i } (1 ᾱ)max i {v i }) > 0. (22) The impact of correlation cannot be directly observed from the above equation. Yet if one remembers, correlation is implicitly brought through via Rf. A natural question arises from Equation (22) is then under what condition do we observe trade, that is when the spread is zero. To to find this threshold, e-express the upper and lower bounds in a specific structure. Define the lower and upper bounds respectively α = α o (1 ) and ᾱ = α o (1 + ), (23) where α o = α+ᾱ 2 is the average of the lowest and highest possible drop in asset values; and the range for α o and are limited to be α o, ɛ(0, 1). The threshold for risky asset j can be express in terms of (refer to appendix A.2 for proof) and arrive at the following necessary condition oj = [max i{v ij } min i {v ij }](1 α o ) [max i {v ij } + min i {v ij }]α o <. (24) The above inequality describes the necessary condition to observe a restore in liquidity. In order for market to remain in a no-trade environment, needs to be strictly greater than oj, otherwise the inequality cannot be satisfied, and trades prevail. The threshold depends on the ratio of the dispersion of traders sentiment between the most optimistic and the most pessimistic. The less the traders are in an agreement with each other the more likely the trades. Further, the ratio of the the average size of drop α o also impacts the threshold, the smaller the impact, the more likely the trades. Interestingly, the threshold is unaffected by correlation, the variance and/or risk tolerance coefficient; this is because the trader are only ambiguous about the expected future value of assets. In case when traders are ambiguous about the expected future values as well as the variance of risky assets, a different type of relationship may be expected. If there exists such a threshold, then intuitively, there is a corresponding trading price. To derive this price for risky asset j, substitute Equations (24) into Equations (20) and 10

11 (21) and have p ask j = p bid j (25) = p o j 2(1 α o)max i {v ij }min i {v ij } R f (max i{v ij } + min i {v ij }). The trading price for risky asset j is always going to be lower than the initial equilibrium price due to an adverse shock to the future value of the asset, where 2(1 αo)max i{v ij }min i {v ij } Rf (max i{v ij }+min i {v ij }) > 0. However, the level of price depends on the diversity of traders belief. The more diverse the beliefs are among traders, the higher the price. Further, the price also depends on the correlation via Rf, when correlation is positive and high, the buyer will always require a lower price with diminishing diversification advantage and for holding market risk. 3 Model solution for special cases - an illustrative analysis of the model This section studies special cases of the general model. Particular structures is imposed and four different cases are considered. Assume the market consists two traders, two risky assets and one riskfree asset. Risky asset 1 is safer than risky asset 2 with a lower market expected payoff and variance. Further, traders may differ in their risk preference. To study the implication of correlation on price, allow traders to be heterogeneous about the first and second moments of risky asset 2 only. Discussion focus on the the implication of correlation of the second scenario during period 1, where risky assets become illiquid. The following cases are considered. Benchmark case) serves as a comparison to the other four cases. Assume traders are homogeneous in all aspects. Case 1) assumes traders to differ in their risk preference τ 1 = τ o (1 γ τ ) and τ 2 = τ o (1 + γ τ ) where γ τ ɛ( 1, 1). They are also heterogeneous about the expected payoff of risky asset 2, v 12 = v 2 (1 γ v ) and v 22 = v 2 (1 + γ v ) where γ v ɛ( 1, 1). All else being homogeneous. Case 2) assumes heterogeneity about the standard deviation of risky asset 2 σ 21 = σ 2 (1 γ σ ) and σ 22 = σ 2 (1+γ σ ) where γ σ ɛ( 1, 1). Case 3) assumes traders to differ in their risk preference and about the standard deviation of risky asset 2. Finally, case 4) assumes traders to differ about expected payoff and variance of risky asset 2. Table 1 shows the cases in tabular form. 3.1 Benchmark: Homogeneous belief Under the Benchmark case, traders are homogeneous in the sense that they have identical belief about both risky assets. The Benchmark case serves as a baseline to the rest of the cases. To investigate the impact of correlation on spreads, consider ρ to vary between (-1,1). With the above assumptions in mind, the market belief at t=0 becomes v abm = v o = (v 1, v 2 ) and Ω abm = Ω o = (ρσ aj σ ak ) 2X2 such that j = k = 1, 2 where v abm and Ω abm are market expected payoff and variance-covariance matrix respectively. Define σaj 2 as the market confidence for risky asset j with respect to the expected 11

12 Table 1: Traders belief for the four cases Risky Pref. Correlation Beliefs of Risky A.1 Beliefs of Risky A.2 Benchmark case Trader 1 τ o ρ o v 1 ; σ 1 v 2 ; σ 2 Trader 2 τ o ρ o v 1 ; σ 1 v 2 ; σ 2 Case1 Trader 1 τ o (1 γ τ ) ρ o v 1 ; σ 1 v 2 (1 γ v ); σ 2 Trader 2 τ o (1 + γ τ ) ρ o v 1 ; σ 1 v 2 (1 + γ v ); σ 2 Case 2 Trader 1 τ o ρ o v 1 ; σ 1 v 2 ; σ 2 (1 γ σ ) Trader 2 τ o ρ o v 1 ; σ 1 v 2 ; σ 2 (1 + γ σ ) Case 3 Trader 1 τ o (1 γ τ ) ρ o v 1 ; σ 1 v 2 ; σ 2 (1 γ σ ) Trader 2 τ o (1 + γ τ ) ρ o v 1 ; σ 1 v 2 ; σ 2 (1 + γ σ ) Case 4 Trader 1 τ o ρ o v 1 ; σ 1 v 2 (1 γ v ); σ 2 (1 γ σ ) Trader 2 τ o ρ o v 1 ; σ 1 v 2 (1 + γ v ); σ 2 (1 + γ σ ) Table 2: The parameters are τ o = 1, v 1 = 0.52, v 2 = 1.64, σ 1 = 0.13, σ 2 = 0.2, ρ o, γ τ, γ v, γ σ, γ ρ ɛ( 1, 1). They are conveniently chosen to avoid negative Rf and to compute realistic prices, demand and equilibrium bid-ask spreads. payoff of that asset. The lower the variance, the greater the market confidence 8. Further, let ρσ a1 σ a2 be the covariance between two risky assets, the higher the ρ, the greater the portfolio is under-diversified. If correlation is zero, there will only be two constant market components, σa1 2 and σ2 a2. Neither one risky asset affects the other. For simplicity, assume traders have the same risk preference and the market portfolio is x m = (x 1, x 2 ) = (1, 1) henceforth unless otherwise stated. The equilibrium price vector at t=0 is thus p o bm = (p o1 bm, p o2 bm ) (26) where R fbm is endogenously derived = ( 1 (v 1 (σ2 1 + ρσ 1σ 2 ) 1 ), (v 2 (σ2 2 + ρσ 1σ 2 ) ) ) ; R fbm τ R fbm τ R fbm = 2 w mo ((v 1 + v 2 ) 1 τ (σ ρσ 1 σ 2 + σ 2 2)). (27) Equation (26) shows equilibrium price vector at t=0. As expected, the price is the expected payoff of risky asset j reduced by a compensation term for holding risks, discounted by R fbm. However, the exact prices depend on the relationship between risky assets. It is important to note, due to the endoge- 8 Assume ρ is the same across all traders. But in general, it may differs from trader to trader. 12

13 nous nature of R fbm from equation (27), equilibrium prices will always depend on the market belief of both risky assets regardless of correlation. On the other hand, Easley and O Hara (2010) assume R f is exogenously given and constant, therefore is unaffected by the market belief at all times. There exists a negative relationship between correlation and R fbm, such that R f bm ρ < 0 shown in Figure 1. It suggests homogeneous traders are more likely to demand for the risk-free asset with diminishing diversification advantage. However, this linear relationship does not always hold under heterogeneous beliefs. The relationship between risk-free rate and correlation is of particular importance in the later development of the model. As it turns out, the impact of correlation, has on the market is brought through via this equilibrium risk-free rate. Figure 1: Equilibrium risk-free rate under the Benchmark case showing correlation impact. At t=1, traders may experience one of the two scenarios. I) an unexpected shock to the expected payoff vectors with a known magnitude; and II) an unexpected shock to the expected payoff vectors with unknown magnitude. Assume the shock has an adverse impact and is systematic, in line with the characteristics of the GFC. According to equation (11), there will be no trades in the former scenario due to lack of heterogeneous beliefs. The focus of this section however is to study the implication of correlation in an ambiguous market in the second scenario. Use the same structure from equation (23) to describe ambiguity. Because v i1 = v 1 and v i2 = v 2, for i = 1, 2; simply find bid and ask prices from equations (20) and (21) respectively ( p bidbm = p o1 bm (1 α)min{v i1}, p o2 R bm (1 α)min{v ) i2} (28) fbm R fbm ( 1 = (α o (1 )v 1 (σ2 1 + ρσ 1σ 2 ) 1 ), (α o (1 )v 2 (σ2 2 + ρσ 1σ 2 ) )), R fbm τ R fbm τ 13

14 and p askbm = ( p o1 bm (1 ᾱ)max{v i1}, p o2 R bm (1 ᾱ)max{v i2} ) (29) fbm R fbm ( 1 = (α o (1 + )v 1 (σ2 1 + ρσ 1σ 2 ) 1 ), (α o (1 + )v 2 (σ2 2 + ρσ 1σ 2 ) )). R fbm τ R fbm τ (a) Risky A.1: ρ = 1, 1, 0 (b) Risky A. 2: ρ = 1, 1, 0 (c) Risky A.1: = 0.8 (d) Risky A. 2: = 0.8 (e) Spread of Risky A: = 0.6 (f) Spread of Risky A: 2 - = 0.6 Figure 2: Equilibrium no-trade spreads of risky asset 1 and 2 are shown in (a) and (b) for some correlation. While, (c) and (d) show the equilibrium no-trade spreads for various of ρ for some particular. Spreads of both assets are shown in (e) and (f). The only difference between the bid and ask prices of risky asset j is the sign of in the above equations (28) and (29), which suggests equilibrium spreads are indeed attributed to ambiguity. Visually, from Figures 2a and 2b, the spreads will always prevail as long as ambiguity is present in the market, 14

15 even when risky assets are not correlated. In particular, when risky assets exhibit strong positive correlation, traders tend to lean toward risky asset 2 that generate a greater expected payoff, that is also riskier. Due to traders demand for risky asset 2, its bid and ask prices seem to be less responsive to correlation impact shown in Figures 2c and 2d; but in general, prices decrease with increasing correlation due to diminishing diversification advantage. Finally, figures 2e and 2f show spread co-movements where risky asset 2 exhibits greater sensitivity to changes in correlation due to its riskier nature and experiences a deeper market freeze with increasing correlation. 3.2 Case 1: Heterogeneous about the payoff and risk preference To understand the impact of correlation under multi-dimensional heterogeneity, two types of heterogeneity can be considered. 1) traders risk preference and 2) traders sentiment of risky asset 2 9. Traders risk preference become τ 1 = τ o (1 γ τ ) andτ 2 = τ o (1 + γ τ ), where γ τ ɛ( 1, 1). Trader 1 is more (less) risk averse if γ τ > 0(< 0). Further, traders expected payoff vectors become v 1 = (v 1, v 2 (1 γ v )) and v 2 = (v 1, v 2 (1 + γ v )), where γ v ɛ( 1, 1). Traders are homogeneous about the expected payoff of risky asset 1, but for risky asset 2, we study the implication of the relationships between sentiment and risk tolerance γ v γ τ > 0(< 0) on the market given correlated risky assets. The market belief at t=0 is then given by v ac1 = (v 1, v 2 (1 + γ τ γ v )) and Ω ac1 = Ω o = (ρσ aj σ ak ) 2X2. The market expected payoff of risky asset 2 differs to that of the Benchmark case. The additional assumption of heterogeneity shows a linear impact on market sentiment of risky asset 2, and this sentiment depends on the sign of γ v γ τ. Trader s opinion dominates market sentiment whom is least risk averse. Therefore, market becomes optimistic (pessimistic) if γ τ γ v > 0(< 0). With the additional assumptions, we derive the equilibrium price vector at t=0 p o case1 = (p o1 case1, p o2 case1 ) (30) = ( 1 (v 1 (σ2 1 + ρσ 1σ 2 ) 1 ), (v 2 (1 + γ τ γ v ) (σ2 2 + ρσ 1σ 2 ) ) ). R fcase1 τ R fcase1 τ where R fcase1 is R fcase1 = 2 ((v 1 + v 2 (1 + γ τ γ v )) (σ ρσ 1σ 2 + σ2 2) ). (31) w mo τ Equation (30) shows a similar correlation impact on the equilibrium prices as in the Benchmark case. However, due to correlation, both equilibrium prices in Case 1 depend on R fcase1, which depends on market sentiment about risky asset 2. If market is optimistic about risky asset 2, that is γ τ γ v > 0; price increases as traders prefer risky asset 2 due to greater expected future payoff given the same risk; while 9 Due to the nature of implied structure, two types of heterogeneity are assumed. Merely taking heterogeneity about expected payoff will results a cancel out and Case 1 reverses to the Benchmark case. 15

16 risky asset 1 decreases in price to reflect a lower demand. However, if market is pessimistic γ τ γ v < 0, risky asset 1 is preferred by traders due to its safer nature. Equation (31) shows the equilibrium riskfree rate for case 1 in terms of γ τ γ v ; and Figure 3 illustrates this fluctuation of demand for risk-free asset when market is optimistic and pessimistic about risky asset 2. As expected, when pessimism (optimism) dominates the market, traders demand less (more) risky assets, and more (less) of risk-free rate. That is, in a pessimistic (optimist) market, traders tend to flight to quality (seek returns). Figure 3: Risk-free rate of case 1 is shown under 3 different types of scenarios. The parameters are γ τ = ±0.05 and γ v = ±0.25. At t=1, traders experience two scenarios. Trader will trade to rebalance their portfolio in scenario I due to heterogeneous beliefs and prices will depend on the market sentiment, risk tolerance and correlation between risky assets. When market is pessimistic about risky asset 2, risky asset 1 s price will lie above the Benchmark case reflecting demand due to its safer nature, while the equilibrium price for risky asset 2 will experience a downward shift reflecting market pessimism. In scenario II, equilibrium bid and ask spreads for both risky assets are derived (see Appendix B.1). Figure 4 shows the case when market is pessimistic γ v γ τ < 0 and risky assets are positively correlated ρ > 0. In particular, Figure4a exhibit a spillover effect due to correlation, there are less willingness to sell risky asset 1. On the other hand, Figure 4b shows there exist single prices for a range of. The condition for trades according to equation (22) is < 1 α o γ v (1 α o ); trading prices can be observed if and only if is smaller than the condition on the right hand side. When traders are diverse in beliefs, i.e. large γ v, spread is harder to prevail, and trades are more likely to occur. Figures 4c and 4d show an asymmetrical changes in the level of bid and ask prices. These changes in price levels determine the size of spreads shown in Figures 4e and 4f. Instead of observing a reduction in spreads, risky asset 1 shows an increase in spread due to market s pessimistic view about risky asset 2. On the contrary, reduction is observed in risky asset 2 due to heterogeneous beliefs about the asset. Finally, spread co-movements can be observed, but the magnitude of the spreads are not always reduced. 16

17 (a) Risky A.1: ρ = 0.8, γ τ = -0.1, γ v=0.6 (b) Risky A.2 : ρ = 0.8, γ τ = -0.1, γ v=0.6 (c) Risky A.1: = 0.8, γ τ = -0.1, γ v=0.6 (d) Risky A.2: = 0.8, γ τ = -0.1, γ v=0.6 (e) Spread of Risky A.1: = 0.8, γ τ = -0.1, γ v=0.6 (f) Spread of Risky A.2: = 0.8, γ τ = -0.1, γ v=0.6 Figure 4: (a) and (b) illustrate the equilibrium no-trade spreads for risky assets 1 and 2 respectively in a pessimistic market. (c) and (d) show the changes in the level of bid and ask prices, and (e) and (f) show the spreads. 3.3 Case 2: Heterogeneous about the variance A natural extension of case 1 is to introduce heterogeneity about variance-covariance matrix. However, before considering both moments of the asset, let traders to only differ in opinion about the second moment of risky asset 2 first. The structures are σ 11 = σ 21 = σ 1 and σ 12 = σ 2 (1 γ σ ) and σ 22 = σ 2 (1 + γ σ ) where γ σ ɛ( 1, 1). Trader 1 is said to be more (less) confident if γ σ > 0(γ σ < 0). 17

18 The market belief at t=0 becomes v ac2 = v o = (v 1, v 2 ), and Ω ac2 = ( (1 γσρ 2 2 )σ 2 1+γσ ρ ρσ 1 σ 2 (1 γ2 σ(2 ρ 2 ) ) 1+γσ ρ 2 2 ρσ 1 σ 2 (1 γ2 σ (2 ρ2 ) ) ( (1 γ2 1+γσ 2 σ )2 (1 ρ 2 ) )σ 2 ρ2 1+γσ 2 ρ2 2 ). (a) Risky A.1: Change in market confidence coe. (b) Risky A.2: Change in market confidence coe. (c) Covariance of Risky A.1 and Risky A.2: Change in covariance coefficient Figure 5: Figures above capture the change in market confidence and covariance of both risky assets. All coefficients are strictly between (0,1]. Difference of opinion about the variance of risky asset 2 has no effect on the market expected payoff vector, however, there is an apparent non-linear impact on the market confidence and covariance of both risky asset, shown above. Figure 5 shows coefficients of market confidence (1 γ2 σρ 2 )σ 2 1+γσ 2 ρ2 1 and ( (1 γ2 σ) 2 (1 ρ 2 ) )σ 2 1+γσ ρ of risky assets 1 and 2, as well as the coefficient of covariance of the assets (1 γ2 σ(2 ρ 2 ) )ρσ 1+γσ ρ σ 2. In particular, risk asset 1 s confidence increases with large ± ρ ; correlation effect is the greater for large dispersion in belief. Further, market seems to be overly confident in risky asset 2 exhibiting great insensitivity to correlation with increasing diversity in beliefs. Market risk is reduced due to diversified beliefs and the market becomes over-confident in risky assets. The 18

19 equilibrium price vector at period 0 becomes where R fcase2 is p o case2 = (p o1 case2, p o2 case2 ) (32) 1 =( (v 1 (1 ρ2 )((1 + γσ)σ (1 γ2 σ)ρσ 1 σ 2 ) R fcase2 τ o (1 + γσ 2 ρ 2 ), ) 1 (v 2 (1 ρ2 )((1 γσ) 2 2 σ2 2 + (1 γ2 σ)ρσ 1 σ 2 ) R fcase2 τ o (1 + γσ 2 ρ 2 ). ) R fcase2 = 2(v 1 + v 2 ) w mo 2(1 ρ2 )((1 + γ 2 σ)σ (1 γ2 σ)ρσ 1 σ 2 + (1 γ 2 σ) 2 σ 2 2 ) (1 + γ 2 σ ρ 2 )τ o w mo. (33) (a) R fcase1 : γ σ = ±0.6 Figure 6: Risky free rate of case 2 (solid red) is shown against the benchmark (dotted red). Equilibrium price vector shown in Equation (32) exhibits non-linearity and difference of opinion of risky asset 2 spills over to risky asset 1, where its price depends on the heterogeneous beliefs of another. Further, risk-free rate exhibits a similar non-linear relationship to correlation shown in Figure 6a. Compare to the Benchmark, case 2 seems to always lie above the Benchmark and becomes increasingly insensitive to diminishing diversification advantage. At t=1, traders again experience one of the two scenarios. According to equation (11), there will be no trades in the former scenario. We expect, with increasing market confidence, price of risky asset 2 will lie above the Benchmark while risky asset 1 will lie below the Benchmark for most case except when correlation is large and positive reflecting market over-confidence in risky assets. In scenario II, equilibrium spreads are derived (refer to Appendix B.2 for bid and ask prices) 10. In an over-confident market and when risky assets are correlated, Figures 7a and 7b show bid and ask prices of both risky assets shift upwards to reflect greater market confidence due to diversified beliefs among traders. However, correlation impact is asymmetrical, there are greater willingness to buy than to sell due to reduced risk. Changes in the level of bid and ask prices are non-linear; and correlation has greater impact on either sides of the correlation spectrum, these are shown in figures 7c and 7d. 10 Due to the structure of the model in Case 2, prices and spreads are unaffected by the sign of γ σ. 19

20 (a) Risky A.1: γ σ = ±0.8; ρ=0.8 (b) Risky A.2: γ σ = ±0.8; ρ=0.8 (c) Risky A.1: γ σ = ±0.8; =0.8 (d) Risky A.2: γ σ = ±0.8; =0.8 (e) Spread of Risky A.1: γ σ = ±0.8; =0.8 (f) Spread of Risky A.2: γ σ = ±0.8; =0.8 Figure 7: (a) and (b) illustrate the equilibrium no-trade spreads for risky assets 1 and 2 respectively in a over-confident market. (c) and (d) show the changes in the level of bid and ask prices, and (e) and (f) show the spreads. Finally, Figures 7e and 7f show the implication of correlation in general has a similar effect on spreads as the Benchmark, but non-linear. Spread reductions are always observed and is most prominent when correlation is large and positive. 3.4 Case 3: Heterogeneous about the variance and risk preference This section focus on the joint impact of heterogeneity about the second moments of risky asset 2 and traders risk preferences, all else remain homogeneous. Traders standard deviations for risky assets 1 and 2 are given by σ 11 = σ 21 = σ 1, and σ 12 = σ 2 (1 γ σ ) and σ 22 = σ 2 (1 + γ σ ) respectively, 20

21 where γ σ ɛ( 1, 1). Further, traders risk preferences become τ 1 = τ o (1 γ τ ) andτ 2 = τ o (1 + γ τ ), where γ τ ɛ( 1, 1). The spreads depend on the relationship between risk tolerance and confidence, γ τ γ σ > 0(< 0). Market belief at t=0 becomes v ac3 = v o = (v 1, v 2 ), and Ω ac3 = (γ 2 σ 2γτ γσ+1)(ρ2 1) γ 2 τ ρ 2 γ 2 σ γ 2 σ 2γ τ (ρ 2 1)γ σ+ρ 2 1 σ2 1 (γ τ γ σ 1)(γ 2 σ 1)(ρ2 1) γ 2 τ ρ 2 γ 2 σ γ 2 σ 2γ τ (ρ 2 1)γ σ+ρ 2 1 ρσ 1σ 2 (γ τ γ σ 1)(γσ 1)(ρ 2 2 1) γτ 2ρ2 γσ 2 γ2 σ 2γτ (ρ2 1)γ σ+ρ 2 1 ρσ (γσ 1) 1σ 2 2 (ρ 2 1) 2 γτ 2ρ2 γσ 2 γ2 σ 2γτ (ρ2 1)γ σ+ρ 2 1 σ2 2. (a) Risky1: in market confidence coe. γ τ = 0.6 (b) Risky2: in market confidence coe. γ τ = 0.6 (c) Covariance of Risky1 and Risky A: in covariance coe. γ τ = 0.6 Figure 8: Figures above capture the change in market confidence and covariance of both risky assets for where γ τ = 0.6. Coefficients are no longer strictly below 1. Difference of opinion about the variance of risky asset 2 and traders risk preferences have no effect on the market expected payoff vector, however, a similar non-linear effect on the market confidence and covariance of both risky assets can be observed. Figure 8 shows coefficients of market confidence (γσ 2γ 2 τ γ σ+1)(ρ 2 1) γτ 2ρ2 γσ 2 γ2 σ 2γτ (ρ2 1)γ σ+ρ 2 1 and (γσ 1) 2 2 (ρ 2 1) γτ 2ρ2 γσ 2 γ2 σ 2γτ (ρ2 1)γ σ+ρ for risky assets 1 and 2, as well as the 2 1 (γ τ γ σ 1)(γσ coefficient of covariance of the two assets 2 1)(ρ2 1) γτ 2 ρ 2 γσ γ 2 σ 2γ 2 τ (ρ 2 1)γ σ+ρ. Market confidence depends

22 on the magnitude of dispersion of γ σ and γ τ, as well as the relationship between the two. Figures 8a and 8b show the coefficients of variance of both risky assets. The market is over-confident when the coefficient of variances lie below the blue plan while market is under-confident when the coefficient lies above. In general, an increasing dispersion in beliefs reduces market risk and increases market confidence. Market is always over-confident of risky asset 1; but not so of risky asset 2 due to dispersion in risk tolerance. Finally, Figure 8c shows the change in covariance for variance of scenarios. (a) R fcase3 : γ τ = 0.6 Figure 9: (a) Risk-free rate of Case 3 is shown in Red and is compared to the Benchmark case in gray. In case 3, risk-free rate exhibits a non-linear relationship to the correlation shown in Figure 9. Traders prefer more risky assets when market is over-confident, that is γ τ γ σ < 0 and some γ τ γ σ > 0. However, traders tend to flight to quality when market is dominated by confident but risk averse trader. However, as traders become overly diverse in their opinions about the variance of risky asset 2, risk aversion effect diminishes and an increase in demand for risky assets is observed. Similar as in other cases, a non-linear and an increasing insensitivity to diminishing diversification advantage with increasing correlation are shown. At t=1, traders again experience one of the two scenarios. In scenario I, no trades are expected. In scenario II, equilibrium spreads are again derived (refer to Appendix B.2 for bid and ask prices). The size of spreads are shown in Figures 10a and 10b. The non-linearity nature of the spreads means whether it is a reduction or an increase in spreads depend on whether the market is over- or underconfident. When γ σ γ τ < 0, market is dominated by trader whom is most confident and least risk averse, an apparent reduction in spreads is shown. As the relationship changes γ σ and γ τ > 0, spreads also change, when dispersion of belief is small, market is dominated by risk aversion, and becomes under-confident; therefore, spreads increase. However, with increasing dispersion of γ σ, market overconfidence yet again dominates the market and spreads are quickly reduced. Finally, an indicative non-linear spread co-movements can be observed like in the previous cases. 22

23 (a) Spread of Risky A.1: γ σ = 0.8; =0.8 (b) Spread of Risky A.2: γ σ = 0.8; =0.8 Figure 10: (a) and (b) show the spreads from case 3 in red is compared to the Benchmark in blue. 3.5 Case 4: Heterogeneous about the payoff and variance This section studies the joint impact of heterogeneity about the first and second moments of risky asset 2; all else remain homogeneous. Traders expected payoff vectors and standard deviations are given by v 2 = (v 1, v 2 (1 γ v )) and v 2 = (v 1, v 2 (1 + γ v )), where γ v ɛ( 1, 1); and σ 11 = σ 21 = σ 1, and σ 12 = σ 2 (1 γ σ ) and σ 22 = σ 2 (1 + γ σ ), where γ σ ɛ( 1, 1), respectively. The relationship between sentiment and confidence, γ v γ σ > 0(< 0) on the market, given risky assets are correlated affects the bid and ask spreads. At period 0, market belief becomes v ac4 = v o = (v 1 γσγvρσ 1 σ 2 (1+γ 2 σ ρ 2 ) v 2, v 2 (1 γσγv(2 ρ2 ) 1+γ 2 σ ρ 2 ) and Ω ac4 = Ω ac2. In case 3, the same variance-covariance matrix is observed but a different market expected payoff vector. Due to correlation, market expected payoff of risky asset 1 is not only affected by market belief of risky asset 2, the impact is also non-linear. While market sentiment of both risky assets depend on the sign γ v γ σ, risky asset 1 also depends on ρ. The equilibrium price vector at period 0 becomes p o case4 = (p o1 case4, p o2 case4 ) (34) where R fcase4 is 1 =( (v 1 γ σγ v ρσ 1 v 2 R fcase4 (1 + γσ 2 ρ 2 (1 ρ 2 ) ((1 + γ2 σ)σ1 2 + (1 γ2 σ)ρσ 1 σ 2 ) )σ 2 τ o (1 + γσ 2 ρ 2 ), ) 1 (v 2 γ σγ v (2 ρ 2 )v 2 R fcase4 1 + γσ 2 ρ 2 (1 ρ 2 ) ((1 γ2 σ) 2 σ2 2 + (1 γ2 σ)ρσ 1 σ 2 ) τ o (1 + γσ 2 ρ 2 )). ) R fcase4 = 2((1 + γ2 σ ρ 2 )σ 2 v 1 (γ σ γ v ρσ 1 (1 + γ 2 σ ρ 2 γ σ γ v (2 ρ 2 )σ 2 ))v 2 ) (1 + γ 2 σ ρ 2 )w mo σ 2 (35) 2(1 ρ2 )((1 + γ 2 σ)σ (1 γ2 σ)ρσ 1 σ 2 + (1 γ 2 σ) 2 σ 2 2 ) (1 + γ 2 σ ρ 2 )τ o w mo 23

24 (a) Risk-free rate: γ v = ±0.05 and γ σ = ±0.05 Figure 11: Risk-free rate of case is compared against the benchmark. We observe both types of relationship between sentiment and confidence. Figure 11 shows the risk-free rate of both types of relationship γ v γ σ > 0(< 0). If market is optimistic about risky asset 2, γ v γ σ < 0, risk-free rate will lie above the Benchmark reflecting traders demand for risky assets. On the other hand, if market is pessimistic, γ v γ σ > 0, risk-free rate will lie below the Benchmark, reflecting traders decreasing demand. In both instance, demand for risk-free rate is more sensitive to positive correlation. This is because, when risky assets are positively correlated and when market s pessimism about risky asset 2 is translated to risky asset 1; since market is pessimistic about both risky assets, demand for risky assets will also decrease, therefore a sharp downturn in risk-free rate can be observed. Similar to the last case, the risk-free rate is affected non-linear. At t=1, traders again experience one of the two scenarios. Trades may occur in the former. When market is pessimistic about risky asset 2, price of risky asset 1 will lie below the Benchmark, this is because for positive correlation, market is also pessimistic about risky asset 1, therefore lower prices prevail. In scenario II, equilibrium spreads are derived (refer to Appendix B.3 for bid and ask prices). In pessimistic market and when risky assets are positively correlated, Figures 12a shows the asset experiences an downwards shift in prices to reflect pessimistic expectation of risky asset 1 due to the spillover effect; while, Figure 12b shows trading prices under the same necessary condition as in case 1. Changes in the level of bid and ask prices are also non-linear, where Figure 12c shows an decreasing in prices with increasing correlation. Further, Figures 12e and 12f show the shape of the spread depends on the market sentiment. When market is pessimistic, there are less willingness to trade and the magnitude of the spread of risky asset 1 increases, where it is particular sensitive to large and positive correlation. As expected, reduction in spread is always observed in risky asset 2 due to dispersion in beliefs and is relative insensitive to correlation impact. Again, spread co-movement in both spreads is non-linear. 24

25 (a) Risky A.1: γ v= 0.1; γ σ= 0.1; ρ=0.8 (b) Risky A.2: γ v= 0.1; γ σ= 0.1; ρ=0.8 (c) Risky A.1: γ v= 0.1; γ σ= 0.1; =0.8 (d) Risky A.2: γ v= 0.1; γ σ= 0.1; =0.8 (e) Risky A.1:γ v= 0.1; γ σ= 0.1; =0.8 (f) Risky A.2: γ v= 0.1; γ σ= 0.1; =0.8 Figure 12: (a) and (b) illustrate the equilibrium no-trade spreads for risky assets 1 and 2 respectively in a pessimistic market. (c) and (d) show the changes in the level of bid and ask price and (e) and (f) show the spreads. 4 Conclusion This paper investigates the implication of correlation on prices in a multi-risky asset market, populated with heterogeneous traders who are ambiguous about the future value of these assets. An extension of the work of Easley and O Hara (2010) by assuming there exist some form of relationship between risky assets, i.e. correlation coefficient is non-zero. In our one risk-free and two risky assets, and two traders market analysis, the size of the bid and ask spreads depend on the correlation between 25