Hedging instruments in multi-asset markets with
heterogeneous expectations
Rens Dekker June 25, 2014
Abstract
This paper investigates the introduction of more hedging instruments to an as-set pricing model with heterogeneous expectations. Previous work by Brock et al. (2009) used a model with only risky asset. This model is generalized to a multi-asset framework. A relation is derived to show why it may be ex-pected that in the presence of additional Arrow securities the primary bifurca-tion may occur earlier. Numerically it is found that more hedging instruments may destabilize markets with more than one risky asset, regardless of the de-pendence structure of those assets.
Acknowledgements. I would like to thank my supervisor Florian Wagener for his great support and constructive comments. I would also like to thank Jan Tuinstra for taking the time to evaluate my thesis.
1
Introduction
The recent global financial crisis has been partly associated with the rapid growth of complex financial instruments. Brock et al. (2009) formalize the idea that more hedging instruments may destabilize markets when traders have heterogeneous expectations. They show that the introduction of additional Arrow securities to a simple asset pricing model results in increased price volatility, and at the same time decreases welfare. In particular they show that the primary bifurcation, marking the onset of instability, occurs earlier when there are more Arrow securities. Their stylized model contains only a single risky asset. It is of interest to generalize this model to a multi-asset framework.
The main contribution of this paper is to show numerically that also in a multi-asset framework additional hedging instrument may destabilize markets. Sec-ondly this paper shows that this destabilization may occur both when stocks are dependent as well as when they are independent. This is done in a setting where
agents have heterogeneous expectations and actively switch between different stra-tegies by a reinforcement learning mechanism based on past performance. Further-more, derivations are presented indicating why it may be expected that more hedg-ing instruments may destabilize a market. The introduction of additional hedghedg-ing instruments may increase the level of reinforcement, as a result of the extra risk that can be hedged, which allows risk-averse investors to increase their leverage. Positions that are on the right side of the market will then gain additional rewards, whereas positions that are on the wrong side of the market suffer potentially huge losses. The main tools of analysis are the theory on non-linear dynamics and bifur-cations such as can be found in Hommes (2013).
Over the last two decades there has been a growing body of literature rejecting the rational expectations hypothesis and embracing the idea of bounded rationality. The existence and persistence of heterogeneous expectations has been empirically validated by the estimation of heterogeneous agent models in a variety of real mar-kets. Examples include stock prices (Boswijk et al., 2007), commodities (Ellen and Zwinkels, 2010), option prices (Chiarella et al., 2010), and exchange rates (Wester-hoff and Reitz, 2003).
Brock and Hommes (1997, 1998) propose simple Adaptive Belief Systems to model stylized facts of economic and financial markets. In these models agents use simple forecasting rules to form their expectations. Agents can switch between dif-ferent forecasting rules based on the observed performance of those rules. Karceski (2002) and Sirri and Tufano (1998) show empirically that mutual fund investors and consumers invest disproportionately more in funds that recently had a high perfor-mance.
This paper expands the simple asset pricing model by Brock and Hommes (1998) to include more than one stock. Next hedging instrument are added to this model, as was done by Brock et al. (2009). These hedging instruments take the form of Arrow securities. Arrow securities are a proxy for real and more complex instruments such as options, futures, and other derivatives. To be more specific, the markets are assumed to be incomplete. In a complete market additional hedging instruments cannot be used to further lower risk.
An asset pricing model with one stock is only a first step in understanding how investors spread their investments among multiple investment opportunities. Once several risky assets are available to investors, then the dependence structure between those assets becomes important. The focus of this paper is a market with two stocks. In this situation the dependence structure of the two stocks is repre-sented only by the correlation of the two stocks. This keeps the numerical analysis somewhat tractable. A topic that is investigated is whether destabilization occurs earlier for positively or negatively correlated stocks. Another question is what the dependence structure of the stocks looks like, and whether this is consistent, un-der the influence of the interaction of heterogeneous expectations and the learning mechanism.
assets and one riskless asset where agents base their beliefs of price movements, volatility, and correlation on past observations. Their main findings are that co-movements of prices and returns may arise, brought about by trend extrapolation and time-varying beliefs about variances and correlation. Chiarella et al. (2007) examined a multi-asset model. They found that in the presence of price fluctuations determined by the interaction of heterogeneous agents, the investors’ anticipated correlation and dynamic portfolio diversification are not always able to stabilize the financial market, but might even act as a further source of complex behavior.
In the current paper, the agents are limited to have heterogeneous beliefs about the stock prices, but homogeneous beliefs about their volatility and correlation. This is a simplifying assumption that allows the dynamics of the system to be in-vestigated as a function of the correlation parameter.
The remainder of this paper is structured as follows. Section 2 extends the asset pricing model with heterogeneous agents to include two stock assets. Next Arrow securities are added to the system. A matrix lemma is used to derive a relation between a system with n and a system with n + 1 Arrow securities. Its proof is provided in the appendices. This relation is used to explain why it may be expected that additional Arrow securities may destabilize a market. Section 3 demonstrates this destabilization by a numerical analysis for two cases of the asset pricing model with different sets of forecasting rules. Additionally the long run dynamics of the systems are investigated as a function of the correlation between the two stocks. Section 4 reflects upon the results and concludes the paper.
2
Asset pricing model
In this section the asset pricing model with heterogeneous beliefs of Brock et al. (2009) is extended by including more than one stock. Let t denote the time period. There exist S possible states s in the next period t + 1. Each state occurs with probability αsthat is common knowledge and independent of time t. Agents can
buy risk free bonds, as well as K different stocks and n Arrow securities. The price of a bond is 1 and it pays R > 1 in the next period. The price of the kth stock in period t is p0k,t. Its payoff in period t + 1 in state s is
qk,t+1s = p0k,t+1+ yks This consists of an uncertain dividend ys
k plus the value of the stock p0k,t+1 that is
independent of the state. The price of an Arrow security for state i is pit. It pays δis,
which is 1 if s = i and 0 otherwise. Let qk,t+1, yk, and δibe stochastic variables with
outcomes qs
k,t+1, ysk, and δsi in state s, respectively.
The focus of this section is on a market where the number of stocks K is 2. It is further assumed that markets are incomplete. Thus the number of instruments needs to be smaller than the number of states. Let Arrow securities be available for states 1, ..., n, such that n < S − 2.
An agent will optimize his utility by deciding to invest part of his wealth in risky assets and invest the remaining in risk free bonds. Let z0k,tbe the demand of the agent for the kth stock and zi
this demand for the ith Arrow security. Finally, let
ζk0be the time-invariant total supply of the kth stock, and ζi the outside supply of the ith Arrow security.
Introduce some vector notation:
zt0 = (z1,t0 , z2,t0 ); z˜t= (zt1, ..., znt); zt= (zt0, ˜zt)
ζt0 = (ζ1,t0 , ζ2,t0 ); ζ˜t= (ζt1, ..., ζtn); ζt= (ζt0, ˜ζt)
p0t = (p01,t, p02,t); p˜t= (p1t, ..., pnt); pt= (p0t, ˜pt)
qt= (q1,t, q2,t); δ = (δ1, ..., δn)
α = (α1, ..., αn)
Also introduce σ2k = Var(qk,t+1); ρ = Cov(q1,t+1, q2,t+1); ηk = Cov(qk,t+1, δ), and
Σ = Cov(δ). Next, let a > 0 denote the risk aversion coefficient and let Vndenote
the symmetric and positive semi-definite (n + 2) × (n + 2) variance-covariance matrix of the uncertain dividends of the stock
Vn= aCov((qt+1, δ)) = a σ12 ρ η1T ρ σ22 η2T η1 η2 Σ (1)
When the market is complete, such that there are n = S − 2 Arrow securities available, then this matrix is singular. A riskless portfolio can then be constructed out of stocks and Arrow securities. Similarly Vn is singular when the two stocks
are perfectly correlated, i.e. have a correlation of ±1, such that ρ = ±σ1σ2. In this
situation the system can be reduced by taking the two separate stocks as one stock. This would correspond to the one stock model.
The elements of Vnread as
σk2= S X s=1 αs(yks− ¯yk)2 ηk,i= αi(yki − ¯yk) Σij = αi(1 − αi) if i = j −αiαj if i 6= j
Let hv, wi denote the inner product of two vectors v and w. If Wtis the wealth
of an agent in period t, then his wealth in period t + 1 in state s is
The excess profit in state s from trading the risky assets equals πt+1s = Wt+1s − RWt= * −Rp0 1,t+ qs1,t+1 −Rp0 2,t+ qs2,t+1 −R˜pt+ δs , zt +
Utility is assumed to be of mean-variance type
Ut= Eπt+1− a 2Varπt+1= * −Rp0 1,t+ Etq1,t+1 −Rp0 2,t+ Etq2,t+1 −R˜pt+ Etδ , zt + −1 2hzt, Vnzti (2) The demand vector can then be seen to be
zt= Vn−1 −Rp0 1,t+ Etq1,t+1 −Rp0 2,t+ Etq2,t+1 −R˜pt+ Etδ (3)
Lastly, the market clearing condition is
zt= z0 1,t z02,t ˜ zt = ζ10 ζ20 ˜ ζ = ζ (4) 2.1 Rational expectations
Consider the situation where agents are homogeneous and use rational expecta-tions. Additionally consider that it is common knowledge that all agents are ratio-nal. This is the fundamental benchmark of the asset pricing model. Arrow securi-ties are endogenous to the system, i.e. there is no outside supply, such that ˜ζ = 0. The demand vector is then denoted by ζ = (ζ0
1, ζ20, 0). Use that Etδ = α, and the fact
that under rational expectations Etqk,t+1 = Etp0k,t+1+ ¯yk. If all markets clear, then
under rational expectations the price dynamics are obtained by pre-multiplying the market clearing condition (4) by Vn, which results in
−Rp01,t+ Etp01,t+1+ ¯y1 = a(σ21ζ10+ ρζ20)
−Rp02,t+ Etp02,t+1+ ¯y2 = a(σ22ζ20+ ρζ10)
−R˜pt+ α = a(η1ζ10+ η2ζ20)
Assuming the transversality condition holds, no explosive bubble solutions ex-ist and prices remain bounded. The above equations are then solved by constant fundamental prices p∗ = (p∗01 , p∗02 , ˜p∗), given as
p∗01 = y − a(σ¯ 2 1ζ10+ ρζ20) R − 1 p∗02 = y − a(σ¯ 2 2ζ20+ ρζ10) R − 1 ˜ p∗= 1 R α − a(η1ζ 0 1 + η2ζ20)
The terms with the risk aversion coefficient a, can be interpreted as the risk premium required to hold the stock. For the stock prices, a positive correlation between the two stocks leads to a larger risk premium, while a negative correlation decreases the risk premium. If there is zero correlation, then the stock prices reduce to the one stock case of Brock et al. (2009). The prices of the Arrow securities do not depend upon ρ, but probably do depend upon the correlation between the two stocks indirectly through η1 and η2.
2.2 Heterogeneous expectations
Assume that agents are heterogeneous in their expectations or beliefs about next period’s price of the two stocks, but that they are homogeneous with respect to everything else, including their expectations on the dividends, and the variance-covariance matrix Vn. Agents are thus in disagreement about the mean, but not the
variance of risky assets. The demand vector is given by
zht= Vn−1 −Rp0 1,t+ Ehtq1,t+1 −Rp0 2,t+ Ehtq2,t+1 −R˜pt+ Etδ = V −1 n Bht (5)
Here the subscript h denotes a belief type, corresponding to one constant set of expectations. Bhtis the belief vector of type h about the excess return of the stocks
and Arrow securities. It will be convenient to work with price deviations xt =
pt− p∗from the fundamental benchmark price p∗. Assume price expectations to be
of the form
Ehp0k,t+1 = p∗0k + fh,k,t= p∗0k + fh,k(x01,t−1, ..., x01,t−I; x02,t−1, ..., x02,t−I)
The technical trading or forecasting rule fh,k,t, models how type h believes that
the future price p0k,t+1 will deviate from the fundamental, given past prices. In general the forecasting rule for stock k may depend on past prices of that stock, but may also depend on past prices of the other stock. Note that each agent of type h has one forecasting rule for the first stock and has another forecasting rule for the second stock. Call this combination of two forecasting rules used by agent type h a strategy. At this point it is not specified that the first forecasting rule is the same rule as the second forecasting rule, but it would make some sense that these would be similar.
2.2.1 Market clearing
Let the fraction of type h agents in period t be denoted by nht. Market clearing for
the stocks and Arrow securities means that X
h
nhtzht= (ζ10, ζ20, 0, ..., 0) (6)
The demand vector given in Equation (5) can be rewritten by subtraction of Equa-tion (3) evaluated in fundamental prices p∗ and addition of Equation (4), which yields zht = ζ10 ζ20 0 + Vn−1 −Rx0 1,t+ fh,1,t −Rx0 2,t+ fh,2,t −R˜xt (7)
The first term equals the fundamental demand and the second term represents the deviation from this fundamental. Plugging in Equation (7) into Equation (6), i.e. summing over the weighted demand, yields the equilibrium dynamics
Rx01,t =X h nhtfh,1,t, Rx02,t = X h nhtfh,2,t, x˜t= 0 (8)
According to Equation (8) the price deviations of the Arrow securities are zero, which implies that they are correctly priced. When all agents are rational, such that f1,t and f2,t are equal to zero, then there will be no deviations of the stock
prices. Also in this case there will be zero demand for each of the Arrow securities, as a result of Equation (7). When agents are heterogeneous, then the stock prices will generally differ from their fundamental prices. Also demand for the Arrow securities will be non-zero.
2.2.2 Fitness
The fractions of agents nhtbelonging to each trader type evolve over time. Given
fitness uh,t−1, the fraction of agents using strategy type h is determined by
nht = eβuh,t−1 Zt , Zt= X h eβuh,t−1 (9)
where Ztis a normalization factor, so that the fractions nhtsum to 1. The fractions
nhtcan be interpreted as the probability of an agent choosing strategy type h, which
is given by a discrete choice model with multinomial logit probabilities. The pa-rameter β is called the intensity of choice. It measures how quickly the agents will change to the optimal forecasting strategy. When β = 0, differences in utility are not observed and each fraction becomes 1/H. When β = +∞, then each period all agents switch to the previous period’s optimal investment strategy. The intensity of choice is assumed to be constant across different trader types.
As a fitness measure for the different strategies, average risk-adjusted profits are used. These profits are corrected for the risks taken when holding risky assets. This measure is consistent with the mean-variance utility given in Equation (2). It is assumed that agents only take into account their performance in the last period, and not any of the prior periods.
uht= * −Rp0 1,t−1+ p01,t+ ¯y1 −Rp0 2,t−1+ p02,t+ ¯y2 −R˜pt−1+ α , zht−1 + −1 2hzht−1, Vnzht−1i Using Bt−1= −Rp0 1,t−1+ p01,t+ ¯y1 −Rp0 2,t−1+ p02,t+ ¯y2 −R˜pt+ α , (10) Bht−1 = −Rp0 1,t−1+ p∗01 + fh,1,t+ ¯y1 −Rp0 2,t−1+ p∗02 + fh,2t,+ ¯y2 −R˜pt+ α , (11)
and zht= Vn−1Bhtrewrite this as
uht= hBt−1, Vn−1Bht−1i −
1
2hBht−1, V
−1 n Bht−1i
For rational agents Bt= Bhtso that their utility reduces to uRt = 12hBt−1, V −1 n Bt−1i.
Then, in deviation from the fundamental, the excess utility becomes uht− uRt = hBt−1, Vn−1Bht−1i − 1 2hBht−1, V −1 n Bht−1i − 1 2hBt−1, V −1 n Bt−1i = −1 2h(Bt−1− Bh,t−1), V −1 n (Bt−1− Bh,t−1)i = −1 2 * x0 1,t− fh,1,t−1 x02,t− fh,2,t−1 0 , V −1 n x0 1,t− fh,1,t−1 x02,t− fh,2,t−1 0 + = −1 2(V −1 n )11(x01,t− fh,1,t−1)2 − (Vn−1)12(x01,t− fh,1,t−1)(x02,t− fh,2,t−1) −1 2(V −1 n )22(x02,t− fh,2,t−1)2
In the last expression, only the first term is present in the one stock case. There is a similar term for the second stock, as well as an interaction term. If (Vn−1)12 = 0
then the utility reduces to a weighted sum of squared prediction errors of the two stocks. One case where this happens is when all Arrow securities are uncorrelated with both stocks and there is no correlation between the two stock (ρ = 0).
prices of the other stock, then the price deviation of one stock will still depend upon the past prices of the other stock through changing fractions nht. This is a
result of the fitness measure and adaptive belief system that was introduced above.
2.2.3 Quadratic form and positive definiteness
In the previous subsection the fitness measure was formulated in deviation from the fundamental. This subsection derives some useful properties of Vn and
intro-duces some new notation, which will help establish how the dynamics of the asset pricing model change as Arrow securities are added in Section 2.3.
Let Sndenote the upper-left 2 × 2 submatrix of Vn−1, and rewrite it in terms of
Vn−1 Sn= (Vn−1)11 (Vn−1)12 (Vn−1)21 (Vn−1)22 = WTVn−1W (12) with W = 1 0 0 1 0 0 .. . ... 0 0
. Next uht− uRt can be rewritten in terms of Snas a quadratic
form uht− uRt = − 1 2hbh,t , W TV−1 n W bh,ti = − 1 2hbh,t, Snbh,ti (13) with bh,t = x01,t− fh,1,t−1 x0 2,t− fh,2,t−1 .
The covariance matrix of a multivariate probability distribution is always pos-itive semi-definite. Also, unless one variable is an exact linear combination of the others, the covariance matrix is positive definite. The case where a stock or Arrow security is perfectly correlated with another risky asset is excluded. For instance, if the two stocks are perfectly correlated, then the system can be reduced by taking the two stocks as one stock. This would correspond to the one stock case.
Thus assume that Vnis positive definite. Then the inverse of a positive definite
matrix is itself positive definite. The inverse covariance matrix Vn−1being positive
definite is equivalent to all its leading principal minors being positive. So then Sn
is positive definite.
2.3 Adding Arrow securities
In this section Arrow securities are added to the asset pricing model with hetero-geneous expectations that has been developed so far. A general relation is derived between a system with n Arrow securities and a system with n + 1 Arrow secu-rities. Specifically a relation is found between inverse covariance submatrices Sn
and Sn+1by applying Lemma 1, which is proven in Appendix A. This lemma
gen-eralizes the similar lemma found in Brock et al. (2006), for K > 1.
Lemma 1. Let Qn+1be a symmetric nonsingular (n + 1) × (n + 1) matrix of the form
Qn+1= Qn r rT s ,
where Qn is a n × n matrix, r is a n × 1 matrix, and s is a scalar. Furthermore let
˜
W = (W, w0T)be a (n + 1) × K matrix, where W has dimensions n × K, and w0T has dimensions 1 × K. Then ˜ WTQ−1n+1W = W˜ TQ−1n W +(w T 0 − rTQ−1n W )T(w0T − rTQ−1n W ) s − rTQ−1 n r . Also det Qn+1= det Qn(s − rTQ−1n r).
Take Qn= Vn, and r = aCov(δn+1, (q1,t0 , q02,t, δ1, ..., δn)), and s = aVar(δn+1). Also
take W = 1 0 0 1 0 0 .. . ... 0 0 , wT
0 = 0 0 , and ˜W = (W, wT0)T. Then using lemma 1
˜ WTVn+1−1W = W˜ TVn−1W + (r TV−1 n W )T(rTVn−1W ) s − rTV−1 n r , which can be rewritten, using Equation (12) as
Sn+1= Sn+
(rTVn−1W )T(rTVn−1W ) s − rTV−1
n r
.
This in turn can be rewritten into something of the form uht−uRt = −12hbh,t , Snbh,ti
by premultiplying by bTh,tand postmultiplying by bh,t
uht− uRt = − 1 2hbh,t, Sn+1bh,ti = −1 2 hbh,t , Snbh,ti + hbh,t , (rTVn−1W )T(rTVn−1W )bh,ti s − rTV−1 n r = −1 2 hbh,t , Snbh,ti + hbh,t , WTVn−1ri2 s − rTV−1 n r
Matrices Sn+1and Snare positive definite, so for bh,t 6= 0, hbh,t , Sn+1bh,ti and
hbh,t , Snbh,ti are positive. Because Vn and Vn+1 are both positive definite, their
determinants are positive. From the second statement of lemma 1 detVn+1=detVn(s − rTVn−1r),
which can be rewritten, such that
s − rTVn−1r = detVn+1 detVn
> 0. The term hbh,t, WTVn−1ri2is nonnegative so then
−1 2hbh,t , Sn+1bh,ti = − 1 2 hbh,t , Snbh,ti + hbh,t , WTVn−1ri2 s − rTV−1 n r ≤ −1 2hbh,t , Snbh,ti (14) This relation states that for any belief vector bh,tthe utility in deviation from the
fundamental in the case with n+1 Arrow securities is smaller or equal to the utility in the case with n Arrow securities. In absolute terms the gap in utility increases. Recall that the utility measure is risk-adjusted profits. When additional hedging instruments are available the utility is expected to increase, given the fact that the portfolio risk can be lowered. Alternatively the additional hedging instruments allow the agents to hold a larger portfolio of stocks for the same amount of risk. This larger portfolio can increase profits, but can similarly increase the losses made by an investor. In a heterogeneous market, agents switch between different com-peting forecasting strategies. Regardless of the fact that different groups of agents may be able to increase their utility, Equation (14) states that the difference in utility between two groups grows as the number of hedging instruments increases.
Before interpreting what the result given in Equation (14) implies for the stabil-ity of the system, first the cases are explored where an additional Arrow securstabil-ity has no effect on the utility in deviation from the fundamental.
2.4 Exploration of hairline cases
The hairline cases where Equation (14) holds with equality will be explored in this section. It is of interest to know how generally the inequality holds strictly and how the hairline cases can be interpreted. For some n, Equation (14) is an equality if hbh,t , WTVn−1ri = 0, i.e. if the belief vector bh,tand the ’relevance’ vector WTVn−1r
are orthogonal.
2.4.1 Rational expectations
Naturally if homogeneous rational expectations are used, then bh,t = 0, such that
2.4.2 Irrelevant Arrow securities
Call the nth Arrow security relevant if WTVn−1r 6= 0. When the nth Arrow security is irrelevant then hbh,t , WTVn−1ri = 0. An interpretation is that an irrelevant
Arrow security cannot be used to hedge additional risk, given the other assets in the market. Next the situations are explored where an Arrow security is in fact irrelevant.
When n = 0, note that both WT = W =
1 0 0 1
and V0−1are invertible, such that their product also is invertible. Then r is in the kernel of the linear map WTV−1
0
if and only if r is the null vector. That is if WTV0−1r = 0, the Arrow security must be uncorrelated with both stocks. When n > 0, then WT is no longer invertible and
the kernel of WTV−1
0 is no longer trivial.
Recall that r = aCov(δn+1, (q01,t, q02,t, δ1, ..., δn)). In Section 2, it was shown that
the covariance between two Arrow securities can be computed and is negative. Therefore r = 0 cannot be a solution to WTV−1
0 r = 0for n > 0. There does exist a
special case where WTVn−1r = 0, but this is not the only solution for n > 0. Lemma 2. The relevance vector WTV−1
n r = 0if all n+1 Arrow security are uncorrelated
with both stocks.
This lemma is proven in Appendix B. The (n + 1)th Arrow security being uncor-related with both stocks is not a sufficient condition for WTVn−1r to be equal to 0. Neither is it sufficient if the (n + 1)th Arrow security is uncorrelated with both stocks and the other n Arrow securities are uncorrelated with one of the stocks.
An interpretation of these facts and Lemma 2 is that if an Arrow security is uncorrelated with the stocks then it can still be used to hedge risk through its cor-relation with other Arrow securities that are themselves correlated with the stocks, directly, or indirectly through yet another combination of Arrow securities.
2.4.3 Time-dependent orthogonality
Let the agent have boundedly rational expectations with regards to both stock. This case has already been explored for the situation where the (n + 1)th Arrow security is irrelevant. So further assume that WTVn−1r 6= 0. This term does not depend upon the time t. However depending upon the strategy h that is used, bh,t may be
time-dependent outside of a steady state equilibrium.
If bh,t does change in time, then WTVn−1r and bh,t may be orthogonal in one
period t, but may not be in following periods. In this case, bh,t and WTVn−1rbeing
2.5 Destabilizing effects
First the previous results are summarized. The utility in deviation from the bench-mark uht− uRt given a price realization is monotonically decreasing in the number
of Arrow securities, i.e.: −1
2hbh,t , Sn+1bh,ti ≤ − 1
2hbh,t , Snbh,ti
If the (n + 1)th Arrow security is relevant, then for heterogeneous expectations fh, typical dividends ysk, and typical probabilities αi, the inequality above holds
strictly.
If for the fundamental steady state there exists a bifurcation in a system with n Arrow securities, let βn∗ be the value of β for which the fundamental steady state
undergoes its first bifurcation.
Let a system with a certain number of Arrow securities n have a bifurcation for β equal to βn∗. When an additional Arrow security is present, the agents are
incen-tivized to update their forecasting strategies more rapidly as a result of an increase in the difference in performance between competing strategies. Thus increasing the number of instruments may effectively increase the intensity of choice. It may then be expected that the bifurcation of the system with n + 1 Arrow securities oc-curs for a smaller value of β, ceteris paribus. At worst it should occur for the same value. This expectation is carefully formulated in the relation
βn+1∗ ≤ β∗n, n = 0, ..., S − K − 2
If the primary bifurcation does come earlier, then it can be said that hedging in-struments such as Arrow securities may be a destabilizing force to the fundamental equilibrium. The more hedging instruments present in the market, the earlier the system may destabilize.
Brock et al. (2009) were able to prove that the bifurcation occurs earlier. An increase in the number of Arrow securities has an effect on the matrix Sn, but does
not change the term bh,t. In the one stock case of Brock et al. (2009), Sn+1is a scalar,
which allows the effect of an increase in n to be separated from the time effects of bh,t. For K > 1 these two are intertwined. This is why I am unable to prove in
a multi-asset framework that the bifurcation comes earlier. However in the next section a numerical analysis for two cases of the asset pricing model indicates that indeed the bifurcation comes earlier.
3
Numerical analysis
This section contains a numerical analysis for two cases of the asset pricing model with differing trader types. Simulations are done using E&F Chaos. E&F Chaos is a freely available software package for simulation of nonlinear dynamic models
(Diks et al., 2008).
The primary goal of this section is to show that, the value of β for which the system undergoes its first bifurcation βn∗, decreases in n. (This property can only be investigated if there exists a bifurcation in the system.) In other words, for larger n the primary bifurcation occurs for a smaller β. Assumptions have to be made about the number of states and the dividends of the two stocks. Therefore con-struct a world with some arbitrary states and dividends. Specifically this paper uses randomly drawn dividends to construct a set of inverse covariance submatri-ces Snfor n = 0, ..., S − K − 1. The same set is used throughout the remainder of
this section. The procedure used to construct such an arbitrary set is described in Appendix C. In total there are 40 states, so that for incomplete markets there is a maximum of 37 Arrow securities.
A secondary goal is to investigate the dynamics of the system as a function of the dependence structure of the two stocks. For two stocks this dependence is characterized by ρ. A related question is whether additional Arrow securities destabilize a system with correlated or uncorrelated stocks. Another is whether destabilization occurs earlier in a system with positive, negative, or uncorrelated stocks. A last question is what the dependence structure of the stocks looks like un-der the influence of the interaction of heterogeneous expectations and the learning mechanism.
Case 1: costly-fundamentalist vs optimist and pessimist
Consider a market with two stocks and three trader types. Each type makes a forecast for each of the stocks. For simplicity assume that the same forecasting rule is used for each stock.
The first trader type is a costly-fundamentalist. The fundamentalist is not com-pletely rational in the sense that they are not perfectly clairvoyant about the future prices. This type uses information about economic fundamentals and predicts that the price of the stocks will be equal to their fundamental value. However to obtain this information the trader pays a price C > 0. This price is subtracted from his utility. Information costs represent an extra effort required to obtain a more sophis-ticated price forecast. Sirri and Tufano (1998) found in their empirical investigation that search costs seem to be an important determinant of which funds consumers invest in.
The fundamental value of a stock would arise as the equilibrium price if all agents had homogeneous rational expectations. The fundamentalist’s forecasting rules in deviation from the fundamental are:
f1,1,t= 0
f1,2,t= 0
above the fundamental:
f2,1,t = b
f2,2,t = b
The third is a pessimist, predicting the price to always be below the fundamental: f3,1,t = −b
f3,2,t = −b
Note that the forecasting rules for the optimist and pessimist are assumed to be symmetric around the fundamental.
The utility for each trader type is given by filling in these forecasting rules into Equation (13). The fractions of agents using each type are given by Equation (9), and the dynamics of the system are given by the equilibrium dynamics Equation (8).
The following set of parameter values is used R = 1.1, b = 0.2, σ2
1 = 1, σ22 =
1. The choice for σ21 = σ22 = 1 is for simplicity purposes. Parameter ρ then lies between -1 and 1, and can be interpreted as the correlation between the first and second stock. A choice for particular σ12, ρ, and σ22 defines the inverse covariance
submatrix Snfor n = 0. The adding of any additional Arrow securities changes the
elements of Sn, potentially changing the dynamics of the system. (Specifically the
diagonal elements are monotonically increasing in n.)
Because the second and third trader types are symmetric, there is a steady state equilibrium in x1 = x2 = 0. The system has dimension 2 and its eigenvalues
at x1 = x2 = 0 can be calculated analytically. Figure 1c shows (for n = 0) for
which combinations of ρ and β the first eigenvalue takes the values +1. The second eigenvalue lies within the unit circle.
When simulating, the system is observed to have a pitchfork bifurcation. At the bifurcation the fundamental equilibrium around x1 = x2 = 0loses its stability and
two new stable steady states are created. This is illustrated in Figure 1b for n = 10 and ρ = 0.9. Using symmetric initial conditions, both the lower and upper half of the ’fork’ are shown in Figure 1b. After the bifurcation, the costly-fundamentalist can no longer compete with the ’cheap’ biased trader types, due to the additional cost C. When initially there are more optimists than pessimists (n2 > n3), then
after the bifurcation the optimists gradually ’take over’ the system, such that the stable steady state will be positive. When the pessimists are in the majority, then the negative steady state is observed.
Figure 2 illustrates the dynamics of the system as a function of the correlation ρ for n = 0. (The used setup of randomly drawn dividends does not enable an n > 0.) The bifurcation diagrams in 2a and 2b show that the fundamental equilibrium destabilizes earlier the more negatively correlated the two stocks are. Also the number of values of ρ for which the fundamental equilibrium is stable increases for larger β.
-50 -40 -30 -20 -10 0 0 10 20 30 40 50 LLE beta (a) (b) 0 10 20 30 40 50 - 1.0 - 0.5 0.0 0.5 1.0 beta rho (c)
Figure 1: Case 1. Simulation results. (a) Largest Lyapunov exponent as a function of β for n = 10, ρ = 0.9. (b) Bifurcation diagram as a function of β for n = 10, ρ = 0.9. Using two sets of initial conditions, both the upper and lower half of a pitchfork bifurcation are shown. (c) The eigenvalues of the steady state around x1= x2= 0for n = 0. Shown is for which values of ρ and β the first eigenvalue takes the value of +1. The second eigenvalue lies within the unit circle.
(a) β = 10 (b) β = 20
(c) β = 10 (d) β = 20
Figure 2: Case 1. Bifurcation diagrams as a function of ρ for n = 0. (a)+(b) use optimist {f2,1,t= b, f2,2,t= b}and pessimist {f3,1,t= −b, f3,2,t= −b}trader types. (c)+(d) use mixed biased trader types {f2,1,t= b, f2,2,t= −b}and {f3,1,t= −b, f3,2,t= b}.
−b, f3,2,t = b} are used then something different is observed. The bifurcation
diagrams in 2c and 2d show that the fundamental equilibrium destabilizes earlier the more positively correlated the two stocks are.
With the first set of trader types the equilibrium after the bifurcation was either positive in both markets or negative in both markets. This paper proposes that the stocks can be considered to be positively correlated. Say that the realized correla-tion is positive, regardless of the actual correlacorrela-tion of the dividends (that may be negative). This is a result of the learning mechanism and heterogeneous expecta-tions. The biased trader types use the same forecasting rule for each stock. Either the optimist will gain the upper hand in both markets or the pessimist will do so. Note that in this situation a more negative correlation seems to be destabilizing.
With the second set of trader types the equilibrium after the bifurcation is pos-itive in one market and negative in the other. Here this paper proposes that the stocks can be considered to be negatively correlated. Again one of the two biased trader types will gain the upper hand. The trader type that does, establishes a pos-itive equilibrium in one market and a negative equilibrium in the other. Observe that here a more positive correlation is destabilizing.
An interpretation of these results could be that it takes the learning mechanism considerably more effort to make negatively correlated stocks move in phase (sit-uation 1). Or similarly to make positively correlated stocks move in counter-phase (situation 2). Thus destabilizing the system earlier. This is by no means the defini-tive answer.
Figure 3 shows nine bifurcation diagrams as a function of β for three values of nand three values of ρ. It is observed that for larger n the fundamental equilibrium becomes unstable for a lower intensity of choice β. Also for smaller ρ the biased trader types take over the system earlier.
Figure 4 shows as a function of n, the values of βn∗ for which the system
un-dergoes its first bifurcation, as approximated by the largest Lyapunov exponent. In accordance with the expectations formulated in Section 2.5, βn∗ is monotonically decreasing in n. The same monotonic decrease is observed, regardless whether the two stocks have a positive, negative, or no correlation.
Case 2: near-fundamentalist vs. trend extrapolator
Consider a two stock market where two different forecasting strategies are used. The first trader type is a near-fundamentalist. This type uses information about economic fundamentals and predicts that the price of the stocks will be equal to their fundamental value. However assume that these traders are unable to observe the true fundamental value and so make a mistake of size .
f1,1,t =
(a) n = 0, ρ = −0.9 (b) n = 0, ρ = 0 (c) n = 0, ρ = 0.9
(d) n = 10, ρ = −0.9 (e) n = 10, ρ = 0 (f) n = 10, ρ = 0.9
(g) n = 20, ρ = −0.9 (h) n = 20, ρ = 0 (i) n = 20, ρ = 0.9 Figure 3: Case 1. Bifurcation diagrams as a function of β for three different values of n and three different values of ρ. Using two different sets of initial conditions, both the upper and lower half of the pitchfork bifurcation are shown.
Figure 4: Case 1. The β∗for which the first bifurcation occurs as a function of n the number of Arrow securities. Results are shown for ρ ∈ {0.9, 0, −0.9}. Note that a bifurcation value for n = 34, ..., 37was not observed
The second trader type is a trend extrapolator. This type makes its forecast by constructing a trend between the prices of the last two periods.
f2,1,t = x1,t−1+ g(x1,t−1− x1,t−2)
f2,2,t = x2,t−1+ g(x2,t−1− x2,t−2)
The following set of parameter values is used R = 1.1, = 1, g = 1.101, σ2
1 =
1, σ22 = 1. This system heavily depends on its initial conditions. The following initial conditions are specified for t = 0: x1,t = 0.2, x1,t−1 = 0.3, x2,t = 0.1, x2,t−1 =
0.2, n1,t = 0.2, n2,t = 0.8.
Once again look at the dynamics of the system. Figure 5 shows simulation results for n = 10 and ρ = 0.9. The bifurcation diagram in Figure 5b shows a bifur-cation diagram using dynamic initial conditions. For each new value of β the initial conditions are the last point plotted for the previous, slightly different parameter value. This is intended to keep the orbit on the ’same’ attractor as the parameter slowly changes, thus following the attractor as long as it exists. The attractor can be ’followed’ from either ’left to right’ or ’right to left’. Both directions are plotted here. Three bifurcations can be seen. Brock et al. (2009) have looked at similar fore-casting rules in the case of 1 stock and found two saddle-node bifurcations and one Hopf bifurcation leading to chaotic dynamics. Their case looks quite analogous to the system here.
The time series of Figure 5c shows that the stocks show cyclical behavior. The phase plot of Figure 5d shows that the prices of the two stocks move in phase in the long run. That is, in the long run the stocks are perfectly correlated even though the dividends of the stock have a correlation of ρ = 0.9. This is similar to what was observed in the analysis of the dependence structure in the previous case.
-0.35 -0.3 -0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 0 10 20 30 40 50 LLE beta (a) (b) -1 -0.5 0 0.5 1 10000 10020 10040 10060 10080 10100 x1 t (c) (d)
Figure 5: Case 2. Simulation results for n = 10, ρ = 0.9. (a) Largest Lyapunov exponent as a function of β. (b) Bifurcation diagram using dynamic initial conditions. (c) Time series for β = 10after 10,000 initial iterations. (d) Phase plot for β = 10 after 10,000 initial iterations.
(a) β = 5 (b) β = 10
(c) β = 5 (d) β = 10
Figure 6: Case 2. Bifurcation diagrams as a function of ρ, with n = 10. These diagrams use dynamic initial conditions. (a)+(b) use near-fundamentalist with positive forecasting error {f1,1,t = , f1,2,t = }. (c)+(d) use near-fundamentalist with mixed forecasting error {f1,1,t= , f1,2,t= −}.
Figure 6 illustrates the dynamics of the system as a function of the correlation ρ for n = 0. The bifurcation diagrams in Figures 6a and 6b show that the system destabilizes earlier the more negatively correlated the two stocks are. The band-width of ρ wherein the system is stable decreases for larger β. As in case 1, this is compared to a different set of trader types. For example, let go the assumption that f1,1t = f2,1t = . Suppose that the near-fundamentalist makes a positive error
for the first stock, but a negative error for the second stock, such that {f1,1t = ,
f2,1t = −}. Then the system destabilizes the more positively correlated the stocks
are, instead of negatively (see 6c and 6d).
With the first set of trader types the equilibrium before the Hopf bifurcation was positive in both markets. Also Figure 5d illustrates how after the Hopf bifurcation the stocks move in phase. The realized correlation of the stocks can be considered to be positive. Note that in this situation a more negative correlation seems to be destabilizing.
With the second set of trader types the equilibrium before the Hopf bifurcation is positive in one market and negative in the other. After the Hopf bifurcation the stocks move in counter-phase. The realized correlation of the stocks can be consid-ered to be negative. Note that here a more positive correlation is destabilizing.
These results can be interpreted in the same way as in case 1. However this is not the definitive answer. It takes the learning mechanism considerably more effort to make negatively correlated stocks move in phase (situation 1). Or similarly to make positively correlated stocks to move in counter-phase (situation 2). Thus destabilizing the system earlier.
Figure 7 shows nine bifurcation diagram as a function of β for three values of nand three values of ρ. It is observed that for larger n the chaotic dynamics occur earlier and the variation in the price deviations is larger. Also for a smaller ρ the chaotic dynamics occur earlier and the absolute price deviations are larger.
Figure 8 shows as a function of n, the values of βn∗ for which the system
un-dergoes its first bifurcation, as approximated by the largest Lyapunov exponent. The same behavior is observed for markets with positively, negatively, as well as uncorrelated stocks: βn∗ is monotonically decreasing in n. Thus an increase in the
number of Arrow securities destabilizes the system.
4-type extension
The previous 2-type model was extended to a 4-type system. The two additional trader types make use of the same forecasting rules as before, but in a new combi-nation. They are
f3,1,t =
f3,2,t = x2,t−1+ g(x2,t−1− x2,t−2)
and
f4,1,t = x1,t−1+ g(x1,t−1− x1,t−2)
(a) n = 0, ρ = −0.9 (b) n = 0, ρ = 0 (c) n = 0, ρ = 0.9
(d) n = 10, ρ = −0.9 (e) n = 10, ρ = 0 (f) n = 10, ρ = 0.9
(g) n = 20, ρ = −0.9 (h) n = 20, ρ = 0 (i) n = 20, ρ = 0.9 Figure 7: Case 2. Bifurcation diagrams as a function of β for three different values of n and three different values of ρ. These diagrams use dynamic initial conditions.
Figure 8: Case 2. The β∗for which the first bifurcation occurs as a function of n the number of Arrow securities. Results are shown for ρ ∈ {0.9, 0, −0.9}.
The same parameter values are used and the initial conditions are supplemented with n3= 0and n4 = 0.
In this 4-type model, for n = 0 and ρ = 0.9, the bifurcation occurs for roughly the same value of β, compared to the 2-type model. The extra two trader types thus do not seem to affect the stability.
The phase plots of Figures 9b-d show that for relatively low β the stocks move in phase, but that for increasing β, the attractor changes into an invariant circle. Thus for some β the stocks are perfectly correlated, but for larger β the stocks lose this property. In fact the stocks jump from being perfectly correlated to being neg-atively correlated.
For n = 10 and ρ = 0.9 the same dynamics are observed, but the changes in behavior occur for smaller values of β (see 9e-h).
4
Conclusion
Brock et al. (2009) showed that additional hedging instruments may destabilize markets with one risky asset. A market with two uncorrelated stocks does not cor-respond to two independent markets where the same theorem can be applied as in Brock et al. (2009). This is because each agent trades in both markets and uses a single set of expectations to make his investment decision for both stocks. A mean-variance optimizing agent will be able to reduce his portfolio risk simply by holding two stocks that do not have perfect positive correlation. This diversifica-tion effect makes the effect of hedging instruments more complex than in the one stock case.
In a numerical analysis, the same result was found as in Brock et al. (2009). Two cases are provided with heterogeneous agents where the fundamental steady state
-0.35 -0.3 -0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 0 10 20 30 40 50 LLE beta
(a) Lyapunov plot with n = 0 (b) n = 0, β = 15
(c) n = 0, β = 45 (d) n = 0, β = 75 -0.35 -0.3 -0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 0 10 20 30 40 50 LLE beta
(e) Lyapunov plot with n = 10 (f) n = 10, β = 15
(g) n = 10, β = 45 (h) n = 10, β = 75
Figure 9: Case 2. 4-type extension with {f3,1,t = , f3,2,t= x2,t−1+ g(x2,t−1− x2,t−2)} and {f4,1,t= x1,t−1+ g(x1,t−1− x1,t−2), f4,2,t= }. Top 4 panels use n = 0 and ρ = 0.9. Bottom 4 panels use n = 10 and ρ = 0.9. (a) Largest Lyapunov exponent as a function of β. (b)-(d) Phase plots after 10,000 initial iterations. (e) Largest Lyapunov exponent as a function
destabilizes earlier as a result of additional hedging instruments. Concretely the primary bifurcation marking the onset of instability occurs for a smaller value of the intensity of choice. In the market with the costly-fundamentalist versus the op-posite biases, the fundamental equilibrium destabilizes and the bifurcation creates two new stable steady states. In the market with the near-fundamentalist versus the trend-extrapolator the bifurcation eventually leads to chaotic or quasi-period dynamics. In both cases it was shown that the system destabilizes earlier as a func-tion of the number of Arrow securities. These results were observed to hold for positively, negatively, as well as uncorrelated stocks.
As for the question whether a positive or negative correlation will lead to an earlier bifurcation, the answer is that it depends. It was found that it depends upon the specific forecasting rules that the agents employ. It is possible that it de-pends upon the correlation between the two stocks that is realized by the interac-tion of the different forecasting rules. This will certainly get more complicated, the more types of different agents are present in the market. In a 4-type case of near-fundamentalists versus trend extrapolators sensitive dependence was observed of the dependence structure of the stock movements on the intensity of choice. Stocks with positively correlated dividends were observed to jump from a perfect positive correlation to being negatively correlated.
It was also found that the interaction of heterogeneous agents may lead to ob-served correlations of the two stocks that are inconsistent with the agents’ expec-tations on the same correlation. In future work it might be of interest to adopt an approach with time-varying expected correlation based on past observations. This may however introduce an additional layer of complexity, such as was found by Chiarella et al. (2007). A different suggestion for further research is to look at the effect of additional hedging instruments on average welfare.
Theoretically I was unable to prove that the primary bifurcation occurs earlier in the presence of additional Arrow securities. However the current paper does provide derivations that indicate why it may be expected that additional hedging instruments may destabilize incomplete markets with more than one risky asset. With respect to the generality of these derivations: The used matrix lemma can also be applied to a general K number of stocks. The mentioned destabilization is the result of the interaction of heterogeneous agents and the reinforcement learning mechanism. This works as follows: One forecasting strategy may perform better than another. In the presence of additional hedging instruments, more risk can be hedged, which allows the agents to hold larger portfolios of stock. The larger portfolio leads to higher profits, but also increases the performance gap between competing forecasting strategies. With a larger absolute difference in utility, agents will be more incentivized to reconsider their investment strategy. Their intensity of choice is effectively increased. Agents switching faster between different fore-casting rules leads to an earlier bifurcation of the system. Thus the introduction of additional hedging instruments may destabilize the market earlier.
highly stylized and unable to explain all complex behavior of real financial mar-kets. Though hedging instruments may possess the useful property that risk can be lowered, they can similarly be used to increase leverage. This type of reinforcement is associated with greater instability. The construction of larger positions generates bigger rewards for investment strategies that are on the right side of the market, but potentially huge losses for strategies that are on the wrong side of the market. Not only do financial innovations need to be watched carefully because products may grow in complexity and may lack transparency (Simkovic, 2009, p.253). They also need to be watched because even if new products are understood perfectly well, they can lead to greater instability of financial markets.
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A
Proof of Lemma 1
The following lemma and its proof follow Brock et al. (2006). In the current paper their lemma is generalized for K > 1.
Lemma 1. Let Qn+1be a symmetric nonsingular (n + 1) × (n + 1) matrix of the form
Qn+1= Qn r rT s ,
where Qnis an n × n matrix, r is an n × 1 matrix, and s is a scalar. Furthermore let
˜
W = (W, w0T)be a (n + 1) × K matrix, where W has dimensions n × K, and wT0 has dimensions 1 × K. Then ˜ WTQ−1n+1W = W˜ TQ−1n W +(w T 0 − rTQ−1n W )T(w0T − rTQ−1n W ) s − rTQ−1 n r . Also det Qn+1= det Qn(s − rTQ−1n r).
Proof.Let ˜ξ = (ξ, ξ0)be a (n + 1) × K matrix, where ξ has dimensions n × K, and ξ0
has dimensions 1 × K, such that ˜W = Qn+1ξ. In the first step ˜˜ ξwill be determined,
after which ˜WTQ−1n+1W˜ is evaluated by computing ˜WTξ. ˜˜ W = Qn+1ξ˜can be split
up into components
W = Qnξ + rξ0
Solving ξ in terms of W and ξ0gives
ξ = Q−1n W − Q−1n rξ0
Substituting this into the second equation yields ξ0 =
w0T − rTQ−1
n W
s − rTQ−1 n r
From this, ξ is obtained as
ξ = Q−1n W − Q−1n rw T 0 − rTQ−1n W s − rTQ−1n r The result is ˜ WTQ−1n+1W = ˜˜ WTξ = W˜ Tξ + w0ξ0 = WTQ−1n W − WTQ−1n rw T 0 − rTQ−1n W s − rTQ−1 n r +w0w T 0 − w0rTQ−1n W s − rTQ−1 n r = WTQ−1n W + (w T 0 − rTQ−1n W )T(wT0 − rTQ−1n W ) s − rTQ−1n r ,
which proves the first statement. This statement can be established by a variation on the use of the formula for the inverse of a partitioned matrix, which uses the Schur complement of a submatrix of a matrix (Skogestad and Postlethwaite, 1996, p.499). The second statement does not have to be generalized. Its proof can be seen in Brock et al. (2006) and is established using Schur’s formula for the determinant of a partitioned matrix (Skogestad and Postlethwaite, 1996, p.500).
B
Proof of Lemma 2
Proof. Let Vn−1 = a A B BT D, where following the notation given in Equation (1): A = σ21 ρ ρ σ22 , B = ηT1 ηT2 , D = Σ , and a > 0
Here B is the only matrix that can be a null matrix. By the partitioned matrix inverse WTVn−1= a
(A − BD−1BT)−1 : (A − BD−1BT)−1BD−1
The last (n − 2) elements of r = aCov(δn+1, (q1,t0 , q2,t0 , δ1, ..., δn))can be computed
and were shown to be negative in Section 2. If the (n + 1)th Arrow security is uncorrelated with both stocks, such that the first two elements of r are 0. Then a solution to WTVn−1r = 0is if B = 0. Meaning that also the first n Arrow securities are uncorrelated with both stocks.
C
Constructing dividends
It is expected that an arbitrary set of states and dividends will be enough to show that β∗nis monotonically decreasing in n. This appendix describes a procedure used
to construct such a set for the purpose of this paper.
Take a world with S = 40 states and specify the chance to be in the ith state of the world as αi= 1/S.
1. Let the dividends for the first stock be ˜ys1= s − 1. 2. Standardize these dividends as ys
1 = ˜ ys 1 √ Var(˜y1s) .
3. Randomly generate numbers ˜φs from a Uniform(0,1) distribution for each state of the world.
4. Standardize these numbers as φs= φ˜s
√
Var(φ˜s)
.
5. Let the dividends for the second stock be ˜ys2= (θ − Cov(y1, φ))ys1+ φs.
6. Standardize these dividends as ys 2 = ˜ ys 2 √ Var(˜ys 2) .
The θ can be varied to obtain the required ρ. For θ = 0, ρ = 0. For ρ ± 0.9, it suffices to use θ ± 2.04. Let the risk aversion coefficient be a = 1, such that
V0= Cov((ys1, y2s)) =
1 ρ ρ 1
All other elements of VS−K−1 can then be derived using the dividends y1sand y2s.
The formulas to do this, were presented in Section 2. From VS−K−1, Sn can be
