Financial Complexity and Trade

Introduction An example Model

Financial Complexity and Trade Spyros Galanis University of Southampton

July 16, 2016

Spyros Galanis University of Southampton

Financial Complexity and Trade


Motivation I

An implicit assumption when modelling financial markets is that each investor can understand and trade any subset of the available securities

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In reality, understanding the payoff structure of complex securities requires time, cognitive effort and special training, usually in short supply “It is hard for investors to evaluate complex financial instruments, because difficult risk modelling is required, and because they are often unaware of the details of the asset pool which backs financial securitisations.” — House of Lords Economic Affairs Committee (2009)




Motivation I

An implicit assumption when modelling financial markets is that each investor can understand and trade any subset of the available securities

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In reality, understanding the payoff structure of complex securities requires time, cognitive effort and special training, usually in short supply “It is hard for investors to evaluate complex financial instruments, because difficult risk modelling is required, and because they are often unaware of the details of the asset pool which backs financial securitisations.” — House of Lords Economic Affairs Committee (2009)




Motivation I

[Célérier and Vallée, 2015] show that payoff structure complexity has increased over time Jayanne 4: “This is a growth product linked to a basket composed of the FTSE Euro First 80, the FTSE 100, the SMI and the NIKKEI 225. The Annual Performance is set at 5% for the first three years. In the following years, if the performance since the start date of the worst-performing index is positive or null, then the Annual Performance for that year is registered at 5%, otherwise 0%. The Basket Performance since the start date is registered every six months. The Final Basket Performance is calculated as the average of all these six-monthly readings, capped at a maximum basket performance of 100%. After 8 years, the product offers a guaranteed capital return of 100%, plus the greater of either the sum of the Annual Performances, or 100% of the Final Basket performance.” Spyros Galanis University of Southampton



Motivation I

A typical investor may understand and bet separately on each index (FTSE Euro First 80, FTSE 100, SMI, NIKKEI 225), but not on their combination

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Cognitive limitations: limited attention, bounded memory, unawareness, costly information, inability to formulate complex plans

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[Polkovnichenko, 2005] reports a persistent pattern from the Survey of Consumer Finances, of investing simultaneously in mutual funds and an individual stock

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[Nieuwerburgh and Veldkamp, 2010] derive optimal under-diversification in a framework with costly information acquisition Spyros Galanis University of Southampton



Motivation I

A typical investor may understand and bet separately on each index (FTSE Euro First 80, FTSE 100, SMI, NIKKEI 225), but not on their combination

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Cognitive limitations: limited attention, bounded memory, unawareness, costly information, inability to formulate complex plans

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[Polkovnichenko, 2005] reports a persistent pattern from the Survey of Consumer Finances, of investing simultaneously in mutual funds and an individual stock

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[Nieuwerburgh and Veldkamp, 2010] derive optimal under-diversification in a framework with costly information acquisition Spyros Galanis University of Southampton



Complexity and market freezes

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Financial complexity is defined as the property of restricting the number of securities an investor can simultaneously trade

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Can it explain market freezes?

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Consider an economy and fix the asset structure, the investors’ preferences and the initial allocation

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If there is trade in the standard environment of no complexity, where each investor can simultaneously trade any subset of the available securities, will there be trade in the presence of maximum complexity, where each investor can trade at most one security?





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Financial complexity is defined as the property of restricting the number of securities an investor can simultaneously trade

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Can it explain market freezes?

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Consider an economy and fix the asset structure, the investors’ preferences and the initial allocation

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If there is trade in the standard environment of no complexity, where each investor can simultaneously trade any subset of the available securities, will there be trade in the presence of maximum complexity, where each investor can trade at most one security?





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We characterize the existence of trading in both environments

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We show that for smooth preferences (no kinks on indifference curves) complexity does not shut down trade, otherwise it does

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For smooth preferences, the result is robust because trade would still occur with all less extreme definitions of complexity




Literature I

[Billot et al., 2000, Billot et al., 2002], [Rigotti et al., 2008] characterize trading in terms of disjoint sets of beliefs in a complete markets setting without complexity and convex preferences

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[Kajii and Ui, 2006] and [Dominiak et al., 2012] characterize trading in the case of only one security, using the MEU and CEU models

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Complex securities: [Amromin et al., 2011], [Henderson and Pearson, 2011], [Simsek, 2013] [Valkanov and Ghent, 2014], [Griffin et al., 2014], [Hens and Rieger, 2014], [Sato, 2014], [Célérier and Vallée, 2015] Spyros Galanis University of Southampton



Literature I


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Literature I


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Outline

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An example

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Model

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Trading in the absence of complexity

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Trading with maximum complexity

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Conclude




The economy

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One-good economy with two agents, i and j

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State space S has three states

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General convex preferences

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Arrow-Debreu security k = 1, 2, 3 pays 1 if si occurs and 0 otherwise




Subjective beliefs

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A key component in characterizing the occurence of trade is the subjective beliefs, defined by [Rigotti et al., 2008]

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These beliefs depend on the investor’s preferences and initial allocation and are “revealed” by potential market behavior

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They do not necessarily coincide with the “priors” of the investor’s utility function, but are convex sets

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The definition applies to general convex preferences, including many models with ambiguity aversion




Subjective beliefs s2

g ei k

h

s1

Figure:Financial Complexity and Trade



Probability simplex

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The triangle in the following figure represents the probability simplex so that each point represents a probability on {s1 , s2 , s3 }

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Given the initial allocation, agent i’s subjective beliefs is given by the rhombus, whereas j’s is given by the circle




Probability simplex s1

a

s2

s3




Trade in the absence of complexity

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There is trade in the absence of complexity if and only if the investors’ subjective beliefs are disjoint

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Because the rhombus and the circle are disjoint, there is trade in the absence of complexity (the initial allocation is not Pareto efficient)




Maximum complexity I

Suppose j’s maximum subjective belief about s is below i’s minimum subjective belief

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p j (s) < c < p i (s)

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Then, agent i bets on s (buys respective A-D security at price c)

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Agent j bets on s not occurring, selling at price c

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The discontinuous lines denote i’s maximum and minimum subjective beliefs about s1 , s2 and s3

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The shaded area containing the rhombus (resp. circle) consists of all probabilities that agree with i’s (resp. j’s ) maximum and minimum subjective beliefs about s1 , s2 and s3 Spyros Galanis University of Southampton



Maximum complexity I

Suppose j’s maximum subjective belief about s is below i’s minimum subjective belief

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p j (s) < c < p i (s)

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Then, agent i bets on s (buys respective A-D security at price c)

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Agent j bets on s not occurring, selling at price c

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The discontinuous lines denote i’s maximum and minimum subjective beliefs about s1 , s2 and s3

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The shaded area containing the rhombus (resp. circle) consists of all probabilities that agree with i’s (resp. j’s ) maximum and minimum subjective beliefs about s1 , s2 and s3 Spyros Galanis University of Southampton



Trade in the presence of maximum complexity

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Theorem: Under no arbitrage, there is trade in the presence of maximum complexity if and only if an investor’s maximum subjective belief about an event sk is lower than another investor’s minimum subjective belief about sk

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Because the shaded areas intersect at a, there is no trade in the presence of maximum complexity

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Under complexity, subjective beliefs “explode”, unless they are singletons (smooth preferences)




Preferences inconsistent with market freezes due to complexity

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The two types of disagreement are equivalent if subjective beliefs are singletons (indifference curves have no kinks), as in

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Subjective Expected Utility, smooth variational preferences [Maccheroni et al., 2006] (including, as special cases, the mean-variance preferences [Markowitz, 1952], [Tobin, 1958] and the multiplier preferences [Hansen and Sargent, 2001]), smooth ambiguity [Klibanoff et al., 2005] and second-order expected utility [Ergin and Gul, 2009]




Preferences consistent with market freezes due to complexity

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Models where subjective beliefs are non singletons and therefore complexity can impede trade

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Choquet Expected Utility with convex capacity [Schmeidler, 1989], Maxmin Expected Utility [Gilboa and Schmeidler, 1989], variational preferences [Maccheroni et al., 2006], confidence preferences [Chateauneuf and Faro, 2009] and uncertainty averse preferences [Cerreia-Vioglio et al., 2011]




Absence of complexity Maximum complexity

Preliminaries

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One-good economy with I agents

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Finite state space S

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R+ : set of consequences

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F = RS : set of acts

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Act or security f ∈ F pays f (s) if s ∈ S occurs

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{e i }i∈I ∈ RSI ++ is the initial allocation

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Preference relation %i on F is complete, transitive, continuous, strongly monotonic and convex





Subjective beliefs

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Agent i’s subjective beliefs at endowment e i , π i (e i ) = {p ∈ ∆S : Ep g ≥ Ep e i for all g %i e i }, consist of the normals of the hyperplanes supporting the agent’s indifference curve at f

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We can interpret them as the set of prices of the Arrow-Debreu securities for which the agent with endowment e i would have zero net demand





Subjective beliefs s2

g ei k

h

s1





Common beliefs I

p i (s) = min p(s) p∈π i (e i )

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p i (s) = max p(s) p∈π i (e i )

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Let π i (e i ) = {q ∈ ∆S : p i (s) ≤ q(s) ≤ p i (s) for all s ∈} be the set of probability measures that are within i’s minimum and maximum subjective beliefs at e i ∈ RS , for each state s∈S

Definition There is an S-common belief at {e i }i∈I if

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π i (e i ) 6= ∅

i∈I Spyros Galanis University of Southampton




Trade in the absence of complexity

There is a common belief at {e i }i∈I if

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π i (e i ) 6= ∅

i∈I

Theorem There is trade in the absence of complexity (allocation is not Pareto efficient) if and only if there is no common belief





Betting on a state s

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Security f ∈ RS is a bet on state s ∈ S if a, if s 0 = s f (s 0 ) = b, otherwise





Agreeable bet

Tuple {f i }i∈I ∈ RSI is P i I a trade if f =0 i∈I I I

agreeable if e i + f i i e i for all i ∈ I P an arbitrage if f i < 0 i∈I i f is

a bet on some s ∈ S

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a bet if each

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an agreeable bet if it is an agreeable trade and a bet






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There is trade in the presence of maximum complexity if there is an agreeable bet

Theorem Under no arbitrage, there is trade in the presence of maximum complexity if and only if there is no S-common belief






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There is trade in the presence of maximum complexity if there is an agreeable bet

Theorem Under no arbitrage, there is trade in the presence of maximum complexity if and only if there is no S-common belief





No arbitrage

Proposition T Suppose that [q(s), q(s)] ≡ [p i (s), p i (s)] 6= ∅ for all s ∈ S. i∈I P P Then, no arbitrage implies q(s) ≤ 1 ≤ q(s). Conversely, s∈S s∈S P P q(s) ≤ 1 ≤ q(s) implies that there are no agreeable s∈S

s∈S

arbitrage bets





Sketch of the proof, step 1 I

Suppose there is no S-common belief

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Case 1: p j (s) < c < p i (s) for some s ∈ S

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Let f pay 1 − c at s and −c otherwise. Then,

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Ep f = p(s)(1 − c) − (1 − p(s))c = p(s) − c > 0 for all p ∈ π i (e i )

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Ep (−f ) = p(s)(−1 + c) + (1 − p(s))c = c − p(s) > 0 for all p ∈ π j (e j )

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By convexity, Ep (f + e i ) > Ep e i for all p ∈ π i (e i ) implies that k(f + e i ) + (1 − k)e i = kf + e i i e i for small enough k, similarly for j







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Sketch of the proof, step 2

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[p i (s), p i (s)] 6= ∅ for each s ∈ S P P q(s) Show that no arbitrage implies q(s) ≤ 1 ≤ s∈S s∈S P Suppose not, so that q(s) < 1 s∈S P Choose as > q(s) so that as < 1

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Let fs,as pay as − 1 at s and as otherwise

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For all as > p i (s) = q(s), Ep (fs,as + e i ) = as − p(s) + Ep e i > Ep e i for all p ∈ π i (e i )

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By convexity, ks fs,as + e i i e i for small enough ks

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Case 2: [q(s), q(s)] ≡

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i∈I

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s∈S






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Repeating for all s ∈ S and setting k = min ks , tuple s∈S

{kfs,as }s∈S is agreeable I

It is also arbitrage because at each ! s∈S P P P kfs 0 ,as 0 (s) = k as 0 + as − 1 = k as − 1 < 0 s 0 ∈S

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s 0 6=s

s∈S

Contradiction






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P

P q(s) ≤ 1 ≤ q(s) implies that there exists a ∈ (0, 1) s∈S s∈S P P such that a q(s) + (1 − a) q(s) = 1 s∈S

s∈S

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Define p(s) = aq(s) + (1 − a)q(s) ∈ [q(s), q(s)] P Then, p(s) = 1 and p is a S-common belief

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Contradiction

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s∈S





Behavioral implications

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Suppose we can control the complexity of the asset structure but we don’t know the preference model

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If, by increasing the complexity, we eventually observe that trading stops, it must be that the agents’ preferences are from the second set





Complexity and market freeze I

If an event does not change the investors’ beliefs about fundamentals but nevertheless freezes the market, we can interpret it as triggering the investors’ perception that the asset structure is now complex

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[Acharya et al., 2009] describe how a series of events that was triggered by an unexpected decrease of the US house prices in 2006 led to the freezing of the market for asset-backed commercial paper in 2007, right after BNP Paribas announced that it was suspending redemptions from its structured investment vehicles, which were trading these types of securities





Concluding remarks

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We interpret financial complexity as a limit on the number of securities that each investor is able to trade simultaneously

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We characterize trading, in terms of beliefs, in an environment with no complexity and with maximum complexity

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We provide a dichotomy of models with convex preferences, in terms of whether complexity can impede trade

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The model can explain market freezes and provides behavioral implications in terms of what is the preference model, or in terms of whether the asset structure is complex





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