Supply and Shorting in Speculative Markets Marcel Nutz Columbia University with Johannes Muhle-Karbe (Part I) and José Scheinkman (Parts II–III) May 2017 Marcel Nutz (Columbia) Supply and Shorting in Speculative Markets 1 / 19
Outline Part I: Resale Option 1 Part II: Supply 2 Part III: Short-Selling 3
Static Model Consider Agents i ∈ { 1 , 2 , . . . , n } Using distributions Q i for the state X ( t ) Trading an asset with a single payoff f ( X ( T )) at time T The asset cannot be shorted and is in supply s > 0 Static case: Suppose the agents trade only once, at time t = 0 Equilibrium: Determine an equilibrium price p for f and portfolios q i ∈ R + such that q i maximizes q ( E i [ f ( X ( T ))] − p ) over q ≥ 0, for all i and the market clears: � i q i = s Marcel Nutz (Columbia) Supply and Shorting in Speculative Markets 2 / 19
Static Equilibrium Solution: The most optimistic agent determines the price (Miller ’77) , p = max E i [ f ( X ( T ))] i Let i ∗ ∈ { 1 , 2 , . . . , n } be the maximizer With portfolios q i ∗ = s and q i = 0 for i � = i ∗ , this in an equilibrium It is unique (modulo having several maximizers) Note: At price p , the optimist is invariant and will accept any portfolio All other agents want to have q i = 0 Price not affected by supply Marcel Nutz (Columbia) Supply and Shorting in Speculative Markets 3 / 19
Preview: The Resale Option (Harrison and Kreps ’78) When there are several trading dates, the relatively most optimistic agent depends on date and state Option to resell the asset to another agent at a later time Adds to the static price: speculative bubble Scheinkman and Xiong ’03, ’04 Marcel Nutz (Columbia) Supply and Shorting in Speculative Markets 4 / 19
A Continuous-Time Model Asset can be traded on [ 0 , T ] . Agents: Risk-neutral agents i ∈ { 1 , . . . , n } using models Q i Here: agent i uses a local vol model Q i for X , dX ( t ) = σ i ( t , X ( t )) dW i ( t ) , X ( 0 ) = x Equilibrium: Find a price process P ( t ) with P ( T ) = f ( X ( T )) Q i -a.s. Agents choose portfolio processes Φ � T such as to optimize expected P&L: E i [ 0 Φ( t ) dP ( t )] Market clearing � i Φ i ( t ) = s Marcel Nutz (Columbia) Supply and Shorting in Speculative Markets 5 / 19
Existence Theorem: There exists a unique equilibrium price P ( t ) = v ( t , X ( t )) , and v is the solution of 1 2 σ 2 v t ( t , x ) + sup i ( t , x ) v xx ( t , x ) = 0 , v ( T , · ) = f . i ∈{ 1 ,..., n } The optimal portfolios Φ i ( t ) = φ i ( t , X ( t )) are given by � s , if i is the maximizer at ( t , x ) φ i ( t , x ) = 0 , else Derivative held by the locally most optimistic agent at any time Agents trade as this role changes Marcel Nutz (Columbia) Supply and Shorting in Speculative Markets 6 / 19
Control Problem and Speculative Bubble v is also characterized as the value function E [ f ( X t , x v ( t , x ) = sup θ ( T ))] θ ∈ Θ ◮ Θ is the set of { 1 , . . . , n } -valued, progressive processes ◮ X t , x θ ( r ) , r ∈ [ t , T ] is the solution of dX ( r ) = σ θ ( r ) ( r , X ( r )) dW ( t ) , X ( t ) = x . Bubble: The control problem (or comparison) shows that P ( 0 ) ≥ max E i [ f ( X ( T ))] i Thus, the price exceeds the static equilibrium This “speculative bubble” can be attributed to the resale option Marcel Nutz (Columbia) Supply and Shorting in Speculative Markets 7 / 19
Remarks Note: Price is again independent of supply No-shorting was essential Comparison with UVM: v is the uncertain volatility (UV) or G -expectation price corresponding to the interval � � � � = inf i σ i ( t , x ) , sup i σ i ( t , x ) σ, σ → In our model, the UV price arises as an equilibrium price of risk-neutral agents, instead of a superhedging price Marcel Nutz (Columbia) Supply and Shorting in Speculative Markets 8 / 19
Outline Part I: Resale Option 1 Part II: Supply 2 Part III: Short-Selling 3
Model Supply: Supply should diminish price, not reflected in the model of Part I Need (risk) aversion against large positions Add Cost-of-Carry: For holding a position y = Φ( t ) at time t , agents must pay an instantaneous cost � 1 2 α + y 2 , y ≥ 0 c ( y ) = ∞ , y < 0 Equilibrium: Agents optimize expected P&L − cost: �� T � T � Φ( t ) dP ( t ) − c (Φ( t )) dt E i 0 0 Marcel Nutz (Columbia) Supply and Shorting in Speculative Markets 9 / 19
Existence Theorem: • There exists a unique equilibrium price P ( t ) = v ( t , X ( t )) , and v is the solution of � � � 1 1 2 σ 2 s v t + sup i v xx − = 0 | J | | J | α + ∅� = J ⊆{ 1 ,..., n } i ∈ J • The optimal portfolios Φ i ( t ) = φ i ( t , X ( t )) are unique and given by � + α + L i v ( t , x ) � φ i ( t , x ) = where L i v ( t , x ) = ∂ t v ( t , x ) + 1 2 σ 2 i ∂ xx v ( t , x ) s Supply: enters as a running cost, κ = | J | α + Marcel Nutz (Columbia) Supply and Shorting in Speculative Markets 10 / 19
Delay Effect Again, one can consider a static version of the equilibrium: price is � � � 1 sT p = max E i [ f ( X ( T ))] − . | J | | J | α + ∅� = J ⊆{ 1 ,..., n } i ∈ J The resale option is still present and increases the dynamic price Novel: Delay Effect If many agents expect to increase positions over time, they may anticipate the increase in the static case The resulting demand pressure raises the static price This effect may dominate, causing a “negative bubble” Marcel Nutz (Columbia) Supply and Shorting in Speculative Markets 11 / 19
Delay Effect Again, one can consider a static version of the equilibrium: price is � � � 1 sT p = max E i [ f ( X ( T ))] − . | J | | J | α + ∅� = J ⊆{ 1 ,..., n } i ∈ J The resale option is still present and increases the dynamic price Novel: Delay Effect If many agents expect to increase positions over time, they may anticipate the increase in the static case The resulting demand pressure raises the static price This effect may dominate, causing a “negative bubble” Marcel Nutz (Columbia) Supply and Shorting in Speculative Markets 11 / 19
Outline Part I: Resale Option 1 Part II: Supply 2 Part III: Short-Selling 3
Short-Selling In securities markets, shorting is often possible, though at a cost Not modeled in the existing literature Asymmetric Cost-of-Carry: For holding a position y = Φ( t ) at time t , instantaneous cost � 1 2 α + y 2 , y ≥ 0 c ( y ) = 2 α − y 2 , 1 y < 0 Short is more costly than long: α − ≤ α + Marcel Nutz (Columbia) Supply and Shorting in Speculative Markets 12 / 19
Existence Theorem: • There exists a unique equilibrium price P ( t ) = v ( t , X ( t )) , and v is the solution of � � 1 2 Σ 2 v t ( t , x ) + sup I ( t , x ) v xx ( t , x ) − κ I ( t , x ) = 0 , v ( T , · ) = f , I ⊆{ 1 ,..., n } where the coefficients are defined as s ( t , x ) κ I ( t , x ) = , | I | α − + | I c | α + α − � α + � Σ 2 σ 2 σ 2 I ( t , x ) = i ( t , x ) + i ( t , x ) | I | α − + | I c | α + | I | α − + | I c | α + i ∈ I i ∈ I c • The optimal portfolios Φ i ( t ) = φ i ( t , X ( t )) are unique and given by φ i ( t , x ) = α sign ( L i v ( t , x )) L i v ( t , x ) , L i v ( t , x ) = ∂ t v ( t , x ) + 1 2 σ 2 i ∂ xx v ( t , x ) . Marcel Nutz (Columbia) Supply and Shorting in Speculative Markets 13 / 19
Control Representation HJB equation of the control problem � T � � f ( X t , x κ I ( r ) ( r , X t , x v ( t , x ) = sup I ( T )) − I ( r )) dr E I∈ Θ 0 ◮ Θ is the set of 2 { 1 ,..., n } -valued, progressive processes ◮ X t , x I ( r ) , r ∈ [ t , T ] is the solution of dX ( r ) = Σ I ( r ) ( r , X ( r )) dW ( t ) , X ( t ) = x . Interpretation? Marcel Nutz (Columbia) Supply and Shorting in Speculative Markets 14 / 19
A Principal Agent Problem At each state ( t , x ) , principal assigns a cost coefficient α i ∈ { α − , α + } to every agent i ∈ { 1 , . . . , n } This assignment will play the role of a contract (Second Best) With these coefficients given, agents maximize �� T � T � Φ( t ) dP ( t ) − c i ( t , X ( t ) , Φ( t )) dt E i 0 0 where c i ( t , x , y ) = α i ( t , x ) y 2 irrespectively of y being long or short. An assignment can be summarized as a set I ( t , x ) = { i ∈ { 1 , . . . , n } : α i ( t , x ) = α − } . I c = { agents with α + } I.e., I = { agents with α − } , Marcel Nutz (Columbia) Supply and Shorting in Speculative Markets 15 / 19
Principal Agent Problem: Solution Theorem: (i) For any assignment I ( t ) = I ( t , X ( t )) of the principal, there exists a unique equilibrium price P I ( t ) = v I ( t , X ( t )) , and � T � � f ( X t , x κ I ( r ) ( r , X t , x v I ( t , x ) = E I ( T )) − I ( r )) dr 0 (ii) If the principal’s aim is to maximize the price, the optimal value is our previous equilibrium price v ( t , x ) the optimal contract assigns, in equilibrium, α − to short positions and α + to long positions → Interpretation for Σ I , κ I in our PDE for v ( t , x ) Marcel Nutz (Columbia) Supply and Shorting in Speculative Markets 16 / 19
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