Motivation Notation Arbitrage Optimal strategies Change of measure Example Summary Optimal trading strategies under arbitrage Johannes Ruf Columbia University, Department of Statistics The Third Western Conference in Mathematical Finance November 14, 2009
Motivation Notation Arbitrage Optimal strategies Change of measure Example Summary How should an investor trade and how much capital does she need? • Imagine an investor who wants to hold the stock S i with price S i (0) of a company in a year. • Surely, she could just buy the stock today for a price S i (0). • This might not be an “optimal strategy”, even under a classical no-arbitrage situation (“no free lunch with vanishing risk”). • There can be other “strategies” which require less initial capital than S i (0) but enable her to hold the stock after one year. • But how much initial capital does she need at least and how should she trade?
Motivation Notation Arbitrage Optimal strategies Change of measure Example Summary Two examples where the stock price does not reflect the hedging price. • Reciprocal of the three-dimensional Bessel process (NFLVR): d ˜ S ( t ) = − ˜ S 2 ( t ) dW ( t ) • Three-dimensional Bessel process: dS ( t ) = 1 S ( t ) dt + dW ( t )
Motivation Notation Arbitrage Optimal strategies Change of measure Example Summary How can a portfolio optimally been hedged? • Optimality in the sense of smallest initial capital. • Solved by Fernholz D. & Karatzas I. for the market portfolio. • “Hedging price” characterized via a PDE which can allow for non-unique solutions. • Change of measure to simplify computations. • In Markovian framework. • No martingale representation theorem used.
Motivation Notation Arbitrage Optimal strategies Change of measure Example Summary We assume a Markovian market model. • Our time is finite: T < ∞ . Interest rates are zero. • The stocks S ( · ) = ( S 1 ( · ) , . . . , S n ( · )) T follow � � � K dS i ( t ) = S i ( t ) β i ( t , S ( t )) dt + σ i , k ( t , S ( t )) dW k ( t ) k =1 with some measurability and integrability conditions. • → Markovian • but not necessarily complete ( K > n allowed). • The covariance process is defined as � K α i , j ( t , S ( t )) := σ i , k ( t , S ( t )) σ j , k ( t , S ( t )) k =1
Motivation Notation Arbitrage Optimal strategies Change of measure Example Summary Two important guys: the market price of risk and its close relative, the stochastic discount factor. • The market price of risk is the function ν satisfying � � �� T �� � ν � 2 dt < ∞ { β = σν ∀ t ∈ [0; T ] } = 1 . P 0 • The market price of risk is not necessarily unique. • Related is the stochastic discount factor � � � t � t ν T ( u , S ( u )) dW ( u ) − 1 Z ν ( t ) := exp � ν ( u , S ( u )) � 2 du − 2 0 0 with dynamics dZ ν ( t ) = − ν T ( t , S ( t )) Z ν ( t ) dW ( t ) • If Z ν is a strict local martingale, arbitrage is possible.
Motivation Notation Arbitrage Optimal strategies Change of measure Example Summary Everything an investor cares about: how and how much? • We focus on Markovian trading strategies π ( t , S ( t )). • Which percentage of the total wealth is invested in the single stocks depends on time and current market situation. • The corresponding wealth process V v ,π follows � n dV v ,π ( t ) π i ( t , S ( t )) dS i ( t ) V v ,π ( t ) = S i ( t ) i =1 and is usually not Markovian. • We are ready to define the π -specific price of risk as ν π ( t , s ) := ν ( t , s ) − σ T ( t , s ) π ( t , s ) . • Then K � d ( V v ,π ( t ) Z ν ( t )) ν π = − k ( t , S ( t )) dW k ( t ) . V v ,π ( t ) Z ν ( t ) k =1
Motivation Notation Arbitrage Optimal strategies Change of measure Example Summary We explicitly allow for arbitrage. It can be shown that excluding arbitrage a priori leads to market models which cannot capture important properties of the real markets very well. (diversity, behavior of market weights) • We call a strategy ρ with P ( V ρ ( T ) ≥ V π ( T )) = 1 and P ( V ρ ( T ) > V π ( T )) > 0 relative arbitrage opportunity with respect to the strategy π . • We call ρ a classical arbitrage opportunity if π invests fully in the bond, that is, if π ( t , s ) ≡ 0 for all ( t , s ) ∈ [0; T ] × R n + .
Motivation Notation Arbitrage Optimal strategies Change of measure Example Summary But we exclude some forms of arbitrage. • Remember: We have assumed that there exists some ν which maps the volatility into the drift, that is σ ( · , · ) ν ( · , · ) = β ( · , · ). • It can be shown that this assumption excludes “unbounded profit with bounded risk”. • Thus “making (a considerable) something out of almost nothing” is not possible. • However, it is still possible to “certainly make something more out of something”.
Motivation Notation Arbitrage Optimal strategies Change of measure Example Summary A new candidate for a hedging price is a risk-adjusted expectation. • Let us define � � V π ( T ) Z ν ( T ) U π,ν ( t , s ) := E T − t , s . V π ( T − t ) Z ν ( T − t ) • U π,ν is non-random. • Not clear at this point how U π,ν depends on market price of risk ν . • We assume that U π,ν satisfies the PDE � n � n ∂ U π,ν ( t , s ) = 1 s i s j α i , j ( T − t , s ) D 2 i , j U π,ν ( t , s ) ∂ t 2 i =1 j =1 � n � n s i α i , j ( T − t , s ) π j ( T − t , s ) D i U π,ν ( t , s ) . + i =1 j =1
Motivation Notation Arbitrage Optimal strategies Change of measure Example Summary Theorem: How does the optimal strategy look like? π ν such that For any strategy π there here exists a new strategy � π ν with initial wealth the corresponding wealth process V ˆ v , � v := U π,ν ( T , S 0 ) ≤ 1 has the same value as V π at time T , that is ˆ V ˆ v , � π ν ( T ) = V π ( T ) . π ν takes the form � π ν i ( t , s ) = s i D i log U π,ν ( T − t , s ) + π i ( t , s ) � and is optimal: There exists no strategy ρ such that V ˜ v ,ρ ( T ) ≥ V π ( T ) = V ˆ v , � π ν ( T ) for some ˜ v < ˆ v .
Motivation Notation Arbitrage Optimal strategies Change of measure Example Summary The proof relies on Itˆ o’s formula. • Define the martingale N π,ν as N π,ν ( t ) := E s [ V π ( T ) Z ν ( T ) |F ( t )] and compute its dynamics as � n � � K � dN π,ν ( t ) D i U π,ν − ν π dW k ( t ) + C π,ν dt , N π,ν ( t ) = S i ( t ) σ i , k k U π,ν k =1 i =1 where C π,ν ( t , s ) disappears because of the assumption that U π,ν satisfies a PDE. • But thus, K � dN π,ν ( t ) ν � π ν N π,ν ( t ) = − k ( t , S ( t )) dW k ( t ) , k =1 which are the dynamics of V ˆ v , � π ν ( t ) Z ν ( t ).
Motivation Notation Arbitrage Optimal strategies Change of measure Example Summary We can change the measure to compute U π,ν • There exists not always an equivalent local martingale measure. • However, after making some technical assumptions on the probability space and the filtration we can construct a new measure Q ν which corresponds to a “removal of the stock price drift”. • Based on the work of F¨ ollmer and Meyer and along the lines of Delbaen and Schachermayer.
Motivation Notation Arbitrage Optimal strategies Change of measure Example Summary Theorem: Under a new measure Q ν the drifts disappear. There exists a measure Q ν such that P ≪ Q ν . More precisely, for all nonnegative F ( T )-measurable random variables Y we have E P [ Z ν ( T ) Y ] = E Q ν � � Y 1 � � . 1 Z ν ( T ) > 0 Under this measure Q ν , the stock price processes follow K � σ i , k ( t , S ( t )) d � dS i ( t ) = S i ( t ) W k ( t ) k =1 up to time τ ν := inf { t ∈ [0; T ] : 1 / Z ν ( t ) = 0 } . Here, � t ∧ τ ν � W k ( t ∧ τ ν ) := W k ( t ∧ τ ν ) + ν k ( u , S ( u )) du 0 is a K -dimensional Q ν -Brownian motion stopped at time τ ν .
Motivation Notation Arbitrage Optimal strategies Change of measure Example Summary What happens in between time 0 and time T : Bayes’ rule. For all nonnegative F ( T )-measurable random variables Y the representation � E Q ν � � 1 � F ( t ) = E P [ Z ν ( T ) Y |F ( t )] Y 1 { 1 / Z ν ( T ) > 0 } Z ν ( t ) 1 { 1 / Z ν ( t ) > 0 } holds Q ν -almost surely (and thus P -almost surely) for all t ∈ [0; T ].
Motivation Notation Arbitrage Optimal strategies Change of measure Example Summary The class of Bessel processes with drift provides interesting arbitrage opportunities. • We begin with defining an auxiliary stochastic process X as � � 1 dX ( t ) = X ( t ) − c dt + dW ( t ) with W denoting a Brownian motion and c ≥ 0 a constant. • X ( t ) is for all t ≥ 0 strictly positive since X is a Bessel process under an equivalent measure. • The stock price process is now defined via � � 1 1 1 dS ( t ) = X ( t ) dt + dW ( t ) = S ( t ) S 2 ( t ) − S ( t ) ct dt + S ( t ) dW ( t ) with S (0) = X (0) > 0.
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