Hedging under arbitrage Johannes Ruf Columbia University, - PowerPoint PPT Presentation

Motivation Notation Hedging (price) Smoothness Change of measure Example Summary Hedging under arbitrage Johannes Ruf Columbia University, Department of Statistics AnStAp10 August 12, 2010

Motivation Notation Hedging (price) Smoothness Change of measure Example Summary Motivation • Usually, there are several trading strategies at one’s disposal to obtain a given wealth at a specified time. • 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 Hedging (price) Smoothness Change of measure Example Summary Two generic examples • 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 Hedging (price) Smoothness Change of measure Example Summary Strict local martingales • A stochastic process X ( · ) is a local martingale if there exists a sequence of stopping times ( τ n ) with lim n →∞ τ n = ∞ such that X τ n ( · ) is a martingale. • Here, in our context, a local martingale is a nonnegative stochastic process X ( · ) which does not have a drift: dX ( t ) = X ( t ) something dW ( t ) . • Strict local martingales (local martingales, which are not martingales) do only appear in continuous time. • Nonnegative local martingales are supermartingales.

Motivation Notation Hedging (price) Smoothness 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 d ( · )) T follow � � K � dS i ( t ) = S i ( t ) µ 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 > d allowed). • The covariance process is defined as K � a i , j ( t , S ( t )) := σ i , k ( t , S ( t )) σ j , k ( t , S ( t )) . k =1 • The underlying filtration is denoted by F = {F ( t ) } 0 ≤ t ≤ T .

Motivation Notation Hedging (price) Smoothness Change of measure Example Summary An important guy: the market price of risk. • A market price of risk is an R K -valued process θ ( · ) satisfying µ ( t , S ( t )) = σ ( t , S ( t )) θ ( t ) . • We assume it exists and � T � θ ( t ) � 2 dt < ∞ . 0 • The market price of risk is not necessarily unique. • We will always use a Markovian version of the form θ ( t , S ( t )). (needs argument!)

Motivation Notation Hedging (price) Smoothness Change of measure Example Summary Related is the stochastic discount factor. • The stochastic discount factor corresponding to θ is denoted by � � t � t � θ T ( u , S ( u )) dW ( u ) − 1 Z θ ( t ) := exp � θ ( u , S ( u )) � 2 du − . 2 0 0 • It has dynamics dZ θ ( t ) = − θ T ( t , S ( t )) Z θ ( t ) dW ( t ) . • If Z θ ( · ) is a martingale, that is, if E [ Z θ ( T )] = 1, then it defines a risk-neutral measure Q with d Q = Z θ ( T ) d P . • Otherwise, Z θ ( · ) is a strict local martingale and classical arbitrage is possible. • From Itˆ o’s rule, we have � � K � Z θ ( t ) S i ( t ) = Z θ ( t ) S i ( t ) ( σ i , k ( t , S ( t )) − θ k ( t , S ( t ))) dW k ( t ) d k =1

Motivation Notation Hedging (price) Smoothness Change of measure Example Summary Everything an investor cares about: how and how much? • We call trading strategy the number of shares held by an investor: η ( t ) = ( η 1 ( t ) , . . . , η d ( t )) T • We assume that η ( · ) is progressively measurable with respect to F and self-financing. • The corresponding wealth process V v ,η ( · ) for an investor with initial wealth V v ,η (0) = v has dynamics � d dV v ,η ( t ) = η i ( t ) dS i ( t ) . i =1 • We restrict ourselves to trading strategies which satisfy V 1 ,η ( t ) ≥ 0

Motivation Notation Hedging (price) Smoothness Change of measure Example Summary The terminal payoff • Let p : R d + → [0 , ∞ ) denote a measurable function. • The investor wants to have the payoff p ( S ( T )) at time T . • For example, p ( s ) = � d • market portfolio: ˜ i =1 s i • money market: p 0 ( s ) = 1 • stock: p 1 ( s ) = s 1 • call: p C ( s ) = ( s 1 − L ) + for some L ∈ R . • We define a candidate for the hedging price as h p ( t , s ) := E t , s � � ˜ Z θ ( T ) p ( S ( T )) , where ˜ Z θ ( T ) = Z θ ( T ) / Z θ ( t ) and S ( t ) = s under the expectation operator E t , s .

Motivation Notation Hedging (price) Smoothness Change of measure Example Summary Prerequisites • We shall call ( t , s ) ∈ [0 , T ] × R d + a point of support for S ( · ) if there exists some ω ∈ Ω such that S ( t , ω ) = s . • We have assumed Markovian stock price dynamics such that S ( t ) is R d -valued, unique and stays in the positive orthant and a square-integrable Markovian market price of risk θ ( t , S ( t )). • We have defined h p ( t , s ) := E t , s � � Z θ ( T ) p ( S ( T )) ˜ , where ˜ Z θ ( T ) = Z θ ( T ) / Z θ ( t ) and S ( t ) = s under the expectation operator E t , s . • In particular, h p ( T , s ) := p ( s ) .

Motivation Notation Hedging (price) Smoothness Change of measure Example Summary A first result: non path-dependent European claims Assume that we have a contingent claim of the form p ( S ( T )) ≥ 0 and that for all points of support ( t , s ) for S ( · ) with t ∈ [0 , T ) we have h p ∈ C 1 , 2 ( U t , s ) for some neighborhood U t , s of ( t , s ). Then, i ( t , s ) := D i h p ( t , s ) and v p := h p (0 , S (0)), we get with η p V v p ,η p ( t ) = h p ( t , S ( t )) . The strategy η p is optimal in the sense that for any ˜ v > 0 and for any strategy ˜ η whose associated wealth process is nonnegative and satisfies V ˜ v , ˜ η ( T ) ≥ p ( S ( T )), we have ˜ v ≥ v p . Furthermore, h p solves the PDE d d � � ∂ t h p ( t , s ) + 1 ∂ s i s j a i , j ( t , s ) D 2 i , j h p ( t , s ) = 0 2 i =1 j =1 at all points of support ( t , s ) for S ( · ) with t ∈ [0 , T ).

Motivation Notation Hedging (price) Smoothness Change of measure Example Summary The proof relies on Itˆ o’s formula. • Define the martingale N p ( · ) as N p ( t ) := E [ Z θ ( T ) p ( S ( T )) |F ( t )] = Z θ ( t ) h p ( t , S ( t )) . • Use a localized version of Itˆ o’s formula to get the dynamics of N p ( · ). Since it is a martingale, its dt term must disappear which yields the PDE. • Then, another application of Itˆ o’s formula yields d � D i h p ( t , S ( t )) dS i ( t ) = dV v p ,η p ( t ) . dh p ( t , S ( t )) = i =1 • This yields directly V v p ,η p ( · ) ≡ h p ( · , S ( · )).

Motivation Notation Hedging (price) Smoothness Change of measure Example Summary Proof (continued) • Next, we prove optimality. • Assume we have some initial wealth ˜ v > 0 and some strategy η with nonnegative associated wealth process such that ˜ V ˜ v , ˜ η ( T ) ≥ p ( S ( T )) is satisfied. • Then, Z θ ( · ) V ˜ v , ˜ η ( · ) is a supermartingale. • This implies v ≥ E [ Z θ ( T ) V ˜ v , ˜ η ( T )] ≥ E [ Z θ ( T ) p ( S ( T ))] ˜ = E [ Z θ ( T ) V v p ,η p ( T )] = v p

Motivation Notation Hedging (price) Smoothness Change of measure Example Summary Non-uniqueness of PDE • Usually, d d � � ∂ t v ( t , s ) + 1 ∂ s i s j a i , j ( t , s ) D 2 i , j v ( t , s ) = 0 2 i =1 j =1 does not have a unique solution. • However, if h p is sufficiently differentiable, it can be characterized as the minimal nonnegative solution of the PDE. • This follows as in the proof of optimality. If ˜ h is another nonnegative solution of the PDE with ˜ h ( T , s ) = p ( s ), then Z θ ( · )˜ h ( · , S ( · )) is a supermartingale.

Motivation Notation Hedging (price) Smoothness Change of measure Example Summary Corollary: Modified put-call parity For any L ∈ R we have the modified put-call parity for the call- and put-options ( S 1 ( T ) − L ) + and ( L − S 1 ( T )) + , respectively, with strike price L : E t , s � Z θ ( T )( L − S 1 ( T )) + � ˜ + h p 1 ( t , s ) = E t , s � Z θ ( T )( S 1 ( T ) − L ) + � + Lh p 0 ( t , s ) , ˜ where p 0 ( · ) ≡ 1 denotes the payoff of one monetary unit and p 1 ( s ) = s 1 the price of the first stock for all s ∈ R d + .

Motivation Notation Hedging (price) Smoothness Change of measure Example Summary A technical definition We shall call a function f : [0 , T ] × R d + → R locally Lipschitz and bounded on R d + if for all s ∈ R d + the function t → f ( t , s ) is right-continuous with left limits and for all M > 0 there exists some C ( M ) < ∞ such that for all t ∈ [0 , T ]. | f ( t , y ) − f ( t , z ) | sup + sup | f ( t , y ) | ≤ C ( M ) . � y − z � 1 1 M ≤� y � , � z �≤ M M ≤� y �≤ M y � = z