On ratchet-cap pricing problems under the Levy LIBOR model Hsuan Ku - PowerPoint PPT Presentation

On ratchet-cap pricing problems under the Levy LIBOR model Hsuan Ku Liu Department of Mathematics and Information Education National Taipei University of Education

Outline Outline Simple forwards 1 Modelling jump 2 Arbitrage-free dynamics 3 The partial integro-differential equations for pricing ratchet 4 caps

Simple forwards Outline Simple forwards 1 Modelling jump 2 Arbitrage-free dynamics 3 The partial integro-differential equations for pricing ratchet 4 caps

Simple forwards Notations δ > 0fixed. L ( t , T ) : the forward rate for the interval from T to T + δ as of time t ≤ T P ( t , T ) : the time-t price of a ZCB maturity at time T , the forward rate satisfies L ( t , T ) = 1 P ( t , T ) δ ( P ( t , T + δ ) − 1 ) f ( t , T ) : the instantaneous forwards of HJM framework satisfies � T + δ L ( t , T ) = 1 δ ( exp { f ( t , s ) ds } − 1 ) T

Modelling jump Outline Simple forwards 1 Modelling jump 2 Arbitrage-free dynamics 3 The partial integro-differential equations for pricing ratchet 4 caps

Modelling jump The jump Process J ( t ) N ( i ) r t H i ( X ( i ) n , τ ( i ) � � J ( t ) = n ) : the jump process i = 1 n = 1 { ( τ ( i ) n , X ( i ) n ) | n = 1 , 2 , . . . } , i = 1 , 2 , . . . , r : the marked point process N ( i ) is the counting process associated with the i-th marked t process H i , i = 1 , 2 , . . . , r : the jump-size function

Modelling jump The random measure µ ( dx , dt ) A marked point process { ( τ n , X n ) } can be described by a random measure µ on the product of the time axis and the mark space: the measure µ assigns unit mass to each point ( τ n , X n ) . This makes it possible to write N ( i ) � t � ∞ t H i ( X ( i ) n , τ ( i ) � n ) = δ i ( x , s ) µ ( dx , ds ) . 0 0 n = 1 For a marked point process in which the points follow a Poission process and marks are i.i.d. random variable.

Modelling jump The intensity has the property that, for all suitably integrable δ i � t � ∞ � t � ∞ δ i ( x , s ) λ ( i ) ( dx , s ) ds δ i ( x , s ) µ i ( dx , s ) ds − 0 0 0 0 is a martingale in t

Modelling jump The intensity takes the form λ ( i ) ( dx , t ) = λ i f i ( x ) dx For the logarithm of the jump sizes to have an asymmetric double distribution, we take the density of the jump size to be f i ( x ) = 1 x p i η 1 , i e − η 1 , i log ( x ) 1 x ≥ 1 + 1 x p i η 2 , i e η 2 , i log ( x ) 1 0 < x < 1

Arbitrage-free dynamics Outline Simple forwards 1 Modelling jump 2 Arbitrage-free dynamics 3 The partial integro-differential equations for pricing ratchet 4 caps

Arbitrage-free dynamics GOAL Define a model of term structure of simple forwards L ( t , T ) to be arbitrage free if it can be embedded in an arbitrage-free model of instantaneous forwards f ( t , T )

Arbitrage-free dynamics Suppose that a bond price is specified through forward rate, ie. for T ∈ R + � T � � P ( t , T ) = exp − f ( t , s ) ds , t ≤ T t where f ( t , T ) is the forward rates of which the dynamics is given by r � df ( t , T ) = α ( t , T ) dt + σ T ( t , T ) · dW t + � δ i ( t , x , T ) µ i ( dt , dx ) R i = 1 Here, W t is a standard wiener process in R n , µ i is a random jump measure with the compensator λ i ( dt , dx ) = F i ( dx ) dt , i = 1 , 2 , ..., r

Arbitrage-free dynamics For all finite t and t ≤ T � T � T t | α ( u , s ) | dsdu < ∞ , 0 � T � T t | σ ( u , s ) | 2 dsdu < ∞ 0 and � T � T � | δ i ( u , x , s ) | 2 ds ν i ( du , dx ) < ∞ , i = 1 , 2 , ..., r 0 R t

Arbitrage-free dynamics It is convenient to extend the definitions of the coefficients by putting them equal to zero for t > T Put � T A ( t , T ) = − α ( t , s ) ds t � T σ ∗ i ( t , T ) = − σ i ( t , s ) ds , i = 1 , 2 , ..., m t σ ∗ ( t , T ) = ( σ ∗ 1 ( t , T ) , ..., σ ∗ m ( t , T )) � T D i ( t , T , α ) = − δ i ( t , x , s ) ds , i = 1 , 2 , ..., r t

Arbitrage-free dynamics Theorem R ( e D i ( t , x , T ) − 1 ) F i ( dx ) < ∞ . The zero couple � Assume that bond (ZCB) price process P ( t , T ) on [ 0 , T ] has the form �� 2 � σ ∗ ( t , T ) � 2 � d P ( t − , T ) r t + A ( t , T ) + 1 dt + σ ∗ ( t , T ) dw t = P ( t , T ) r � R ( e D i ( t , x , T ) − 1 ) µ i ( dt , dx ) � + � i = 1

Arbitrage-free dynamics Proof Let � T Y t = ln P ( t , T ) = − 0 f ( t , s ) ds � t � t 0 σ ∗ ( u , T ) dW u = ln P ( 0 , T ) + 0 A ( u , T ) du + � t � t + � r � R D i ( u , x , T ) µ i ( du , dx ) + 0 r u du . i = 1 0 Then, applying Ito Lemma to P ( t , T ) = P ( 0 , T ) e Y t , we get the theorem.

Arbitrage-free dynamics Girsanov Theorem Let Γ be an m-dimensional predictable process and Ψ = Ψ( w , t , x ) be a strictly positive measurable function such that for finite t � t � t � || Γ s || 2 ds < ∞ , | Ψ( s , x ) | λ ( s , dx ) ds < ∞ . 0 0 R Define � dL t = L t Γ t dW t + L t − (Ψ( t , x ) − 1 ) { µ ( dt , dx ) − ν ( dt , dx ) } , L 0 = 1 , R and suppose that for all finite t E P [ L t ] = 1 .

Arbitrage-free dynamics Then there exists a probability measure Q on F locally equivalent to P with dQ t = L t dP t such that we have dW t = Γ t dt + dW Q t , where W Q lsi a Q -Wiener process. The point process µ has a Q -intensity, given by λ Q ( t , dx ) = Ψ( t , x ) λ ( t , dx ) .

Arbitrage-free dynamics Martingale measure Under the martingale measure Q , the dynamics of bond prices are follows. � d P ( t − , T ) = P ( t , T ) r t dt + σ ∗ ( t , T ) dw Q t � r R ( e D i ( t , x , T ) − 1 )( µ i − ν Q � � + i )( dt , dx ) i − 1

Arbitrage-free dynamics Proof A measure Q is a martingale measure if and only if the bond dynamics under Q is of the form dP ( t , T ) = r t P ( t , T ) dt + dM Q P , where M Q P is a Q -local martingale. Thus, using Girsonov theorem to dP ( t , T ) and comparing with the above equation, we get the theorem.

Arbitrage-free dynamics Remark The forward rate dynamics, under the martingale measure Q , are follows: df ( t , T ) = − σ ( t , T ) σ ∗ ( t , T ) dt + σ ( t , T ) T · dw Q t r R δ i ( t , x , T )( µ i ( dt , dx ) − ν Q � + � i ( dt , dx )) i = 1

Arbitrage-free dynamics 1 The maturity T is restricted to a finite set of dates 0 = T 0 < T 1 < · · · < T M < T M + 1 . 2 Let L n ( t ) = L ( t , T n ) so that L n is the forward rate for the accrual period [ T n , T n + 1 ] . 3 Let P n ( t ) = P ( t , T n ) denote the price of a ZCB maturing at T n .

Arbitrage-free dynamics For each n = 1 , 2 , . . . , M let γ n ( . ) be a bounded, adapted, R d -valued process. Let H ni , i = 1 , 2 , . . . , r be deterministic functions from [ 0 , ∞ ) to [ − 1 , ∞ ) . The model dL n ( t ) L n ( t ) = α n ( t ) dt + γ n ( t ) dw ( t ) + dJ n ( t ) is arbitrage free if

Arbitrage-free dynamics Theorem n δγ k ( t ) T L k ( t ) α n ( t ) = γ n ( t ) ψ 0 ( t ) + γ n ( t ) � 1 + δ L k ( t ) k =[ t ] r n � ∞ 1 + δ L k ( t − ) ( 1 + δ L k ( t − ))( 1 + H ki ( x )) ψ i ( x , t ) λ ( i ) ( dx , t ) − � H ni ( x ) � 0 i = 1 k =[ t ] for some R d -valued process ψ 0 and strictly positive scalar process ψ i ( x , . ) x ∈ ( 0 , ∞ ) , i = 1 , 2 , . . . , r satisfying � t � t 0 || ψ 0 ( s ) || 2 ds < ∞ and 0 ψ i ( s ) λ ( i ) ( dx , s ) ds < ∞ , i = 1 , 2 , . . . , r

Arbitrage-free dynamics Proof For two maturities T and U with T < U , the forward process is defined by F B ( t , T , U ) = P ( t , T ) P ( t , U ) . Let δ > 0, the δ -LIBOR rates are defined by L ( t , T n ) = 1 δ ( F B ( t , T n , T n + 1 ) − 1 ); that is dF B ( t , T n , T n + 1 ) = δ dL ( t , T n ) . Applying Ito Lemma to the above equation, we get the theorem.

Arbitrage-free dynamics Theorem Under the forward measure P M + 1 , if for each n = 1 , 2 , . . . , M we have dL n ( t ) L n ( t ) = α n ( t ) dt + γ n ( t ) dw n + 1 ( t ) + dJ n ( t ) is arbitrage-free if M δγ n ( t ) γ k ( t ) T L k ( t ) α n ( t ) = � 1 + δ L k ( t ) k = n + 1 r M � ∞ λ ( i ) 1 + δ L k ( t − )( 1 + H ki ( x )) � � − H ni ( x ) M + 1 ( dx , t ) 0 1 + δ L k ( t − ) i = 1 k = n + 1

Arbitrage-free dynamics Proof B ( t , T ) Define Z ( t , T ) = 0 r ( s ) ds ) , the the dynamics � t exp ( dZ ( t , T + δ ) = σ ∗ ( t , T + δ ) dW Q Z ( t , T + δ ) � ∞ 0 ( e D ( t , T + δ ) − 1 )( µ i ( dx , dt ) − λ Q + � r i ( dx , t ) dt ) . i = 1 is martingale. Define the measure P T + δ through the likelihood ratio � dP T + δ � = Z ( t , T + δ ) P ( 0 , T + δ ) . dQ Applying Girsanov’s theorem, the intensity of µ ( i ) ( dx , dt ) is given by λ ( i ) T + δ ( dx , t ) = e D ( t , T + δ ) λ ( i ) Q ( dx , t ) and the process W T + δ = dW Q ( t ) + σ ∗ ( t , T + δ ) is a standard Brownian motion.

Arbitrage-free dynamics In this paper, we assume that each LIBOR is affected by the same rare events, that is the dynamics of LIBOR rate is written as N t dL n ( t ) � L n ( t ) = α n ( t ) dt + γ n ( t ) dw n + 1 ( t ) + H n ( X k , τ k ) k = 1

The partial integro-differential equations for pricing ratchet caps Outline Simple forwards 1 Modelling jump 2 Arbitrage-free dynamics 3 The partial integro-differential equations for pricing ratchet 4 caps