Completeness and hedging in a Lvy bond market. Jos M. Corcuera, - PowerPoint PPT Presentation

Completeness and hedging in a Lévy bond market. José M. Corcuera, University of Barcelona Faculty of Mathematics University of Barcelona Workshop on Stochastic Analysis and Finance, Hong Kong, June 30, 2009 J.M. Corcuera (University of Barcelona) Completeness and hedging in a Lévy bond market. 1 / 49

Completeness and hedging in a Lévy bond market. José M. Corcuera, University of Barcelona Faculty of Mathematics University of Barcelona Workshop on Stochastic Analysis and Finance, Hong Kong, June 30, 2009 J.M. Corcuera (University of Barcelona) Completeness and hedging in a Lévy bond market. 2 / 49

Outline The model 1 The forward rates J.M. Corcuera (University of Barcelona) Completeness and hedging in a Lévy bond market. 3 / 49

Outline The model 1 The forward rates Completeness of the market 2 J.M. Corcuera (University of Barcelona) Completeness and hedging in a Lévy bond market. 3 / 49

Outline The model 1 The forward rates Completeness of the market 2 L 2 -completeness 3 Power-Jump Processes Proof Power payoffs Completion with bonds Proof The hedging portfolios Another set of basic bonds Hedgeable claims J.M. Corcuera (University of Barcelona) Completeness and hedging in a Lévy bond market. 3 / 49

Outline The model 1 The forward rates Completeness of the market 2 L 2 -completeness 3 Power-Jump Processes Proof Power payoffs Completion with bonds Proof The hedging portfolios Another set of basic bonds Hedgeable claims References 4 J.M. Corcuera (University of Barcelona) Completeness and hedging in a Lévy bond market. 3 / 49

The model We assume that only the bank account is fixed exogenously and under the historical probability P . There is not any other primary asset in the market. So, any equivalent measure can serve to price derivatives, and a zero coupon bond is a derivative where the underlying is the bank account. Then we have a bank account process B that evolves as � t B t = exp { r ( s ) ds } 0 where r ( t ) , the so-called short-rate interest. We consider a dynamics of the form � t r ( t ) = µ ( t ) + γ ( s , t ) dL s , (1) 0 J.M. Corcuera (University of Barcelona) Completeness and hedging in a Lévy bond market. 4 / 49

The model where L is a non-homogeneous Lévy process, or a process with independent increments and absolutely continuous characteristics ( σ 2 s , υ s , a s ) : L t � t � t � t � = a s ds + σ s dW s + x 1 {| x |≤ 1 } ( J ( dx , ds ) − υ s ( dx ) ds ) 0 0 0 R � + ∆ L s 1 {| ∆ L s | > 1 } s ≤ t � t � t � t � x d � M P = a s ds + σ s dW s + s 0 0 0 | x |≤ 1 � + ∆ L s 1 {| ∆ L s | > 1 } s ≤ t The coefficients µ ( t ) and γ ( s , t ) are assumed to be deterministic and càdlàg. We also assume that the filtration generated by L and r is the same. J.M. Corcuera (University of Barcelona) Completeness and hedging in a Lévy bond market. 5 / 49

The model Fix T > 0 and consider a T -zero-coupon bond . This is a contract that guarantees the holder 1 monetary unit at time T . Write P ( t , T ) , 0 ≤ t ≤ T , for the price of this contract. We know that P ( T , T ) = 1 , take Q equivalent to P (the historical probability) and structure preserving (with respect to L ) and define � T ˜ P ( t , T ) = E Q ( exp {− r ( s ) ds }|F t ) 0 this discounted price, whatever Q , equivalent to P , we choose, does not produce arbitrage. J.M. Corcuera (University of Barcelona) Completeness and hedging in a Lévy bond market. 6 / 49

The model From here we have that ˜ P ( t , T ) � � � � � T = exp − r ( s ) ds | F t E Q 0 � � � � �� T � � � � T � T = exp − µ ( s ) ds × E Q exp − γ ( u , s ) ds dL u | F t 0 0 u � � � �� T � � � T = exp − µ ( s ) ds × E Q exp Γ( T ) u dL u | F t 0 0 where � T Γ( T ) t = − γ ( t , s ) ds . t J.M. Corcuera (University of Barcelona) Completeness and hedging in a Lévy bond market. 7 / 49

The model � · Under Q , 0 Γ( T ) u dL u is still a process with independent increments so � �� T � � exp Γ( T ) u dL u | F t E Q 0 � �� T �� �� t � = exp Γ( T ) u dL u exp Γ( T ) u dL u , E Q 0 t J.M. Corcuera (University of Barcelona) Completeness and hedging in a Lévy bond market. 8 / 49

The model Therefore �� t � ˜ P ( t , T ) ∝ exp Γ( T ) u dL u , 0 and we can write �� t � exp 0 Γ( T ) u dL u ˜ � �� t �� . P ( t , T ) = P ( 0 , T ) (2) exp 0 Γ( T ) u dL u E Q J.M. Corcuera (University of Barcelona) Completeness and hedging in a Lévy bond market. 9 / 49

The model The forward rates The previous dynamics of ˜ P ( t , T ) for all maturities, T > t , can be described in terms of the instantaneous forward rates, by using the definition � t � T ˜ P ( t , T ) = exp {− r ( s ) ds − f ( t , s ) ds } . 0 t Then − ∂ T log ˜ f ( t , T ) = P ( t , T ) � t = f ( 0 , T ) + A ( T ) t + γ ( s , T ) dL s 0 where � �� t �� ∂ T E Q exp 0 Γ( T ) u dL u A ( T ) t = � �� t �� E Q exp 0 Γ( T ) u dL u J.M. Corcuera (University of Barcelona) Completeness and hedging in a Lévy bond market. 10 / 49

The model The forward rates In Eberlein, Jacod and Raible (2005), the authors assume that the forward rates are given and then study the arbitrage and completeness problems. In fact they start by assuming that (under P ) � t � t f ( t , T ) = f ( 0 , T ) + α ( s , T ) d s + γ ( s , T ) d L s , 0 0 and they look for martingale measures. They obtain that their � t existence implies some constrains for the process 0 α ( s , T ) d s . This was already observed by Heath, Jarrow and Morton in 1992 in the case when L is continuous. The constraint for the existence of ”structure preserving” martingale measures is then � t α ( s , T ) d s = A ( T ) t . 0 for some Q structure preserving. J.M. Corcuera (University of Barcelona) Completeness and hedging in a Lévy bond market. 11 / 49

The model The forward rates We have considered that Q is a given structure preserving equivalent measure, we construct the bond market with this Q and deduce the form of the forward rates which are compatible with this prices. So the relevant problem is the completeness of the market. We take Q and we check if this market is complete by trying to demonstrate that any (squared integrable) contingent claim can be replicated by a self-financing portfolio. J.M. Corcuera (University of Barcelona) Completeness and hedging in a Lévy bond market. 12 / 49

Completeness of the market According with Eberlein, Jacod and Raible (2005) ([EJR05]) completeness is important by two reasons: Completeness amounts to the fact that any claim can be priced in a unique way: by taking the expectation of the discounted value of the claim with respect to the (unique) equivalent martingale measure. This means, in our context, that if we choose different, structure preserving equivalent measures, Q we obtain different bond prices. Completeness is also related with hedging and is in fact ”equivalent” to the property that any square integrable (discounted) contingent claim Y can be written as � T � H j s d ˜ Y = E ( Y ) + P ( s , T j ) , (3) 0 T j ∈ J and this is equivalent to the martingale representation property � t � H j s d ˜ E ( Y |F t ) = E ( Y ) + P ( s , T j ) . 0 T j ∈ J J.M. Corcuera (University of Barcelona) Completeness and hedging in a Lévy bond market. 13 / 49

Completeness of the market Instead of looking if a system of bond prices implies a unique Q we can use the martingale representation theorems for studying if and how we can replicate any contingent claim. We have seen that P ( t , T ) = P ( 0 , T ) exp { ¯ ˜ Z t } � ¯ � where Z t 0 ≤ t < T is a Q non-homogeneous Lévy process (we shall omit the dependency on T ) with characteristic triplet given say � ¯ � c 2 t , ¯ ν t , ¯ γ t . Note that ¯ c t = Γ( T ) t σ t B ¯ ν t ( B ) = υ t ( ) Γ( T ) t J.M. Corcuera (University of Barcelona) Completeness and hedging in a Lévy bond market. 14 / 49

Completeness of the market Then, by Itô’s formula and the Lévy-Itô representation of ¯ Z d ˜ P ( t , T ) � c 2 γ t + ¯ � � e x − 1 − x 1 {| x | < 1 ˜ c t d ¯ t ν t ( dx )) d t + ¯ = P ( t − , T )((¯ 2 + ¯ W t R � ( e x − 1 ) d � ¯ + M t ) R � � � ( e x − 1 ) d � ˜ c t d ¯ ¯ ¯ = P ( t − , T ) W t + M t R � � � � � e Γ( T ) t x − 1 ˜ Γ( T ) t σ t d ¯ d � M Q = P ( t − , T ) W t + t R � � � � � e Γ( T ) t x − 1 ˜ d � M P = P ( t − , T ) Γ( T ) t σ t d W t + t R � � � � � e Γ( T ) t x − 1 +˜ P ( t − , T ) Γ( T ) t σ t G t + H t ( x ) d υ t d t R J.M. Corcuera (University of Barcelona) Completeness and hedging in a Lévy bond market. 15 / 49