ON THE ESSCHER TRANSFORM AND OTHER EQUIVALENT MARTINGALE MEASURES - PowerPoint PPT Presentation

ON THE ESSCHER TRANSFORM AND OTHER EQUIVALENT MARTINGALE MEASURES FOR THE BNS STOCHASTIC VOLATILITY MODELS WITH JUMPS Friedrich HUBALEK, TUWIEN Carlo SGARRA, POLITECNICO di MILANO SPECIAL SEMESTER ON STOCHASTICS WITH EMPHASIS ON FINANCE, CON- CLUDING WORKSHOP – DECEMBER 3 2008 1

PLAN OF THE TALK • THE BNS MODEL • THE ESSCHER TRANSFORMS FOR THE BNS MODEL • THE ”STRUCTURE PRESERVING” MARTINGALE MEA- SURES • THE MINIMAL ENTROPY MARTINGALE MEASURE • THE MINIMAL MARTINGALE MEASURE AND THE ”NO LEVERAGE” CASE 2

THE BNS MODEL Given a probability space (Ω , , P ) carrying a standard Brown- ian motion W and an independent increasing pure jump L´ evy process Z , We assume that the discounted stock price is given by S t = S 0 e X t , where S 0 > 0 is a constant, logarithmic returns satisfy � dX t = ( µ + βV t − ) dt + V t − dW t + ρdZ λt , dV t = − λV t − dt + dZ λt 3

The parameter range is µ ∈ , β ∈ , ρ ≤ 0, λ > 0. We denote the cumulant function and the L´ evy measure of Z by k ( z ) resp. U ( dx ). Since Z is increasing we have � ∞ ( e zx − 1) U ( dx ) . k ( z ) = 0 We will work with the usual natural filtration generated by the pair ( W t , Z λt ). 4

The semimartingale characteristics of X with respect to the zero truncation function are ( B, C, ν ), which satisfy dB t = b t dt , dC t = c t dt, ν ( dt, dx ) = F ( t, dx ) dt , b t = µ + βV t − , c t = V t − , F ( t, dx )= λU ρ ( dx ) . U ρ ( dx ) = U ( ρ − 1 dx ) 5

THE ESSCHER MARTINGALE TRANSFORM FOR THE EXPONENTIAL PROCESS From the characteristics computed above, provided that θ · X is exponentially special, the modified Laplace cumulant process of X in θ ∈ L ( X ) is given by � t K X ( θ ) t = κ X ( θ ) s ds, 0 ˜ Where κ X ( θ ) t = b t θ t + 1 2 c t θ 2 ˜ t + λk ( ρθ t ) . The general result by Kallsen and Shiryaev specializes in the following form: 6

THEOREM 1: Suppose there is θ ♯ ∈ L ( X ), such that θ ♯ · X is exponentially special, K X ( θ ♯ + 1) − K X ( θ ♯ ) = 0 , and G ♯ N ♯ ) t , with t = E ( ˜ � t � t � N ♯ 0 ψ ♯ ( Y ♯ ( s, x ) − 1)( µ X − ν )( dx, ds ) , ˜ t = s dW s + 0 Y ♯ ( t, x ) = e θ ♯ ψ ♯ t = θ ♯ � t ρx V t − , t defines a martingale ( G ♯ t ) 0 ≤ t ≤ T . 7

Then dP ♯ N ♯ ) T defines a probability measure P ♯ ∼ P on dP = E ( ˜ F T . The process ( X t ) 0 ≤ t ≤ T is a semimartingale under P ♯ and its semimartingale characteristics with respect to the zero trun- cation function are ( B ♯ , C ♯ , ν ♯ ) which are given by dB ♯ t = b ♯ t dt, dC ♯ t = c ♯ t dt, ν ♯ ( dt, dx ) = F ♯ ( t, dx ) dt b ♯ t = µ + ( β + θ ♯ c ♯ t ) V t − , t = V t − , F ♯ ( t, dx ) = Y ♯ ( t, x ) λU ρ ( dx ) . 8

The measure P ♯ is then called the Esscher martingale trans- form for the exponential process e X . If there is no solution with the required properties, we say the Esscher martingale transform for the exponential process does not exist. Now we give sufficient conditions, that the solution θ ♯ ex- ists, which is then of the form θ ♯ t = φ ♯ ( V t − ) for some Borel function φ ♯ , and G ♯ is a proper martingale and thus a density process. PROPOSITION 1: Let ξ 1 = sup { ξ ≥ 0 : E [ e ξZ 1 ] < ∞} , ℓ 0 = θ>ξ 1 /ρ [ k ( ρ ( θ + 1)) − k ( ρθ )] . inf 9

If one of the four conditions • ξ 1 = + ∞ , or • ξ 1 < + ∞ and ℓ 0 = −∞ , • ξ 1 < + ∞ and ℓ 0 > −∞ , β +1 / 2+ ξ 1 /ρ = 0, and µ + λℓ 0 ≤ 0, or • ξ 1 < + ∞ and ℓ 0 > −∞ , β + 1 / 2 + ξ 1 /ρ < 0, and V 0 e − λT ≥ µ + λℓ 0 − β +1 / 2+ ξ 1 /ρ , holds, then there is a measurable function φ , such that ϑ ♯ t = φ ( V t − ) is a solution to the given equation.

PROPOSITION 2: Suppose θ ♯ is a solution for the equation before. Let � t N ♯ 0 θ ♯ s dX s − K X ( θ ♯ ) t , t = t = e N ♯ and G ♯ t . If E [ Z 1 e ρ Θ ♯ 2 (Θ ♯ 1 1 ) 2 Z 1 ] < ∞ 0 Z 1 ] < ∞ , E [ e where 10

� ( µ + λk ( ρ )) + � Θ ♯ + ˜ 0 = − β , V 0 e − λT + � ( µ + λk ( ρ )) + � Θ ♯ + ˜ = max { β , 1 V 0 e − λT + � � − ( µ + λk ( ρ )) − + ˜ β } V 0 e − λT − then ( G ♯ t ) 0 ≤ t ≤ T is a martingale.

THE ESSCHER MARTINGALE TRANSFORM FOR THE LINEAR PROCESS It follows from the characteristics of X given above that the semimartingale characteristics of ˜ X with respect to the zero truncation function are given by (˜ B, ˜ C, ˜ ν ) which satisfy B t = ˜ d ˜ b t dt, d ˜ ν ( dt, dx ) = ˜ C t = ˜ c t dt, ˜ F ( t, dx ) dt ˜ b t = µ + ˜ F ( t, dx ) = λ ˜ ˜ βV t − ˜ c t = V t − U ρ ( dx ) , U ρ = U ◦ g − 1 with ˜ β = β + 1 / 2 , and ˜ is the image measure of ρ under the mapping g ρ ( x ) = e ρx − 1 . U 11

It is convenient to introduce the corresponding cumulant func- tion � ∞ ( e z ( e ρx − 1) − 1) U ( dx ) . ˜ k ρ ( z ) = 0 REMARK: In the following we must be careful not to con- K X of X and the modified fuse the Laplace cumulant process ˜ Laplace cumulant process K ˜ X of ˜ X 12

˜ The modified Laplace cumulant process of X in θ and its derivative are then given by � t � t K ˜ κ ˜ X ( θ ) s ds, DK ˜ κ ˜ X ( θ ) t = X ( θ ) = X ( θ ) s ds, 0 ˜ 0 D ˜ b t θ t + 1 κ ˜ c t θ 2 X ( θ ) t = ˜ t + λ ˜ ˜ 2˜ k ρ ( θ t ) , κ ˜ X ( θ ) t = ˜ k ′ c t θ t + λ ˜ D ˜ b t + ˜ ρ ( θ t ) . 13

THEOREM 2: Suppose there is θ ∗ ∈ L ( ˜ X ) , such that θ ∗ · ˜ X is exponentially special, DK ˜ X (˜ θ ∗ ) t = 0 , N ∗ ) t and suppose G t = E ( ˜ with � t � t � N ∗ 0 ψ ∗ ( Y ∗ ( s, x ) − 1)( µ X − ν )( dx, ds ) , ˜ t = s dW s + 0 Y ∗ ( t, x ) = e θ ∗ t ( e x − 1) � ψ ∗ t = θ ∗ V t − , t defines a martingale ( G ∗ t ) 0 ≤ t ≤ T . 14

Then dP ∗ N ∗ ) T defines a probability measure P ∗ ∼ P dP = E ( ˜ on F T . is a semimartingale under P ∗ The process ( X t ) 0 ≤ t ≤ T with ( B ∗ , C ∗ , ν ∗ ) semimartingale characteristics given by dB ∗ t = b ∗ t dt , dC ∗ t = c ∗ t dt , ν ∗ ( dt, dx ) = F ∗ ( t, dx ) dt , b ∗ t = µ + ( β + θ ∗ c ∗ t ) V t − , t = V t − , F ∗ ( t, dx ) = Y ∗ ( t, x ) λU ρ ( dx ) . 15

The measure P ∗ is then called the Esscher martingale trans- form for the linear process ˜ X . If there is no solution with the required properties, we say the Esscher martingale transform for the linear process does not exist. There exists always a measurable function φ ∗ , such that ϑ ∗ t = φ ∗ ( V t − ) is a solution to eq. given before, and sufficient con- ditions are given that G ∗ is a proper martingale and thus a density process. 16

PROPOSITION 3: Let � t X s − K ˜ N ∗ 0 θ ∗ X ( θ ∗ ) t , s d ˜ t = t = e N ∗ t , where θ ∗ is as above. If and G ∗ 1 2 (Θ ∗ 1 ) 2 Z 1 ] < ∞ E [ e with � ( µ + λk ( ρ )) + � Θ ∗ + ˜ 1 = max { β , V 0 e − λT + � � − ( µ + λk ( ρ )) − + ˜ β } V 0 e − λT − then ( G ∗ t ) t ∈ [0 ,T ] is a martingale. 17

EXAMPLES THE POISSON TOY EXAMPLE This model is used for illustrative purposes, since all calcula- tions are explicitly possible. EXPONENTIAL ESSCHER MARTINGALE TRANSFORM Suppose Z t = δN t where δ > 0 is the jump size and N is a standard Poisson process with intensity parameter γ > 0. Then k ( θ ) = γ ( e δθ − 1) and the solution of equation before is 18

− µ + ˜ βV t − − 1 θ ♯ = ρδw × t V t − δρλγ ( e δρ − 1) � � �� − δρµ + βV t − × exp , V t − V t − where w is known as (the principal branch of) the Lambert W (or polylogarithm) function. 19

The function w is available in Mathematica, Maple, and many other computer packages and libraries. Basically it is the in- verse function of xe x . For this model we have E [ e ξZ 1 ] < ∞ for all ξ ∈ , so the condi- tion given in Proposition 1 before is satisfied, and the expo- nential Esscher martingale transform exists. 20

LINEAR ESSCHER MARTINGALE TRANSFORM X t = e ρ ∆ X t − 1 and since we have in The jumps of ˜ X are ∆ ˜ the Poisson toy model only one jump size, this implies, that δ ρ = e ρδ − 1. we can write ˜ X t = ˜ δ ρ N t where ˜ So the cumulant function is of the same form we have seen k ρ ( z ) = γ ( e ˜ δ ρ z − 1), and its in the previous section, namely ˜ δ ρ e ˜ k ′ δ ρ z . derivative is ˜ ρ ( z ) = γ ˜ 21

For the linear Esscher transform we have to solve the second δ ρ e ˜ δ ρ θ = 0. algebraic equation given, which becomes ˜ c t θ + λγ ˜ b t +˜ The solution will be given, again using the Lambert w function, as  � δ 2  λγ ˜ t = − µ + ˜ µ + ˜ � βV t − − 1 βV t − ρ θ ∗  . − ˜ exp w δ ρ ˜ V t − δ ρ V t − V t − As we have E [ e ξZ 1 ] < ∞ for all ξ ∈ , the condition in Proposi- tion 3 is satisfied, and the linear Esscher martingale transform exists. 22