university of perugia · university of perugia 30 april - 5 may 2007, ii amamef conference flavio...
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![Page 1: University of Perugia · University of Perugia 30 April - 5 May 2007, II AMAMEF Conference Flavio Angelini, Stefano Herzel Hedging strategies in discrete time. Outline The problem](https://reader036.vdocuments.us/reader036/viewer/2022070107/6024b6fc26c76256032a2447/html5/thumbnails/1.jpg)
OutlineThe problem
The optimal strategyThe Delta strategy
Transaction costsApplicationsConclusions
Hedging strategies in discrete time
Flavio Angelini, Stefano Herzel
University of Perugia
30 April - 5 May 2007, II AMAMEF Conference
Flavio Angelini, Stefano Herzel Hedging strategies in discrete time
![Page 2: University of Perugia · University of Perugia 30 April - 5 May 2007, II AMAMEF Conference Flavio Angelini, Stefano Herzel Hedging strategies in discrete time. Outline The problem](https://reader036.vdocuments.us/reader036/viewer/2022070107/6024b6fc26c76256032a2447/html5/thumbnails/2.jpg)
OutlineThe problem
The optimal strategyThe Delta strategy
Transaction costsApplicationsConclusions
The problem
The optimal strategy
The Delta strategy
Transaction costs
Applications
Conclusions
Flavio Angelini, Stefano Herzel Hedging strategies in discrete time
![Page 3: University of Perugia · University of Perugia 30 April - 5 May 2007, II AMAMEF Conference Flavio Angelini, Stefano Herzel Hedging strategies in discrete time. Outline The problem](https://reader036.vdocuments.us/reader036/viewer/2022070107/6024b6fc26c76256032a2447/html5/thumbnails/3.jpg)
OutlineThe problem
The optimal strategyThe Delta strategy
Transaction costsApplicationsConclusions
The problem
The optimal strategy
The Delta strategy
Transaction costs
Applications
Conclusions
Flavio Angelini, Stefano Herzel Hedging strategies in discrete time
![Page 4: University of Perugia · University of Perugia 30 April - 5 May 2007, II AMAMEF Conference Flavio Angelini, Stefano Herzel Hedging strategies in discrete time. Outline The problem](https://reader036.vdocuments.us/reader036/viewer/2022070107/6024b6fc26c76256032a2447/html5/thumbnails/4.jpg)
OutlineThe problem
The optimal strategyThe Delta strategy
Transaction costsApplicationsConclusions
The problem
The optimal strategy
The Delta strategy
Transaction costs
Applications
Conclusions
Flavio Angelini, Stefano Herzel Hedging strategies in discrete time
![Page 5: University of Perugia · University of Perugia 30 April - 5 May 2007, II AMAMEF Conference Flavio Angelini, Stefano Herzel Hedging strategies in discrete time. Outline The problem](https://reader036.vdocuments.us/reader036/viewer/2022070107/6024b6fc26c76256032a2447/html5/thumbnails/5.jpg)
OutlineThe problem
The optimal strategyThe Delta strategy
Transaction costsApplicationsConclusions
The problem
The optimal strategy
The Delta strategy
Transaction costs
Applications
Conclusions
Flavio Angelini, Stefano Herzel Hedging strategies in discrete time
![Page 6: University of Perugia · University of Perugia 30 April - 5 May 2007, II AMAMEF Conference Flavio Angelini, Stefano Herzel Hedging strategies in discrete time. Outline The problem](https://reader036.vdocuments.us/reader036/viewer/2022070107/6024b6fc26c76256032a2447/html5/thumbnails/6.jpg)
OutlineThe problem
The optimal strategyThe Delta strategy
Transaction costsApplicationsConclusions
The problem
The optimal strategy
The Delta strategy
Transaction costs
Applications
Conclusions
Flavio Angelini, Stefano Herzel Hedging strategies in discrete time
![Page 7: University of Perugia · University of Perugia 30 April - 5 May 2007, II AMAMEF Conference Flavio Angelini, Stefano Herzel Hedging strategies in discrete time. Outline The problem](https://reader036.vdocuments.us/reader036/viewer/2022070107/6024b6fc26c76256032a2447/html5/thumbnails/7.jpg)
OutlineThe problem
The optimal strategyThe Delta strategy
Transaction costsApplicationsConclusions
The problem
The optimal strategy
The Delta strategy
Transaction costs
Applications
Conclusions
Flavio Angelini, Stefano Herzel Hedging strategies in discrete time
![Page 8: University of Perugia · University of Perugia 30 April - 5 May 2007, II AMAMEF Conference Flavio Angelini, Stefano Herzel Hedging strategies in discrete time. Outline The problem](https://reader036.vdocuments.us/reader036/viewer/2022070107/6024b6fc26c76256032a2447/html5/thumbnails/8.jpg)
OutlineThe problem
The optimal strategyThe Delta strategy
Transaction costsApplicationsConclusions
Introduction
I Most mathematical models for financial markets assumecontinuous-time trading
I Strategies that are optimal in continuous time (e.g. BS deltahedging) need not to be optimal in discrete time
I Can we effectively compute a discrete time optimal strategy?
I Can we measure the error due to time discretization?
Flavio Angelini, Stefano Herzel Hedging strategies in discrete time
![Page 9: University of Perugia · University of Perugia 30 April - 5 May 2007, II AMAMEF Conference Flavio Angelini, Stefano Herzel Hedging strategies in discrete time. Outline The problem](https://reader036.vdocuments.us/reader036/viewer/2022070107/6024b6fc26c76256032a2447/html5/thumbnails/9.jpg)
OutlineThe problem
The optimal strategyThe Delta strategy
Transaction costsApplicationsConclusions
Introduction
I Most mathematical models for financial markets assumecontinuous-time trading
I Strategies that are optimal in continuous time (e.g. BS deltahedging) need not to be optimal in discrete time
I Can we effectively compute a discrete time optimal strategy?
I Can we measure the error due to time discretization?
Flavio Angelini, Stefano Herzel Hedging strategies in discrete time
![Page 10: University of Perugia · University of Perugia 30 April - 5 May 2007, II AMAMEF Conference Flavio Angelini, Stefano Herzel Hedging strategies in discrete time. Outline The problem](https://reader036.vdocuments.us/reader036/viewer/2022070107/6024b6fc26c76256032a2447/html5/thumbnails/10.jpg)
OutlineThe problem
The optimal strategyThe Delta strategy
Transaction costsApplicationsConclusions
Introduction
I Most mathematical models for financial markets assumecontinuous-time trading
I Strategies that are optimal in continuous time (e.g. BS deltahedging) need not to be optimal in discrete time
I Can we effectively compute a discrete time optimal strategy?
I Can we measure the error due to time discretization?
Flavio Angelini, Stefano Herzel Hedging strategies in discrete time
![Page 11: University of Perugia · University of Perugia 30 April - 5 May 2007, II AMAMEF Conference Flavio Angelini, Stefano Herzel Hedging strategies in discrete time. Outline The problem](https://reader036.vdocuments.us/reader036/viewer/2022070107/6024b6fc26c76256032a2447/html5/thumbnails/11.jpg)
OutlineThe problem
The optimal strategyThe Delta strategy
Transaction costsApplicationsConclusions
Introduction
I Most mathematical models for financial markets assumecontinuous-time trading
I Strategies that are optimal in continuous time (e.g. BS deltahedging) need not to be optimal in discrete time
I Can we effectively compute a discrete time optimal strategy?
I Can we measure the error due to time discretization?
Flavio Angelini, Stefano Herzel Hedging strategies in discrete time
![Page 12: University of Perugia · University of Perugia 30 April - 5 May 2007, II AMAMEF Conference Flavio Angelini, Stefano Herzel Hedging strategies in discrete time. Outline The problem](https://reader036.vdocuments.us/reader036/viewer/2022070107/6024b6fc26c76256032a2447/html5/thumbnails/12.jpg)
OutlineThe problem
The optimal strategyThe Delta strategy
Transaction costsApplicationsConclusions
The setting
I Let H be the payoff of a contingent claimI Let S = (Sk)Nk=0 be the asset price process and ϑ = (ϑk)Nk=1
be the hedging ratio of a self-financing trading strategyI Assume zero interest rateI The gains process is
GN(ϑ) =N∑
j=1
ϑj∆Sj
where ∆Sj = Sj − Sj−1
I Given an initial value c , follow the strategy ϑ. The hedgingerror is
ε(ϑ, c) = H − c − GN(ϑ)
Flavio Angelini, Stefano Herzel Hedging strategies in discrete time
![Page 13: University of Perugia · University of Perugia 30 April - 5 May 2007, II AMAMEF Conference Flavio Angelini, Stefano Herzel Hedging strategies in discrete time. Outline The problem](https://reader036.vdocuments.us/reader036/viewer/2022070107/6024b6fc26c76256032a2447/html5/thumbnails/13.jpg)
OutlineThe problem
The optimal strategyThe Delta strategy
Transaction costsApplicationsConclusions
The setting
I Let H be the payoff of a contingent claimI Let S = (Sk)Nk=0 be the asset price process and ϑ = (ϑk)Nk=1
be the hedging ratio of a self-financing trading strategyI Assume zero interest rateI The gains process is
GN(ϑ) =N∑
j=1
ϑj∆Sj
where ∆Sj = Sj − Sj−1
I Given an initial value c , follow the strategy ϑ. The hedgingerror is
ε(ϑ, c) = H − c − GN(ϑ)
Flavio Angelini, Stefano Herzel Hedging strategies in discrete time
![Page 14: University of Perugia · University of Perugia 30 April - 5 May 2007, II AMAMEF Conference Flavio Angelini, Stefano Herzel Hedging strategies in discrete time. Outline The problem](https://reader036.vdocuments.us/reader036/viewer/2022070107/6024b6fc26c76256032a2447/html5/thumbnails/14.jpg)
OutlineThe problem
The optimal strategyThe Delta strategy
Transaction costsApplicationsConclusions
The setting
I Let H be the payoff of a contingent claimI Let S = (Sk)Nk=0 be the asset price process and ϑ = (ϑk)Nk=1
be the hedging ratio of a self-financing trading strategyI Assume zero interest rateI The gains process is
GN(ϑ) =N∑
j=1
ϑj∆Sj
where ∆Sj = Sj − Sj−1
I Given an initial value c , follow the strategy ϑ. The hedgingerror is
ε(ϑ, c) = H − c − GN(ϑ)
Flavio Angelini, Stefano Herzel Hedging strategies in discrete time
![Page 15: University of Perugia · University of Perugia 30 April - 5 May 2007, II AMAMEF Conference Flavio Angelini, Stefano Herzel Hedging strategies in discrete time. Outline The problem](https://reader036.vdocuments.us/reader036/viewer/2022070107/6024b6fc26c76256032a2447/html5/thumbnails/15.jpg)
OutlineThe problem
The optimal strategyThe Delta strategy
Transaction costsApplicationsConclusions
The setting
I Let H be the payoff of a contingent claimI Let S = (Sk)Nk=0 be the asset price process and ϑ = (ϑk)Nk=1
be the hedging ratio of a self-financing trading strategyI Assume zero interest rateI The gains process is
GN(ϑ) =N∑
j=1
ϑj∆Sj
where ∆Sj = Sj − Sj−1
I Given an initial value c , follow the strategy ϑ. The hedgingerror is
ε(ϑ, c) = H − c − GN(ϑ)
Flavio Angelini, Stefano Herzel Hedging strategies in discrete time
![Page 16: University of Perugia · University of Perugia 30 April - 5 May 2007, II AMAMEF Conference Flavio Angelini, Stefano Herzel Hedging strategies in discrete time. Outline The problem](https://reader036.vdocuments.us/reader036/viewer/2022070107/6024b6fc26c76256032a2447/html5/thumbnails/16.jpg)
OutlineThe problem
The optimal strategyThe Delta strategy
Transaction costsApplicationsConclusions
The setting
I Let H be the payoff of a contingent claimI Let S = (Sk)Nk=0 be the asset price process and ϑ = (ϑk)Nk=1
be the hedging ratio of a self-financing trading strategyI Assume zero interest rateI The gains process is
GN(ϑ) =N∑
j=1
ϑj∆Sj
where ∆Sj = Sj − Sj−1
I Given an initial value c , follow the strategy ϑ. The hedgingerror is
ε(ϑ, c) = H − c − GN(ϑ)
Flavio Angelini, Stefano Herzel Hedging strategies in discrete time
![Page 17: University of Perugia · University of Perugia 30 April - 5 May 2007, II AMAMEF Conference Flavio Angelini, Stefano Herzel Hedging strategies in discrete time. Outline The problem](https://reader036.vdocuments.us/reader036/viewer/2022070107/6024b6fc26c76256032a2447/html5/thumbnails/17.jpg)
OutlineThe problem
The optimal strategyThe Delta strategy
Transaction costsApplicationsConclusions
The discretization error in the BS model
−10 −8 −6 −4 −2 0 2 4 6 8 100
0.05
0.1
0.15
0.2
0.25BS hedging
N=12, N=180
Flavio Angelini, Stefano Herzel Hedging strategies in discrete time
![Page 18: University of Perugia · University of Perugia 30 April - 5 May 2007, II AMAMEF Conference Flavio Angelini, Stefano Herzel Hedging strategies in discrete time. Outline The problem](https://reader036.vdocuments.us/reader036/viewer/2022070107/6024b6fc26c76256032a2447/html5/thumbnails/18.jpg)
OutlineThe problem
The optimal strategyThe Delta strategy
Transaction costsApplicationsConclusions
The problem
I Measurement of hedging error
I Minimization Problem:
minϑ∈Θ
E[ε(ϑ, c)2
]for fixed c ∈ IR
I Computation ofE
[ε(ϑ, c)2
]for given c , ϑ
Flavio Angelini, Stefano Herzel Hedging strategies in discrete time
![Page 19: University of Perugia · University of Perugia 30 April - 5 May 2007, II AMAMEF Conference Flavio Angelini, Stefano Herzel Hedging strategies in discrete time. Outline The problem](https://reader036.vdocuments.us/reader036/viewer/2022070107/6024b6fc26c76256032a2447/html5/thumbnails/19.jpg)
OutlineThe problem
The optimal strategyThe Delta strategy
Transaction costsApplicationsConclusions
The problem
I Measurement of hedging error
I Minimization Problem:
minϑ∈Θ
E[ε(ϑ, c)2
]for fixed c ∈ IR
I Computation ofE
[ε(ϑ, c)2
]for given c , ϑ
Flavio Angelini, Stefano Herzel Hedging strategies in discrete time
![Page 20: University of Perugia · University of Perugia 30 April - 5 May 2007, II AMAMEF Conference Flavio Angelini, Stefano Herzel Hedging strategies in discrete time. Outline The problem](https://reader036.vdocuments.us/reader036/viewer/2022070107/6024b6fc26c76256032a2447/html5/thumbnails/20.jpg)
OutlineThe problem
The optimal strategyThe Delta strategy
Transaction costsApplicationsConclusions
The problem
I Measurement of hedging error
I Minimization Problem:
minϑ∈Θ
E[ε(ϑ, c)2
]for fixed c ∈ IR
I Computation ofE
[ε(ϑ, c)2
]for given c , ϑ
Flavio Angelini, Stefano Herzel Hedging strategies in discrete time
![Page 21: University of Perugia · University of Perugia 30 April - 5 May 2007, II AMAMEF Conference Flavio Angelini, Stefano Herzel Hedging strategies in discrete time. Outline The problem](https://reader036.vdocuments.us/reader036/viewer/2022070107/6024b6fc26c76256032a2447/html5/thumbnails/21.jpg)
OutlineThe problem
The optimal strategyThe Delta strategy
Transaction costsApplicationsConclusions
The existence of the optimal strategy
I Assume that process X satisfies the following Non-Degeneracy(ND) condition:
(Ek−1∆Sk)2
vark−1∆Sk< M
for all ω and k.
I Then there exists a unique optimal trading strategy θc thatsolves the basic problem (Schweizer (1995))
I A counterexample shows that the ND condition is necessaryfor the existence of a solution
I If X is a (non degenerate) martingale then condition ND isobviously satisfied
Flavio Angelini, Stefano Herzel Hedging strategies in discrete time
![Page 22: University of Perugia · University of Perugia 30 April - 5 May 2007, II AMAMEF Conference Flavio Angelini, Stefano Herzel Hedging strategies in discrete time. Outline The problem](https://reader036.vdocuments.us/reader036/viewer/2022070107/6024b6fc26c76256032a2447/html5/thumbnails/22.jpg)
OutlineThe problem
The optimal strategyThe Delta strategy
Transaction costsApplicationsConclusions
The structure of the optimal strategy
I In general, the explicit computation of the optimal strategy isa very hard task
I Schweizer (1995): ”The optimal hedge ratio can bedecomposed in 3 pieces: locally optimal (pure hedgingdemand) ξH , demand for mean-variance purposes, demand forhedging against mean-variance ratio stochastic changes (notpresent if the m.v.r. is deterministic)”.
Flavio Angelini, Stefano Herzel Hedging strategies in discrete time
![Page 23: University of Perugia · University of Perugia 30 April - 5 May 2007, II AMAMEF Conference Flavio Angelini, Stefano Herzel Hedging strategies in discrete time. Outline The problem](https://reader036.vdocuments.us/reader036/viewer/2022070107/6024b6fc26c76256032a2447/html5/thumbnails/23.jpg)
OutlineThe problem
The optimal strategyThe Delta strategy
Transaction costsApplicationsConclusions
The Laplace transform approach
I Hubalek, Kallsen and Krawczyk (2006), Cerny (2007)
I Let (Ω,F , (Fn)n∈(0,1,...,N),P) be a filtered probability space.Consider a one-dimensional process
Sn = S0 exp(Xn),
where the process X = (Xn) for n = 0, 1, . . . ,N, satisfies
1. X is adapted to the filtration Fn, n ∈ (0, 1, . . . ,N),2. X0 = 0,3. ∆Xn = Xn − Xn−1 n = 1, . . . ,N are i.i.d.
Flavio Angelini, Stefano Herzel Hedging strategies in discrete time
![Page 24: University of Perugia · University of Perugia 30 April - 5 May 2007, II AMAMEF Conference Flavio Angelini, Stefano Herzel Hedging strategies in discrete time. Outline The problem](https://reader036.vdocuments.us/reader036/viewer/2022070107/6024b6fc26c76256032a2447/html5/thumbnails/24.jpg)
OutlineThe problem
The optimal strategyThe Delta strategy
Transaction costsApplicationsConclusions
The Laplace transform approach
I Hubalek, Kallsen and Krawczyk (2006), Cerny (2007)
I Let (Ω,F , (Fn)n∈(0,1,...,N),P) be a filtered probability space.Consider a one-dimensional process
Sn = S0 exp(Xn),
where the process X = (Xn) for n = 0, 1, . . . ,N, satisfies
1. X is adapted to the filtration Fn, n ∈ (0, 1, . . . ,N),2. X0 = 0,3. ∆Xn = Xn − Xn−1 n = 1, . . . ,N are i.i.d.
Flavio Angelini, Stefano Herzel Hedging strategies in discrete time
![Page 25: University of Perugia · University of Perugia 30 April - 5 May 2007, II AMAMEF Conference Flavio Angelini, Stefano Herzel Hedging strategies in discrete time. Outline The problem](https://reader036.vdocuments.us/reader036/viewer/2022070107/6024b6fc26c76256032a2447/html5/thumbnails/25.jpg)
OutlineThe problem
The optimal strategyThe Delta strategy
Transaction costsApplicationsConclusions
The Laplace transform approach
I Hubalek, Kallsen and Krawczyk (2006), Cerny (2007)
I Let (Ω,F , (Fn)n∈(0,1,...,N),P) be a filtered probability space.Consider a one-dimensional process
Sn = S0 exp(Xn),
where the process X = (Xn) for n = 0, 1, . . . ,N, satisfies
1. X is adapted to the filtration Fn, n ∈ (0, 1, . . . ,N),2. X0 = 0,3. ∆Xn = Xn − Xn−1 n = 1, . . . ,N are i.i.d.
Flavio Angelini, Stefano Herzel Hedging strategies in discrete time
![Page 26: University of Perugia · University of Perugia 30 April - 5 May 2007, II AMAMEF Conference Flavio Angelini, Stefano Herzel Hedging strategies in discrete time. Outline The problem](https://reader036.vdocuments.us/reader036/viewer/2022070107/6024b6fc26c76256032a2447/html5/thumbnails/26.jpg)
OutlineThe problem
The optimal strategyThe Delta strategy
Transaction costsApplicationsConclusions
The Laplace transform approach
I Hubalek, Kallsen and Krawczyk (2006), Cerny (2007)
I Let (Ω,F , (Fn)n∈(0,1,...,N),P) be a filtered probability space.Consider a one-dimensional process
Sn = S0 exp(Xn),
where the process X = (Xn) for n = 0, 1, . . . ,N, satisfies
1. X is adapted to the filtration Fn, n ∈ (0, 1, . . . ,N),2. X0 = 0,3. ∆Xn = Xn − Xn−1 n = 1, . . . ,N are i.i.d.
Flavio Angelini, Stefano Herzel Hedging strategies in discrete time
![Page 27: University of Perugia · University of Perugia 30 April - 5 May 2007, II AMAMEF Conference Flavio Angelini, Stefano Herzel Hedging strategies in discrete time. Outline The problem](https://reader036.vdocuments.us/reader036/viewer/2022070107/6024b6fc26c76256032a2447/html5/thumbnails/27.jpg)
OutlineThe problem
The optimal strategyThe Delta strategy
Transaction costsApplicationsConclusions
The Laplace transform approach
I Hubalek, Kallsen and Krawczyk (2006), Cerny (2007)
I Let (Ω,F , (Fn)n∈(0,1,...,N),P) be a filtered probability space.Consider a one-dimensional process
Sn = S0 exp(Xn),
where the process X = (Xn) for n = 0, 1, . . . ,N, satisfies
1. X is adapted to the filtration Fn, n ∈ (0, 1, . . . ,N),2. X0 = 0,3. ∆Xn = Xn − Xn−1 n = 1, . . . ,N are i.i.d.
Flavio Angelini, Stefano Herzel Hedging strategies in discrete time
![Page 28: University of Perugia · University of Perugia 30 April - 5 May 2007, II AMAMEF Conference Flavio Angelini, Stefano Herzel Hedging strategies in discrete time. Outline The problem](https://reader036.vdocuments.us/reader036/viewer/2022070107/6024b6fc26c76256032a2447/html5/thumbnails/28.jpg)
OutlineThe problem
The optimal strategyThe Delta strategy
Transaction costsApplicationsConclusions
The Laplace transform approach
I Can compute the moment generating function of ∆X
m(z) = E [ez∆X ],
assuming that it exists for 0 ≤ Re(z) ≤ 2
I A rather general class of models, including Black-Scholes,Merton jump-diffusion, NIG, etc.
Flavio Angelini, Stefano Herzel Hedging strategies in discrete time
![Page 29: University of Perugia · University of Perugia 30 April - 5 May 2007, II AMAMEF Conference Flavio Angelini, Stefano Herzel Hedging strategies in discrete time. Outline The problem](https://reader036.vdocuments.us/reader036/viewer/2022070107/6024b6fc26c76256032a2447/html5/thumbnails/29.jpg)
OutlineThe problem
The optimal strategyThe Delta strategy
Transaction costsApplicationsConclusions
The Laplace transform approach
I Can compute the moment generating function of ∆X
m(z) = E [ez∆X ],
assuming that it exists for 0 ≤ Re(z) ≤ 2
I A rather general class of models, including Black-Scholes,Merton jump-diffusion, NIG, etc.
Flavio Angelini, Stefano Herzel Hedging strategies in discrete time
![Page 30: University of Perugia · University of Perugia 30 April - 5 May 2007, II AMAMEF Conference Flavio Angelini, Stefano Herzel Hedging strategies in discrete time. Outline The problem](https://reader036.vdocuments.us/reader036/viewer/2022070107/6024b6fc26c76256032a2447/html5/thumbnails/30.jpg)
OutlineThe problem
The optimal strategyThe Delta strategy
Transaction costsApplicationsConclusions
The idea of the Laplace transform approach
I An exponential claim is a contingent claim with payoff
H(z) = SzN = Sz
0 ezXN
I The i.i.d. assumption for ∆Xn makes computations easy forexponential claims.
I Consider a contingent claim which is a ”linear combination”of exponential claims (let us call it a simple claim)
I Use linearity properties and Fubini
Flavio Angelini, Stefano Herzel Hedging strategies in discrete time
![Page 31: University of Perugia · University of Perugia 30 April - 5 May 2007, II AMAMEF Conference Flavio Angelini, Stefano Herzel Hedging strategies in discrete time. Outline The problem](https://reader036.vdocuments.us/reader036/viewer/2022070107/6024b6fc26c76256032a2447/html5/thumbnails/31.jpg)
OutlineThe problem
The optimal strategyThe Delta strategy
Transaction costsApplicationsConclusions
The idea of the Laplace transform approach
I An exponential claim is a contingent claim with payoff
H(z) = SzN = Sz
0 ezXN
I The i.i.d. assumption for ∆Xn makes computations easy forexponential claims.
I Consider a contingent claim which is a ”linear combination”of exponential claims (let us call it a simple claim)
I Use linearity properties and Fubini
Flavio Angelini, Stefano Herzel Hedging strategies in discrete time
![Page 32: University of Perugia · University of Perugia 30 April - 5 May 2007, II AMAMEF Conference Flavio Angelini, Stefano Herzel Hedging strategies in discrete time. Outline The problem](https://reader036.vdocuments.us/reader036/viewer/2022070107/6024b6fc26c76256032a2447/html5/thumbnails/32.jpg)
OutlineThe problem
The optimal strategyThe Delta strategy
Transaction costsApplicationsConclusions
The idea of the Laplace transform approach
I An exponential claim is a contingent claim with payoff
H(z) = SzN = Sz
0 ezXN
I The i.i.d. assumption for ∆Xn makes computations easy forexponential claims.
I Consider a contingent claim which is a ”linear combination”of exponential claims (let us call it a simple claim)
I Use linearity properties and Fubini
Flavio Angelini, Stefano Herzel Hedging strategies in discrete time
![Page 33: University of Perugia · University of Perugia 30 April - 5 May 2007, II AMAMEF Conference Flavio Angelini, Stefano Herzel Hedging strategies in discrete time. Outline The problem](https://reader036.vdocuments.us/reader036/viewer/2022070107/6024b6fc26c76256032a2447/html5/thumbnails/33.jpg)
OutlineThe problem
The optimal strategyThe Delta strategy
Transaction costsApplicationsConclusions
The idea of the Laplace transform approach
I An exponential claim is a contingent claim with payoff
H(z) = SzN = Sz
0 ezXN
I The i.i.d. assumption for ∆Xn makes computations easy forexponential claims.
I Consider a contingent claim which is a ”linear combination”of exponential claims (let us call it a simple claim)
I Use linearity properties and Fubini
Flavio Angelini, Stefano Herzel Hedging strategies in discrete time
![Page 34: University of Perugia · University of Perugia 30 April - 5 May 2007, II AMAMEF Conference Flavio Angelini, Stefano Herzel Hedging strategies in discrete time. Outline The problem](https://reader036.vdocuments.us/reader036/viewer/2022070107/6024b6fc26c76256032a2447/html5/thumbnails/34.jpg)
OutlineThe problem
The optimal strategyThe Delta strategy
Transaction costsApplicationsConclusions
The Laplace transform approach
I More precisely consider contingent claims whose payoff is aninverse Laplace Transform
H =
∫Sz
NΠ(dz)
I A European call is a simple claim!
(SN − K )+ =1
2πi
∫ R+i∞
R−i∞Sz
N
K 1−z
z(z − 1)dz ,
with R > 1 arbitrary
Flavio Angelini, Stefano Herzel Hedging strategies in discrete time
![Page 35: University of Perugia · University of Perugia 30 April - 5 May 2007, II AMAMEF Conference Flavio Angelini, Stefano Herzel Hedging strategies in discrete time. Outline The problem](https://reader036.vdocuments.us/reader036/viewer/2022070107/6024b6fc26c76256032a2447/html5/thumbnails/35.jpg)
OutlineThe problem
The optimal strategyThe Delta strategy
Transaction costsApplicationsConclusions
The Laplace transform approach
I More precisely consider contingent claims whose payoff is aninverse Laplace Transform
H =
∫Sz
NΠ(dz)
I A European call is a simple claim!
(SN − K )+ =1
2πi
∫ R+i∞
R−i∞Sz
N
K 1−z
z(z − 1)dz ,
with R > 1 arbitrary
Flavio Angelini, Stefano Herzel Hedging strategies in discrete time
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OutlineThe problem
The optimal strategyThe Delta strategy
Transaction costsApplicationsConclusions
The results
I Locally optimal hedge ratio at 0
ξ1 =
∫Sz−1
0 g(z)h(z)N−1Π(dz)
I Value at 0 of the optimal portfolio
V0 =
∫Sz
0 h(z)N−1Π(dz)
I Optimal variance
Var0 =
∫ ∫J0(y , z)Π(dy)Π(dz)
Flavio Angelini, Stefano Herzel Hedging strategies in discrete time
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OutlineThe problem
The optimal strategyThe Delta strategy
Transaction costsApplicationsConclusions
The results
I Locally optimal hedge ratio at 0
ξ1 =
∫Sz−1
0 g(z)h(z)N−1Π(dz)
I Value at 0 of the optimal portfolio
V0 =
∫Sz
0 h(z)N−1Π(dz)
I Optimal variance
Var0 =
∫ ∫J0(y , z)Π(dy)Π(dz)
Flavio Angelini, Stefano Herzel Hedging strategies in discrete time
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OutlineThe problem
The optimal strategyThe Delta strategy
Transaction costsApplicationsConclusions
The results
I Locally optimal hedge ratio at 0
ξ1 =
∫Sz−1
0 g(z)h(z)N−1Π(dz)
I Value at 0 of the optimal portfolio
V0 =
∫Sz
0 h(z)N−1Π(dz)
I Optimal variance
Var0 =
∫ ∫J0(y , z)Π(dy)Π(dz)
Flavio Angelini, Stefano Herzel Hedging strategies in discrete time
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OutlineThe problem
The optimal strategyThe Delta strategy
Transaction costsApplicationsConclusions
A relevant question
I Everyday market practice adopts classical Delta hedging
I Other easily implementable hedging strategies have also beenproposed (e.g. Willmott)
I Can we measure the variance of a given (non-optimal)strategy?
Flavio Angelini, Stefano Herzel Hedging strategies in discrete time
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OutlineThe problem
The optimal strategyThe Delta strategy
Transaction costsApplicationsConclusions
A relevant question
I Everyday market practice adopts classical Delta hedging
I Other easily implementable hedging strategies have also beenproposed (e.g. Willmott)
I Can we measure the variance of a given (non-optimal)strategy?
Flavio Angelini, Stefano Herzel Hedging strategies in discrete time
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OutlineThe problem
The optimal strategyThe Delta strategy
Transaction costsApplicationsConclusions
A relevant question
I Everyday market practice adopts classical Delta hedging
I Other easily implementable hedging strategies have also beenproposed (e.g. Willmott)
I Can we measure the variance of a given (non-optimal)strategy?
Flavio Angelini, Stefano Herzel Hedging strategies in discrete time
![Page 42: University of Perugia · University of Perugia 30 April - 5 May 2007, II AMAMEF Conference Flavio Angelini, Stefano Herzel Hedging strategies in discrete time. Outline The problem](https://reader036.vdocuments.us/reader036/viewer/2022070107/6024b6fc26c76256032a2447/html5/thumbnails/42.jpg)
OutlineThe problem
The optimal strategyThe Delta strategy
Transaction costsApplicationsConclusions
Delta hedging, previous results
I Let ϑ = ∆N be the BS-delta and S is the BS-process.
I Hayashi and Mykland (2005) showed that
ε(∆N , c)−√
T
2N
∫ T
0Γuσ
2S2udW ∗
u → 0
Flavio Angelini, Stefano Herzel Hedging strategies in discrete time
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OutlineThe problem
The optimal strategyThe Delta strategy
Transaction costsApplicationsConclusions
Delta hedging, previous results
I Let ϑ = ∆N be the BS-delta and S is the BS-process.
I Hayashi and Mykland (2005) showed that
ε(∆N , c)−√
T
2N
∫ T
0Γuσ
2S2udW ∗
u → 0
Flavio Angelini, Stefano Herzel Hedging strategies in discrete time
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OutlineThe problem
The optimal strategyThe Delta strategy
Transaction costsApplicationsConclusions
Delta hedging, previous results
I Toft (1996) provides an asymptotic approximation for thevariance
1
2σ4
(T
N
)2 N−1∑k=0
E0
[(ΓkS2
k )2]
I Kamal and Derman (1999) propose a more trader-friendlyapproximation
π
4Nσ2Vega2
Flavio Angelini, Stefano Herzel Hedging strategies in discrete time
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OutlineThe problem
The optimal strategyThe Delta strategy
Transaction costsApplicationsConclusions
Delta hedging, previous results
I Toft (1996) provides an asymptotic approximation for thevariance
1
2σ4
(T
N
)2 N−1∑k=0
E0
[(ΓkS2
k )2]
I Kamal and Derman (1999) propose a more trader-friendlyapproximation
π
4Nσ2Vega2
Flavio Angelini, Stefano Herzel Hedging strategies in discrete time
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OutlineThe problem
The optimal strategyThe Delta strategy
Transaction costsApplicationsConclusions
Crucial observation
I The Delta of a simple claim is an inverse Laplace transform
∆n =
∫f (z)nS
z−1n−1Π(dz),
where f (z)n = zm0(z)N−n+1 does not depend on Sn−1.I The price at time tn−1 of a simple claim is
Pn−1 = EQn−1
[∫Sz
NΠ(dz)
]=
∫EQ
n−1[SzN ]Π(dz) =
∫Sz
n−1m0(z)N−n+1Π(dz),
where EQn−1 is the pricing measure and m0(z) is the m.g.f. of
corresponding ∆X .
Flavio Angelini, Stefano Herzel Hedging strategies in discrete time
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OutlineThe problem
The optimal strategyThe Delta strategy
Transaction costsApplicationsConclusions
Crucial observation
I The Delta of a simple claim is an inverse Laplace transform
∆n =
∫f (z)nS
z−1n−1Π(dz),
where f (z)n = zm0(z)N−n+1 does not depend on Sn−1.I The price at time tn−1 of a simple claim is
Pn−1 = EQn−1
[∫Sz
NΠ(dz)
]=
∫EQ
n−1[SzN ]Π(dz) =
∫Sz
n−1m0(z)N−n+1Π(dz),
where EQn−1 is the pricing measure and m0(z) is the m.g.f. of
corresponding ∆X .
Flavio Angelini, Stefano Herzel Hedging strategies in discrete time
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OutlineThe problem
The optimal strategyThe Delta strategy
Transaction costsApplicationsConclusions
Delta hedging error
I The hedging error of a simple claim is an inverse Laplacetransform
H − c − GN(∆) =
∫(H(z)− c − GN(∆(z)))Π(dz)
I Can compute expected value and variance of the hedging errorif I am able to compute things inside the integral
Flavio Angelini, Stefano Herzel Hedging strategies in discrete time
![Page 49: University of Perugia · University of Perugia 30 April - 5 May 2007, II AMAMEF Conference Flavio Angelini, Stefano Herzel Hedging strategies in discrete time. Outline The problem](https://reader036.vdocuments.us/reader036/viewer/2022070107/6024b6fc26c76256032a2447/html5/thumbnails/49.jpg)
OutlineThe problem
The optimal strategyThe Delta strategy
Transaction costsApplicationsConclusions
Delta hedging error
I The hedging error of a simple claim is an inverse Laplacetransform
H − c − GN(∆) =
∫(H(z)− c − GN(∆(z)))Π(dz)
I Can compute expected value and variance of the hedging errorif I am able to compute things inside the integral
Flavio Angelini, Stefano Herzel Hedging strategies in discrete time
![Page 50: University of Perugia · University of Perugia 30 April - 5 May 2007, II AMAMEF Conference Flavio Angelini, Stefano Herzel Hedging strategies in discrete time. Outline The problem](https://reader036.vdocuments.us/reader036/viewer/2022070107/6024b6fc26c76256032a2447/html5/thumbnails/50.jpg)
OutlineThe problem
The optimal strategyThe Delta strategy
Transaction costsApplicationsConclusions
Expected value
I
E [H] =
∫E [Sz
N ]Π(dz) =
∫Sz
0 m(z)NΠ(dz)
I
E [∆n∆Sn] =
∫E [f (z)nS
z−1n−1∆Sn]Π(dz)
=
∫Sz
0 f (z)n(m(1)− 1)m(z)n−1Π(dz),
for n = 1, . . . ,N
Flavio Angelini, Stefano Herzel Hedging strategies in discrete time
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OutlineThe problem
The optimal strategyThe Delta strategy
Transaction costsApplicationsConclusions
Expected value
I
E [H] =
∫E [Sz
N ]Π(dz) =
∫Sz
0 m(z)NΠ(dz)
I
E [∆n∆Sn] =
∫E [f (z)nS
z−1n−1∆Sn]Π(dz)
=
∫Sz
0 f (z)n(m(1)− 1)m(z)n−1Π(dz),
for n = 1, . . . ,N
Flavio Angelini, Stefano Herzel Hedging strategies in discrete time
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OutlineThe problem
The optimal strategyThe Delta strategy
Transaction costsApplicationsConclusions
Variance of Delta hedging
I Need to compute all the covariances
I
E [H(y)H(z)] = E [SyNSz
N ] = Sy+z0 m(y + z)N
I
E [H(y)Szn−1∆Sn] = E [Sy
NSzn−1∆Sn] = Sy+z
0 v2(y , z)n,
where v2(y , z)n depends on m.g.f., N and n = 1, . . . ,N
I
E [Syn−1S
zm−1∆Sn∆Sm] = Sy+z
0 v3(y , z)n,m,
where v3(y , z)n,m depends on m.g.f., N and n,m = 1, . . . ,N
Flavio Angelini, Stefano Herzel Hedging strategies in discrete time
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OutlineThe problem
The optimal strategyThe Delta strategy
Transaction costsApplicationsConclusions
Variance of Delta hedging
I Need to compute all the covariances
I
E [H(y)H(z)] = E [SyNSz
N ] = Sy+z0 m(y + z)N
I
E [H(y)Szn−1∆Sn] = E [Sy
NSzn−1∆Sn] = Sy+z
0 v2(y , z)n,
where v2(y , z)n depends on m.g.f., N and n = 1, . . . ,N
I
E [Syn−1S
zm−1∆Sn∆Sm] = Sy+z
0 v3(y , z)n,m,
where v3(y , z)n,m depends on m.g.f., N and n,m = 1, . . . ,N
Flavio Angelini, Stefano Herzel Hedging strategies in discrete time
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OutlineThe problem
The optimal strategyThe Delta strategy
Transaction costsApplicationsConclusions
Variance of Delta hedging
I Need to compute all the covariances
I
E [H(y)H(z)] = E [SyNSz
N ] = Sy+z0 m(y + z)N
I
E [H(y)Szn−1∆Sn] = E [Sy
NSzn−1∆Sn] = Sy+z
0 v2(y , z)n,
where v2(y , z)n depends on m.g.f., N and n = 1, . . . ,N
I
E [Syn−1S
zm−1∆Sn∆Sm] = Sy+z
0 v3(y , z)n,m,
where v3(y , z)n,m depends on m.g.f., N and n,m = 1, . . . ,N
Flavio Angelini, Stefano Herzel Hedging strategies in discrete time
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OutlineThe problem
The optimal strategyThe Delta strategy
Transaction costsApplicationsConclusions
Variance of Delta hedging
I Need to compute all the covariances
I
E [H(y)H(z)] = E [SyNSz
N ] = Sy+z0 m(y + z)N
I
E [H(y)Szn−1∆Sn] = E [Sy
NSzn−1∆Sn] = Sy+z
0 v2(y , z)n,
where v2(y , z)n depends on m.g.f., N and n = 1, . . . ,N
I
E [Syn−1S
zm−1∆Sn∆Sm] = Sy+z
0 v3(y , z)n,m,
where v3(y , z)n,m depends on m.g.f., N and n,m = 1, . . . ,N
Flavio Angelini, Stefano Herzel Hedging strategies in discrete time
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OutlineThe problem
The optimal strategyThe Delta strategy
Transaction costsApplicationsConclusions
The main result
I For a simple claim and any ϑ given by
ϑn =
∫f (z)nS
z−1n−1Π(dz),
I
E [ε(ϑ, 0)] =
∫Sz
0
[m(z)N − (m(1)− 1)
N∑k=1
f (z)km(z)k−1
]Π(dz)
E [ε(ϑ, 0)2] =
∫ ∫Sy+z
0 V (y , z)Π(dz)Π(dy),
I Examples of strategies: BS-Delta, Wilmott ”improved delta”,local optimal (already in Cerny (2007))
Flavio Angelini, Stefano Herzel Hedging strategies in discrete time
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OutlineThe problem
The optimal strategyThe Delta strategy
Transaction costsApplicationsConclusions
The main result
I For a simple claim and any ϑ given by
ϑn =
∫f (z)nS
z−1n−1Π(dz),
I
E [ε(ϑ, 0)] =
∫Sz
0
[m(z)N − (m(1)− 1)
N∑k=1
f (z)km(z)k−1
]Π(dz)
E [ε(ϑ, 0)2] =
∫ ∫Sy+z
0 V (y , z)Π(dz)Π(dy),
I Examples of strategies: BS-Delta, Wilmott ”improved delta”,local optimal (already in Cerny (2007))
Flavio Angelini, Stefano Herzel Hedging strategies in discrete time
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OutlineThe problem
The optimal strategyThe Delta strategy
Transaction costsApplicationsConclusions
The main result
I For a simple claim and any ϑ given by
ϑn =
∫f (z)nS
z−1n−1Π(dz),
I
E [ε(ϑ, 0)] =
∫Sz
0
[m(z)N − (m(1)− 1)
N∑k=1
f (z)km(z)k−1
]Π(dz)
E [ε(ϑ, 0)2] =
∫ ∫Sy+z
0 V (y , z)Π(dz)Π(dy),
I Examples of strategies: BS-Delta, Wilmott ”improved delta”,local optimal (already in Cerny (2007))
Flavio Angelini, Stefano Herzel Hedging strategies in discrete time
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OutlineThe problem
The optimal strategyThe Delta strategy
Transaction costsApplicationsConclusions
Comments
I With same technique may in principle compute moments ofhigher order
I Can choose a strategy (”f (z)n”) and a model (”m(z)”)
I In particular, one can measure the hedging error of the Deltastrategy for different data generating processes
I The result can be extended to the case of ∆Xn not identicallydistributed
Flavio Angelini, Stefano Herzel Hedging strategies in discrete time
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OutlineThe problem
The optimal strategyThe Delta strategy
Transaction costsApplicationsConclusions
Comments
I With same technique may in principle compute moments ofhigher order
I Can choose a strategy (”f (z)n”) and a model (”m(z)”)
I In particular, one can measure the hedging error of the Deltastrategy for different data generating processes
I The result can be extended to the case of ∆Xn not identicallydistributed
Flavio Angelini, Stefano Herzel Hedging strategies in discrete time
![Page 61: University of Perugia · University of Perugia 30 April - 5 May 2007, II AMAMEF Conference Flavio Angelini, Stefano Herzel Hedging strategies in discrete time. Outline The problem](https://reader036.vdocuments.us/reader036/viewer/2022070107/6024b6fc26c76256032a2447/html5/thumbnails/61.jpg)
OutlineThe problem
The optimal strategyThe Delta strategy
Transaction costsApplicationsConclusions
Comments
I With same technique may in principle compute moments ofhigher order
I Can choose a strategy (”f (z)n”) and a model (”m(z)”)
I In particular, one can measure the hedging error of the Deltastrategy for different data generating processes
I The result can be extended to the case of ∆Xn not identicallydistributed
Flavio Angelini, Stefano Herzel Hedging strategies in discrete time
![Page 62: University of Perugia · University of Perugia 30 April - 5 May 2007, II AMAMEF Conference Flavio Angelini, Stefano Herzel Hedging strategies in discrete time. Outline The problem](https://reader036.vdocuments.us/reader036/viewer/2022070107/6024b6fc26c76256032a2447/html5/thumbnails/62.jpg)
OutlineThe problem
The optimal strategyThe Delta strategy
Transaction costsApplicationsConclusions
Comments
I With same technique may in principle compute moments ofhigher order
I Can choose a strategy (”f (z)n”) and a model (”m(z)”)
I In particular, one can measure the hedging error of the Deltastrategy for different data generating processes
I The result can be extended to the case of ∆Xn not identicallydistributed
Flavio Angelini, Stefano Herzel Hedging strategies in discrete time
![Page 63: University of Perugia · University of Perugia 30 April - 5 May 2007, II AMAMEF Conference Flavio Angelini, Stefano Herzel Hedging strategies in discrete time. Outline The problem](https://reader036.vdocuments.us/reader036/viewer/2022070107/6024b6fc26c76256032a2447/html5/thumbnails/63.jpg)
OutlineThe problem
The optimal strategyThe Delta strategy
Transaction costsApplicationsConclusions
Transaction costs
I With same technique can include transaction costs.
I
TCn =1
2κSn|∆n+1 −∆n|,
for n = 1, . . . ,N − 1
TCN =1
2κSN |∆N |
I
ε(ϑ, c) = H − c − GN(ϑ) +N∑
n=1
TCn.
I No results about optimal strategy
Flavio Angelini, Stefano Herzel Hedging strategies in discrete time
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OutlineThe problem
The optimal strategyThe Delta strategy
Transaction costsApplicationsConclusions
Transaction costs
I With same technique can include transaction costs.
I
TCn =1
2κSn|∆n+1 −∆n|,
for n = 1, . . . ,N − 1
TCN =1
2κSN |∆N |
I
ε(ϑ, c) = H − c − GN(ϑ) +N∑
n=1
TCn.
I No results about optimal strategy
Flavio Angelini, Stefano Herzel Hedging strategies in discrete time
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OutlineThe problem
The optimal strategyThe Delta strategy
Transaction costsApplicationsConclusions
Transaction costs
I With same technique can include transaction costs.
I
TCn =1
2κSn|∆n+1 −∆n|,
for n = 1, . . . ,N − 1
TCN =1
2κSN |∆N |
I
ε(ϑ, c) = H − c − GN(ϑ) +N∑
n=1
TCn.
I No results about optimal strategy
Flavio Angelini, Stefano Herzel Hedging strategies in discrete time
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OutlineThe problem
The optimal strategyThe Delta strategy
Transaction costsApplicationsConclusions
Transaction costs
I With same technique can include transaction costs.
I
TCn =1
2κSn|∆n+1 −∆n|,
for n = 1, . . . ,N − 1
TCN =1
2κSN |∆N |
I
ε(ϑ, c) = H − c − GN(ϑ) +N∑
n=1
TCn.
I No results about optimal strategy
Flavio Angelini, Stefano Herzel Hedging strategies in discrete time
![Page 67: University of Perugia · University of Perugia 30 April - 5 May 2007, II AMAMEF Conference Flavio Angelini, Stefano Herzel Hedging strategies in discrete time. Outline The problem](https://reader036.vdocuments.us/reader036/viewer/2022070107/6024b6fc26c76256032a2447/html5/thumbnails/67.jpg)
OutlineThe problem
The optimal strategyThe Delta strategy
Transaction costsApplicationsConclusions
Transaction costs
I For a simple (and increasing in the underlying as the Delta)strategy can be written too as an inverse Laplace transformbecause
|∆n+1 −∆n| = 1∆Xn>02(∆n+1 −∆n)− (∆n+1 −∆n)
I Can compute the variance of transaction costs and thecovariance with other terms by computing
m+(z) = E [1∆X>0ez∆X ]
Flavio Angelini, Stefano Herzel Hedging strategies in discrete time
![Page 68: University of Perugia · University of Perugia 30 April - 5 May 2007, II AMAMEF Conference Flavio Angelini, Stefano Herzel Hedging strategies in discrete time. Outline The problem](https://reader036.vdocuments.us/reader036/viewer/2022070107/6024b6fc26c76256032a2447/html5/thumbnails/68.jpg)
OutlineThe problem
The optimal strategyThe Delta strategy
Transaction costsApplicationsConclusions
Transaction costs
I For a simple (and increasing in the underlying as the Delta)strategy can be written too as an inverse Laplace transformbecause
|∆n+1 −∆n| = 1∆Xn>02(∆n+1 −∆n)− (∆n+1 −∆n)
I Can compute the variance of transaction costs and thecovariance with other terms by computing
m+(z) = E [1∆X>0ez∆X ]
Flavio Angelini, Stefano Herzel Hedging strategies in discrete time
![Page 69: University of Perugia · University of Perugia 30 April - 5 May 2007, II AMAMEF Conference Flavio Angelini, Stefano Herzel Hedging strategies in discrete time. Outline The problem](https://reader036.vdocuments.us/reader036/viewer/2022070107/6024b6fc26c76256032a2447/html5/thumbnails/69.jpg)
OutlineThe problem
The optimal strategyThe Delta strategy
Transaction costsApplicationsConclusions
Applications
I Assess precision of existing approximations of the variance
I How worse than optimal-variance is Delta hedging?
I Compare performances of various strategies for differentmodels
I To measure performances use Sharpe ratio
s(ϑ, c) =−E [ε(ϑ, c)]√var(ε(ϑ, c))
Flavio Angelini, Stefano Herzel Hedging strategies in discrete time
![Page 70: University of Perugia · University of Perugia 30 April - 5 May 2007, II AMAMEF Conference Flavio Angelini, Stefano Herzel Hedging strategies in discrete time. Outline The problem](https://reader036.vdocuments.us/reader036/viewer/2022070107/6024b6fc26c76256032a2447/html5/thumbnails/70.jpg)
OutlineThe problem
The optimal strategyThe Delta strategy
Transaction costsApplicationsConclusions
Applications
I Assess precision of existing approximations of the variance
I How worse than optimal-variance is Delta hedging?
I Compare performances of various strategies for differentmodels
I To measure performances use Sharpe ratio
s(ϑ, c) =−E [ε(ϑ, c)]√var(ε(ϑ, c))
Flavio Angelini, Stefano Herzel Hedging strategies in discrete time
![Page 71: University of Perugia · University of Perugia 30 April - 5 May 2007, II AMAMEF Conference Flavio Angelini, Stefano Herzel Hedging strategies in discrete time. Outline The problem](https://reader036.vdocuments.us/reader036/viewer/2022070107/6024b6fc26c76256032a2447/html5/thumbnails/71.jpg)
OutlineThe problem
The optimal strategyThe Delta strategy
Transaction costsApplicationsConclusions
Applications
I Assess precision of existing approximations of the variance
I How worse than optimal-variance is Delta hedging?
I Compare performances of various strategies for differentmodels
I To measure performances use Sharpe ratio
s(ϑ, c) =−E [ε(ϑ, c)]√var(ε(ϑ, c))
Flavio Angelini, Stefano Herzel Hedging strategies in discrete time
![Page 72: University of Perugia · University of Perugia 30 April - 5 May 2007, II AMAMEF Conference Flavio Angelini, Stefano Herzel Hedging strategies in discrete time. Outline The problem](https://reader036.vdocuments.us/reader036/viewer/2022070107/6024b6fc26c76256032a2447/html5/thumbnails/72.jpg)
OutlineThe problem
The optimal strategyThe Delta strategy
Transaction costsApplicationsConclusions
Applications
I Assess precision of existing approximations of the variance
I How worse than optimal-variance is Delta hedging?
I Compare performances of various strategies for differentmodels
I To measure performances use Sharpe ratio
s(ϑ, c) =−E [ε(ϑ, c)]√var(ε(ϑ, c))
Flavio Angelini, Stefano Herzel Hedging strategies in discrete time
![Page 73: University of Perugia · University of Perugia 30 April - 5 May 2007, II AMAMEF Conference Flavio Angelini, Stefano Herzel Hedging strategies in discrete time. Outline The problem](https://reader036.vdocuments.us/reader036/viewer/2022070107/6024b6fc26c76256032a2447/html5/thumbnails/73.jpg)
OutlineThe problem
The optimal strategyThe Delta strategy
Transaction costsApplicationsConclusions
Relative errors of approximations of standard deviation
0 10 20 30 40 50 60 70−0.05
0
0.05
0.1
0.15
0.2
N
kdvegakd
Flavio Angelini, Stefano Herzel Hedging strategies in discrete time
![Page 74: University of Perugia · University of Perugia 30 April - 5 May 2007, II AMAMEF Conference Flavio Angelini, Stefano Herzel Hedging strategies in discrete time. Outline The problem](https://reader036.vdocuments.us/reader036/viewer/2022070107/6024b6fc26c76256032a2447/html5/thumbnails/74.jpg)
OutlineThe problem
The optimal strategyThe Delta strategy
Transaction costsApplicationsConclusions
Model risk
I Data generating process: Merton jump-diffusion process withnormally distributed jumps, with returns with annual meanµ ≈ 0.14 and volatility σ ≈ 0.44
I Hedging ratios: B-S Delta and B-S locally optimal computedwith parameters µ and σ, Merton optimal
I ATM Call option (K = 100), T = 3 months, number oftrading dates from 1 to 65
Flavio Angelini, Stefano Herzel Hedging strategies in discrete time
![Page 75: University of Perugia · University of Perugia 30 April - 5 May 2007, II AMAMEF Conference Flavio Angelini, Stefano Herzel Hedging strategies in discrete time. Outline The problem](https://reader036.vdocuments.us/reader036/viewer/2022070107/6024b6fc26c76256032a2447/html5/thumbnails/75.jpg)
OutlineThe problem
The optimal strategyThe Delta strategy
Transaction costsApplicationsConclusions
Model risk
I Data generating process: Merton jump-diffusion process withnormally distributed jumps, with returns with annual meanµ ≈ 0.14 and volatility σ ≈ 0.44
I Hedging ratios: B-S Delta and B-S locally optimal computedwith parameters µ and σ, Merton optimal
I ATM Call option (K = 100), T = 3 months, number oftrading dates from 1 to 65
Flavio Angelini, Stefano Herzel Hedging strategies in discrete time
![Page 76: University of Perugia · University of Perugia 30 April - 5 May 2007, II AMAMEF Conference Flavio Angelini, Stefano Herzel Hedging strategies in discrete time. Outline The problem](https://reader036.vdocuments.us/reader036/viewer/2022070107/6024b6fc26c76256032a2447/html5/thumbnails/76.jpg)
OutlineThe problem
The optimal strategyThe Delta strategy
Transaction costsApplicationsConclusions
Model risk
I Data generating process: Merton jump-diffusion process withnormally distributed jumps, with returns with annual meanµ ≈ 0.14 and volatility σ ≈ 0.44
I Hedging ratios: B-S Delta and B-S locally optimal computedwith parameters µ and σ, Merton optimal
I ATM Call option (K = 100), T = 3 months, number oftrading dates from 1 to 65
Flavio Angelini, Stefano Herzel Hedging strategies in discrete time
![Page 77: University of Perugia · University of Perugia 30 April - 5 May 2007, II AMAMEF Conference Flavio Angelini, Stefano Herzel Hedging strategies in discrete time. Outline The problem](https://reader036.vdocuments.us/reader036/viewer/2022070107/6024b6fc26c76256032a2447/html5/thumbnails/77.jpg)
OutlineThe problem
The optimal strategyThe Delta strategy
Transaction costsApplicationsConclusions
Model risk
Sharpe index of different strategies
0 10 20 30 40 50 60 700
0.01
0.02
0.03
0.04
0.05
0.06
0.07
N
optdeltaloc opt bs
Flavio Angelini, Stefano Herzel Hedging strategies in discrete time
![Page 78: University of Perugia · University of Perugia 30 April - 5 May 2007, II AMAMEF Conference Flavio Angelini, Stefano Herzel Hedging strategies in discrete time. Outline The problem](https://reader036.vdocuments.us/reader036/viewer/2022070107/6024b6fc26c76256032a2447/html5/thumbnails/78.jpg)
OutlineThe problem
The optimal strategyThe Delta strategy
Transaction costsApplicationsConclusions
Model mispecification
I Data generating process: Black-Scholes model with given µand σ
I Hedging ratios: B-S Delta and B-S local optimal computedwith parameters µ0 and σ0 = 0.3
I ATM Call option (K = 100), T = 3 months, number oftrading dates from 1 to 65
I If σ < σ0 expect a gain from trading strategy
Flavio Angelini, Stefano Herzel Hedging strategies in discrete time
![Page 79: University of Perugia · University of Perugia 30 April - 5 May 2007, II AMAMEF Conference Flavio Angelini, Stefano Herzel Hedging strategies in discrete time. Outline The problem](https://reader036.vdocuments.us/reader036/viewer/2022070107/6024b6fc26c76256032a2447/html5/thumbnails/79.jpg)
OutlineThe problem
The optimal strategyThe Delta strategy
Transaction costsApplicationsConclusions
Model mispecification
I Data generating process: Black-Scholes model with given µand σ
I Hedging ratios: B-S Delta and B-S local optimal computedwith parameters µ0 and σ0 = 0.3
I ATM Call option (K = 100), T = 3 months, number oftrading dates from 1 to 65
I If σ < σ0 expect a gain from trading strategy
Flavio Angelini, Stefano Herzel Hedging strategies in discrete time
![Page 80: University of Perugia · University of Perugia 30 April - 5 May 2007, II AMAMEF Conference Flavio Angelini, Stefano Herzel Hedging strategies in discrete time. Outline The problem](https://reader036.vdocuments.us/reader036/viewer/2022070107/6024b6fc26c76256032a2447/html5/thumbnails/80.jpg)
OutlineThe problem
The optimal strategyThe Delta strategy
Transaction costsApplicationsConclusions
Model mispecification
I Data generating process: Black-Scholes model with given µand σ
I Hedging ratios: B-S Delta and B-S local optimal computedwith parameters µ0 and σ0 = 0.3
I ATM Call option (K = 100), T = 3 months, number oftrading dates from 1 to 65
I If σ < σ0 expect a gain from trading strategy
Flavio Angelini, Stefano Herzel Hedging strategies in discrete time
![Page 81: University of Perugia · University of Perugia 30 April - 5 May 2007, II AMAMEF Conference Flavio Angelini, Stefano Herzel Hedging strategies in discrete time. Outline The problem](https://reader036.vdocuments.us/reader036/viewer/2022070107/6024b6fc26c76256032a2447/html5/thumbnails/81.jpg)
OutlineThe problem
The optimal strategyThe Delta strategy
Transaction costsApplicationsConclusions
Model mispecification
I Data generating process: Black-Scholes model with given µand σ
I Hedging ratios: B-S Delta and B-S local optimal computedwith parameters µ0 and σ0 = 0.3
I ATM Call option (K = 100), T = 3 months, number oftrading dates from 1 to 65
I If σ < σ0 expect a gain from trading strategy
Flavio Angelini, Stefano Herzel Hedging strategies in discrete time
![Page 82: University of Perugia · University of Perugia 30 April - 5 May 2007, II AMAMEF Conference Flavio Angelini, Stefano Herzel Hedging strategies in discrete time. Outline The problem](https://reader036.vdocuments.us/reader036/viewer/2022070107/6024b6fc26c76256032a2447/html5/thumbnails/82.jpg)
OutlineThe problem
The optimal strategyThe Delta strategy
Transaction costsApplicationsConclusions
Sharpe index as a function of realized volatility σ, withµ0 = µ = 0.1 and N = 10
0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5−2
−1
0
1
2
3
4
5
σ
deltaloc. opt.
Flavio Angelini, Stefano Herzel Hedging strategies in discrete time
![Page 83: University of Perugia · University of Perugia 30 April - 5 May 2007, II AMAMEF Conference Flavio Angelini, Stefano Herzel Hedging strategies in discrete time. Outline The problem](https://reader036.vdocuments.us/reader036/viewer/2022070107/6024b6fc26c76256032a2447/html5/thumbnails/83.jpg)
OutlineThe problem
The optimal strategyThe Delta strategy
Transaction costsApplicationsConclusions
s(∆)− s(ξH) as a function of σ, for different µ (σ0 = 0.3,µ0 = 0.1)
0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5−0.08
−0.07
−0.06
−0.05
−0.04
−0.03
−0.02
−0.01
0
0.01
µ = 0µ =0.1µ =−0.1
Flavio Angelini, Stefano Herzel Hedging strategies in discrete time
![Page 84: University of Perugia · University of Perugia 30 April - 5 May 2007, II AMAMEF Conference Flavio Angelini, Stefano Herzel Hedging strategies in discrete time. Outline The problem](https://reader036.vdocuments.us/reader036/viewer/2022070107/6024b6fc26c76256032a2447/html5/thumbnails/84.jpg)
OutlineThe problem
The optimal strategyThe Delta strategy
Transaction costsApplicationsConclusions
Sharpe index of local optimal strategy as a function of σ
and µ (σ0 = 0.3, µ0 = 0, S = K = 100, N = 10)
−0.2
−0.1
0
0.1
0.2
0.1
0.2
0.3
0.4
0.5−5
−4
−3
−2
−1
0
1
2
µσ
Flavio Angelini, Stefano Herzel Hedging strategies in discrete time
![Page 85: University of Perugia · University of Perugia 30 April - 5 May 2007, II AMAMEF Conference Flavio Angelini, Stefano Herzel Hedging strategies in discrete time. Outline The problem](https://reader036.vdocuments.us/reader036/viewer/2022070107/6024b6fc26c76256032a2447/html5/thumbnails/85.jpg)
OutlineThe problem
The optimal strategyThe Delta strategy
Transaction costsApplicationsConclusions
Conclusions
I We have an efficient way to compute moments of hedgingerrors of strategies in presence of transaction costs for simpleclaims and for a wide class of data generating process
I This allows to measure the performance of hedging strategiesin different settings, for instance under model mispecification
I Need to study the effect of transaction costs
I Is it possible to find optimal strategy? Or best hedgingvolatility?
I Robust hedging
I Multi-dimensional
Flavio Angelini, Stefano Herzel Hedging strategies in discrete time
![Page 86: University of Perugia · University of Perugia 30 April - 5 May 2007, II AMAMEF Conference Flavio Angelini, Stefano Herzel Hedging strategies in discrete time. Outline The problem](https://reader036.vdocuments.us/reader036/viewer/2022070107/6024b6fc26c76256032a2447/html5/thumbnails/86.jpg)
OutlineThe problem
The optimal strategyThe Delta strategy
Transaction costsApplicationsConclusions
Conclusions
I We have an efficient way to compute moments of hedgingerrors of strategies in presence of transaction costs for simpleclaims and for a wide class of data generating process
I This allows to measure the performance of hedging strategiesin different settings, for instance under model mispecification
I Need to study the effect of transaction costs
I Is it possible to find optimal strategy? Or best hedgingvolatility?
I Robust hedging
I Multi-dimensional
Flavio Angelini, Stefano Herzel Hedging strategies in discrete time
![Page 87: University of Perugia · University of Perugia 30 April - 5 May 2007, II AMAMEF Conference Flavio Angelini, Stefano Herzel Hedging strategies in discrete time. Outline The problem](https://reader036.vdocuments.us/reader036/viewer/2022070107/6024b6fc26c76256032a2447/html5/thumbnails/87.jpg)
OutlineThe problem
The optimal strategyThe Delta strategy
Transaction costsApplicationsConclusions
Conclusions
I We have an efficient way to compute moments of hedgingerrors of strategies in presence of transaction costs for simpleclaims and for a wide class of data generating process
I This allows to measure the performance of hedging strategiesin different settings, for instance under model mispecification
I Need to study the effect of transaction costs
I Is it possible to find optimal strategy? Or best hedgingvolatility?
I Robust hedging
I Multi-dimensional
Flavio Angelini, Stefano Herzel Hedging strategies in discrete time
![Page 88: University of Perugia · University of Perugia 30 April - 5 May 2007, II AMAMEF Conference Flavio Angelini, Stefano Herzel Hedging strategies in discrete time. Outline The problem](https://reader036.vdocuments.us/reader036/viewer/2022070107/6024b6fc26c76256032a2447/html5/thumbnails/88.jpg)
OutlineThe problem
The optimal strategyThe Delta strategy
Transaction costsApplicationsConclusions
Conclusions
I We have an efficient way to compute moments of hedgingerrors of strategies in presence of transaction costs for simpleclaims and for a wide class of data generating process
I This allows to measure the performance of hedging strategiesin different settings, for instance under model mispecification
I Need to study the effect of transaction costs
I Is it possible to find optimal strategy? Or best hedgingvolatility?
I Robust hedging
I Multi-dimensional
Flavio Angelini, Stefano Herzel Hedging strategies in discrete time
![Page 89: University of Perugia · University of Perugia 30 April - 5 May 2007, II AMAMEF Conference Flavio Angelini, Stefano Herzel Hedging strategies in discrete time. Outline The problem](https://reader036.vdocuments.us/reader036/viewer/2022070107/6024b6fc26c76256032a2447/html5/thumbnails/89.jpg)
OutlineThe problem
The optimal strategyThe Delta strategy
Transaction costsApplicationsConclusions
Conclusions
I We have an efficient way to compute moments of hedgingerrors of strategies in presence of transaction costs for simpleclaims and for a wide class of data generating process
I This allows to measure the performance of hedging strategiesin different settings, for instance under model mispecification
I Need to study the effect of transaction costs
I Is it possible to find optimal strategy? Or best hedgingvolatility?
I Robust hedging
I Multi-dimensional
Flavio Angelini, Stefano Herzel Hedging strategies in discrete time
![Page 90: University of Perugia · University of Perugia 30 April - 5 May 2007, II AMAMEF Conference Flavio Angelini, Stefano Herzel Hedging strategies in discrete time. Outline The problem](https://reader036.vdocuments.us/reader036/viewer/2022070107/6024b6fc26c76256032a2447/html5/thumbnails/90.jpg)
OutlineThe problem
The optimal strategyThe Delta strategy
Transaction costsApplicationsConclusions
Conclusions
I We have an efficient way to compute moments of hedgingerrors of strategies in presence of transaction costs for simpleclaims and for a wide class of data generating process
I This allows to measure the performance of hedging strategiesin different settings, for instance under model mispecification
I Need to study the effect of transaction costs
I Is it possible to find optimal strategy? Or best hedgingvolatility?
I Robust hedging
I Multi-dimensional
Flavio Angelini, Stefano Herzel Hedging strategies in discrete time
![Page 91: University of Perugia · University of Perugia 30 April - 5 May 2007, II AMAMEF Conference Flavio Angelini, Stefano Herzel Hedging strategies in discrete time. Outline The problem](https://reader036.vdocuments.us/reader036/viewer/2022070107/6024b6fc26c76256032a2447/html5/thumbnails/91.jpg)
OutlineThe problem
The optimal strategyThe Delta strategy
Transaction costsApplicationsConclusions
Appendix: Inverse Laplace transform
I The (one-dimensional) Inverse Laplace transform is
f (t) = L−1F (s) =1
2πi
∫ R+i∞
R−i∞estF (s)ds
I There are several algorithms that perform numerical inversionof the Laplace transform in an efficient way
I The computation of variance involve a double dimensionalinversion. This is usually a harder task.
Flavio Angelini, Stefano Herzel Hedging strategies in discrete time
![Page 92: University of Perugia · University of Perugia 30 April - 5 May 2007, II AMAMEF Conference Flavio Angelini, Stefano Herzel Hedging strategies in discrete time. Outline The problem](https://reader036.vdocuments.us/reader036/viewer/2022070107/6024b6fc26c76256032a2447/html5/thumbnails/92.jpg)
OutlineThe problem
The optimal strategyThe Delta strategy
Transaction costsApplicationsConclusions
Appendix: Inverse Laplace transform
I The (one-dimensional) Inverse Laplace transform is
f (t) = L−1F (s) =1
2πi
∫ R+i∞
R−i∞estF (s)ds
I There are several algorithms that perform numerical inversionof the Laplace transform in an efficient way
I The computation of variance involve a double dimensionalinversion. This is usually a harder task.
Flavio Angelini, Stefano Herzel Hedging strategies in discrete time
![Page 93: University of Perugia · University of Perugia 30 April - 5 May 2007, II AMAMEF Conference Flavio Angelini, Stefano Herzel Hedging strategies in discrete time. Outline The problem](https://reader036.vdocuments.us/reader036/viewer/2022070107/6024b6fc26c76256032a2447/html5/thumbnails/93.jpg)
OutlineThe problem
The optimal strategyThe Delta strategy
Transaction costsApplicationsConclusions
Appendix: Inverse Laplace transform
I The (one-dimensional) Inverse Laplace transform is
f (t) = L−1F (s) =1
2πi
∫ R+i∞
R−i∞estF (s)ds
I There are several algorithms that perform numerical inversionof the Laplace transform in an efficient way
I The computation of variance involve a double dimensionalinversion. This is usually a harder task.
Flavio Angelini, Stefano Herzel Hedging strategies in discrete time
![Page 94: University of Perugia · University of Perugia 30 April - 5 May 2007, II AMAMEF Conference Flavio Angelini, Stefano Herzel Hedging strategies in discrete time. Outline The problem](https://reader036.vdocuments.us/reader036/viewer/2022070107/6024b6fc26c76256032a2447/html5/thumbnails/94.jpg)
OutlineThe problem
The optimal strategyThe Delta strategy
Transaction costsApplicationsConclusions
Appendix: Numerical implementation
I The formulas we wish to compute involve one- andtwo-dimensional Laplace transforms.
I There are at least two possible approaches: numericalintegration and inversion of Laplace transform.
I Second approach, implementing the algorithms in MATLAB.
I One-dimensional case: we used ”invlap.m” constructed byHollenbeck (1998), very accurate
I Bi-dimensional case: we wrote a code based on Choudhury,Lucantoni, Whitt (1994), quite accurate.
Flavio Angelini, Stefano Herzel Hedging strategies in discrete time