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

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

OutlineThe problem



The problem

The optimal strategy

The Delta strategy

Transaction costs

Applications

Conclusions


OutlineThe problem



The problem


The Delta strategy

Transaction costs

Applications

Conclusions


OutlineThe problem



The problem


The Delta strategy

Transaction costs

Applications

Conclusions


OutlineThe problem



The problem


The Delta strategy

Transaction costs

Applications

Conclusions


OutlineThe problem



The problem


The Delta strategy

Transaction costs

Applications

Conclusions


OutlineThe problem



The problem


The Delta strategy

Transaction costs

Applications

Conclusions


OutlineThe problem



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?


OutlineThe problem



Introduction






OutlineThe problem



Introduction






OutlineThe problem



Introduction






OutlineThe problem



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)


OutlineThe problem



The setting



GN(ϑ) =N∑

j=1

ϑj∆Sj



ε(ϑ, c) = H − c − GN(ϑ)


OutlineThe problem



The setting



GN(ϑ) =N∑

j=1

ϑj∆Sj



ε(ϑ, c) = H − c − GN(ϑ)


OutlineThe problem



The setting



GN(ϑ) =N∑

j=1

ϑj∆Sj



ε(ϑ, c) = H − c − GN(ϑ)


OutlineThe problem



The setting



GN(ϑ) =N∑

j=1

ϑj∆Sj



ε(ϑ, c) = H − c − GN(ϑ)


OutlineThe problem



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


OutlineThe problem



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


OutlineThe problem



The problem



minϑ∈Θ

E[ε(ϑ, c)2

]for fixed c ∈ IR

I Computation ofE

[ε(ϑ, c)2

]for given c , ϑ


OutlineThe problem



The problem



minϑ∈Θ

E[ε(ϑ, c)2

]for fixed c ∈ IR

I Computation ofE

[ε(ϑ, c)2

]for given c , ϑ


OutlineThe problem



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


OutlineThe problem



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)”.


OutlineThe problem



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.


OutlineThe problem






Sn = S0 exp(Xn),




OutlineThe problem






Sn = S0 exp(Xn),




OutlineThe problem






Sn = S0 exp(Xn),




OutlineThe problem






Sn = S0 exp(Xn),




OutlineThe problem




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.


OutlineThe problem




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.


OutlineThe problem



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


OutlineThe problem





H(z) = SzN = Sz

0 ezXN





OutlineThe problem





H(z) = SzN = Sz

0 ezXN





OutlineThe problem





H(z) = SzN = Sz

0 ezXN





OutlineThe problem




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


OutlineThe problem




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


OutlineThe problem



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)


OutlineThe problem



The results


ξ1 =

∫Sz−1



V0 =

∫Sz

0 h(z)N−1Π(dz)

I Optimal variance

Var0 =



OutlineThe problem



The results


ξ1 =

∫Sz−1



V0 =

∫Sz

0 h(z)N−1Π(dz)

I Optimal variance

Var0 =



OutlineThe problem



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?


OutlineThe problem



A relevant question





OutlineThe problem



A relevant question





OutlineThe problem



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


OutlineThe problem




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


OutlineThe problem




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


OutlineThe problem




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


OutlineThe problem



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 .


OutlineThe problem



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 .


OutlineThe problem



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


OutlineThe problem



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


OutlineThe problem



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


OutlineThe problem



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


OutlineThe problem



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


OutlineThe problem





I


N ] = Sy+z0 m(y + z)N

I



0 v2(y , z)n,


I

E [Syn−1S


0 v3(y , z)n,m,



OutlineThe problem





I


N ] = Sy+z0 m(y + z)N

I



0 v2(y , z)n,


I

E [Syn−1S


0 v3(y , z)n,m,



OutlineThe problem





I


N ] = Sy+z0 m(y + z)N

I



0 v2(y , z)n,


I

E [Syn−1S


0 v3(y , z)n,m,



OutlineThe problem



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))


OutlineThe problem



The main result


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




OutlineThe problem



The main result


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




OutlineThe problem



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


OutlineThe problem



Comments






OutlineThe problem



Comments






OutlineThe problem



Comments






OutlineThe problem



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


OutlineThe problem



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.



OutlineThe problem



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.



OutlineThe problem



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.



OutlineThe problem



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 ]


OutlineThe problem



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 ]


OutlineThe problem



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))


OutlineThe problem



Applications





s(ϑ, c) =−E [ε(ϑ, c)]√var(ε(ϑ, c))


OutlineThe problem



Applications





s(ϑ, c) =−E [ε(ϑ, c)]√var(ε(ϑ, c))


OutlineThe problem



Applications





s(ϑ, c) =−E [ε(ϑ, c)]√var(ε(ϑ, c))


OutlineThe problem



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


OutlineThe problem



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


OutlineThe problem



Model risk





OutlineThe problem



Model risk





OutlineThe problem



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


OutlineThe problem



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 If σ < σ0 expect a gain from trading strategy


OutlineThe problem









OutlineThe problem









OutlineThe problem









OutlineThe problem



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.


OutlineThe problem



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


OutlineThe problem



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


OutlineThe problem



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


OutlineThe problem



Conclusions





I Robust hedging

I Multi-dimensional


OutlineThe problem



Conclusions





I Robust hedging

I Multi-dimensional


OutlineThe problem



Conclusions





I Robust hedging

I Multi-dimensional


OutlineThe problem



Conclusions





I Robust hedging

I Multi-dimensional


OutlineThe problem



Conclusions





I Robust hedging

I Multi-dimensional


OutlineThe problem



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.


OutlineThe problem





f (t) = L−1F (s) =1

2πi

∫ R+i∞

R−i∞estF (s)ds




OutlineThe problem





f (t) = L−1F (s) =1

2πi

∫ R+i∞

R−i∞estF (s)ds




OutlineThe problem



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.


Documents

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