parametrix of static hedge (of a timing risk)...i will introduce a parametrix of barrier type...
TRANSCRIPT
. . . . . .
Parametrix of Static Hedge (of a Timing Risk)
Jiro Akahori
Ritsumeikan University
Dec. 1, 2014
joint work with Yuri Imamura1 and Flavia Barsotti 2
1Ritsumeikan University2Risk Methodologies, Group Financial Risks, UniCredit SpA, Milan.
The views presented in this paper are solely those of the author and do notnecessarily represent those of UniCredit Spa.
. . . . . .
Overview, Parametrix?
• A parametrix is an approximation to a fundamental solutionof a PDE (as Wikipedia says) by an easier one (sometimes itis a Gaussian kernel).
• We may understand it as an approximation of an option pricein a general diffusion environment by a Black-Scholes one. (A.Pascucci, ...)
• In the latter, it is known that a perfect static replication of abarrier option is possible. (P. Carr,...)
• I will introduce a parametrix of barrier type options in ageneral diffusion environment by the static hedge in BS.
. . . . . .
Overview, Parametrix?
• A parametrix is an approximation to a fundamental solutionof a PDE (as Wikipedia says) by an easier one (sometimes itis a Gaussian kernel).
• We may understand it as an approximation of an option pricein a general diffusion environment by a Black-Scholes one. (A.Pascucci, ...)
• In the latter, it is known that a perfect static replication of abarrier option is possible. (P. Carr,...)
• I will introduce a parametrix of barrier type options in ageneral diffusion environment by the static hedge in BS.
. . . . . .
Overview, Parametrix?
• A parametrix is an approximation to a fundamental solutionof a PDE (as Wikipedia says) by an easier one (sometimes itis a Gaussian kernel).
• We may understand it as an approximation of an option pricein a general diffusion environment by a Black-Scholes one. (A.Pascucci, ...)
• In the latter, it is known that a perfect static replication of abarrier option is possible. (P. Carr,...)
• I will introduce a parametrix of barrier type options in ageneral diffusion environment by the static hedge in BS.
. . . . . .
Overview, Parametrix?
• A parametrix is an approximation to a fundamental solutionof a PDE (as Wikipedia says) by an easier one (sometimes itis a Gaussian kernel).
• We may understand it as an approximation of an option pricein a general diffusion environment by a Black-Scholes one. (A.Pascucci, ...)
• In the latter, it is known that a perfect static replication of abarrier option is possible. (P. Carr,...)
• I will introduce a parametrix of barrier type options in ageneral diffusion environment by the static hedge in BS.
. . . . . .
Timing Risk?
• Timing risk is a risk associated with a payment time.
• Barrier options, defaultable bonds, or American options havea timing risk.
• European claims do not have any timing risk since thepayment occurs only at the prescribed time.
. . . . . .
Timing Risk?
• Timing risk is a risk associated with a payment time.
• Barrier options, defaultable bonds, or American options havea timing risk.
• European claims do not have any timing risk since thepayment occurs only at the prescribed time.
. . . . . .
Timing Risk?
• Timing risk is a risk associated with a payment time.
• Barrier options, defaultable bonds, or American options havea timing risk.
• European claims do not have any timing risk since thepayment occurs only at the prescribed time.
. . . . . .
(Semi-) Static Hedge?
• We do not want to take a position with a timing risk. Werather want to exchange the position to the one without atiming risk.
• “Without a timing risk” means a portfolio composed only ofunderlyings and European type options. We call such anexchange technique semi-static hedge.
. . . . . .
(Semi-) Static Hedge?
• We do not want to take a position with a timing risk. Werather want to exchange the position to the one without atiming risk.
• “Without a timing risk” means a portfolio composed only ofunderlyings and European type options. We call such anexchange technique semi-static hedge.
. . . . . .
Static Hedge of a knock out option [underBS]
It has been widely known since the paper by P. Carr and J. Bowie(1994) that under Black-Scholes assumptions, simple Barrieroptions can be hedged by a static position of two options.
. . . . . .
Static Hedge of a knock OUT optionunder BS
When the option to be hedged is a call option knocked out whenthe underlying hits a boundary, it can be hedged by
• Start with long position of a call option and a short positionof a put.
• When the boundary is hit, the value of the two optioncoincides (by the BS assumption) and so we can liquidate theposition with no extra cost.
• If the boundary is never hit until the maturity, the call optionat hand hedges the barrier option.
. . . . . .
Static Hedge of a knock OUT optionunder BS
When the option to be hedged is a call option knocked out whenthe underlying hits a boundary, it can be hedged by
• Start with long position of a call option and a short positionof a put.
• When the boundary is hit, the value of the two optioncoincides (by the BS assumption) and so we can liquidate theposition with no extra cost.
• If the boundary is never hit until the maturity, the call optionat hand hedges the barrier option.
. . . . . .
Static Hedge of a knock OUT optionunder BS
When the option to be hedged is a call option knocked out whenthe underlying hits a boundary, it can be hedged by
• Start with long position of a call option and a short positionof a put.
• When the boundary is hit, the value of the two optioncoincides (by the BS assumption) and so we can liquidate theposition with no extra cost.
• If the boundary is never hit until the maturity, the call optionat hand hedges the barrier option.
. . . . . .
Static Hedge of a knock OUT optionunder BS
When the option to be hedged is a call option knocked out whenthe underlying hits a boundary, it can be hedged by
• Start with long position of a call option and a short positionof a put.
• When the boundary is hit, the value of the two optioncoincides (by the BS assumption) and so we can liquidate theposition with no extra cost.
• If the boundary is never hit until the maturity, the call optionat hand hedges the barrier option.
. . . . . .
Static Hedge of a knock IN option underBS
When the option to be hedged is a call option knocked in when theunderlying hits a boundary, it can be hedged by
• Start with long position of a put.
• When the boundary is hit, the value of put option coincides(by the BS assumption) with that of call option and so we canexchange them with no extra cost.
• If the boundary is never hit until the maturity, both theknock-in option and the put option at hand are worth zero.
. . . . . .
Static Hedge of a knock IN option underBS
When the option to be hedged is a call option knocked in when theunderlying hits a boundary, it can be hedged by
• Start with long position of a put.
• When the boundary is hit, the value of put option coincides(by the BS assumption) with that of call option and so we canexchange them with no extra cost.
• If the boundary is never hit until the maturity, both theknock-in option and the put option at hand are worth zero.
. . . . . .
Static Hedge of a knock IN option underBS
When the option to be hedged is a call option knocked in when theunderlying hits a boundary, it can be hedged by
• Start with long position of a put.
• When the boundary is hit, the value of put option coincides(by the BS assumption) with that of call option and so we canexchange them with no extra cost.
• If the boundary is never hit until the maturity, both theknock-in option and the put option at hand are worth zero.
. . . . . .
Static Hedge of a knock IN option underBS
When the option to be hedged is a call option knocked in when theunderlying hits a boundary, it can be hedged by
• Start with long position of a put.
• When the boundary is hit, the value of put option coincides(by the BS assumption) with that of call option and so we canexchange them with no extra cost.
• If the boundary is never hit until the maturity, both theknock-in option and the put option at hand are worth zero.
. . . . . .
Static Hedge of the simplest timing risk
P. Carr and J. Picron (1999), under a Black-Scholes environment,tried to apply the semi-static hedging formula of barrier options tohedge a constant payment at a stopping time (which actually is ahitting time).
. . . . . .
Static Hedge of the simplest timing risk
• P. Carr and J. Picron found that an integration of thesemi-static hedging formula for barrier options provides asemi-static hedge of the timing risk of constant payment.
• Under Black-Scholes economy, the integral of the semi-statichedging formula of barrier options of Bowie and Carr typeprovides a perfect static hedge of the timing risk.
• The integral (with respect to maturities) implies that thestatic hedging portfolio consists of (infinitesimal amount of)options with different (continuum of) maturities, which shouldbe discretized in practice.
. . . . . .
Static Hedge of the simplest timing risk
• P. Carr and J. Picron found that an integration of thesemi-static hedging formula for barrier options provides asemi-static hedge of the timing risk of constant payment.
• Under Black-Scholes economy, the integral of the semi-statichedging formula of barrier options of Bowie and Carr typeprovides a perfect static hedge of the timing risk.
• The integral (with respect to maturities) implies that thestatic hedging portfolio consists of (infinitesimal amount of)options with different (continuum of) maturities, which shouldbe discretized in practice.
. . . . . .
Static Hedge of the simplest timing risk
• P. Carr and J. Picron found that an integration of thesemi-static hedging formula for barrier options provides asemi-static hedge of the timing risk of constant payment.
• Under Black-Scholes economy, the integral of the semi-statichedging formula of barrier options of Bowie and Carr typeprovides a perfect static hedge of the timing risk.
• The integral (with respect to maturities) implies that thestatic hedging portfolio consists of (infinitesimal amount of)options with different (continuum of) maturities, which shouldbe discretized in practice.
. . . . . .
Iteration of Static Hedges
• In a general diffusion model, the static hedge of B-S of barrieroptions cause an error at the knock-out/in time.
• The error part is also a timing risk, therefore a static hedge ofthe error (the timing risk) is provided by an integral of thebarrier option formula.
• Again the barrier option in the integral is statically hedgedwith error. The error which is again a timing risk, etc ...
• This procedure gives a (Taylor-like) series expansion ofsemi-static hedge of a timing risk.
• This is our Parametrix.
. . . . . .
Iteration of Static Hedges
• In a general diffusion model, the static hedge of B-S of barrieroptions cause an error at the knock-out/in time.
• The error part is also a timing risk, therefore a static hedge ofthe error (the timing risk) is provided by an integral of thebarrier option formula.
• Again the barrier option in the integral is statically hedgedwith error. The error which is again a timing risk, etc ...
• This procedure gives a (Taylor-like) series expansion ofsemi-static hedge of a timing risk.
• This is our Parametrix.
. . . . . .
Iteration of Static Hedges
• In a general diffusion model, the static hedge of B-S of barrieroptions cause an error at the knock-out/in time.
• The error part is also a timing risk, therefore a static hedge ofthe error (the timing risk) is provided by an integral of thebarrier option formula.
• Again the barrier option in the integral is statically hedgedwith error. The error which is again a timing risk, etc ...
• This procedure gives a (Taylor-like) series expansion ofsemi-static hedge of a timing risk.
• This is our Parametrix.
. . . . . .
Iteration of Static Hedges
• In a general diffusion model, the static hedge of B-S of barrieroptions cause an error at the knock-out/in time.
• The error part is also a timing risk, therefore a static hedge ofthe error (the timing risk) is provided by an integral of thebarrier option formula.
• Again the barrier option in the integral is statically hedgedwith error. The error which is again a timing risk, etc ...
• This procedure gives a (Taylor-like) series expansion ofsemi-static hedge of a timing risk.
• This is our Parametrix.
. . . . . .
Iteration of Static Hedges
• In a general diffusion model, the static hedge of B-S of barrieroptions cause an error at the knock-out/in time.
• The error part is also a timing risk, therefore a static hedge ofthe error (the timing risk) is provided by an integral of thebarrier option formula.
• Again the barrier option in the integral is statically hedgedwith error. The error which is again a timing risk, etc ...
• This procedure gives a (Taylor-like) series expansion ofsemi-static hedge of a timing risk.
• This is our Parametrix.
. . . . . .
From the next slide, I will use mathematics explicitly.
We will work on a filtered probabilty space (Ω,F ,P, Ft), and...etc.
. . . . . .
From the next slide, I will use mathematics explicitly.
We will work on a filtered probabilty space (Ω,F ,P, Ft), and...etc.
. . . . . .
Carr and Bowie’s static hedge
Let X be an adapted process, and τ = inft : Xt ∈ D, whereD ⊂ Rd is a region.
Definition
Let T > 0 be fixed. Let Θ be an FT -measurable random variable.Another FT -measurable random variable Θ′ is called in (abstract)
Put-Call Symmetry of Θ with respect to (X ,D) if
1τ<TE [Θ1X∈D|Fτ ] = 1τ<TE [Θ′1X ∈D|Fτ ]. (1)
If it is the case, the knock option whose pay-off at maturity T isΘ1X∈D1τ>T, is hedged by long of Θ1X∈D and short ofΘ′1X ∈D of non-knock out options since...
. . . . . .
Carr and Bowie’s static hedge
Let X be an adapted process, and τ = inft : Xt ∈ D, whereD ⊂ Rd is a region.
Definition
Let T > 0 be fixed. Let Θ be an FT -measurable random variable.Another FT -measurable random variable Θ′ is called in (abstract)
Put-Call Symmetry of Θ with respect to (X ,D) if
1τ<TE [Θ1X∈D|Fτ ] = 1τ<TE [Θ′1X ∈D|Fτ ]. (1)
If it is the case, the knock option whose pay-off at maturity T isΘ1X∈D1τ>T, is hedged by long of Θ1X∈D and short ofΘ′1X ∈D of non-knock out options since...
. . . . . .
Carr and Bowie’s static hedge
Let X be an adapted process, and τ = inft : Xt ∈ D, whereD ⊂ Rd is a region.
Definition
Let T > 0 be fixed. Let Θ be an FT -measurable random variable.Another FT -measurable random variable Θ′ is called in (abstract)
Put-Call Symmetry of Θ with respect to (X ,D) if
1τ<TE [Θ1X∈D|Fτ ] = 1τ<TE [Θ′1X ∈D|Fτ ]. (1)
If it is the case, the knock option whose pay-off at maturity T isΘ1X∈D1τ>T, is hedged by long of Θ1X∈D and short ofΘ′1X ∈D of non-knock out options since...
. . . . . .
Carr and Bowie’s static hedge
If X = exp(σB + rt − σ2t/2), a Geometric Brownian motion,
• A put option (K −XT )+ is in put-call symmetry of call option(XT − K )+ with respect to the region x > K when r = 0.
• More generally, (XTK )1−
2rσ2 f (K
2
XT) is in put-call symmetry of
f (XT ).
. . . . . .
Carr and Bowie’s static hedge
If X = exp(σB + rt − σ2t/2), a Geometric Brownian motion,
• A put option (K −XT )+ is in put-call symmetry of call option(XT − K )+ with respect to the region x > K when r = 0.
• More generally, (XTK )1−
2rσ2 f (K
2
XT) is in put-call symmetry of
f (XT ).
. . . . . .
Carr and Bowie’s static hedge
If X = exp(σB + rt − σ2t/2), a Geometric Brownian motion,
• A put option (K −XT )+ is in put-call symmetry of call option(XT − K )+ with respect to the region x > K when r = 0.
• More generally, (XTK )1−
2rσ2 f (K
2
XT) is in put-call symmetry of
f (XT ).
. . . . . .
Carr and Picron’s static hedge
We are interested in rewriting
E[e−rτ ] = the value of the simplest timing risk.
We have that
E[e−rτ ] =
∫ ∞
0e−rtP(τ ∈ dt)
= [e−rtP(τ < t)]∞0 + r
∫ ∞
0e−rtP(τ < t)dt
= r
∫ ∞
0e−rtP(τ < t)dt
= r
∫ ∞
0e−rt(1− E[Iτ>t])dt
= r
∫ ∞
0e−rtE[(1 + (
Xt
K)1−
2rσ2 )IXt≤K|Fτ ]dt.
. . . . . .
Carr and Picron’s static hedge
We are interested in rewriting
E[e−rτ ] = the value of the simplest timing risk.
We have that
E[e−rτ ] =
∫ ∞
0e−rtP(τ ∈ dt)
= [e−rtP(τ < t)]∞0 + r
∫ ∞
0e−rtP(τ < t)dt
= r
∫ ∞
0e−rtP(τ < t)dt
= r
∫ ∞
0e−rt(1− E[Iτ>t])dt
= r
∫ ∞
0e−rtE[(1 + (
Xt
K)1−
2rσ2 )IXt≤K|Fτ ]dt.
. . . . . .
Carr and Picron’s static hedge
In short,
E[e−rτ ] = r
∫ ∞
0e−rt(1− E[Iτ>t])dt
= r
∫ ∞
0e−rtE[(1 + (
Xt
K)1−
2rσ2 )IXt≤K|Fτ ]dt.
Note that
• The timing risk is decomposed into the integral of the digitalknock-in options with exercise price K and maturityt ∈ (0,∞) with the volume rdt.
• With a put-call symmetry, the knock-in option is rewritten asoptions without timing risk.
. . . . . .
Carr and Picron’s static hedge
In short,
E[e−rτ ] = r
∫ ∞
0e−rt(1− E[Iτ>t])dt
= r
∫ ∞
0e−rtE[(1 + (
Xt
K)1−
2rσ2 )IXt≤K|Fτ ]dt.
Note that
• The timing risk is decomposed into the integral of the digitalknock-in options with exercise price K and maturityt ∈ (0,∞) with the volume rdt.
• With a put-call symmetry, the knock-in option is rewritten asoptions without timing risk.
. . . . . .
Carr and Picron’s static hedge
In short,
E[e−rτ ] = r
∫ ∞
0e−rt(1− E[Iτ>t])dt
= r
∫ ∞
0e−rtE[(1 + (
Xt
K)1−
2rσ2 )IXt≤K|Fτ ]dt.
Note that
• The timing risk is decomposed into the integral of the digitalknock-in options with exercise price K and maturityt ∈ (0,∞) with the volume rdt.
• With a put-call symmetry, the knock-in option is rewritten asoptions without timing risk.
. . . . . .
Imperfect Semi-Static HedgeLet us consider the static hedge a knock-out option with pay-off Θwithout put-call symmetry.The hedge is:
• buys a European option with pay-off Θ1XT∈D,
• sells a European option with pay-off Θ′1XT ∈D.
The value of the portfolio at time t is :
e−r(T−t)E[Θ1XT∈D −Θ′1XT ∈D|Ft ].
Hedging Error
Errt := −e−r(T−t)E[Θ1τ>T|Ft ]
+ e−r(T−t)E[Θ1XT∈D −Θ′1XT ∈D|Ft ]
= e−r(T−t)E[1τ≤T(Θ1XT∈D −Θ′1XT ∈D
)|Ft ].
Put-Call Symmetry ⇒ Errt = 0
. . . . . .
Imperfect Semi-Static HedgeLet us consider the static hedge a knock-out option with pay-off Θwithout put-call symmetry.The hedge is:
• buys a European option with pay-off Θ1XT∈D,
• sells a European option with pay-off Θ′1XT ∈D.
The value of the portfolio at time t is :
e−r(T−t)E[Θ1XT∈D −Θ′1XT ∈D|Ft ].
Hedging Error
Errt := −e−r(T−t)E[Θ1τ>T|Ft ]
+ e−r(T−t)E[Θ1XT∈D −Θ′1XT ∈D|Ft ]
= e−r(T−t)E[1τ≤T(Θ1XT∈D −Θ′1XT ∈D
)|Ft ].
Put-Call Symmetry ⇒ Errt = 0
. . . . . .
Imperfect Semi-Static HedgeLet us consider the static hedge a knock-out option with pay-off Θwithout put-call symmetry.The hedge is:
• buys a European option with pay-off Θ1XT∈D,
• sells a European option with pay-off Θ′1XT ∈D.
The value of the portfolio at time t is :
e−r(T−t)E[Θ1XT∈D −Θ′1XT ∈D|Ft ].
Hedging Error
Errt := −e−r(T−t)E[Θ1τ>T|Ft ]
+ e−r(T−t)E[Θ1XT∈D −Θ′1XT ∈D|Ft ]
= e−r(T−t)E[1τ≤T(Θ1XT∈D −Θ′1XT ∈D
)|Ft ].
Put-Call Symmetry ⇒ Errt = 0
. . . . . .
Hedging error as a timing risk
The error is in fact realized at the time of knock-out/in. In thatsense, it is with a timing risk.
From now on I will explain why and how the error is represented asan integral of knock-in options, just as Carr-Picron’s, whenΘ = f (XT ), and Θ′ = (πf )(X ′
T ) is in put-call symmetry of f (X ′T )
with respect to another diffusion process X ′.
As announced, the Paramatrix is a key ingredient.
. . . . . .
Hedging error as a timing risk
The error is in fact realized at the time of knock-out/in. In thatsense, it is with a timing risk.
From now on I will explain why and how the error is represented asan integral of knock-in options, just as Carr-Picron’s, whenΘ = f (XT ), and Θ′ = (πf )(X ′
T ) is in put-call symmetry of f (X ′T )
with respect to another diffusion process X ′.
As announced, the Paramatrix is a key ingredient.
. . . . . .
Hedging error as a timing risk
The error is in fact realized at the time of knock-out/in. In thatsense, it is with a timing risk.
From now on I will explain why and how the error is represented asan integral of knock-in options, just as Carr-Picron’s, whenΘ = f (XT ), and Θ′ = (πf )(X ′
T ) is in put-call symmetry of f (X ′T )
with respect to another diffusion process X ′.
As announced, the Paramatrix is a key ingredient.
. . . . . .
Parametrix
Let pt and qt be transition densities (assuming they exist) ofprocess Y and Y ′, respectively. We consider a “parametrix” of qtby pt using the following relation:
Lemma
It holds that
qt(x , y)− pt(x , y)
=
∫ t
0ds
∫Rd
dz qs(x , z)(Lz − L′z)pt−s(z , y)
where L = Lx and L′ = L′x denote the infinitesimal generator of Yand Y ′, respectively, acting on the variable x.
. . . . . .
Parametrix
Let pt and qt be transition densities (assuming they exist) ofprocess Y and Y ′, respectively. We consider a “parametrix” of qtby pt using the following relation:
Lemma
It holds that
qt(x , y)− pt(x , y)
=
∫ t
0ds
∫Rd
dz qs(x , z)(Lz − L′z)pt−s(z , y)
where L = Lx and L′ = L′x denote the infinitesimal generator of Yand Y ′, respectively, acting on the variable x.
. . . . . .
Parametrix, proof of the lemma
(Proof) We have that
∂sqs(x , z)pt−s(z , y) = ∂sqs(x , z)pt−s(z , y)− qs(x , z)∂sqt−s(x , z)
= (L∗zqs)(x , z)pt−s(z , y)− qs(x , z)(L′zpt−s)(z , y),
where L∗ denotes the adjoint operator of L.
By integrating the equation, we obtain the RHS of the lemma;
limϵ↓0
∫ t−ϵ
ϵ∂sqs(x , z)pt−s(z , y)ds
= limϵ↓0
∫ t−ϵ
ϵds
∫Rd
dz(L∗zqs)(x , z)pt−s(z , y)− qs(x , z)(Lyzpt−s)(z , y)
=
∫ t
0ds
∫dz qs(x , z)(Lz − Lyz )pt−s(z , y).
. . . . . .
Parametrix, proof of the lemma
(Proof) We have that
∂sqs(x , z)pt−s(z , y) = ∂sqs(x , z)pt−s(z , y)− qs(x , z)∂sqt−s(x , z)
= (L∗zqs)(x , z)pt−s(z , y)− qs(x , z)(L′zpt−s)(z , y),
where L∗ denotes the adjoint operator of L.By integrating the equation, we obtain the RHS of the lemma;
limϵ↓0
∫ t−ϵ
ϵ∂sqs(x , z)pt−s(z , y)ds
= limϵ↓0
∫ t−ϵ
ϵds
∫Rd
dz(L∗zqs)(x , z)pt−s(z , y)− qs(x , z)(Lyzpt−s)(z , y)
=
∫ t
0ds
∫dz qs(x , z)(Lz − Lyz )pt−s(z , y).
. . . . . .
Parametrix, proof of the lemma
On the other hand, since
lims↓0
∫qs(x , z)pt−s(z , y) dz = pt(x , y)
and
lims↑t
∫qs(x , z)pt−s(z , y) dz = qt(x , y),
we obtain the LHS;
limϵ↓0
∫ t−ϵ
ϵ∂sqs(x , z)pt−s(z , y)ds = qt(x , y)− pt(x , y).
q.e.d.
. . . . . .
Expression of the Error
Now we are back in the specific situation where Y = X andY ′ = X ′, etc. We put, for y ∈ D,
h0(t, z , y) := (Lz − L′z)pt(z , y),
andh(t, z ; y) := h0(t, z , y)− π∗
yh0(t, z , y),
where π∗y is the adjoint of π acting on y .
Define a one-parameterfamily of integral operators S(t) by
St f (x) =
∫Dh(t, x , y)f (y) dy , t ≥ 0.
. . . . . .
Expression of the Error
Now we are back in the specific situation where Y = X andY ′ = X ′, etc. We put, for y ∈ D,
h0(t, z , y) := (Lz − L′z)pt(z , y),
andh(t, z ; y) := h0(t, z , y)− π∗
yh0(t, z , y),
where π∗y is the adjoint of π acting on y . Define a one-parameter
family of integral operators S(t) by
St f (x) =
∫Dh(t, x , y)f (y) dy , t ≥ 0.
. . . . . .
Expression of the Error
Then, we have the following
Theorem
The error is equal to the value of the integral of knock-in optionswith pay-off ST−s f (Xs)ds;
Errt = e−r(T−t)E [
∫ T
t1τ≤sST−s f (Xs) ds|Ft ].
. . . . . .
Expression of the Error, proof of thetheorem
(Proof) Observe that
Errt = e−r(T−t)E [1τ≤Tf (XT )1XT∈D − πf (XT )1XT ∈D|Ft ]
= e−r(T−t)E [1τ≤TE [f (XT )1XT∈D − πf (XT )1XT ∈D|Fτ ]|Ft ].
Thanks to the optional sampling theorem, the expectationconditioned by Fτ turns into∫
Df (y)qT−τ (Xτ , y) dy −
∫Dc
πf (y)qT−τ (Xτ , y) dy
on the set τ ≤ T.
. . . . . .
Expression of the Error, proof of thetheorem
By applying the lemma (of parametrix), we have∫Df (y)qT−τ (Xτ , y) dy =
∫Df (y)pT−τ (Xτ , y) dy
+
∫ T
τds
∫Rd
dz qs−τ (Xτ , z)
∫Dh0(T − s, z , y)f (y) dy ,
. . . . . .
Expression of the Error, proof of thetheorem
which is also valid for πf ;∫Dc
πf (y)qT−τ (Xτ , y) dy =
∫Dc
πf (y)pT−τ (Xτ , y) dy
+
∫ T
τds
∫Rd
dz qs−τ (Xτ , z)
∫Dc
h0(T − s, z , y)πf (y) dy .
Since πf is in pcs of f w.r.t. pt , we know that∫Dc
πf (y)pT−τ (Xτ , y) dy =
∫Df (y)pT−τ (Xτ , y) dy .
. . . . . .
Expression of the Error, proof of thetheorem
which is also valid for πf ;∫Dc
πf (y)qT−τ (Xτ , y) dy =
∫Dc
πf (y)pT−τ (Xτ , y) dy
+
∫ T
τds
∫Rd
dz qs−τ (Xτ , z)
∫Dc
h0(T − s, z , y)πf (y) dy .
Since πf is in pcs of f w.r.t. pt , we know that∫Dc
πf (y)pT−τ (Xτ , y) dy =
∫Df (y)pT−τ (Xτ , y) dy .
. . . . . .
Expression of the Error, proof of thetheorem
Now we see that the expectation conditioned by Fτ∫Df (y)qT−τ (Xτ , y) dy −
∫Dc
πf (y)qT−τ (Xτ , y) dy
is equal to∫ T
τds
∫Rd
dz qs−τ (Xτ , z)
·∫
Dh0(T − s, z , y)f (y) dy −
∫Dc
h0(T − s, z , y)πf (y) dy
=
∫ T
τds
∫Rd
dz qs−τ (Xτ , z)
·∫D
h0(T − s, z , y)− π∗
yh0(T − s, z , y)f (y) dy
=
∫ T
τds
∫Rd
dz qs−τ (Xτ , z)ST−s f (z).
. . . . . .
Expression of the Error, proof of thetheorem
Noting that∫Rd
dz qs−τ (Xτ , z)ST−s f (z) = E [1τ≤sST−s f (Xs)|Fτ ],
we have
Errt = e−r(T−t)E [1τ≤T
∫ T
τE [1τ≤sST−s f (Xs)|Fτ ] ds|Ft ]
= e−r(T−t)
∫ T
tE [1τ≤sST−s f (Xs)|Ft ] ds.
. . . . . .
Second Order Static HedgeThe formula of the previous theorem
Errt = e−r(T−t)
∫ T
tE [ST−s f (Xs)− 1τ≥sST−s f (Xs)|Ft ] ds,
claims that the Errt can be understood as a timing risk. We set
ηg(x) := g(x)1x∈D − πg(x)1x ∈D.
Then Errt is hedged by a portfolio composed of options withpay-off
(1− η)ST−s f (Xs) = ST−s f (Xs) + πST−s f (Xs)1Xs ∈D,
and the volume “e−r(T−s)ds”.Note that the value at time t of the portfolio is given by
e−r(T−t)E [
∫ T
t(1− π)ST−s f (Xs)ds|Ft ]ds.
. . . . . .
Second Order Static HedgeThe hedge error coincides with the one in the correspondingknock-out case. So the error of the static hedge for the optionmaturing s is given by
Errs2,tds := e−r(T−t)E [1τ≤sπST−s f (Xs)|Ft ]ds.
Now we can apply Theorem to obtain that
Errs2,tds = e−r(T−t)
∫ s
tE [1τ≤uSs−uST−s f (Xu)|Ft ] duds,
We know that Errs2,t is integrable in s on [t,T ] almost surely.Thus the totality of the error is obtained as∫ T
tErrs2,tds = e−r(T−t)
∫ T
tds
∫ s
tduE [1τ≤uSs−uST−s f (Xu)|Ft ]
=: Err2,t .
. . . . . .
n-th Order Static HedgeBy repeating this procedure, we obtain the n-th order static hedgesand the n-th error for any n:
Theorem
We have that
e−r(T−t)− E [f (XT )1τ>T|Ft ] + E [πf (XT )|Ft ]
−n∑
k=1
∫ T
tds E [(1− η)S∗k
T−s(Xs)|Ft ]
= e−r(T−t)
∫ T
tdu E [1τ≤uS
∗nT−uf (Xu)|Ft ] =: Errn,t ,
where
S∗1t = St , S∗k
t =
∫ t
0SsS
∗(k−1)t−s ds, k = 2, 3, · · · .
. . . . . .
Perfect Static Hedge
The previous formula reads: the semi-static hedge of theknock-out option with pay-off f , by the option with pay-offηf (XT ) and the options with pay-off
∑nk=1(1− η)S∗k
T−uf (Xu)dufor u < T , has the error∫ T
tdu E [S∗n
T−uf (Xu)|Fτ ] (2)
at the knock-out time τ .
. . . . . .
Perfect Static Hedge
Under suitable conditions, it converges to zero as n goes infinity sothat we have the following
Theorem
The series∑∞
k=1(1− η)S∗kT−uf (Xu) is absolutely convergent almost
surely, and the option with pay-off ηf (XT ) and the options withpay-off
∑∞k=1(1− η)S∗k
T−uf (Xu)e−r(T−u)du for each u < T gives a
perfect semi-static hedge (the error is zero almost surely).
. . . . . .
Hint
Hint: the lemma can be rephrased as
qt(x , y)
= pt(x , y) +
∫ t
0ds
∫dz qs(x , z)h0(z , y)
by which q can be understood as a solution to a Volterra typeequation.
. . . . . .
Thank you for your kind attentions.