Parametrix of Static Hedge (of a Timing Risk)...I will introduce a parametrix of barrier type...

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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.

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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.

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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.

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Timing Risk?

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Timing Risk?

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(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.

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(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.

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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.

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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.

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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.

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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).

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• 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.

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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.

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From the next slide, I will use mathematics explicitly.

We will work on a filtered probabilty space (Ω,F ,P, Ft), and...etc.

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From the next slide, I will use mathematics explicitly.

We will work on a filtered probabilty space (Ω,F ,P, Ft), and...etc.

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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...

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Definition

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Definition

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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

XT) is in put-call symmetry of

f (XT ).

. . . . . .

2rσ2 f (K

f (XT ).

. . . . . .

2rσ2 f (K

f (XT ).

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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

∫ ∞

0e−rtP(τ < t)dt

∫ ∞

0e−rt(1− E[Iτ>t])dt

∫ ∞

0e−rtE[(1 + (

K)1−

2rσ2 )IXt≤K|Fτ ]dt.

. . . . . .

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

∫ ∞

0e−rtP(τ < t)dt

∫ ∞

0e−rtE[(1 + (

K)1−

. . . . . .

In short,

E[e−rτ ] = r

∫ ∞

0e−rtE[(1 + (

K)1−

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.

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In short,

E[e−rτ ] = r

∫ ∞

0e−rtE[(1 + (

K)1−

Note that

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In short,

E[e−rτ ] = r

∫ ∞

0e−rtE[(1 + (

K)1−

Note that

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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

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Hedging Error

)|Ft ].

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Hedging Error

)|Ft ].

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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.

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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:

It holds that

qt(x , y)− pt(x , y)

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.

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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:

It holds that

qt(x , y)− pt(x , y)

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.

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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

dz(L∗zqs)(x , z)pt−s(z , y)− qs(x , z)(Lyzpt−s)(z , y)

∫dz qs(x , z)(Lz − Lyz )pt−s(z , y).

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(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

dz(L∗zqs)(x , z)pt−s(z , y)− qs(x , z)(Lyzpt−s)(z , y)

∫dz qs(x , z)(Lz − Lyz )pt−s(z , y).

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On the other hand, since

lims↓0

∫qs(x , z)pt−s(z , y) dz = pt(x , y)

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.

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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.

. . . . . .

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.

. . . . . .

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 [

t1τ≤sST−s f (Xs) ds|Ft ].

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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 −

πf (y)qT−τ (Xτ , y) dy

on the set τ ≤ T.

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By applying the lemma (of parametrix), we have∫Df (y)qT−τ (Xτ , y) dy =

∫Df (y)pT−τ (Xτ , y) dy

dz qs−τ (Xτ , z)

∫Dh0(T − s, z , y)f (y) dy ,

. . . . . .

which is also valid for πf ;∫Dc

πf (y)qT−τ (Xτ , y) dy =

πf (y)pT−τ (Xτ , y) dy

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 .

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which is also valid for πf ;∫Dc

πf (y)qT−τ (Xτ , y) dy =

πf (y)pT−τ (Xτ , y) dy

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 .

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Now we see that the expectation conditioned by Fτ∫Df (y)qT−τ (Xτ , y) dy −

πf (y)qT−τ (Xτ , y) dy

is equal to∫ T

Dh0(T − s, z , y)f (y) dy −

h0(T − s, z , y)πf (y) dy

·∫D

h0(T − s, z , y)− π∗

yh0(T − s, z , y)f (y) dy

dz qs−τ (Xτ , z)ST−s f (z).

. . . . . .

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

τE [1τ≤sST−s f (Xs)|Fτ ] ds|Ft ]

= e−r(T−t)

tE [1τ≤sST−s f (Xs)|Ft ] ds.

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Second Order Static HedgeThe formula of the previous theorem

Errt = e−r(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(1− π)ST−s f (Xs)ds|Ft ]ds.

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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)

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)

tduE [1τ≤uSs−uST−s f (Xu)|Ft ]

=: Err2,t .

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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∑

tds E [(1− η)S∗k

T−s(Xs)|Ft ]

= e−r(T−t)

tdu E [1τ≤uS

∗nT−uf (Xu)|Ft ] =: Errn,t ,

S∗1t = St , S∗k

∗(k−1)t−s ds, k = 2, 3, · · · .

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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 .

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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).

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Hint: the lemma can be rephrased as

qt(x , y)

= pt(x , y) +

∫dz qs(x , z)h0(z , y)

by which q can be understood as a solution to a Volterra typeequation.　

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Thank you for your kind attentions.

Parametrix of Static Hedge (of a Timing Risk)...I will introduce a parametrix of barrier type...

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