Pathwise analysis and robustness of hedging strategies for path … · 2019-01-10 · Motivation...

Motivation Pathwise Functional Calculus Pathwise Continuous-Time Trading Pathwise analysis of Hedging Examples

Pathwise analysis and robustnessof hedging strategies forpath-dependent options

Candia Riga(joint work with Rama Cont)

University of Zurich

7th General Advanced Mathematical Methods in Financeand Swissquote Conference 2015

Lausanne, September 7-10


Outline

1 Motivation

2 Pathwise Calculus for Non-AnticipativeFunctionals

3 A Pathwise Approach to Continuous-TimeTrading

4 A Pathwise Analysis of Hedging Strategies:Hedging Error and Robustness

5 Examples


Model Ambiguity and Hedging Issues

Classical Framework: Traded assets X “ pX ptqqtPr0,T s modeled as

Rd`-valued semimartingale on pΩ,F , pFtqtPr0,T s,Pq

The choice of P may be challenged ‘a la De Finetti’(Knightian uncertainty)Ñ Our approach: we set up a probability-free financial model

The gain process is a stochastic integral, thus

it is not necessarily defined for a given path/price scenarioscenario analysis and stress tests cannot be performed

Ñ In our setting: for a certain class of trading strategies, thegain process is well-defined path-by-path (as a limit ofRiemann sums)

Robustness analyses are based on the existence of a ‘truemodel’, and study the performance of a ‘mis-specified model’Ñ Our analysis: we study the performance and robustness ofhedging strategies in given sets of scenarios.


Robustness of Hedging Strategies

Consider a market participant who sells an (exotic) option withpayoff H and maturity T on some underlying asset, at a modelprice given by

V ptq “ EQrH|Fts

and hedges the resulting Profit/Loss using the hedging strategyderived from the same model (say, Black-Scholes delta hedge).The actual dynamics of the underlying asset may, of course, bedifferent from the assumed dynamics.

How good is the result of the hedging strategy?

How ‘robust’ is it to model mis-specification?

How does the hedging error relate to model parameters andoption characteristics?


Robustness of Hedging Strategies

El Karoui, Jeanblanc & Shreve (1998) provided an answer to theseimportant questions, for non-path-dependent options, when theunderlying dynamics is

dSptq “ Sptqrptqdt ` SptqσptqdW ptq under Qsuch that S is square-integrable. Then a hedging strategy,computed in a (mis-specified) Markovian model

dSptq “ Sptqrptqdt ` Sptqσ0pt, SptqqdW ptq

with local volatility σ0 leads to a profit

ż T

0

σ20pt,Sptqq ´ σ

2ptq

2Sptq2e

şTt rpsqds

Γptqhkkkkkkikkkkkkj

B2xx f pt, Sptqqdt

where f is the unique solution of the PDE

Bt f ` rptqxB2x f ` σ2

0pt, xqx2B2

xx f 2 “ rptqf f pT , xq “ Hpxq


Notation: Non-Anticipative Functionals

Given x P Dpr0,T s,Rdq, for all t P r0,T s we denote:

xptq P Rd the value of x at t

xt “ xpt ^ ¨q P Dpr0,T s,Rdq the path stopped at t

xt´ “ x1r0,tq ` xpt´q1rt,T s P Dpr0,T s,Rdq

for δ P Rd , xδt “ xt ` δ1rt,T s P Dpr0,T s,Rdq

We define the space of stopped paths:

ΛT :“´

r0,T s ˆ Dpr0,T s,Rdq

¯

„,

where pt, xq „ pt 1, x 1q ðñ t “ t 1 and xt “ x 1t , and the metric

d8ppt, xq, pt1, x 1qq “ sup

uPr0,T s

ˇ

ˇxpu ^ tq ´ x 1pu ^ t 1qˇ

ˇ` |t ´ t 1|

Definition (Non-anticipative functional)

A non-anticipative functional on Dpr0,T s,Rdq is a mapF : pΛT , d8q Ñ R.


Notation: Regularity of F : Λ Ñ R

A non-anticipative functional F : ΛT Ñ R is said to be:

continuous at fixed times if for all t P r0,T s

F pt, ¨q :´´

ttu ˆ Dpr0,T s,Rdq

¯

„, || ¨ ||8

¯

ÞÑ R

is continuous

left-continuous, F P C0,0l pΛT q, if

@pt, xq P ΛT , @ε ą 0, Dη ą 0 : @h P r0, ts, @pt ´ h, x 1q P ΛT ,d8ppt, xq, pt ´ h, x 1qq ă η ñ |Ftpxq ´ Ft´hpx

1q| ă ε

boundedness-preserving, F P BpΛT q, if, @K Ă Rd compact,@t0 P r0,T s, DCK ,t0 ą 0 s.t. @t P r0, t0s, @pt, xq P ΛT

xpr0, tsq Ă K ñ |Ftpxq| ă CK ,t0


Notation: Differentiability of F : Λ Ñ R

Definition (Horizontal derivative)

A non-anticipative functional F is horizontally differentiable atpt, xq P ΛT if t ÞÑ F pt, xtq is right-differentiable, with derivativedenoted by DFpt, xq; if it holds @pt, xq P ΛT , then we denoteDF :“ pDF pt, ¨qqtPr0,T q

Definition (Vertical derivative)

A non-anticipative functional F is vertically differentiable atpt, xq P ΛT if the map Rd Q e ÞÑ F pt, xe

t q is differentiable at 0; inthis case we denote ∇ωFpt, xq :“ pBiF pt, xqqi“1,...,d , where

BiF pt, xq :“ limhÑ0`

F pt, xheit q ´ F pt, xtq

h.

If this holds for all pt, xq P ΛT , then ∇ωF :“ p∇ωF pt, ¨qqtPr0,T s


Sets of Smooth Non-Anticipative Functionals

Definition (C1,2b pΛq and C1,2

locpΛq)

Denote by C1,2b pΛTq the set of non-anticipative functionals

F P C0,0l pΛT q such that:

DDF continuous at fixed times,

D∇jωF P C0,0

l pΛT q j “ 1, 2,

DF ,∇ωF ,∇2ωF P BpΛT q

(

.

Denote by C1,2locpΛTq the set of non-anticipative functionals

F P C0,0l pΛT q such that there exists a sequence of stopping times

pτkqkě1 going to 8 and a sequence pF k P C1,2b pΛT qqkě1,

F pt, xtq “ÿ

kě1

F kpt, xtq1rτk pxq,τk`1pxqqptq @pt, xq P ΛT


Quadratic Variation for Cadlag Paths

Fixed sequence of time partitions: Π “ tπnuně1,πn “ pt

ni qi“0,...,mpnq, 0 “ tn0 ă . . . ă tnmpnq “ T , |πn| ÝÝÝÑ

nÑ80

Definition (Paths of finite quadratic variation)

x P Dpr0,T s,Rq is of finite quadratic variation along Π if

@t P r0,T s, rxsptq :“ limnÑ8

ÿ

tni ďt

`

xptni`1q ´ xptni q˘2ă 8

x P Dpr0,T s,Rdq is of finite quadratic variation along Π if,@1 ď i , j ď d, x i , x i ` x j are so. In this case:

rxsi ,jptq ” rxi , x j sptq “

1

2

`

rx i ` x j sptq ´ rx i sptq ´ rx j sptq˘

Denote QpU,Πq :“tx P U Ă Dpr0,T s,Rdq, x is of f.q.v. along Πu

For every cadlag path ω P QpDpr0,T s,Rdq,Πq, we can alwaysassume that suptPr0,T s´πn |∆ωptq| ÝÝÝÑ

nÑ80.


Change of Variable Formula for Functionals

The piecewise constant approximation

ωn :“

mpnq´1ÿ

i“0

ωptni`1´q1rtni ,tni`1q

` ωpT q1tTu,

converges uniformly to ω

Theorem (Cont, Fournie (2010))

If F P C1,2locpΛT q and ω P QpDpr0,T s,Rdq,Πq, then the limit

ż T

0∇ωFtpωt´qd

Πω :“ limnÑ8

mpnq´1ÿ

i“0

∇ωFtnipω

n,∆ωptni qtni ´

q¨pωptni`1q´ωptni qq

exists and

FT pωT q “ F0pω0q `

ż T

0∇ωFtpωt´qd

Πω

`

ż T

0DtF pωt´qdt `

1

2

ż

p0,T str`

∇2ωFtpωt´qdrωs

cptq˘

`ÿ

tPp0,T s

pFtpωtq ´ Ftpωt´q ´∇ωFtpωt´q ¨∆ωptqq


Functional Ito Formula for Functionals ofSemimartingales

Theorem (Cont, Fournie (2013))

If F P C1,2locpΛT q and X is an Rd -valued semimartingale, then a.s.

F pT ,XT q “ F p0,X0q `

ż T

0∇ωF pt,Xt´q ¨ dX

`

ż T

0DF pt,Xt´qdt `

1

2

ż

p0,T str`

∇2ωF pt,Xt´qdrX s

cptq˘

`ÿ

tPp0,T s

pF pt,Xtq ´ F pt,Xt´q ´∇ωF pt,Xt´q ¨∆X ptqq

In particular, Y defined by Y ptq “ F pt,Xtq for all t P r0,T s, is asemimartingale


A Pathwise Setting for Continuous-TimeTrading

Financial market: pΩ, || ¨ ||8q metric space, Ω :“ Dpr0,T s,Rd`q,

ω P Ω is a possible trajectory of the (forward) asset prices,F is the Borel σ-field and F “ pFtqtPr0,T s the canonical filtration

Trading strategies: pV0, φ, ψq, where

V0 Ω ÞÑ R, F0-measurable (initial investment)φ Rd -valued F-adapted caglad process (asset position)ψ R-valued F-adapted caglad process (bond position)

Portfolio value at time t P r0,T s along the price pathω P Dpr0,T s,Rd

`q:

V pt, ω;φ, ψq “ φpt, ωq ¨ ωptq ` ψpt, ωq

When is a strategy self-financing?When can we explicitly define its gain?


Simple Trading Strategies

Definition (Simple self-financing trading strategies)

pV0, φ, ψq is a simple trading strategy if φ P ΣpRd ,Πq andψ P ΣpR,Πq, where ΣpU,Πq :“ Y

ně1ΣpU, πnq,

ΣpU, πnq :“

"

φ : @i “ 0, . . . ,mpnq ´ 1, Dλi : Ω Ñ U

Ftni-measurable, φpt, ωq “

mpnq´1ÿ

i“0

λi pωq1ptni ,tni`1s

*

A simple trading strategy pV0, φ, ψq is a self-financing if thereexists n ě 1 such that φ, ψ P Σpπnq and

ψpt, ωq :“V0´φp0`, ωq¨ωp0q´

mpnqÿ

i“1

ωptni ^tq¨pφptni`1^t, ωq´φptni ^t, ωqq

Equivalently: V pt, ω;φq “ V0 ` G pt, ω;φq, where

G pt, ω;φq :“

mpnqÿ

i“1

φptni ^ t, ωq ¨ pωptni ^ tq ´ ωptni´1 ^ tqq


Self-financing Trading Strategies

Definition (Self-financing Trading Strategies on U)

pV0, φq is a self-financing trading strategy on U Ă Dpr0,T s,Rd`q if

there exists a sequence tpV0, φn, ψnq, n P Nu of self-financing

simple trading strategies, such that

@ω P U, @t P r0,T s, φnpt, ωq ÝÝÝÑnÑ8

φpt, ωq,

and any of the following equivalent conditions is satisfied:

(i) DF-adapted cadlag process G p¨, ¨;φq, @t P r0,T s, ω P U

G pt, ω;φnq ÝÝÝÑnÑ8

G pt, ω;φq, ∆G pt, ω;φq “ φpt, ωq∆ωptq;

(ii) DF-adapted cadlag process ψp¨, ¨;φq, such that@t P r0,T s, ω P U ψnpt, ωq ÝÝÝÑ

nÑ8ψpt, ω;φq and

ψpt`, ω;φq ´ ψpt, ω;φq “ ´ωptq pφpt`, ωq ´ φpt, ωqq ;

(iii) DF-adapted cadlag process V p¨, ¨;φq, @t P r0,T s, ω P U

V pt, ω;φnq ÝÝÝÑnÑ8

V pt, ω;φq, ∆V pt, ω;φq “ φpt, ωq∆ωptq.


Gain: Pathwise Construction

Proposition

If there exists F P C1,2locpΛT q X C0,0

r pΛT q, ∇ωF P C0,0pΛT q,

φpt, ωq “ ∇ωF pt, ωt´q @ω P QpΩ,Πq, t P r0,T s,

Then, φ is the asset position of a self-financing trading strategy onQpΩ,Πq with gain

G pt, ω;φq “

ż t

0φpu, ωuqd

Πω

“ limnÑ8

ÿ

tni ăt

∇ωF ptni , ωntni ´q ¨ pωptni`1q ´ ωpt

ni qq

and bond position

ψpt, ω;φq :“V0 ´ φ`p0, ωq`

´ limnÑ8

mnÿ

i“1

ωptni ^ tq ¨ pφnptni`1 ^ t, ωq ´ φnptni ^ t, ωqq


Hedging Error and Super-Strategies

Definition (Hedging error in specific scenarios)

The hedging error of a self-financing trading strategy pV0, φq onU Ă Dpr0,T s,Rd

`q for a path-dependent derivative with payoff Hin a scenario ω P U is the value

V pT , ω;φq ´ Hpωq “ V0pωq ` G pT , ω;φq ´ Hpωq.

pV0, φq is called a super-strategy for H on U if its hedging error forH is non-negative on U, i.e.

V0pωq ` G pT , ω;φq ě HpωT q @ω P U.

Given A P Dpr0,T s,Sq, S :“ tM P Rdˆd ,M ą 0 symmetricu,

QApΠq :“

"

ω P QpΩ,Πq such that rωsptq “

ż t

0Apsqds @t P r0,T s

*


Pathwise Replication of (Exotic) Derivatives

Proposition (Pathwise replication of exotic derivatives)

If H : pΩ, ‖¨‖8q ÞÑ R is continuous and F P C1,2locpWT q XC0,0pWT q

solves"

DF pt, ωtq `12tr

`

∇2ωF pt, ωtq ¨ Aptq

˘

“ 0, t P r0,T qF pT , ωq “ Hpωq, @ω P QApΠq

Then, the hedging error of the trading strategy pF0pω0q,∇ωF q,self-financing on QpC pr0,T s,Rdq, in any scenario ω P QApΠq is

1

2

ż T

0tr´

∇2ωF pt, ωtq ¨ pAptq ´ Aptqq

¯

dt

In particular, in all scenarios ω P QApΠq, pF p0, ω0q,∇ωF q replicatesH and its portfolio value at any time t P r0,T s is given by F pt, ωtq.


A Hedging Formula for Path-DependentOptions

Let Ω :“ C pr0,T s,R` and S denote the coordinate process on thecanonical space pΩ,F ,Fq, i.e. Spt, ωq “ ωptq, @ω P Ω, t P r0,T s.

Assumption (Hedger’s model assumption)

The market participant assumes that the underlying asset price Sevolves according to dSptq “ σptqSptqdW ptq, i.e.

Sptq “ Sp0qeşt

0 σpuqdW puq´ 12

şt0 σpuq

2du, t P r0,T s, (1)

where W is a standard Brownian motion on pΩ,F ,F,Pq and thevolatility σ is a non-negative F-adapted process such that σ ‰ 0dt ˆ dP-almost surely and S is a square-integrable P-martingale.

The discounted price at time t of a path-dependent derivative withpayoff HpST q P L1pΩ,F ,Pq is given by

Y ptq “ EPrHpST q|Fts P-a.s.


A Universal Hedging Equation

Theorem (Universal hedging formula (Cont-Fournie, 2013))

If EPr|HpST q|2s ă 8, then Y PM2pFq and the following

martingale representation formula holds:

HpST q “ Y p0q `

ż T

0∇SY puq ¨ dSpuq P-a.s.

Moreover, if Y P C1,2b pSq, where

C1,2b pSq :“tY : DF P C1,2

b pWT q, Y ptq“F pt,Stq P-a.s.@t P r0,T su,

then the hedging strategy for H is pathwise defined by∇SY “ ∇ωF p¨, S¨q dt ˆ dP´ a.s.

In case Y PM2pFq but Y R C1,2b pSq, the hedging strategy ∇SY

for H is a ‘weak vertical derivative’, yet it can be uniformlyapproximated by regular functionals (i.e. Lu-Cont 2015)


A Universal Pricing Equation

Theorem (Pricing equation for path-dependent derivatives)

Consider a path-dependent derivative with payoff HpST q. If

DF P C1,2b , F pt, Stq “ EPrHpST q|Fts

such that DF P C0,0l pWT q, then F is the unique solution of the

pricing equation

DF pt, ωtq `1

2tr`

∇2ωF pt, ωtq ¨ σ

2ptqω2ptq˘

“ 0, (2)

with the terminal condition F pT , ωq “ Hpωq, on the topologicalsupport of pS ,Pq in pC pr0,T s,R`q, ‖¨‖8q, i.e.

@ω P supppSq :“

ω P Ω, such that PpST P V q ą 0

@V neighborhood of ω in pΩ, ‖¨‖8q(


Pathwise Analysis of Hedging Performance

Problem:

The hedger sells a path-dependent option with maturity Tand payoff HpST q such that EPr|HpST q|

2s ă 8

He computes the price and hedging strategy according to PHe trades according to that strategy, but taking in input therealized market prices ñ in a scenario ω, the final value ofthe hedging portfolio will differ from HpωT q

What is the performance of the hedging strategypY p0q,∇SY q with respect to H?


Robustness of Delta-Hedging Strategies

Definition (Robust delta hedge)

The delta-hedging strategy pY p0q,∇SY q is said to be robust for Hon U Ă Ω if it is a super-strategy for H on U.

Notation: For paths of absolutely continuous finite quadraticvariation along Π, we define the local realized volatility as

σreal : r0,T s ˆA Ñ R,pt, ωq ÞÑ σrealpt, ωq “ 1

ωptq

b

ddt rωsptq,

where

A :“ tω P QpΩ,Πq, t ÞÑ rωsptq is absolutely continuousu.


The Hedging Error Formula and Robustness

Proposition (The hedging error formula and robustness ofdelta-hedging)

If there exists F P C1,2b pWT q X C0,0pWT q such that

DF P C0,0l pWT q, and

F pt, Stq “ EPrHpST q|Fts dt ˆ dP-a.s.

Then, the hedging error of pF p0, ¨q,∇ωF q for H is explicitly givenby 1

2

ż T

0

´

σpt, ωq2 ´ σrealpt, ωq2¯

ω2ptq∇2ωF pt, ωqdt

In particular, if for all ω P U and Lebesgue-a.e. t P r0,T q

∇2ωF pt, ωq ě 0 and σpt, ωq ě σrealpt, ωq

then the delta hedge for H is robust on U.


The Impact of Jumps

Proposition (Impact of jumps on delta hedging)

If there exists a non-anticipative functional F : ΛT Ñ R such that

F P C1,2b pΛT q X C0,0pΛT q, ∇ωF P C0,0pΛT q, DF P C0,0

l pWT q

F pt,Stq “ EPrHpST q|Fts dt ˆ dP-a.s.

Then, for any ω P QpDpr0,T s,R`q,Πq such that rωsc is absolutelycontinuous, the hedging error of the delta hedge pF p0, ¨q,∇ωF q forH is explicitly given by

1

2

ż T

0

´

σpt, ωq2 ´ σrealpt, ωq2¯

ω2ptq∇2ωF pt, ωqdt

´ÿ

tPp0,T s

pF pt, ωtq ´ F pt, ωt´q ´∇ωF pt, ωt´q ¨∆ωptqq .


Vertical Smoothness

The previous results extend El Karoui, Jeanblanc & Shreve (1998):- to path-dependent options- to a pathwise setting: it gives the pathwise P&L of the strategyand removes unnecessary probabilistic assumptions

Definition (Vertical smoothness)

A functional h : Dpr0,T s,Rq ÞÑ R is vertically smooth onU Ă Dpr0,T s,Rq if, @pt,wq P r0,T s ˆ U, the real map

ghpt, ω; ¨q : RÑ R, e ÞÑ hpω ` e1rt,T sq

is of class C 2 on a neighborhood V of 0 and there existK , c , β ą 0 such that, for all ω, ω1 P U, t, t 1 P r0,T s,∣∣∣Beghpe; t, ωq

∣∣∣` ∣∣∣B2eeghpe; t, ωq

∣∣∣ ď K , e P V ,∣∣Beghp0; t, ωq ´ Beghp0; t 1, ω1q∣∣` ∣∣B2

eeghp0; t, ωq ´ B2eeghp0; t 1, ω1q

∣∣ď cp‖ω ´ ω1‖8 ` |t ´ t 1|βq.


Pricing Functional: Existence andRegularity

Proposition (Pricing functional: existence and regularity)

Let H : pDpr0,T s,Rq, ‖¨‖8q Ñ R a locally-Lipschitz payofffunctional such that EPr|HpST q|s ă 8 and define

h : pDpr0,T s,Rq Ñ R, hpωT q “ HpexpωT q,

where expωT ptq :“ eωptq for all t P r0,T s.

If h is vertically smooth on Cpr0,T s,R`q, then

DF P C0,2b pWT q X C0,0pWT q

such that

F pt,Stq “ EPrHpST q|Fts dt ˆ dP-a.s.


Propagation of ‘Vertical Convexity’

Definition (Vertical convexity of non-anticipative functionals)

A non-anticipative functional G : ΛT Ñ R is called verticallyconvex on U Ă ΛT if, for all pt, ωq P U, there exists aneighborhood V Ă R of 0 such that the map

V Ñ R, e ÞÑ G`

t, ω ` e1rt,T s˘

is convex.

Proposition (Propagation of vertical convexity)

Assume that, for all pt, ωq P Tˆ supppS ,Pq, there exists aninterval I Ă R, 0 P I, such that the map

vHp¨; t, ωq : I Ñ R, e ÞÑ vHpe; t, ωq “ H`

ωp1` e1rt,T sq˘

is convex. If

DF P C0,2b pWT q, F pt,Stq “ EPrHpST q|Fts dt ˆ dP-a.s.,

then F is vertically convex on Tˆ supppS ,Pq. In particular:

∇2ωF pt, ωq ě 0, @pt, ωq P Tˆ supppS ,Pq.


Example: Discretely-Monitored ExoticDerivatives with Black-Scholes

Lemma (Discretely-monitored path-dependent derivativeswith Black-Scholes)

Let σ : r0,T s Ñ R` such thatşT

0 σ2ptqdt ă 8.

Assume that H : Dpr0,T s,R`q and there exist a partition0 “ t0 ă t1 ă . . . ă tn ď T and a function h P C 2

b pRn;R`q suchthat

@ω P Dpr0,T s,R`q, HpωT q “ hpωpt1q, ωpt2q, . . . , ωptnqq.

Then,

DF P C1,2locpWT q, F pt,Stq “ EPrHpST q|Fts dt ˆ dP-a.s.,

and the horizontal and vertical derivatives of F are given in aclosed form.


Example: Hedging Asian Options withBlack-Scholes

Consider an arithmetic Asian call option

HpST q “

ˆ

1

T

ż T

0Spuqdu ´ K

˙`

and assume σ : r0,T s Ñ R` such thatşT

0 σ2ptqdt ă 8.

The pricing functional F is given by: @pt, ωq PWT ,

F pt, ωtq “ f pt, aptq, ωptqq, aptq “

ż t

0ωpsqds,

where f P C1,1,2pr0.T q ˆ R` ˆ R`q X Cpr0.T s ˆ R` ˆ R`q solves

#

σ2ptqx2

2 B2xx f pt, a, xq ` xBaf pt, a, xq ` Bt f pt, a, xq “ 0,

f pT , a, xq “ g`

aT

˘ (3)

Note that (3) turns out to be a particular case of the universalpricing equation.


Example: Robustness of Black-ScholesDelta-Hedging for Asian Options

Corollary (Robustness of delta-hedging for Asian options)

If the Black-Scholes volatility term structure over-estimates thelocal realized volatility on AX supppS ,Pq, i.e.

σptq ě σrealpt, ωq @ω P AX supppS ,Pq,

Then the Black-Scholes-delta hedge for the arithmetic Asian calloption is robust on AX supppS ,Pq. Moreover, the hedging erroris given by

1

2

ż T

t

´

σpuq2 ´ σrealpt, ωq2¯

ω2puqB2xx f pu, apuq, ωpuqqdu

where f is the solution to the Cauchy problem (3).


Example: Hedging Asian Options withHobson-Rogers

dSptq “ SptqσnpOptqqdW ptq,where

dOptq “ σnpOptqqdW ptq ´1

2pσnpOptqq2 ` λOptqqdt.

Consider a geometric Asian call option

HpST q “

´

eMpT q ´ K¯`

, MpT q “1

T

ż T

0log Spuqdu

The pricing functional F is given by: @pt, ωq PWT ,

F pt, ωq “ upT ´ t, logωptq, logωptq ´ opt, ωq, gpt, ωqq,

where u is the classical solution of the following Cauchy problemon r0.T q ˆ Rˆ Rˆ R#

12σ

npx1 ´ x2q2pB2

x1x1u ´ Bx1uq ` λpx1 ´ x2qBx2u ` x1Bx3u ´ Btu “ 0,

up0, x1, x2, x3q “ ΨG pex1 , x3T q.

(4)(4) is another particular case of the universal pricing equation


Hedging Asian Options with Path-DependentModels

Corollary

If σnpopt, ωqq ě σrealpt, ωq @ω P AX supppS ,Pq, then theHobson-Rogers delta hedge for H is robust on AX supppS ,Pq.Moreover, the hedging error at maturity is given by

12

şT0

`

σnpopt, ωqq2 ´ σmktpt, ωq2˘

ω2ptqB2xupT ´ t, logωptq, logωptq ´ opt, ωq, gpt, ωqqdt,

where u is the solution of the Cauchy problem (4).

Other models that generalize Hobson-Rogers and allow to derivefinite-dimensional Markovian representation for S and itsarithmetic mean are given by

Pascucci,Foschi 2006,Salvatore, Tankov 2014.

They thus guarantee the existence of a smooth pricing functionalfor arithmetic Asian options, then robustness of the delta hedgecan be proved the same way as we showed in the Black-Scholesand Hobson-Rogers cases.

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