Robust framework for valuation and hedging: theory and ...people.maths.ox.ac.uk/obloj/Obloj_2011_QMF_shorter.pdf · Theory of robust valuation and hedging Discrete time models Continuous

Robust framework for valuation and hedging:theory and practice

Jan Ob lojUniversity of Oxford

parts of this talk are based on joint works with

Alexander Cox (University of Bath), Mark Davis (Imperial College London),

Sergey Nadtochiy (Oxford), Vimal Raval and Frederik Ulmer.

The Bruti-Liberati Lecture

QMF Conference, Sydney, 14–17 Dec 2011

Modelling in Mathematical Finance Theory Building beliefs Practice

Modelling: Classical approachThe prevailing approach in math finance proceeds as follows:

• Write down a plausible and well behaved stochastic model.• Compute prices of (liquid) financial instruments as function of

model parameters.• Calibrate the model: chose the parameters to match the

prices already observed in the market.• Use it: sell and hedge derivatives.

Robust approach to Mathematical Finance Bruti-Liberati Lecture, QMF 2011 Jan Ob loj


Recently, its critique has been vocal ...

the economics profession went astray because economists, as agroup, mistook beauty, clad in impressive-looking mathematics, fortruth (Paul Krugman ’09)

Misplaced reliance on sophisticated maths [...] inherent ‘Knightian’uncertainty. (The Turner Review ’09)

we need insights into the omnipresent model risk. [...] the programof introducing a Robust framework is one the greatest challengesalso from the mathematical perspective (Hans Follmer ’09)

For banks, the only way to avoid a repetition of the current crisis isto measure and control all their risks, including the risk that theirmodels give incorrect results (Steven Shreve ’08)

the problem is that we don’t have enough math. [...] Frictions arejust bloody hard with the mathematical tools we have now (JohnCochrane ’09)







This approach has important drawbacks:

• It is exposed to model risk which may be hard to quantify.• Inevitably, it ignores information present in the market.

Models are re-calibrated daily: theoretically inconsistent.• It is a first order approximation which ignores market frictions.

A robust approach aims to address these weaknesses.It does not replace the classical framework but offers an alternativeallowing for a comprehensive understanding and control of risk.







This approach has important drawbacks:

• It is exposed to model risk which may be hard to quantify.• Inevitably, it ignores information present in the market.

Models are re-calibrated daily: theoretically inconsistent.• It is a first order approximation which ignores market frictions.

A robust approach aims to address these weaknesses.It does not replace the classical framework but offers an alternativeallowing for a comprehensive understanding and control of risk.



Outline

Modelling in Mathematical FinanceClassical modelling frameworkToward a Robust modelling framework

Theory of robust valuation and hedgingDiscrete time modelsContinuous time models with discrete tradingContinuous time models with continuous trading

Beliefs based on historic time seriesConfidence intervals for realised volatilityBounds on implied volatility

Robust hedging in practice



Outline







Modelling: Key Ingredients

Inputs:

• Beliefs (about the future dynamics of the stock price process)

• Information (market prices)

• Rules (self-financing trading strategies, frictions)

Reasoning principles:

• Markets are efficient – there are no arbitrages.

Outputs:

• Prices and hedges

• Portofolio optimisation

• Risk management



Modelling: Classical approachInputs:

• Beliefs (about the future dynamics of the stock price process):Prices of risky assets (S i

t) have given dynamics and aresemimartingales on a fixed (Ω,F , (Ft),P).

• Information (market prices): S i0

Market quotes used to calibrate the above by choosing freeparameters via reverse engineering.

• Rules (self-financing trading strategies, frictions):s-f trading strategy = stochastic integral; no frictions


• Markets are efficient – there are no arbitrages:Fundamental theorem of asset pricing (FTAP) asserts thatno-arbitrage is equivalent to existence of Q equivalent to Punder which (discounted) S i

t are all martingales.

Outputs:

• Prices and hedges – via cost of replication; computingexpectations of functionals of solutions to SDEs

• Portofolio optimisation – stochastic control theory• Risk management – duality, convex and functional analysis












Outputs:














Outputs:














Outputs:














Outputs:





Relaxing Beliefs: Model uncertaintyInputs:

• Beliefs (about the future dynamics of the stock price process):Prices of the assets (S i

t) are adapted to (Ω,F , (Ft)) and wehave a set of possible measures (Pα).

• Information (market prices): only S i0.

• Rules (self-financing trading strategies, frictions):no frictions; trading strategy = stochastic integral: classicaland its extensions (quasi-sure analysis, G-expectation...)


• Markets are efficient – there are no arbitrages:Essentially link back to the classical FTAP.

Outputs:

• Robust Portofolio optimisation and Risk management• Price bounds and super/sub-hedges of liabilities

see e.g. Rogers (01), Maccheroni et al. (06), Schied (07); Lyons (95),

Avellaneda et al. (95), Denis and Martini (06), Peng (07), Soner et al.

(11); Bick and Willinger (94), Cassese (08), Vovk (12)Robust approach to Mathematical Finance Bruti-Liberati Lecture, QMF 2011 Jan Ob loj


Including Information: Market ModelsInputs:

• Beliefs (about the future dynamics of the stock price process):A probability space (Ω,F , (Ft),Q) is fixed and we specify the

joint dynamics of the asset St and the option prices PK ,Tt .

• Information (market prices):full use – market prices simply specify the initial values ofS0,P

K ,T0 .

• Rules (self-financing trading strategies, frictions):no frictions; trading strategies are stochastic integrals.


• Markets are efficient – there are no arbitrages:The classical FTAP.

Outputs:

• In principle all prices/hedges are specified clearly if one cancompute them.

see Schonbucher (99), Bergomi (05), Schweizer and Wissel (08), Jacod

and Protter (10), Carmona and Natochiy (09,11),Robust approach to Mathematical Finance Bruti-Liberati Lecture, QMF 2011 Jan Ob loj


B & I: the Robust Approach (Cox & O. (11))

Inputs:


t)t≤T belong to some path space P.• Information (market prices): set of payoffs X ,X 3 X : P→ R, with given prices P : X → R

• Rules (self-financing trading strategies, frictions): no frictions,trading simple trading; pathwise stochastic integrals


• New FTAP needed:no-arbitrage ⇐⇒ exists a (P,X ,P)–market model i.e. aclassical setup (Ω,F , (Ft),Q, (St)) with (St) ∈ P a.s., St aQ–martingale and with PX = EQX , X ∈ X .no-arbitrage ⇐⇒ restrictions on P and on P.

Outputs: Consider an option OT : P→ R and investigate

P admits no arbitrage on X ∪ OT ⇔ LB ≤ POT ≤ UB

⇔ there exists a (P,P,X ∪ OT)–market model

& derive robust hedging strategies which enforce LB and UB.




Inputs:













Inputs:












Outline







One period model – a first passInputs:

• Beliefs: (S0,S1) ∈ P = S0 × P, P ⊂ R.• Information: set of payoffs X = X1, . . . ,Xn, Xn : P→ R,

with given prices P : X → R.• Rules: no frictions, self-financing trading given by π(S1 − S0)

i.e. P(S1) = S0.


(Model–independent arbitrage (MIA))

We say that (P,X ,P) admits no model–independent arbitrage iffor any traded X , i.e. X =

∑ni=1 πi (Xi − PXi ) + π(S1 − S0), with

X (S1) ≥ 0 for all S1 ∈ P, we have PX ≥ 0.

A (P,P,X )–market model simply corresponds to a probability µon R, µ(P) = 1,

∫sµ(ds) = S0,

∫Xi (s)µ(ds) = P(Xi ).

Outputs: Assume the Inputs are consistent with no-arbitrage andconsider an option with a payoff O(S1)...






i.e. P(S1) = S0.







∫sµ(ds) = S0,








i.e. P(S1) = S0.







∫sµ(ds) = S0,





One period model – a first passOutputs:

Let M(P,P,X ) =MP be the set of market models and assumeMP 6= ∅. Consider an option with a payoff O(S1).

Then no MIA =⇒ LB ≤ PO(S1) ≤ UB with

supµ∈MP

∫O(s)µ(ds) ≤ UB = inf

P(Y ) : Y =

n∑i=1

πiXi (S1) + πS1 ≥ O(S1)

infµ∈MP

∫O(s)µ(ds) ≥ LB = sup

P(Y ) : Y =

n∑i=1

πiXi (S1) + πS1 ≤ O(S1)

Davis, O. and Raval (12): for O ′′ ≥ 0, Xi = (S1 − Ki )+ the above

boils down to a semi-infinite linear programming.UB is trivial, for LB the RHS always has a maximiser but dualminimiser mail fail to exist.

Absence of MIA is to weak to grant existence of the dual minimiserIn fact, it is too weak to grant MP 6= ∅ in the first place!






supµ∈MP


P(Y ) : Y =

n∑i=1

πiXi (S1) + πS1 ≥ O(S1)

infµ∈MP


P(Y ) : Y =

n∑i=1

πiXi (S1) + πS1 ≤ O(S1)









supµ∈MP


P(Y ) : Y =

n∑i=1

πiXi (S1) + πS1 ≥ O(S1)

infµ∈MP


P(Y ) : Y =

n∑i=1

πiXi (S1) + πS1 ≤ O(S1)









supµ∈MP


P(Y ) : Y =

n∑i=1

πiXi (S1) + πS1 ≥ O(S1)

infµ∈MP


P(Y ) : Y =

n∑i=1

πiXi (S1) + πS1 ≤ O(S1)






One period model – a first passAn Instructive Example

The following example by Davis and Hobson (07) shows that thearbitrage may require further specification of beliefs...Consider three call options with prices:

10 ATM 15 160

1

5

Then the arbitrage is

• If we believe that S1 ≤ 16 a.s. then sell Call(16).• If we believe that S1 > 16 with a positive p-ty then sell

Call(16) and buy Call(15).



One period model – a second pass

(Robust (sequential) arbitrage)

We say that (P,X ,P) admits a robust arbitrage if there existP1 ⊂ . . . ⊂ Pn = P such that P admits a MIA arbitrage on(P1,X ,P) and for each i = 2, . . . , n there exists a traded portoflioXi with PXi ≤ 0, Xi ≥ 0 for all paths in Pi−1 and X > 0 for allpaths in Pi \Pi−1.

In the example above it suffices to take n = 2 andP1 = S ∈ P : S1 ≤ 16.






Theorem (One-step Robust FTAP)

Consider a one-step model with (S0, S1) ∈ P = S0 × P. AssumeX is finite and all X ∈ X are continuous, piecewise polynomialfunctions of S1 with at most M segments. Then (P,X ,P) admitsno Robust Arbitrage iff there exists a (P,X ,P) – market model,i.e. a (Ω,F , (Ft),Q, (St)) with (St) ∈ P a.s., (St : t = 0, 1) aQ–martingale and with PX = EQX , X ∈ X .Furthermore, if all X ∈ X are piecewise linear, then it suffices totake n = 2 in the definition of RA.






Theorem (One-step Robust FTAP)

Consider a one-step model with (S0, S1) ∈ P = S0 × P. AssumeX is finite and all X ∈ X are continuous, piecewise polynomialfunctions of S1 with at most M segments. Then (P,X ,P) admitsno Robust Arbitrage iff there exists a (P,X ,P) – market model,i.e. a (Ω,F , (Ft),Q, (St)) with (St) ∈ P a.s., (St : t = 0, 1) aQ–martingale and with PX = EQX , X ∈ X .Furthermore, if all X ∈ X are piecewise linear, then it suffices totake n = 2 in the definition of RA.



One period model – a second passSome remarks about the one-step Robust FTAP:

• The essence of the proof is to show that:RA ⇐⇒ for any model (Ω,F , (Ft),P, (St)), (St) ∈ P a.s.,there exists an arbitrage X : PX ≤ 0, X ≥ 0 a.s. andP(X > 0) > 0.The rest then follows from the DMW version of the FTAP.

• Note that the other property was taken as the definition of“weak arbitrage” in Davis and Hobson (07). Their thm is aspecial case of the above with X as set of put options.

• The essence of Davis and Hobson (07) however was to giveequivalent algebraic properties of PX : X ∈ X. In generalthis seems too much to hope for!

• Crucially there is no “full support on P” assumption: marketinformation may imply that part of P is inaccessible!Under “full support” and some topological structure thingssimplify greatly, see e.g. Riedel (12).
























One period model – a third passInputs:

• Beliefs: P = R+

• Information: set of payoffs X = (S1 − K )+ : K ≥ 0 withgiven prices P(S1 − K )+ = C (K ).

• Rules: no frictions, self-financing trading given by π(S1 − S0)i.e. P(S1) = S0.

Reasoning principles: no MIA =⇒ C (0) = S0, C ′(0+) ≥ −1and C (K ) is convex and non-increasing.FTAP (Cox & O. (11)): no WFLVR ⇔ MP 6= ∅ ⇔ C (K ) asabove and C (K )→ 0 as K →∞.

Outputs: Any option O(S1) can be replicated by an infiniteportfolio of calls and puts – has a unique price. Perfect hedge mayfail to exist.



One period model – a third passInputs:

• Beliefs: P = R+

• Information: set of payoffs X = (S1 − K )+ : K ≥ 0 withgiven prices P(S1 − K )+ = C (K ).

• Rules: no frictions, self-financing trading given by π(S1 − S0)i.e. P(S1) = S0.

Reasoning principles: no MIA =⇒ C (0) = S0, C ′(0+) ≥ −1and C (K ) is convex and non-increasing.FTAP (Cox & O. (11)): no WFLVR ⇔ MP 6= ∅ ⇔ C (K ) asabove and C (K )→ 0 as K →∞.

Outputs: Any option O(S1) can be replicated by an infiniteportfolio of calls and puts – has a unique price. Perfect hedge mayfail to exist.



Discrete time modelsInputs:

• Beliefs: P = Rn+

• Information: set of payoffsX = (St − K )+ : K ≥ 0, t = 1, . . . , n with given prices.

• Rules: no frictions, discrete time trading in S , static in calls.

Reasoning principles: Assume MP 6= ∅.Outputs: Using Moge-Kantorovich (martingale) mass transportapproach Beiglbock, Henry-Labordere and Penkner (12) show thatduality holds:

supMP

(payoff expectation) = infP(superhedge)

and existence holds on the LHS.



Discrete time modelsInputs:

• Beliefs: P = Rn+

• Information: set of payoffsX = (St − K )+ : K ≥ 0, t = 1, . . . , n with given prices.

• Rules: no frictions, discrete time trading in S , static in calls.

Reasoning principles: Assume MP 6= ∅.Outputs: Using Moge-Kantorovich (martingale) mass transportapproach Beiglbock, Henry-Labordere and Penkner (12) show thatduality holds:

supMP

(payoff expectation) = infP(superhedge)

and existence holds on the LHS.



Up-and-out put example (Brown, Hobson, Rogers (01))

Consider an up-and-out put option with strike K and barrierB > K . Assume prices of vanilla put options are known withstrikes Ki . For L = Ki < K we have, with S t := supu≤t Su,

(K−ST )+1ST <B ≤B − K

B − L(L− ST )+ − K − L

B − K(ST − B)+

K − L

B − K(ST−B)1ST≥B





(K−ST )+1ST <B ≤B − K

B − L(L− ST )+ − K − L

B − K(ST − B)+

K − L


0 0.5 1 1.5 2 2.5

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.20.3

0.4

0.5

Initial PositionPut Payoff

K=0.6L=0.3 B=1.2





(K−ST )+1ST <B ≤B − K

B − L(L− ST )+ − K − L

B − K(ST − B)+

K − L


0 0.5 1 1.5 2 2.5

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

Put PayoffAfter B hit

K=0.6L=0.3 B=1.2



No-arbitrage pricing & the Skorokhod EmbeddingsThe methodology used in:

Hobson (98); Brown, Hobson and Rogers (01), Dupire (05), Lee (07),

Cox, Hobson and O. (08), Cox and Wang (12); cf. O. (10), Hobson (10).



No-arbitrage pricing & the Skorokhod Embeddings

• P = C ([0,T ]), we are given call prices and want to price OT .• Suppose we have a market model: a classical framework with

• (St) is a continuous martingale under P = Q,• we see market prices C (K ) = E(ST − K )+, K ≥ 0.

• Equivalently (St : t ≤ T ) is a UI martingale, ST ∼ µ,

µ(dx) = C ′′(dx).

• Via Dubins-Schwarz St = Bτt is a time-changed Brownianmotion. Say we have OT = O(S)T = O(B)τT .

• We are led then to investigate the bounds

LB = infτ

EO(B)τ , and UB = supτ

EO(B)τ ,

for all stopping times τ : Bτ ∼ µ and (Bt∧τ ) a UI martingale,i.e. for all solutions to the Skorokhod Embedding problem.

• The bounds are tight: the process St := Bτ∗∧ tT−t

defines an

asset model which matches the market data.







µ(dx) = C ′′(dx).



LB = infτ


EO(B)τ ,



defines an








µ(dx) = C ′′(dx).



LB = infτ


EO(B)τ ,



defines an








µ(dx) = C ′′(dx).



LB = infτ


EO(B)τ ,



defines an




From BM to MI superhedging

• Consider UB. The idea is to devise inequalities of the form

O(B)u ≤ Nu + F (Bu), u ≥ 0,

and where Nu is a is a martingale , with equality for τ∗ withBτ∗ ∼ µ: then E[O(B)τ∗ ] = UB = E[F (Bτ∗)].

• If can make sense of Ht = Nτ−1t

as a self-financing tradingstrategy then we have the cheapest superhedge andPO(S)T ≤ UB = PF (ST ).

• Case 1: Ht =∑k

i=1 πi (ST − Sρi )1ρi≤t , for some first hittingtimes ρi – i.e. we simply buy/sell forwards when the pricereaches predefined levels. Makes sense pathwise.




• Consider UB. The idea is to devise super-hedges of the form

O(S)T ≤ Nτ−1T

+ F (ST )

and where Nτ−1t

is a ?• If can make sense of Ht = Nτ−1

tas a self-financing trading

strategy then we have the cheapest superhedge andPO(S)T ≤ UB = PF (ST ).







O(S)T ≤ Nτ−1T

+ F (ST )

and where Nτ−1t










O(S)T ≤ Nτ−1T

+ F (ST )

and where Nτ−1t








Example: double barriers options

Theorem (Cox and O. (’11))

Let P = C ([0,T ]). Suppose P admits no WFLVR onX = 1, 1ST>b, 1ST≥b, (ST − K )+ : K ≥ 0.Then the following are equivalent

• P admits no WFLVR on X ∪ 1ST≥b, ST≤b,• there exists a (P,P,X ∪ 1ST≥b, ST≤b) market model,

•

P(1ST≥b, ST≤b) ≤ infP(H

I),P(H

II(K2)),P(H

III(K3))

,

P(1ST≥b, ST≤b) ≥ supP(H I ),P(H II (K1,K2))

.

where H·, H · are explicit super- and sub- hedging strategies.



Robust hedging and pathwise stochastic integralsCase 2: Nu =

∫ u0 hsdBs Ht =

∫ t0 hτ−1

udSu.

• If hs = h(Bs ,As) for a ‘nice’ A, e.g. Au = sups≤u Bs then we

have Ht =∫ t

0 h(Su,A(S)u)dSu. (Extensions of) Follmer’spathwise stochastic calculus allows to make sense of suchtrading strategies. Gives pathwise Ito or Tanaka. Applies toe.g. options on variance or on local time, see Cox, Hobson and

O. (08), Davis, O. and Raval (12), Cox and Wang (12).

Requires pathspace P with well defined quadratic variation orpathwise local time.We can either assume (‘believe’) it or try to rule out otherpaths by arbitrage, see Vovk (12).

Admissibility is an interesting question (again)!• Otherwise, if OT is arbitrary, we look at generic

∫hudSu

defined on (Ω,F , (Ft)) simultaneously for a large class ofmeasures Pα, see e.g. Denis and Martini (06), Peng (07), Soner,

Touzi and Zhang (11), Nutz (12), Galichon, Henry-Labordere and

Touzi (12).




∫ u0 hsdBs Ht =

∫ t0 hτ−1

udSu.


have Ht =∫ t





∫hudSu



Touzi (12).




∫ u0 hsdBs Ht =

∫ t0 hτ−1

udSu.


have Ht =∫ t





∫hudSu



Touzi (12).




∫ u0 hsdBs Ht =

∫ t0 hτ−1

udSu.


have Ht =∫ t





∫hudSu



Touzi (12).




∫ u0 hsdBs Ht =

∫ t0 hτ−1

udSu.


have Ht =∫ t





∫hudSu



Touzi (12).



Outline







Beliefs about realised volatility

So far we only had market information = prices of liquid options.

What about time-series of past data? Way may want to use it toshrink P and narrow down the super/sub-hedging bounds derivedso far (Mykland (03,05)).

Consider PΞ the set of continuous functions (St) which admitquadratic variation and with [ln S ]T ≤ Ξ.We can solve the problem of robust pricing and hedging of aEuropean option with a convex payoff λ(ST ) given prices of nco-maturing put options.















Beliefs about realised volatilityConsider PΞ the set of continuous functions (St) which admitquadratic variation and with [ln S ]T ≤ Ξ.We can solve the problem of robust pricing and hedging of aEuropean option with a convex payoff λ(ST ) given prices of nco-maturing put options. Consider a classical setup withdSt = σtStdWt . Then

(ST , [ln S ]T ) = (ST ,

∫ T

0σ2

t dt) = (YτT , τT ), dYu = YudBu.

The superhedging problem becomes

supEλ(ST ) : [ln S ]T ≤ Ξ,E(Ki − ST )+ = Pi= supEλ(YτT ) : τT ≤ Ξ,E(Ki − YτT )+ = Pi

which is solved by Root’s stopping time (cf. Rost ’76).The superhedge is obtained by solving a Variational Inequality(FBP) with obstacle of the form λ(S)−

∑ni=1 πi (Ki − S)+.



Beliefs about realised volatilityConsider PΞ the set of continuous functions (St) which admitquadratic variation and with [ln S ]T ≤ Ξ.We can solve the problem of robust pricing and hedging of aEuropean option with a convex payoff λ(ST ) given prices of nco-maturing put options. Consider a classical setup withdSt = σtStdWt . Then

(ST , [ln S ]T ) = (ST ,

∫ T

0σ2

t dt) = (YτT , τT ), dYu = YudBu.

The superhedging problem becomes

supEλ(ST ) : [ln S ]T ≤ Ξ,E(Ki − ST )+ = Pi= supEλ(YτT ) : τT ≤ Ξ,E(Ki − YτT )+ = Pi

which is solved by Root’s stopping time (cf. Rost ’76).The superhedge is obtained by solving a Variational Inequality(FBP) with obstacle of the form λ(S)−

∑ni=1 πi (Ki − S)+.



Robust Static hedging (with S. Nadtochiy)

Suppose we believe the implied volatility IVt(K ) over the next yearwill stay within two curves: IV ≤ IVt ≤ IV .

To accommodate this within the framework we consider as P thespace of possible paths of a market model: the underlying & itsEuropean options (coded via IV or LV).

Consider an up-and-out put with strike K and barrier B and itsexact static hedge in a time-homogenous local volatility modelwhere IV (K ) ≈ IV (K )1K≤B + IV (K )1K>B , see Carr andNadtochiy (11). It is given by

(K−ST )+−g(ST ), s.t. g |[0,B] = 0 and E[(K−ST )+−g(ST )|St = B] = 0.

Then, for any “path” in P, (K − St)+ − g(ST ) is a superhedge,where g is the smallest convex majorant of g .




























We now combine this with the robust hedge of Brown, Hobson andRoger (01). Consider K1 < K and its associated g1. Then

B − K

B − K1(K1−ST )+−K − K1

B − K(ST−B)+

K − K1

B − K(ST−B)1supt≤T≥B−g1(ST )

is a superhedge for any path in P. Finally, we minimise the initialcost over K1 < K to obtain the Robust Static hedge.

Theorem

If IV → 0 and IV →∞ then g1 → 0, the robust static hedge andits cost converge to the robust hedge.

work in progress...




We now combine this with the robust hedge of Brown, Hobson andRoger (01). Consider K1 < K and its associated g1. Then

B − K

B − K1(K1−ST )+−K − K1

B − K(ST−B)+

K − K1

B − K(ST−B)1supt≤T≥B−g1(ST )

is a superhedge for any path in P. Finally, we minimise the initialcost over K1 < K to obtain the Robust Static hedge.

Theorem

If IV → 0 and IV →∞ then g1 → 0, the robust static hedge andits cost converge to the robust hedge.

work in progress...



Robust Static hedging – examples

Up-and-out put with K = 0.6, B = 1.2, L = 0.3 and IV /IV = 3.

0 2 4 6 8 10 12 14 16 18 20−12

−10

−8

−6

−4

−2

0

2

Put PayoffOriginal RobustStatic Robust hedge



Robust Static hedging – examples

Up-and-out put with K = 0.6, B = 1.2, L = 0.3 and IV /IV = 6.

0 2 4 6 8 10 12 14 16 18 20−7

−6

−5

−4

−3

−2

−1

0

1

Robust static hedgePut PayoffOriginal Robust hedge



Outline







Performance of robust hedges (with F. Ulmer (11))

• Traders A & B sell a digital double barrier option, e.g.1ST≥b,ST≤b, 1ST≥b, ST≤b, for an initial price p.

• Underlying dynamics are unknown but calibrated to the initialsurface. We will use BS, Heston, Bates and VGSV models.

• Trader A will use BS delta or delta/vega hedging with theATM IV. Rebalancing is done

• daily or every six hours• optimally: based on a bandwith around his delta and vega

positions (Whalley and Wilmott (97))

• Trader B will use the robust hedges. More precisely, for aprice p1 > p Trader B buys the superhedge H. His final payoffis given as

p − p1 − 1ST≥b,ST≤b + H

which is bounded below by p − p1 if the path is inP = C ([0,T ]). The hitting times of barrier are observedexactly or monitored every six hours or daily.










p − p1 − 1ST≥b,ST≤b + H











p − p1 − 1ST≥b,ST≤b + H




Initial datacorresponds to176 quotes onAUD/USD on 14Jan 2010: from5∆ put to 35∆put, ATM andfrom 35∆ call to5∆ call for 16maturities.We assumetrading in Scarries a 4bpstransaction costsand in options a100bps costs.Spot S0 = 0.9308.

(a) Heston; EURUSD (b) Heston; AUDUSD

(c) Bates; EURUSD (d) Bates; AUDUSD

(e) VGSV; EURUSD (f) VGSV; AUDUSD

Figure 4.2: EURUSD and AUDUSD Market implied bid and ask volatility surfaces and the correspondingcalibrated model-implied volatility surfaces.

28




We consider 24 scenarios:

• 4 positions (long/short in DNT, DT)• each under 6 market scenarios.

For each scenario we run MC and generate hedging errors for

• no hedging• Traders A using 8 hedges• Traders B using 3 hedges.

For each out of 288 combinations, we report

• Mean, SD, Skew, Kurtosis• Minimum, Maximum• VaR, CVaR both at 99%• EUM, EUH – expected exponential utility with γ = 1, 2• CDF plot

We say that hedge H1 outperforms H2 if

• H1 achieves lower VaR, CVaR and max loss than H2• and expected return (H1) ≥ expected return (H2).




































Hedge errors CDF, long position in a DNT option, barriers (0.85, 1.01), BS model:

1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

No hedgingDelta 250Delta/vega 250Opt. delta 1000Opt. delta/vega 1000Robust Hedge 250Robust Hedge 1000Robust Hedge Exact



Hedge errors CDF, short position in a DT option, barriers (0.85, 1.01), BATES:

1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1




Hedge errors CDF, long position in a DT option, barriers (0.875, 0.985), VGSV:

1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1




Comparative performance of hedging methods• Overwhelmingly, we find that the robust hedges outperform

traditional hedging methods. Often lead to a dramaticreduction of risk while achieving similar or higher returns.

• This remains true even in the models with jumps (Bates,VGSV).

• “Robust hedging errors” are typically positively skewed:frequent small losses compared with some large gains.“Traditional hedging errors” are typically negatively skewed.

• SD of robust hedging errors typically 2–3 larger than of thetraditional hedges.

• We assumed no interest rates: applies for currency pairs withsimilar domestic interest rates.• Makes sense for singularly large position in a barrier option in

uncertain market conditions.• Robust hedges akin to the methods used by “old-school

traders”Robust approach to Mathematical Finance Bruti-Liberati Lecture, QMF 2011 Jan Ob loj
























































Conclusions and further research• I advocate a new robust framework for pricing and hedging. It

is a pathwise approach which combines beliefs about possiblepaths together with market information.

• New notions of no-arbitrage are introduced and examples ofFTAP obtained.

• Estimates on e.g. realised total volatility or IVol surfaceobtained from past data can be incorporated.

• The outputs are: no-arbitrage price bounds and the super-and sub- hedging strategies which enforce them.

• Robust hedging methods may greatly outperform delta/vegahedging in presence of transaction costs and/or modeluncertainty.

• So far one-step models are worked out, many examples forcontinuous time but a general theory is missing.

• Pathwise stochastic calculus and its link to no-arbitrage needfurther development.

• Incorporating econometrics to reduce P has only been started,no exotics treated so far.

























Thank You!



This talk was based on

• A.M.G. Cox and J. Ob loj. Robust hedging of double no-touchbarrier options. Finance and Stochastics, 15(3): 573605,2011.

• A.M.G. Cox and J. Ob loj. Robust hedging of double touchbarrier options. SIAM Journal on Financial Mathematics, 2:141–182, 2011.

• M. Davis, J. Ob loj and V. Raval. Arbitrage bounds forweighted variance swap prices. 2009–2011. arXiv: 1001.2678

• F. Ulmer and J. Ob loj. Performance of robust hedging ofdouble barrier options. International Journal of Theoreticaland Applied Finance, to appear, 2011.(DOI: 10.1142/S0219024911006516)

• work in progress with Sergey Nadtochiy, 2012.• my own ideas and not-yet-finished preprints.



(Weak free lunch with vanishing risk (Cox & O. ’11))

We say that (P,XP) admits a weak free lunch with vanishing riskif there exists Xn,Z ∈ Lin(X ) such that Xn → X (pointwise onP), Xn ≥ Z , X ≥ 0 and limPXn < 0.


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