model-free pricing and hedging of options (via …...principal questions and answers double barrier...

Principal Questions and Answers Double barrier options Applications

Model-free Pricing and Hedging of Options

(via Skorokhod Embeddings)

Jan Ob loj

Department of Mathematics, Imperial College London

Joint work with Alexander Cox from University of Bath

Finance and Related Mathematical and Statistical IssuesKyoto, 3-6 September 2008

(Very) Robust Pricing and Hedging of Options

(via Skorokhod Embeddings)

Jan Ob loj

Department of Mathematics, Imperial College London

Joint work with Alexander Cox from University of Bath

Finance and Related Mathematical and Statistical IssuesKyoto, 3-6 September 2008

Outline

Principal Questions and AnswersFinancial Problem (2 questions)

Methodology (2 answers)

Double barrier optionsIntroduction and types of barriersPricing and hedging of double touch options

ApplicationsHedging comparison under transaction costsSummary

Two Main Questions

The general setting and challenge is as follows:

• Observe prices of some liquid instruments which admit noarbitrage.

• Q1: (very) robust pricingGiven a new product, determine its feasible price, i.e. a pricewhich does not introduce any arbitrage in the market.

• Q2: (very) robust hedgingFurthermore, derive tight super-/sub- hedging strategieswhich always work.

E.g.: Put-Call parity, Up-and-in put.

Two Main Questions

Q1 and the Skorokhod Embedding ProblemQ1: What is the range of no-arbitrage prices of an option OT

given prices of European calls?• Suppose:

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

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

µ(dx) = C ′′(x)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 ′′(x)dx .

LB = infτ

EO(B)τ ,

defines an

µ(dx) = C ′′(x)dx .

LB = infτ

EO(B)τ ,

defines an

µ(dx) = C ′′(x)dx .

LB = infτ

EO(B)τ ,

defines an

Q2 and pathwise inequalities

Q2: if we see a price outside the bounds (LB ,UB) can we (andhow) realise a risk-less profit?

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

O(B)t ≤ Nt + F (Bt), t ≥ 0,

with equality for some τ∗ with Bτ∗ ∼ µ, and where is amartingale (i.e. trading strategy), ENτ∗ = 0 and F ′′ > 0.

• Then UB = EF (ST ) and + F (St) is a validsuperhedge. It involves dynamic trading anda static position in calls F (ST ).

• Furthermore, we want (Nτt ) explicitly. We are naturallyrestricted to the family of martingales Nt = N(Bt ,At), forsome process (At) related to the option Ot , e.g. maximumand minimum processes for barrier options.

O(B)t ≤ Nt + F (Bt), t ≥ 0,

with equality for some τ∗ with Bτ∗ ∼ µ, and where Nt is amartingale (i.e. trading strategy), ENτ∗ = 0 and F ′′ > 0.

• Then UB = EF (ST ) and Nτt + F (St) is a validsuperhedge. It involves dynamic trading (Nτt ) anda static position in calls F (ST ).

O(B)t ≤ Nt + F (Bt), t ≥ 0,

O(B)t ≤ N(Bt ,At) + F (Bt), t ≥ 0,

with equality for some τ∗ with Bτ∗ ∼ µ, and where N(Bt ,At)is a martingale (i.e. trading strategy), ENτ∗ = 0 and F ′′ > 0.

• Then UB = EF (ST ) and N(St ,ASt ) + F (St) is a valid

superhedge. It involves dynamic trading N(St ,ASt ) and

a static position in calls F (ST ).

Scope of applications

• Answer to Q1 and pricing: in practice LB << UB , the boundsare too wide to be of any use for pricing.

• Answer to Q2 and hedging: say an agent sells OT for price p.She then can set up our super-hedge for UB . At the expiryshe holds

X = p − UB + F (ST ) + N(ST ,AST ) − OT .

We have EX = 0 and X ≥ p − UB . The hedge might have aconsiderable variance but the loss is bounded below (for allt ≤ T ). The hedge is very robust as we make virtually nomodelling assumptions and only use market input. This canbe advantageous in presence of

• model uncertainty• transaction costs• illiquid markets.

Scope of applications

• Answer to Q1 and pricing: in practice LB << UB , the boundsare too wide to be of any use for pricing.

• Answer to Q2 and hedging: say an agent sells OT for price p.She then can set up our super-hedge for UB . At the expiryshe holds

X = p − UB + F (ST ) + N(ST ,AST ) − OT .

We have EX = 0 and X ≥ p − UB . The hedge might have aconsiderable variance but the loss is bounded below (for allt ≤ T ). The hedge is very robust as we make virtually nomodelling assumptions and only use market input. This canbe advantageous in presence of

• model uncertainty• transaction costs• illiquid markets.

Recipe: how to answer Q2

Consider super-hedging of O(S)T :

1. Answer Q1: devise τ solutions to the Skorokhod Embeddingproblem, Bτ ∼ ST , which maximise the functional O(B)τ .Identify scenarios in the extreme model Bτ∧ t

T−tunder which

the option pays off.

2. Choose At and describe martingales of the form N(Bt ,At),t ≥ 0.

3. Devise inequalities O(B)t ≤ N(Bt ,At) + F (Bt), t ≥ 0, in sucha way that equality is attained in the extreme model Bτ∧ t

T−t.

We will illustrate this with the example of a one-touch optionO(S)T = 1ST≥b, where ST = supt≤T St .

T−tunder which

T−t.

T−tunder which

T−t.

T−tunder which

T−t.

One-touch option: Answer to Q1

• Ψµ(y) =∫ ∞y

uµ(du)µ([y ,∞))

τ = inf{t ≥ 0 : sups≤t

Bs ≥ Ψµ(Bt)}

• We have Bτ ∼ µ and τ

maximises

P(sups≤τ

Bs ≥ b)

over all UI embeddings,for any b > B0.

supu≤t Bu

• Azema-Yor (1979)

Note that ST ≥ b if and only if ST ≥ Ψ−1µ (b), for St = Bτ∧ t

T−t.

• Ψµ(y) =∫ ∞y

uµ(du)µ([y ,∞))

Bs ≥ Ψµ(Bt)}

maximises

P(sups≤τ

Bs ≥ b)

supu≤t Bu

T−t.

• Ψµ(y) =∫ ∞y

uµ(du)µ([y ,∞))

Bs ≥ Ψµ(Bt)}

maximises

P(sups≤τ

Bs ≥ b)

supu≤t Bu

T−t.

• Consider pathwise inequality:

1{S t≥b} ≤(St − K )+

b − K+

(b − St)

b − K1{S t>b}

where b > S0, and K < b.

• When S t < b , we get:

0 ≤ (St − K )+

b − K

1{S t≥b} ≤(St − K )+

b − K+

(b − St)

b − K1{S t>b}

• When S t ≥ b, we get:

1 ≤ b − K

b − K+

(K − St)+

b − K

1{S t≥b} ≤(St − K )+

b − K︸︷︷︸

a call

+(b − St)

b − K1{S t≥b}

︸︷︷︸

a forward transaction

Hence, we have a superhedging strategy for any K < b and, lettingP be the pricing operator, we deduce

P1ST≥b ≤ infK<b

P(ST − K )+

b − K=

P(ST − K ∗)+

b − K ∗,

for K ∗ = Ψ−1µ (b), bound attained in the Azema-Yor model.

1{S t≥b} ≤(St − K )+

b − K︸︷︷︸

a call

+(b − St)

b − K1{S t≥b}

︸︷︷︸

a forward transaction

Hence, we have a superhedging strategy for any K < b and, lettingP be the pricing operator, we deduce

P1ST≥b ≤ infK<b

P(ST − K )+

b − K=

P(ST − K ∗)+

b − K ∗,

for K ∗ = Ψ−1µ (b), bound attained in the Azema-Yor model.

References and current worksPrevious works adapting the strategy:

• (lookback options) D. G. Hobson. Robust hedging of thelookback option. Finance Stoch., 2(4):329347, 1998.

• (one-sided barriers) H. Brown, D. Hobson, and L. C. G.Rogers. Robust hedging of barrier options. Math.Finance,11(3):285314, 2001.

• (local-time related options) A. M. G. Cox, D. G. Hobson, andJ. Ob loj. Pathwise inequalities for local time: applications toSkorokhod embeddings and optimal stopping. Ann. Appl.Probab., 2008. to appear.

As well as:

• forward starting options (D. Hobson and R. Norberg, ...)

• volatility derivatives (B. Dupire, R. Lee, ...)

• double barrier options (A. Cox and J.O., arxiv:0808.4012 andmore...)

Double barriers - introductionWe want to apply the above methodology to derivatives withdigital payoff conditional on the stock price reaching/not reachingtwo levels.Continuity of paths implies level crossings (i.e. payoffs) are notaffected by time-changing.

An example is given by adouble touch:

1supt≤T St≥b and inft≤T St≤b.

1supu≤τBu≥b and infu≤τ

Bu≤b.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

In general the option pays 1 on the event{

supt≤T

(≤≥

inft≤T

(≤≥

Double barriers - introductionWe want to apply the above methodology to derivatives withdigital payoff conditional on the stock price reaching/not reachingtwo levels.Continuity of paths implies level crossings (i.e. payoffs) are notaffected by time-changing.

An example is given by adouble touch:

1supt≤T St≥b and inft≤T St≤b.

1supu≤τBu≥b and infu≤τ

Bu≤b.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

In general the option pays 1 on the event{

supt≤T

(≤≥

inft≤T

(≤≥

Double barriers - introduction

There are 8 possible digital double barrier options. However usingcomplements and symmetry, it suffices to consider 3 types:

• double no-touch (range) option maximised by Perkins’construction and minimised by the tilted-Jacka (A. Cox)construction.

• double touch option new solutions to the SEP.

• double touch/no-touch option new solutions to the SEP.May seem artificial but note that

1{ST≥b, ST≤b} = 1ST≥b − 1{ST≥b,ST≤b}

= 1ST≥b − 1ST≤b,ST≥b,

and if one-touches are liquid we deduce super-/sub- hedgesfor double touch or double no-touch options.

Double touch: Answer to Q1We construct τ with Bτ ∼ µ maximising P

(Bτ ≥ b and Bτ ≤ b

What is the best way of getting as much mass as possible to gofrom b to b and vice-versa?

• Run paths to b and b — might need to stop some in themiddle

• From b, want to run as much as possible to b — needs tobalance out, so run rest to tails (biggest ‘push’ in otherdirection)

• Similarly from b. . .

• Run to remaining gaps.

• In each step we keep uniform integrability, i.e. embeddingmeasure on [α, β] we stop before (or when) hitting α or β.

Double touch: Answer to Q2

We now want to construct an inequality which attains equality inthe above model.

• Initially, we stop in the centre, or run to b, b.

• If we hit b, we run to the upper tail or b — if we hit b now,the inequality needs to be ≥ 1.

• After hitting b, we embed near b.

• If we hit b, we run to the lower tail or b — if we hit b now,the inequality needs to be ≥ 1.

Writing Hz = inf{t : St = z} and S t = supu≤t Su, St = infu≤t Su,the inequality can then be represented as follows:

1ST≥b, ST≤b ≤α1(ST − K1)+ + α2(ST − K2)+