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
Principal Questions and Answers Double barrier options Applications
(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
Principal Questions and Answers Double barrier options Applications
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
Principal Questions and Answers Double barrier options Applications
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.
Principal Questions and Answers Double barrier options Applications
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.
Principal Questions and Answers Double barrier options Applications
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.
Principal Questions and Answers Double barrier options Applications
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.
Principal Questions and Answers Double barrier options Applications
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.
Principal Questions and Answers Double barrier options Applications
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.
Principal Questions and Answers Double barrier options Applications
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.
Principal Questions and Answers Double barrier options Applications
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.
Principal Questions and Answers Double barrier options Applications
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 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 ).
• 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.
Principal Questions and Answers Double barrier options Applications
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 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 ).
• 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.
Principal Questions and Answers Double barrier options Applications
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 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 ).
• 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.
Principal Questions and Answers Double barrier options Applications
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 ≤ 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 ).
• 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.
Principal Questions and Answers Double barrier options Applications
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.
Principal Questions and Answers Double barrier options Applications
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.
Principal Questions and Answers Double barrier options Applications
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 .
Principal Questions and Answers Double barrier options Applications
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 .
Principal Questions and Answers Double barrier options Applications
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 .
Principal Questions and Answers Double barrier options Applications
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 .
Principal Questions and Answers Double barrier options Applications
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.
Bt
supu≤t Bu
• Azema-Yor (1979)
Note that ST ≥ b if and only if ST ≥ Ψ−1µ (b), for St = Bτ∧ t
T−t.
Principal Questions and Answers Double barrier options Applications
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.
Bt
supu≤t Bu
• Azema-Yor (1979)
Note that ST ≥ b if and only if ST ≥ Ψ−1µ (b), for St = Bτ∧ t
T−t.
Principal Questions and Answers Double barrier options Applications
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.
Bt
supu≤t Bu
• Azema-Yor (1979)
Note that ST ≥ b if and only if ST ≥ Ψ−1µ (b), for St = Bτ∧ t
T−t.
Principal Questions and Answers Double barrier options Applications
One-touch option: Answer to Q2
• 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
Stb
Principal Questions and Answers Double barrier options Applications
One-touch option: Answer to Q2
• 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:
1 ≤ b − K
b − K+
(K − St)+
b − K
Stb
Principal Questions and Answers Double barrier options Applications
One-touch option: Answer to Q2
• Consider pathwise inequality:
1{S t≥b} ≤(St − K )+
b − K︸ ︷︷ ︸
a call
+(b − St)
b − K1{S t≥b}
︸ ︷︷ ︸
a forward transaction
where b > S0, and K < b.
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.
Principal Questions and Answers Double barrier options Applications
One-touch option: Answer to Q2
• Consider pathwise inequality:
1{S t≥b} ≤(St − K )+
b − K︸ ︷︷ ︸
a call
+(b − St)
b − K1{S t≥b}
︸ ︷︷ ︸
a forward transaction
where b > S0, and K < b.
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.
Principal Questions and Answers Double barrier options Applications
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...)
Principal Questions and Answers Double barrier options Applications
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
50
60
70
80
90
100
110
120
130
In general the option pays 1 on the event{
supt≤T
St
(≤≥
)
b
(and
or
)
inft≤T
St
(≤≥
)
b
}
Principal Questions and Answers Double barrier options Applications
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
50
60
70
80
90
100
110
120
130
In general the option pays 1 on the event{
supt≤T
St
(≤≥
)
b
(and
or
)
inft≤T
St
(≤≥
)
b
}
Principal Questions and Answers Double barrier options Applications
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.
Principal Questions and Answers Double barrier options Applications
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.
Principal Questions and Answers Double barrier options Applications
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.
Principal Questions and Answers Double barrier options Applications
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?
b bS0
• 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.
Principal Questions and Answers Double barrier options Applications
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?
b bS0
• 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.
Principal Questions and Answers Double barrier options Applications
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?
b bS0
• 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.
Principal Questions and Answers Double barrier options Applications
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?
b bS0
• 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.
Principal Questions and Answers Double barrier options Applications
Double touch: Answer to Q2
We now want to construct an inequality which attains equality inthe above model.
b bS0
• 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.
Principal Questions and Answers Double barrier options Applications
Double touch: Answer to Q2
We now want to construct an inequality which attains equality inthe above model.
b bS0
1
• 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.
Principal Questions and Answers Double barrier options Applications
Double touch: Answer to Q2
We now want to construct an inequality which attains equality inthe above model.
b bS0
1
• 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.
Principal Questions and Answers Double barrier options Applications
Double touch: Answer to Q2
We now want to construct an inequality which attains equality inthe above model.
b bS0
• Initially, we stop in the centre, or run to b, b.
• If we hit b, we run to the lower tail or b — if we hit b now,the inequality needs to be ≥ 1.
• After hitting b, we embed near b.
Principal Questions and Answers Double barrier options Applications
Double touch: Answer to Q2
We now want to construct an inequality which attains equality inthe above model.
b bS0
1
• Initially, we stop in the centre, or run to b, b.
• If we hit b, we run to the lower tail or b — if we hit b now,the inequality needs to be ≥ 1.
• After hitting b, we embed near b.
Principal Questions and Answers Double barrier options Applications
Double touch: Answer to Q2
We now want to construct an inequality which attains equality inthe above model.
b bS0
1
• Initially, we stop in the centre, or run to b, b.
• If we hit b, we run to the lower tail or b — if we hit b now,the inequality needs to be ≥ 1.
• After hitting b, we embed near b.
Principal Questions and Answers Double barrier options Applications
Double touch: Answer to Q2
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)+
+ α3(K3 − ST )+ + α4(K4 − ST )+
− β1(ST − b)1Hb<Hb∧τ + β2(ST − b)1Hb<Hb∧τ
+ β3(ST − b)1Hb<Hb≤τ − β4(ST − b)1Hb<Hb≤τ
=: G I (K1,K2,K3,K4),
Principal Questions and Answers Double barrier options Applications
Double touch: Answer to Q2
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)+
+ α3(K3 − ST )+ + α4(K4 − ST )+
− β1(ST − b)1Hb<Hb∧τ + β2(ST − b)1Hb<Hb∧τ
+ β3(ST − b)1Hb<Hb≤τ − β4(ST − b)1Hb<Hb≤τ
=: G I (K1,K2,K3,K4),
Function of ST (portfolio of calls and puts)
Principal Questions and Answers Double barrier options Applications
Double touch: Answer to Q2
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)+
+ α3(K3 − ST )+ + α4(K4 − ST )+
− β1(ST − b)1Hb<Hb∧τ + β2(ST − b)1Hb<Hb∧τ
+ β3(ST − b)1Hb<Hb≤τ − β4(ST − b)1Hb<Hb≤τ
=: G I (K1,K2,K3,K4),
Forward transactions (Martingale terms)
Principal Questions and Answers Double barrier options Applications
Double touch: Answer to Q2
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)+
+ α3(K3 − ST )+ + α4(K4 − ST )+
− β1(ST − b)1Hb<Hb∧τ + β2(ST − b)1Hb<Hb∧τ
+ β3(ST − b)1Hb<Hb≤τ − β4(ST − b)1Hb<Hb≤τ
=: G I (K1,K2,K3,K4),
Note: Can derive αi , βj in terms of K1, . . . K4 using constraints.
Principal Questions and Answers Double barrier options Applications
Double touch: Answer to Q2
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)+
+ α3(K3 − ST )+ + α4(K4 − ST )+
− β1(ST − b)1Hb<Hb∧τ + β2(ST − b)1Hb<Hb∧τ
+ β3(ST − b)1Hb<Hb≤τ − β4(ST − b)1Hb<Hb≤τ
=: G I (K1,K2,K3,K4),
α3 =(K1 − K2)(b − K4)(b − b) − (K1 − b)(b − K2)(b − K4)
(K1 − K2)(K3 − K4)(b − b)2− (K3 − b)(K1 − b)(b − K2)(b − K4)
8
>
<
>
:
α1 =`
1 − α3K3−K4b−K4
(b − b)´
(K1 − b)−1
α2 =`
1 − α3K3−K4b−K4
(b − b)´
(b − K2)−1
α4 = K2−b
b−K4α3
β1 = α1 + α2
β2 = α3 + α4.
Principal Questions and Answers Double barrier options Applications
Double touch: retrieve Q1 via Q2
• We need to show that for our embedding τ we have perfecthedging using G I (K1,K2,K3,K4) for some choice of strikes.
• In fact, the above construction does not always work — thereare another three cases we need to consider:
• If both b and b are close to zero, we might not need to stopbefore hitting b or b.
• If both b is large, and b is close to S0, constraint becomes‘how much mass can we embed at b.’ Construction becomessame as in Azema-Yor case.
• The case where b is small (close to zero), and b is close to S0
is similar.
Principal Questions and Answers Double barrier options Applications
Double touch: retrieve Q1 via Q2
• We need to show that for our embedding τ we have perfecthedging using G I (K1,K2,K3,K4) for some choice of strikes.
• In fact, the above construction does not always work — thereare another three cases we need to consider:
• If both b and b are close to zero, we might not need to stopbefore hitting b or b.
• If both b is large, and b is close to S0, constraint becomes‘how much mass can we embed at b.’ Construction becomessame as in Azema-Yor case.
• The case where b is small (close to zero), and b is close to S0
is similar.
Principal Questions and Answers Double barrier options Applications
Double touch: retrieve Q1 via Q2
• We need to show that for our embedding τ we have perfecthedging using G I (K1,K2,K3,K4) for some choice of strikes.
• In fact, the above construction does not always work — thereare another three cases we need to consider:
• If both b and b are close to zero, we might not need to stopbefore hitting b or b.
• If both b is large, and b is close to S0, constraint becomes‘how much mass can we embed at b.’ Construction becomessame as in Azema-Yor case.
• The case where b is small (close to zero), and b is close to S0
is similar.
Principal Questions and Answers Double barrier options Applications
Double touch: retrieve Q1 via Q2
• We need to show that for our embedding τ we have perfecthedging using G I (K1,K2,K3,K4) for some choice of strikes.
• In fact, the above construction does not always work — thereare another three cases we need to consider:
• If both b and b are close to zero, we might not need to stopbefore hitting b or b.
• If both b is large, and b is close to S0, constraint becomes‘how much mass can we embed at b.’ Construction becomessame as in Azema-Yor case.
• The case where b is small (close to zero), and b is close to S0
is similar.
Principal Questions and Answers Double barrier options Applications
Double touch: retrieve Q1 via Q2
• We need to show that for our embedding τ we have perfecthedging using G I (K1,K2,K3,K4) for some choice of strikes.
• In fact, the above construction does not always work — thereare another three cases we need to consider:
• If both b and b are close to zero, we might not need to stopbefore hitting b or b.
• If both b is large, and b is close to S0, constraint becomes‘how much mass can we embed at b.’ Construction becomessame as in Azema-Yor case.
• The case where b is small (close to zero), and b is close to S0
is similar.
Principal Questions and Answers Double barrier options Applications
Double touch: superhedging
Write P for the pricing operator. No arbitrage implies:
P1{ST≥b,ST≤b} ≤ (†)
inf{
PG I (K1,K2,K3,K4),PG II (K1,K4),PG III (K2),PG IV (K3)}
where the infimum is taken over values of K1, . . . ,K4 withK1 < b < K2 ≤ K3 < b < K4.
Theorem
For any (arbitrage-free) curve of call prices there exists a stockprice process for which (†) is the price of a double touch option.
In fact we specify which hedge and strikes realise the inf in (†).We have analogue theorems for upper and lower bounds on pricesof all digital double barrier options.
Principal Questions and Answers Double barrier options Applications
Double touch: superhedging
Write P for the pricing operator. No arbitrage implies:
P1{ST≥b,ST≤b} ≤ (†)
inf{
PG I (K1,K2,K3,K4),PG II (K1,K4),PG III (K2),PG IV (K3)}
where the infimum is taken over values of K1, . . . ,K4 withK1 < b < K2 ≤ K3 < b < K4.
Theorem
For any (arbitrage-free) curve of call prices there exists a stockprice process for which (†) is the price of a double touch option.
In fact we specify which hedge and strikes realise the inf in (†).We have analogue theorems for upper and lower bounds on pricesof all digital double barrier options.
Principal Questions and Answers Double barrier options Applications
Double touch: superhedging
Write P for the pricing operator. No arbitrage implies:
P1{ST≥b,ST≤b} ≤ (†)
inf{
PG I (K1,K2,K3,K4),PG II (K1,K4),PG III (K2),PG IV (K3)}
where the infimum is taken over values of K1, . . . ,K4 withK1 < b < K2 ≤ K3 < b < K4.
Theorem
For any (arbitrage-free) curve of call prices there exists a stockprice process for which (†) is the price of a double touch option.
In fact we specify which hedge and strikes realise the inf in (†).We have analogue theorems for upper and lower bounds on pricesof all digital double barrier options.
Principal Questions and Answers Double barrier options Applications
Double touch subhedging: Q2 (inequality)
Again, construct a suitable inequality:
b b
K1K2
• Initially, we run to b, b.
• If we hit b, we run to the right of b, or to b or ‘near’ b — ifwe hit b now, the inequality needs to be ≤ 1.
• After hitting b, we embed in the tails.
• A similar pattern can be followed it we hit b first.
Principal Questions and Answers Double barrier options Applications
Double touch subhedging: Q2 (inequality)
Again, construct a suitable inequality:
b b
K1K2
• Initially, we run to b, b.
• If we hit b, we run to the right of b, or to b or ‘near’ b — ifwe hit b now, the inequality needs to be ≤ 1.
• After hitting b, we embed in the tails.
• A similar pattern can be followed it we hit b first.
Principal Questions and Answers Double barrier options Applications
Double touch subhedging: Q2 (inequality)
Again, construct a suitable inequality:
b b
1
K1K2
• Initially, we run to b, b.
• If we hit b, we run to the right of b, or to b or ‘near’ b — ifwe hit b now, the inequality needs to be ≤ 1.
• After hitting b, we embed in the tails.
• A similar pattern can be followed it we hit b first.
Principal Questions and Answers Double barrier options Applications
Double touch subhedging: Q2 (inequality)
Again, construct a suitable inequality:
b b
K1K2
• Initially, we run to b, b.
• If we hit b, we run to the right of b, or to b or ‘near’ b — ifwe hit b now, the inequality needs to be ≤ 1.
• After hitting b, we embed in the tails.
• A similar pattern can be followed it we hit b first.
Principal Questions and Answers Double barrier options Applications
Double touch subhedging: Q2 (inequality)
Again, construct a suitable inequality:
b b
1
K1K2
• Initially, we run to b, b.
• If we hit b, we run to the right of b, or to b or ‘near’ b — ifwe hit b now, the inequality needs to be ≤ 1.
• After hitting b, we embed in the tails.
• A similar pattern can be followed it we hit b first.
Principal Questions and Answers Double barrier options Applications
Double touch subhedging: Q2 (inequality)Inequality is now of the form:
1{ST≥b,ST≤b} ≥α0 + α1ST − α2(ST − K2)+ + α3(ST − b)+
− α3(ST − K3)+ + α3(ST − b)+ − α4(ST − K1)+
− γ11{ST >b} + γ21{ST≥b}
+ β1(ST − b)1{Hb<Hb∧T} − β2(ST − b)1{Hb<Hb<T}
− β3(ST − b)1{Hb<Hb∧T} + β4(ST − b)1{Hb<Hb<T}
=: g I (K1,K2)
• Note the step functions — still functions of ST with specifiedprices.
• Again, depending on the relative position of the barriers, othersubhedges might be optimal. We derive g II (K1,K2,K3) andg III (K1,K2,K3), where b < K3 < b.
Principal Questions and Answers Double barrier options Applications
Double touch subhedging: Q2 (inequality)Inequality is now of the form:
1{ST≥b,ST≤b} ≥α0 + α1ST − α2(ST − K2)+ + α3(ST − b)+
− α3(ST − K3)+ + α3(ST − b)+ − α4(ST − K1)+
− γ11{ST >b} + γ21{ST≥b}
+ β1(ST − b)1{Hb<Hb∧T} − β2(ST − b)1{Hb<Hb<T}
− β3(ST − b)1{Hb<Hb∧T} + β4(ST − b)1{Hb<Hb<T}
=: g I (K1,K2)
• Note the step functions — still functions of ST with specifiedprices.
• Again, depending on the relative position of the barriers, othersubhedges might be optimal. We derive g II (K1,K2,K3) andg III (K1,K2,K3), where b < K3 < b.
Principal Questions and Answers Double barrier options Applications
Double touch subhedging: from Q2 to Q1
• No arbitrage implies:
P1{ST≥b,ST≤b} ≥
sup{
Pg I (K1,K2),Pg II (K1,K2,K3),Pg III (K1,K2,K3), 0}
.
As before, the hard bit is to show that the four cases cover allpossibilities and that the lower bound is tight ( pricing).
• Also, as before, can give complete characterisation of whichcase is appropriate ( robust sub-hedging).
Principal Questions and Answers Double barrier options Applications
Double touch subhedging: from Q2 to Q1
• No arbitrage implies:
P1{ST≥b,ST≤b} ≥
sup{
Pg I (K1,K2),Pg II (K1,K2,K3),Pg III (K1,K2,K3), 0}
.
As before, the hard bit is to show that the four cases cover allpossibilities and that the lower bound is tight ( pricing).
• Also, as before, can give complete characterisation of whichcase is appropriate ( robust sub-hedging).
Principal Questions and Answers Double barrier options Applications
Double touch: what have we assumed?
The super-/sub- hedges (and hence the implied price bounds) holdassuming only that:
• pricing of positions in calls, forward transactions and doubletouches is linear
• we can trade forwards at the barriers and the forward strike isthen equal to the spot,
• market prices of European calls are given and do not admitstatic arbitrage and P(ST − K )+ → 0 as K → ∞.
Principal Questions and Answers Double barrier options Applications
Double touch: what have we NOT assumed?
We do not need to assume that:
• a risk-neutral measure exists s.t. (St) is a martingale,
• we are in a semimartinagle setting, (depends on N(St ,ASt ))
• sample paths are continuous (only barriers are crossedcontinuously).
We can add fixed or proportional transaction costs and still have anon-trivial super-hedge.We can assume only a finite family of call prices is given in bounds like (†) we have inf over traded strikes.
Principal Questions and Answers Double barrier options Applications
Double touch: what have we NOT assumed?
We do not need to assume that:
• a risk-neutral measure exists s.t. (St) is a martingale,
• we are in a semimartinagle setting, (depends on N(St ,ASt ))
• sample paths are continuous (only barriers are crossedcontinuously).
We can add fixed or proportional transaction costs and still have anon-trivial super-hedge.We can assume only a finite family of call prices is given in bounds like (†) we have inf over traded strikes.
Principal Questions and Answers Double barrier options Applications
Double touch: what have we NOT assumed?
We do not need to assume that:
• a risk-neutral measure exists s.t. (St) is a martingale,
• we are in a semimartinagle setting, (depends on N(St ,ASt ))
• sample paths are continuous (only barriers are crossedcontinuously).
We can add fixed or proportional transaction costs and still have anon-trivial super-hedge.We can assume only a finite family of call prices is given in bounds like (†) we have inf over traded strikes.
Principal Questions and Answers Double barrier options Applications
Applications
• The bounds for prices are theoretically satisfying but they aretypically too wide to be of practical use.
• So we need to sell pricing via Monte Carlo under our favourite(calibrated) model. And then what?Delta/Vega-hedging can be
• expensive (transaction costs)• inaccurate (discrete monitoring)• sensitive to model error
• Our superhedges provide two alternatives:• Enforce our superhedge (for given call prices, b and b).
The loss is bounded below by the difference in prices and wehave no model-error risk.
• Enforce the superhedge and delta-hedge the remainingportfolio. This should reduce the variance of hedging errors.This is similar in spirit to delta&vega-hedging but hopefullyperforms better?
Principal Questions and Answers Double barrier options Applications
Applications
• The bounds for prices are theoretically satisfying but they aretypically too wide to be of practical use.
• So we need to sell pricing via Monte Carlo under our favourite(calibrated) model. And then what?Delta/Vega-hedging can be
• expensive (transaction costs)• inaccurate (discrete monitoring)• sensitive to model error
• Our superhedges provide two alternatives:• Enforce our superhedge (for given call prices, b and b).
The loss is bounded below by the difference in prices and wehave no model-error risk.
• Enforce the superhedge and delta-hedge the remainingportfolio. This should reduce the variance of hedging errors.This is similar in spirit to delta&vega-hedging but hopefullyperforms better?
Principal Questions and Answers Double barrier options Applications
Applications
• The bounds for prices are theoretically satisfying but they aretypically too wide to be of practical use.
• So we need to sell pricing via Monte Carlo under our favourite(calibrated) model. And then what?Delta/Vega-hedging can be
• expensive (transaction costs)• inaccurate (discrete monitoring)• sensitive to model error
• Our superhedges provide two alternatives:• Enforce our superhedge (for given call prices, b and b).
The loss is bounded below by the difference in prices and wehave no model-error risk.
• Enforce the superhedge and delta-hedge the remainingportfolio. This should reduce the variance of hedging errors.This is similar in spirit to delta&vega-hedging but hopefullyperforms better?
Principal Questions and Answers Double barrier options Applications
Robust hedging: a historical note
Of course a variety of hedging strategies (typically known as static,semi-static or robust) have been suggested in the literature, undera variety of more or less restrictive assumptions on the priceprocess, and mostly for single barrier options, and variants such asknock-out calls. We mention Follmer and Schweizer (1990), Carrwith Bowie or Chou or Ellis and Gupta (1994-98), Derman,Ergener and Kani (1995), Brown, Hobson and Rogers (2001),Andersen, Andreasen and Eliezer (2002), Fink (2003), Nalholm(2006), GieseMaruhn (2007), Ilhan, Jonsson and Sircar (2006).
However, all of these approaches require a model for the underlyingto be specified whilst our approach is model-free.
Principal Questions and Answers Double barrier options Applications
Hedging comparison under transaction costsScenario 1 (not realistic): after all options are sold/bought andhedges set-up, market model changes drastically (volatility ր) super-hedging makes money as we bought calls/puts cheap! Delta/vega loses money as double-touch sold too cheap.
Scenario 2: We compare the use of our super-/sub- hedges withdelta-vega hedging using BS ATM deltas.
• All hedges are daily monitored.• Transactions in options carry 1% costs, 0.5% for the
underlying.
We use the Heston mdel{
dSt =√
vtStdW 1t ,
dvt = κ(θ − vt)dt + ξ√
vtdW 2t ,
S0 = S0, v0 = σ0
d〈W 1,W 2〉t = ρdt,
with parameters
S0 = 100, σ0 = 0.5, κ = 0.6, θ = 1, ξ = 1.3 and ρ = 0.15 .
Principal Questions and Answers Double barrier options Applications
Hedging comparison under transaction costsScenario 1 (not realistic): after all options are sold/bought andhedges set-up, market model changes drastically (volatility ր) super-hedging makes money as we bought calls/puts cheap! Delta/vega loses money as double-touch sold too cheap.
Scenario 2: We compare the use of our super-/sub- hedges withdelta-vega hedging using BS ATM deltas.
• All hedges are daily monitored.• Transactions in options carry 1% costs, 0.5% for the
underlying.
We use the Heston mdel{
dSt =√
vtStdW 1t ,
dvt = κ(θ − vt)dt + ξ√
vtdW 2t ,
S0 = S0, v0 = σ0
d〈W 1,W 2〉t = ρdt,
with parameters
S0 = 100, σ0 = 0.5, κ = 0.6, θ = 1, ξ = 1.3 and ρ = 0.15 .
Principal Questions and Answers Double barrier options Applications
Cumulative distributions of hedging errors
of a short position in double touch at b = 83 and b = 117.
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.40
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Delta/Vega HedgeSuper−Hedge
Delta/Vega hedge and Super hedge monitored daily.
Principal Questions and Answers Double barrier options Applications
Cumulative distributions of hedging errors
of a short position in double touch at b = 83 and b = 117.
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.40
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Delta/Vega HedgeSuper−Hedge
Delta/Vega hedge monitored daily, Super hedge monitored exactly.
Principal Questions and Answers Double barrier options Applications
Details of the dist of hedging errors
−1 −0.9 −0.8 −0.7 −0.6 −0.50
0.002
0.004
0.006
0.008
0.01
0.012Cumulative distribution of Hedging Errors with Transaction costs (mean adjusted)
Delta/Vega HedgeSuper−Hedge
Delta/Vega Superhedge Superhedge (exact)
Av TC 0.122 0.022 0.022Var of errors w TC 0.012 0.147 0.142Exp utility (λ = 1) −0.137 −0.065 −0.062Exp utility premium 0.128 0.063 0.060
where the price of double no-touch under our Heston was 0.294,and the implied range was [0.083, 0.476].
Principal Questions and Answers Double barrier options Applications
Cumulative distributions of hedging errors
of a long position in double touch at b = 83 and b = 117.
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.40
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Delta/Vega HedgeSub−Hedge
Delta/Vega hedge and Super hedge monitored daily.
Principal Questions and Answers Double barrier options Applications
Cumulative distributions of hedging errors
of a long position in double touch at b = 83 and b = 117.
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.40
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Delta/Vega HedgeSub−Hedge
Delta/Vega hedge monitored daily, Super hedge monitored exactly.
Principal Questions and Answers Double barrier options Applications
Details of the dist of hedging errors
Delta/Vega Subhedge Subhedge (exact)
Av TC 0.122 0.032 0.032Var of errors w TC 0.014 0.104 0.103Exp utility (λ = 1) −0.139 −0.078 −0.077Exp utility premium 0.130 0.075 0.074
where the price of double no-touch under our Heston was 0.294,and the implied range was [0.083, 0.476].
Principal Questions and Answers Double barrier options Applications
Hedges for other barrier levels
Types of super−hedge used as barriers vary
110 120 130 140 150 160
45
50
55
60
65
70
75
80
85
90
95
I
III
IV
Types of sub−hedge used as barriers vary
110 120 130 140 150 160
45
50
55
60
65
70
75
80
85
90
95I
II
III
IV
Types of super- and sub- hedges used as a function of barriers b, b.
Principal Questions and Answers Double barrier options Applications
Summary• We described a general approach to robust pricing and
hedging of derivatives and applied it to digital double barrieroptions.
• Our hedges have a much larger variance but would bepreferred by an agent with an exponential utility. We observethis consistently for various barrier levels.Hedges provide a reasonable alternative to standarddelta/vega hedging when market frictions and/or model riskare important.
• Similar results for double no-touch options.• Techniques for double touch/no-touch options can improve
hedges of double touch and double no-touch options.• The results for use of our hedges in conjunction with delta
hedging inconclusive.• How to extend the method to books of barrier options?• What to do with interest rates?
Principal Questions and Answers Double barrier options Applications
Summary• We described a general approach to robust pricing and
hedging of derivatives and applied it to digital double barrieroptions.
• Our hedges have a much larger variance but would bepreferred by an agent with an exponential utility. We observethis consistently for various barrier levels.Hedges provide a reasonable alternative to standarddelta/vega hedging when market frictions and/or model riskare important.
• Similar results for double no-touch options.• Techniques for double touch/no-touch options can improve
hedges of double touch and double no-touch options.• The results for use of our hedges in conjunction with delta
hedging inconclusive.• How to extend the method to books of barrier options?• What to do with interest rates?
Principal Questions and Answers Double barrier options Applications
Summary• We described a general approach to robust pricing and
hedging of derivatives and applied it to digital double barrieroptions.
• Our hedges have a much larger variance but would bepreferred by an agent with an exponential utility. We observethis consistently for various barrier levels.Hedges provide a reasonable alternative to standarddelta/vega hedging when market frictions and/or model riskare important.
• Similar results for double no-touch options.• Techniques for double touch/no-touch options can improve
hedges of double touch and double no-touch options.• The results for use of our hedges in conjunction with delta
hedging inconclusive.• How to extend the method to books of barrier options?• What to do with interest rates?
Principal Questions and Answers Double barrier options Applications
Summary• We described a general approach to robust pricing and
hedging of derivatives and applied it to digital double barrieroptions.
• Our hedges have a much larger variance but would bepreferred by an agent with an exponential utility. We observethis consistently for various barrier levels.Hedges provide a reasonable alternative to standarddelta/vega hedging when market frictions and/or model riskare important.
• Similar results for double no-touch options.• Techniques for double touch/no-touch options can improve
hedges of double touch and double no-touch options.• The results for use of our hedges in conjunction with delta
hedging inconclusive.• How to extend the method to books of barrier options?• What to do with interest rates?