Robust Pricing and Hedging of Options on Variance · Introduction Free Boundary Problem Optimality...
Transcript of Robust Pricing and Hedging of Options on Variance · Introduction Free Boundary Problem Optimality...
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Introduction Free Boundary Problem Optimality Numerical Examples Conclusions
Robust Pricing and Hedging of Options onVariance
Alexander Cox Jiajie Wang
University of Bath
Bachelier 2010, Toronto
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Introduction Free Boundary Problem Optimality Numerical Examples Conclusions
Financial Setting
• Option priced on an underlying asset St
• Dynamics of St unspecified, but suppose paths arecontinuous, and we see prices of call options at all strikesK and at maturity time T
• Assume for simplicity that all prices are discounted — thiswon’t affect our main results
• Under risk-neutral measure, St should be a(local-)martingale, and we can recover the law of ST attime T from call prices C(K ). (Breeden-Litzenberger)
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Introduction Free Boundary Problem Optimality Numerical Examples Conclusions
Financial Setting
• Given these constraints, what can we say about marketprices of other options?
• Two questions:• What prices are consistent with a model?• If there is no model, is there an arbitrage which works for
every model in our class — robust!
• Intuitively, understanding ‘worst-case’ model should giveinsight into any corresponding arbitrage.
• Insight into hedge likely to be more important than pricing
• But... prices will indicate size of ‘model-risk’
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Introduction Free Boundary Problem Optimality Numerical Examples Conclusions
Connection to Skorokhod Embeddings
• Under a risk-neutral measure, expect St to be alocal-martingale with known law at time T , say µ.
• Since St is a continuous local martingale, we can write itas a time-change of a Brownian motion: St = BAt
• Now the law of BAT is known, and AT is a stopping time forBt
• Correspondence between possible price processes for St
and stopping times τ such that Bτ ∼ µ.
• Problem of finding τ given µ is Skorokhod EmbeddingProblem
• Commonly look for ‘worst-case’ or ‘extremal’ solutions
• Surveys: Obłój, Hobson,. . .
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Introduction Free Boundary Problem Optimality Numerical Examples Conclusions
Variance Options• We may typically suppose a model for asset prices of the
form:dSt
St= σtdWt ,
where Wt a Brownian motion.
• the volatility, σt , is a predictable process• Recent market innovations have led to asset volatility
becoming an object of independent interest
• For example, a variance swap pays:
∫ T
0
(
σ2t − σ̄2
)
dt
where σ̄t is the ‘strike’. Dupire (1993) and Neuberger(1994) gave a simple replication strategy for such anoption.
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Introduction Free Boundary Problem Optimality Numerical Examples Conclusions
Variance Call
• A variance call is an option paying:
(〈ln S〉T − K )+
• Let dXt = Xt dW̃t for a suitable BM W̃t
• Can find a time change τt such that St = Xτt , and so:
dτt =σ2
t S2t
S2t
dt
• And hence
(XτT , τT ) =
(
ST ,
∫ T
0σ2
u du
)
= (ST , 〈ln S〉T )
• More general options of the form: F (〈ln S〉T ).
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Introduction Free Boundary Problem Optimality Numerical Examples Conclusions
Variance Call
• This suggests finding lower bound on price of variance callwith given call prices is equivalent to:
minimise: E(τ − K )+ subject to: L(Xτ ) = µ
where µ is a given law.
• Is there a Skorokhod Embedding which does this?
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Introduction Free Boundary Problem Optimality Numerical Examples Conclusions
Root Construction
• β ⊆ R× R+ a barrier if:
(x , t) ∈ β =⇒ (x , s) ∈ β
for all s ≥ t
• Given µ, exists β and astopping time
τ = inf{t ≥ 0 : (Bt , t) ∈ β}
which is an embedding.
• Minimises E(τ − K )+over all (UI) embeddings
• Construction andoptimality are subject ofthis talk
Bt
t
• Root (1969)
• Rost (1976)
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Introduction Free Boundary Problem Optimality Numerical Examples Conclusions
Root Construction
• β ⊆ R× R+ a barrier if:
(x , t) ∈ β =⇒ (x , s) ∈ β
for all s ≥ t
• Given µ, exists β and astopping time
τ = inf{t ≥ 0 : (Bt , t) ∈ β}
which is an embedding.
• Minimises E(τ − K )+over all (UI) embeddings
• Construction andoptimality are subject ofthis talk
Bt
t
• Root (1969)
• Rost (1976)
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Introduction Free Boundary Problem Optimality Numerical Examples Conclusions
Variance Call
• Finding lower bound on price of variance call with givencall prices is equivalent to:
minimise: E(τ − K )+ subject to: L(Xτ ) = µ
where µ is a given law.
• This is (almost) the problem solved by Root’s Barrier!
• Root proved this for Xt a Brownian motion. Rost (1976)extended his solution to much more general processes,and proved optimality, which was conjectured by Kiefer.
• This connection to Variance options has been observed bya number of authors: Dupire (’05), Carr & Lee (’09),Hobson (’09).
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Introduction Free Boundary Problem Optimality Numerical Examples Conclusions
Variance Call
• Finding lower bound on price of variance call with givencall prices is equivalent to:
minimise: E(τ − K )+ subject to: L(Xτ ) = µ
where µ is a given law.
• This is (almost) the problem solved by Root’s Barrier!
• Root proved this for Xt a Brownian motion. Rost (1976)extended his solution to much more general processes,and proved optimality, which was conjectured by Kiefer.
• This connection to Variance options has been observed bya number of authors: Dupire (’05), Carr & Lee (’09),Hobson (’09).
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Introduction Free Boundary Problem Optimality Numerical Examples Conclusions
Variance Call
• Finding lower bound on price of variance call with givencall prices is equivalent to:
minimise: E(τ − K )+ subject to: L(Xτ ) = µ
where µ is a given law.
• This is (almost) the problem solved by Root’s Barrier!
• Root proved this for Xt a Brownian motion. Rost (1976)extended his solution to much more general processes,and proved optimality, which was conjectured by Kiefer.
• This connection to Variance options has been observed bya number of authors: Dupire (’05), Carr & Lee (’09),Hobson (’09).
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Introduction Free Boundary Problem Optimality Numerical Examples Conclusions
Questions
Question
This known connection leads to two important questions:
1. How do we find the Root stopping time?
2. Is there a corresponding hedging strategy?
• Dupire has given a connected free boundary problem
• Dupire, Carr & Lee have given strategies whichsub/super-replicate the payoff, but are not necessarilyoptimal
• Hobson has given a formal, but not easily solved, conditiona hedging strategy must satisfy.
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Introduction Free Boundary Problem Optimality Numerical Examples Conclusions
Questions
Question
This known connection leads to two important questions:
1. How do we find the Root stopping time?
2. Is there a corresponding hedging strategy?
• Dupire has given a connected free boundary problem
• Dupire, Carr & Lee have given strategies whichsub/super-replicate the payoff, but are not necessarilyoptimal
• Hobson has given a formal, but not easily solved, conditiona hedging strategy must satisfy.
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Introduction Free Boundary Problem Optimality Numerical Examples Conclusions
Questions
Question
This known connection leads to two important questions:
1. How do we find the Root stopping time?
2. Is there a corresponding hedging strategy?
• Dupire has given a connected free boundary problem
• Dupire, Carr & Lee have given strategies whichsub/super-replicate the payoff, but are not necessarilyoptimal
• Hobson has given a formal, but not easily solved, conditiona hedging strategy must satisfy.
![Page 16: Robust Pricing and Hedging of Options on Variance · Introduction Free Boundary Problem Optimality Numerical Examples Conclusions Variance Options • We may typically suppose a model](https://reader034.fdocuments.us/reader034/viewer/2022043005/5f8aed90ea14bb55bc7eb001/html5/thumbnails/16.jpg)
Introduction Free Boundary Problem Optimality Numerical Examples Conclusions
Questions
Question
This known connection leads to two important questions:
1. How do we find the Root stopping time?
2. Is there a corresponding hedging strategy?
• Dupire has given a connected free boundary problem
• Dupire, Carr & Lee have given strategies whichsub/super-replicate the payoff, but are not necessarilyoptimal
• Hobson has given a formal, but not easily solved, conditiona hedging strategy must satisfy.
![Page 17: Robust Pricing and Hedging of Options on Variance · Introduction Free Boundary Problem Optimality Numerical Examples Conclusions Variance Options • We may typically suppose a model](https://reader034.fdocuments.us/reader034/viewer/2022043005/5f8aed90ea14bb55bc7eb001/html5/thumbnails/17.jpg)
Introduction Free Boundary Problem Optimality Numerical Examples Conclusions
Root’s Problem
We want to connect Root’s solution and the solution of afree-boundary problem. We will consider the case whereXt = σ(Xt)dBt and σ is nice (smooth, Lipschitz, strictly positiveon (0,∞)). To make explicit the first, we define:
Root’s Problem (RP)
Find an open set D ⊂ R×R+ such that (R×R+)/D is a barriergenerating a UI stopping time τD and XτD ∼ µ.
Here we denote the exit time from D as τD.
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Introduction Free Boundary Problem Optimality Numerical Examples Conclusions
Free Boundary Problem
Free Boundary Problem (FBP)
To find a continuous function u : R× [0,∞) → R and aconnected open set D : {(x , t), 0 < t < R(x)} whereR : R → R+ = [ 0,∞ ] is a lower semi-continuous function, and
u ∈ C0(R× [0,∞)) and u ∈ C
2,1(D) ;
∂u∂t
=12σ(x)2 ∂2u
∂x2 , on D ; u(x , 0) = −|x − S0| ;
u(x , t) = Uµ(x) = −∫
|x − y |µ(dy), if t ≥ R(x),
u(x , t) is concave with respect to x ∈ R .
∂2u∂x2 ‘disappears’ on ∂D.
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Introduction Free Boundary Problem Optimality Numerical Examples Conclusions
(RP) is equivalent to (FBP)
An easy connection is then the following:
Theorem
Under some conditions on D, if D is a solution to (RP), we canfind a solution to (FBP). In addition, this solution is unique.
Sketch Proof of (RP) =⇒ (FBP)
Simply takeu(x , t) = −E|Xt∧τD − x |.
Resulting properties are mostly straightforward/follow fromregularity of DC , and fact that, for (x , t) ∈ DC :
−E|Xt∧τD − x | = −E|XτD − x |.
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Introduction Free Boundary Problem Optimality Numerical Examples Conclusions
(RP) is equivalent to (FBP)
An easy connection is then the following:
Theorem
Under some conditions on D, if D is a solution to (RP), we canfind a solution to (FBP). In addition, this solution is unique.
Sketch Proof of (RP) =⇒ (FBP)
Simply takeu(x , t) = −E|Xt∧τD − x |.
Resulting properties are mostly straightforward/follow fromregularity of DC , and fact that, for (x , t) ∈ DC :
−E|Xt∧τD − x | = −E|XτD − x |.
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Introduction Free Boundary Problem Optimality Numerical Examples Conclusions
Optimality of Root’s Barrier
Rost’s Result
Given a function F which is convex, increasing, Root’s barriersolves:
minimise EF (τ)subject to: Xτ ∼ µ
τ a (UI) stopping time
Want:
• A simple proof of this. . .
• . . . that identifies a ‘financially meaningful’ hedging strategy.
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Introduction Free Boundary Problem Optimality Numerical Examples Conclusions
Optimality of Root’s Barrier
Rost’s Result
Given a function F which is convex, increasing, Root’s barriersolves:
minimise EF (τ)subject to: Xτ ∼ µ
τ a (UI) stopping time
Want:
• A simple proof of this. . .
• . . . that identifies a ‘financially meaningful’ hedging strategy.
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Introduction Free Boundary Problem Optimality Numerical Examples Conclusions
Optimality
Write f (t) = F ′(t), define
M(x , t) = E(x ,t)f (τD),
and
Z (x) = 2∫ x
0
∫ y
0
M(z, 0)σ2(z)
dz dy ,
so that in particular, Z ′′(x) = 2σ2(x)M(x , 0). And finally, let:
G(x , t) =∫ t
0M(x , s) ds − Z (x).
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Introduction Free Boundary Problem Optimality Numerical Examples Conclusions
Optimality
Write f (t) = F ′(t), define
M(x , t) = E(x ,t)f (τD),
and
Z (x) = 2∫ x
0
∫ y
0
M(z, 0)σ2(z)
dz dy ,
so that in particular, Z ′′(x) = 2σ2(x)M(x , 0). And finally, let:
G(x , t) =∫ t
0M(x , s) ds − Z (x).
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Introduction Free Boundary Problem Optimality Numerical Examples Conclusions
Optimality
Write f (t) = F ′(t), define
M(x , t) = E(x ,t)f (τD),
and
Z (x) = 2∫ x
0
∫ y
0
M(z, 0)σ2(z)
dz dy ,
so that in particular, Z ′′(x) = 2σ2(x)M(x , 0). And finally, let:
G(x , t) =∫ t
0M(x , s) ds − Z (x).
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Introduction Free Boundary Problem Optimality Numerical Examples Conclusions
OptimalityThen there are two key results:
Proposition ( Proof )
For all (x , t) ∈ R× R+:
G(x , t) +∫ R(x)
0(f (s)− M(x , s)) ds + Z (x) ≤ F (t).
Theorem ( Proof )
We have:G(Xt , t) is a submartingale,
andG(Xt∧τD , t ∧ τD) is a martingale.
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Introduction Free Boundary Problem Optimality Numerical Examples Conclusions
OptimalityThen there are two key results:
Proposition ( Proof )
For all (x , t) ∈ R× R+:
G(x , t) +∫ R(x)
0(f (s)− M(x , s)) ds + Z (x) ≤ F (t).
Theorem ( Proof )
We have:G(Xt , t) is a submartingale,
andG(Xt∧τD , t ∧ τD) is a martingale.
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Introduction Free Boundary Problem Optimality Numerical Examples Conclusions
Optimality
We can now show optimality. Recall we had:
G(x , t) +∫ R(x)
0(f (s)− M(x , s)) ds + Z (x) ≤ F (t).
But∫ R(x)
0 (f (s)− M(x , s)) ds + Z (x) is just a function of x , so
G(Xt , t) + H(Xt) ≤ F (t).
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Introduction Free Boundary Problem Optimality Numerical Examples Conclusions
Optimality
We can now show optimality. Recall we had:
G(x , t) +∫ R(x)
0(f (s)− M(x , s)) ds + Z (x) ≤ F (t).
But∫ R(x)
0 (f (s)− M(x , s)) ds + Z (x) is just a function of x , so
G(Xt , t) + H(Xt) ≤ F (t).
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Introduction Free Boundary Problem Optimality Numerical Examples Conclusions
Hedging Strategy
Since G(Xt , t) is a submartingale, there is a trading strategywhich sub-replicates G(Xt , t):
G(St , 〈ln S〉t) ≥∫ t
0
Gx(Sr , 〈ln S〉r )
σ2r
dSr
and H(Xt) can be replicated using the traded calls; moreover, inthe case where τ = τD, we get equality, so this is the best wecan do.
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Introduction Free Boundary Problem Optimality Numerical Examples Conclusions
Hedging Strategy
Since G(Xt , t) is a submartingale, there is a trading strategywhich sub-replicates G(Xt , t):
G(St , 〈ln S〉t) ≥∫ t
0
Gx(Sr , 〈ln S〉r )
σ2r
dSr
and H(Xt) can be replicated using the traded calls; moreover, inthe case where τ = τD, we get equality, so this is the best wecan do.
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Introduction Free Boundary Problem Optimality Numerical Examples Conclusions
Numerical implementation
• How ‘good’ is the subhedge in practice?
• Take an underlying Heston process:
dSt
St= r dt +
√vtdB1
t
dvt = κ(θ − vt)dt + ξ√
vtdB2t
where B1t ,B
2t are correlated Brownian motions, correlation
ρ.
• Compute Barrier and hedging strategies based on thecorresponding call prices.
• How does the subhedging strategy behave under the ‘true’model?
• How does the strategy perform under another model?
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Introduction Free Boundary Problem Optimality Numerical Examples Conclusions
Numerical Implementation• Payoff: 1
2
(
∫ T0 σt dt
)2. Parameters: T = 1, r = 0.05,S0 =
0.2, σ20 = 0.4, κ = 10, θ = 0.4, ξ = 1.0, ρ = −1.0. Prices:
actual 9.80 × 10−4, subhedge 5.463 × 10−4.
0 0.02 0.04 0.06 0.08 0.10.1
0.12
0.14
0.16
0.18
0.2
0.22
0.24
0.26
0.28
Integrated Variance
Ass
et P
rice
Asset Price and Exit from Barrier
BarrierAsset Price
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Introduction Free Boundary Problem Optimality Numerical Examples Conclusions
Numerical Implementation• Payoff: 1
2
(
∫ T0 σt dt
)2. Parameters: T = 1, r = 0.05,S0 =
0.2, σ20 = 0.4, κ = 10, θ = 0.4, ξ = 1.0, ρ = −1.0. Prices:
actual 9.80 × 10−4, subhedge 5.463 × 10−4.
0 0.005 0.01 0.015 0.02 0.025 0.03 0.035−10
−5
0
5x 10
−4
Integrated Variance
Val
ue
Payoff and Hedge against Integrated Variance
Derivative PayoffSubhedging Portfolio
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Introduction Free Boundary Problem Optimality Numerical Examples Conclusions
Numerical Implementation• Payoff: 1
2
(
∫ T0 σt dt
)2. Parameters: T = 1, r = 0.05,S0 =
0.2, σ20 = 0.4, κ = 10, θ = 0.4, ξ = 1.0, ρ = −1.0. Prices:
actual 9.80 × 10−4, subhedge 5.463 × 10−4.
0 0.2 0.4 0.6 0.8 1−10
−5
0
5x 10
−4
Time
Val
ue
Integrated Variance, Payoff and Hedge against Time
Derivative PayoffSubhedging Portfolio
![Page 36: Robust Pricing and Hedging of Options on Variance · Introduction Free Boundary Problem Optimality Numerical Examples Conclusions Variance Options • We may typically suppose a model](https://reader034.fdocuments.us/reader034/viewer/2022043005/5f8aed90ea14bb55bc7eb001/html5/thumbnails/36.jpg)
Introduction Free Boundary Problem Optimality Numerical Examples Conclusions
Numerical Implementation• Payoff: 1
2
(
∫ T0 σt dt
)2. Parameters: T = 1, r = 0.05,S0 =
0.2, σ20 = 0.4, κ = 10, θ = 0.4, ξ = 1.0, ρ = −1.0. Prices:
actual 9.80 × 10−4, subhedge 5.463 × 10−4.
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
1x 10
−3
Time
Val
ueHedging Gap
Derivative Value − Subhedge
![Page 37: Robust Pricing and Hedging of Options on Variance · Introduction Free Boundary Problem Optimality Numerical Examples Conclusions Variance Options • We may typically suppose a model](https://reader034.fdocuments.us/reader034/viewer/2022043005/5f8aed90ea14bb55bc7eb001/html5/thumbnails/37.jpg)
Introduction Free Boundary Problem Optimality Numerical Examples Conclusions
Numerical Implementation• Payoff: 1
2
(
∫ T0 σt dt
)2. Parameters: T = 1, r = 0.05,S0 =
0.2, σ20 = 0.4, κ = 10, θ = 0.4, ξ = 1.0, ρ = −1.0. Prices:
actual 9.80 × 10−4, subhedge 5.463 × 10−4.
0 0.2 0.4 0.6 0.8 1−1
−0.5
0
0.5
1
1.5x 10
−3
Time
Val
ue
Dynamic and Static Hedge Components
Dynamic HedgeStatic Hedge
![Page 38: Robust Pricing and Hedging of Options on Variance · Introduction Free Boundary Problem Optimality Numerical Examples Conclusions Variance Options • We may typically suppose a model](https://reader034.fdocuments.us/reader034/viewer/2022043005/5f8aed90ea14bb55bc7eb001/html5/thumbnails/38.jpg)
Introduction Free Boundary Problem Optimality Numerical Examples Conclusions
Numerical Implementation• Payoff: 1
2
(
∫ T0 σt dt
)2. Parameters: T = 1, r = 0.05,S0 =
0.2, σ20 = 0.4, κ = 10, θ = 0.4, ξ = 1.0, ρ = −1.0. Prices:
actual 9.80 × 10−4, subhedge 5.463 × 10−4.
−4 −2 0 2 4 6 8 10
x 10−4
0
500
1000
1500
2000
2500Distribution of Underhedge
Underhedge (Truncated at 0.001)
Fre
quen
cy
![Page 39: Robust Pricing and Hedging of Options on Variance · Introduction Free Boundary Problem Optimality Numerical Examples Conclusions Variance Options • We may typically suppose a model](https://reader034.fdocuments.us/reader034/viewer/2022043005/5f8aed90ea14bb55bc7eb001/html5/thumbnails/39.jpg)
Introduction Free Boundary Problem Optimality Numerical Examples Conclusions
Numerical Implementation: ‘Incorrect model’
• Payoff: 12
(
∫ T0 σt dt
)2. Parameters: T = 1, r = 0.05,S0 =
0.2, σ20 = 0.4, κ = 10, θ = 0.4, ξ = 4.0, ρ = −0.5.
0 0.02 0.04 0.06 0.08 0.10.1
0.12
0.14
0.16
0.18
0.2
0.22
0.24
0.26
0.28
Integrated Variance
Ass
et P
rice
Asset Price and Exit from Barrier
BarrierAsset Price
![Page 40: Robust Pricing and Hedging of Options on Variance · Introduction Free Boundary Problem Optimality Numerical Examples Conclusions Variance Options • We may typically suppose a model](https://reader034.fdocuments.us/reader034/viewer/2022043005/5f8aed90ea14bb55bc7eb001/html5/thumbnails/40.jpg)
Introduction Free Boundary Problem Optimality Numerical Examples Conclusions
Numerical Implementation: ‘Incorrect model’
• Payoff: 12
(
∫ T0 σt dt
)2. Parameters: T = 1, r = 0.05,S0 =
0.2, σ20 = 0.4, κ = 10, θ = 0.4, ξ = 4.0, ρ = −0.5.
0 0.01 0.02 0.03 0.04 0.05 0.06−1
−0.5
0
0.5
1
1.5
2x 10
−3
Integrated Variance
Val
uePayoff and Hedge against Integrated Variance
Derivative PayoffSubhedging Portfolio
![Page 41: Robust Pricing and Hedging of Options on Variance · Introduction Free Boundary Problem Optimality Numerical Examples Conclusions Variance Options • We may typically suppose a model](https://reader034.fdocuments.us/reader034/viewer/2022043005/5f8aed90ea14bb55bc7eb001/html5/thumbnails/41.jpg)
Introduction Free Boundary Problem Optimality Numerical Examples Conclusions
Numerical Implementation: ‘Incorrect model’
• Payoff: 12
(
∫ T0 σt dt
)2. Parameters: T = 1, r = 0.05,S0 =
0.2, σ20 = 0.4, κ = 10, θ = 0.4, ξ = 4.0, ρ = −0.5.
0 0.2 0.4 0.6 0.8 1−1
−0.5
0
0.5
1
1.5
2x 10
−3
Time
Val
ueIntegrated Variance, Payoff and Hedge against Time
Derivative PayoffSubhedging Portfolio
![Page 42: Robust Pricing and Hedging of Options on Variance · Introduction Free Boundary Problem Optimality Numerical Examples Conclusions Variance Options • We may typically suppose a model](https://reader034.fdocuments.us/reader034/viewer/2022043005/5f8aed90ea14bb55bc7eb001/html5/thumbnails/42.jpg)
Introduction Free Boundary Problem Optimality Numerical Examples Conclusions
Numerical Implementation: ‘Incorrect model’
• Payoff: 12
(
∫ T0 σt dt
)2. Parameters: T = 1, r = 0.05,S0 =
0.2, σ20 = 0.4, κ = 10, θ = 0.4, ξ = 4.0, ρ = −0.5.
0 0.2 0.4 0.6 0.8 1−2
0
2
4
6
8
10x 10
−4
Time
Val
ueHedging Gap
Derivative Value − Subhedge
![Page 43: Robust Pricing and Hedging of Options on Variance · Introduction Free Boundary Problem Optimality Numerical Examples Conclusions Variance Options • We may typically suppose a model](https://reader034.fdocuments.us/reader034/viewer/2022043005/5f8aed90ea14bb55bc7eb001/html5/thumbnails/43.jpg)
Introduction Free Boundary Problem Optimality Numerical Examples Conclusions
Numerical Implementation: ‘Incorrect model’
• Payoff: 12
(
∫ T0 σt dt
)2. Parameters: T = 1, r = 0.05,S0 =
0.2, σ20 = 0.4, κ = 10, θ = 0.4, ξ = 4.0, ρ = −0.5.
0 0.2 0.4 0.6 0.8 1−1
−0.5
0
0.5
1
1.5
2x 10
−3
Time
Val
ueDynamic and Static Hedge Components
Dynamic HedgeStatic Hedge
![Page 44: Robust Pricing and Hedging of Options on Variance · Introduction Free Boundary Problem Optimality Numerical Examples Conclusions Variance Options • We may typically suppose a model](https://reader034.fdocuments.us/reader034/viewer/2022043005/5f8aed90ea14bb55bc7eb001/html5/thumbnails/44.jpg)
Introduction Free Boundary Problem Optimality Numerical Examples Conclusions
Numerical Implementation: ‘Incorrect model’
• Payoff: 12
(
∫ T0 σt dt
)2. Parameters: T = 1, r = 0.05,S0 =
0.2, σ20 = 0.4, κ = 10, θ = 0.4, ξ = 4.0, ρ = −0.5.
−0.02 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.160
500
1000
1500
2000
2500
3000
3500
4000
4500
5000Distribution of Underhedge
Underhedge (Truncated at 0.15)
Fre
quen
cy
![Page 45: Robust Pricing and Hedging of Options on Variance · Introduction Free Boundary Problem Optimality Numerical Examples Conclusions Variance Options • We may typically suppose a model](https://reader034.fdocuments.us/reader034/viewer/2022043005/5f8aed90ea14bb55bc7eb001/html5/thumbnails/45.jpg)
Introduction Free Boundary Problem Optimality Numerical Examples Conclusions
Numerical Implementation: ‘Variance Call’
• Payoff:(
∫ T0 σ2
t dt − K)
+. Prices: actual = 0.0106,
subhedge = 0.0076.
• Parameters: T = 1, r = 0.05, S0 = 0.2, σ20 = 0.0174,
κ = 1.3253, θ = 0.0354, ξ = 0.3877, ρ = −0.7165,K = 0.02.
![Page 46: Robust Pricing and Hedging of Options on Variance · Introduction Free Boundary Problem Optimality Numerical Examples Conclusions Variance Options • We may typically suppose a model](https://reader034.fdocuments.us/reader034/viewer/2022043005/5f8aed90ea14bb55bc7eb001/html5/thumbnails/46.jpg)
Introduction Free Boundary Problem Optimality Numerical Examples Conclusions
Numerical Implementation: ‘Variance Call’
• Payoff:(
∫ T0 σ2
t dt − K)
+. Prices: actual = 0.0106,
subhedge = 0.0076.
0 0.05 0.1 0.150.1
0.12
0.14
0.16
0.18
0.2
0.22
0.24
0.26
0.28
Integrated Variance
Ass
et P
rice
Asset Price and Exit from Barrier
BarrierAsset Price
![Page 47: Robust Pricing and Hedging of Options on Variance · Introduction Free Boundary Problem Optimality Numerical Examples Conclusions Variance Options • We may typically suppose a model](https://reader034.fdocuments.us/reader034/viewer/2022043005/5f8aed90ea14bb55bc7eb001/html5/thumbnails/47.jpg)
Introduction Free Boundary Problem Optimality Numerical Examples Conclusions
Numerical Implementation: ‘Variance Call’
• Payoff:(
∫ T0 σ2
t dt − K)
+. Prices: actual = 0.0106,
subhedge = 0.0076.
0 0.005 0.01 0.015 0.02 0.025−15
−10
−5
0
5x 10
−3
Integrated Variance
Val
uePayoff and Hedge against Integrated Variance
Derivative PayoffSubhedging Portfolio
![Page 48: Robust Pricing and Hedging of Options on Variance · Introduction Free Boundary Problem Optimality Numerical Examples Conclusions Variance Options • We may typically suppose a model](https://reader034.fdocuments.us/reader034/viewer/2022043005/5f8aed90ea14bb55bc7eb001/html5/thumbnails/48.jpg)
Introduction Free Boundary Problem Optimality Numerical Examples Conclusions
Numerical Implementation: ‘Variance Call’
• Payoff:(
∫ T0 σ2
t dt − K)
+. Prices: actual = 0.0106,
subhedge = 0.0076.
0 0.2 0.4 0.6 0.8 1−0.015
−0.01
−0.005
0
0.005
0.01
0.015
0.02
0.025
Time
Val
ue
Integrated Variance, Payoff and Hedge against Time
Integrated VarianceDerivative PayoffSubhedging Portfolio
![Page 49: Robust Pricing and Hedging of Options on Variance · Introduction Free Boundary Problem Optimality Numerical Examples Conclusions Variance Options • We may typically suppose a model](https://reader034.fdocuments.us/reader034/viewer/2022043005/5f8aed90ea14bb55bc7eb001/html5/thumbnails/49.jpg)
Introduction Free Boundary Problem Optimality Numerical Examples Conclusions
Numerical Implementation: ‘Variance Call’
• Payoff:(
∫ T0 σ2
t dt − K)
+. Prices: actual = 0.0106,
subhedge = 0.0076.
0 0.2 0.4 0.6 0.8 1−2
0
2
4
6
8
10
12
14
16x 10
−3
Time
Val
ueHedging Gap
Derivative Value − Subhedge
![Page 50: Robust Pricing and Hedging of Options on Variance · Introduction Free Boundary Problem Optimality Numerical Examples Conclusions Variance Options • We may typically suppose a model](https://reader034.fdocuments.us/reader034/viewer/2022043005/5f8aed90ea14bb55bc7eb001/html5/thumbnails/50.jpg)
Introduction Free Boundary Problem Optimality Numerical Examples Conclusions
Numerical Implementation: ‘Variance Call’
• Payoff:(
∫ T0 σ2
t dt − K)
+. Prices: actual = 0.0106,
subhedge = 0.0076.
0 0.2 0.4 0.6 0.8 1−0.015
−0.01
−0.005
0
0.005
0.01
0.015
0.02
Time
Val
ue
Dynamic and Static Hedge Components
Dynamic HedgeStatic Hedge
![Page 51: Robust Pricing and Hedging of Options on Variance · Introduction Free Boundary Problem Optimality Numerical Examples Conclusions Variance Options • We may typically suppose a model](https://reader034.fdocuments.us/reader034/viewer/2022043005/5f8aed90ea14bb55bc7eb001/html5/thumbnails/51.jpg)
Introduction Free Boundary Problem Optimality Numerical Examples Conclusions
Numerical Implementation: ‘Variance Call’
• Payoff:(
∫ T0 σ2
t dt − K)
+. Prices: actual = 0.0106,
subhedge = 0.0076.
−0.01 0 0.01 0.02 0.03 0.04 0.050
500
1000
1500
2000
2500
3000
3500
4000Distribution of Underhedge
Underhedge (Truncated at 0.05)
Fre
quen
cy
![Page 52: Robust Pricing and Hedging of Options on Variance · Introduction Free Boundary Problem Optimality Numerical Examples Conclusions Variance Options • We may typically suppose a model](https://reader034.fdocuments.us/reader034/viewer/2022043005/5f8aed90ea14bb55bc7eb001/html5/thumbnails/52.jpg)
Introduction Free Boundary Problem Optimality Numerical Examples Conclusions
Conclusion
• Lower bounds on Pricing Variance options ∼ finding Root’sbarrier
• Equivalence between Root’s Barrier and a Free BoundaryProblem
• New proof of optimality, which allows explicit constructionof a pathwise inequality
• Financial Interpretation: model-free sub-hedges forvariance options.
![Page 53: Robust Pricing and Hedging of Options on Variance · Introduction Free Boundary Problem Optimality Numerical Examples Conclusions Variance Options • We may typically suppose a model](https://reader034.fdocuments.us/reader034/viewer/2022043005/5f8aed90ea14bb55bc7eb001/html5/thumbnails/53.jpg)
Introduction Free Boundary Problem Optimality Numerical Examples Conclusions
Proof of Proposition
If t ≤ R(x) then the left-hand side is:
∫ t
0f (s) ds −
∫ R(x)
tM(x , s) ds = F (t)−
∫ R(x)
tM(x , s) ds
And M(x , s) ≥ f (s) ≥ 0.
If t ≥ R(x), we get:
∫ t
R(x)M(x , s) ds +
∫ R(x)
0f (s) ds =
∫ t
R(x)f (s) ds +
∫ R(x)
0f (s) ds
= F (t).
![Page 54: Robust Pricing and Hedging of Options on Variance · Introduction Free Boundary Problem Optimality Numerical Examples Conclusions Variance Options • We may typically suppose a model](https://reader034.fdocuments.us/reader034/viewer/2022043005/5f8aed90ea14bb55bc7eb001/html5/thumbnails/54.jpg)
Introduction Free Boundary Problem Optimality Numerical Examples Conclusions
Proof of Proposition
If t ≤ R(x) then the left-hand side is:
∫ t
0f (s) ds −
∫ R(x)
tM(x , s) ds = F (t)−
∫ R(x)
tM(x , s) ds
And M(x , s) ≥ f (s) ≥ 0.
If t ≥ R(x), we get:
∫ t
R(x)M(x , s) ds +
∫ R(x)
0f (s) ds =
∫ t
R(x)f (s) ds +
∫ R(x)
0f (s) ds
= F (t).
![Page 55: Robust Pricing and Hedging of Options on Variance · Introduction Free Boundary Problem Optimality Numerical Examples Conclusions Variance Options • We may typically suppose a model](https://reader034.fdocuments.us/reader034/viewer/2022043005/5f8aed90ea14bb55bc7eb001/html5/thumbnails/55.jpg)
Introduction Free Boundary Problem Optimality Numerical Examples Conclusions
Optimality
Recalling that M(x , t) = E(x ,t)f (τD), we have:
E [M(Xt , u)|Fs] ≥{
M(Xs, s − t + u) u ≥ t − s
E [M(Xt−u, 0)|Fs] u ≤ t − s.
And by Itô:
E [Z (Xt)− Z (Xs)|Fs] =
∫ t
sM(Xr , 0) dr , s ≤ t .
Then it can be shown:
E[G(Xt , t)|Fs] ≥ G(Xs, s).
![Page 56: Robust Pricing and Hedging of Options on Variance · Introduction Free Boundary Problem Optimality Numerical Examples Conclusions Variance Options • We may typically suppose a model](https://reader034.fdocuments.us/reader034/viewer/2022043005/5f8aed90ea14bb55bc7eb001/html5/thumbnails/56.jpg)
Introduction Free Boundary Problem Optimality Numerical Examples Conclusions
Optimality
Recalling that M(x , t) = E(x ,t)f (τD), we have:
E [M(Xt , u)|Fs] ≥{
M(Xs, s − t + u) u ≥ t − s
E [M(Xt−u, 0)|Fs] u ≤ t − s.
And by Itô:
E [Z (Xt)− Z (Xs)|Fs] =
∫ t
sM(Xr , 0) dr , s ≤ t .
Then it can be shown:
E[G(Xt , t)|Fs] ≥ G(Xs, s).
![Page 57: Robust Pricing and Hedging of Options on Variance · Introduction Free Boundary Problem Optimality Numerical Examples Conclusions Variance Options • We may typically suppose a model](https://reader034.fdocuments.us/reader034/viewer/2022043005/5f8aed90ea14bb55bc7eb001/html5/thumbnails/57.jpg)
Introduction Free Boundary Problem Optimality Numerical Examples Conclusions
Proof of Submartingale Condition
E [G(Xt , t)|Fs] =
∫ t
0E [M(Xt , u)|Fs] du − E [Z (Xt)|Fs]
= G(Xs, s) +∫ t
0E [M(Xt , u)|Fs] du
−∫ s
0M(Xs, u) du − E [Z (Xt)− Z (Xs)|Fs]
≥ G(Xs, s) +∫ t−s
0E [M(Xt−u, 0)|Fs] du
−∫ s
0M(Xs, u) du −
∫ t
sE [M(Xu, 0)|Fs] du
+
∫ t
t−sM(Xs, s − t + u) du
![Page 58: Robust Pricing and Hedging of Options on Variance · Introduction Free Boundary Problem Optimality Numerical Examples Conclusions Variance Options • We may typically suppose a model](https://reader034.fdocuments.us/reader034/viewer/2022043005/5f8aed90ea14bb55bc7eb001/html5/thumbnails/58.jpg)
Introduction Free Boundary Problem Optimality Numerical Examples Conclusions
Proof of Submartingale Condition
E [G(Xt , t)|Fs] =
∫ t
0E [M(Xt , u)|Fs] du − E [Z (Xt)|Fs]
= G(Xs, s) +∫ t
0E [M(Xt , u)|Fs] du
−∫ s
0M(Xs, u) du − E [Z (Xt)− Z (Xs)|Fs]
≥ G(Xs, s) +∫ t−s
0E [M(Xt−u, 0)|Fs] du
−∫ s
0M(Xs, u) du −
∫ t
sE [M(Xu, 0)|Fs] du
+
∫ t
t−sM(Xs, s − t + u) du
![Page 59: Robust Pricing and Hedging of Options on Variance · Introduction Free Boundary Problem Optimality Numerical Examples Conclusions Variance Options • We may typically suppose a model](https://reader034.fdocuments.us/reader034/viewer/2022043005/5f8aed90ea14bb55bc7eb001/html5/thumbnails/59.jpg)
Introduction Free Boundary Problem Optimality Numerical Examples Conclusions
Proof of Submartingale Condition
E [G(Xt , t)|Fs] =
∫ t
0E [M(Xt , u)|Fs] du − E [Z (Xt)|Fs]
= G(Xs, s) +∫ t
0E [M(Xt , u)|Fs] du
−∫ s
0M(Xs, u) du − E [Z (Xt)− Z (Xs)|Fs]
≥ G(Xs, s) +∫ t−s
0E [M(Xt−u, 0)|Fs] du
−∫ s
0M(Xs, u) du −
∫ t
sE [M(Xu, 0)|Fs] du
+
∫ t
t−sM(Xs, s − t + u) du
![Page 60: Robust Pricing and Hedging of Options on Variance · Introduction Free Boundary Problem Optimality Numerical Examples Conclusions Variance Options • We may typically suppose a model](https://reader034.fdocuments.us/reader034/viewer/2022043005/5f8aed90ea14bb55bc7eb001/html5/thumbnails/60.jpg)
Introduction Free Boundary Problem Optimality Numerical Examples Conclusions
Proof of Submartingale Condition
E [G(Xt , t)|Fs] ≥ G(Xs, s) +∫ t
sE [M(Xu, 0)|Fs] du
−∫ t
sE [M(Xu, 0)|Fs] du +
∫ s
0M(Xs, u) du
−∫ s
0M(Xs, u) du
≥ G(Xs, s).
A somewhat similar computation gives:
E [G(Xt∧τD , t ∧ τD)|Fs] = G(Xs, s)
on {s ≤ τD}.
![Page 61: Robust Pricing and Hedging of Options on Variance · Introduction Free Boundary Problem Optimality Numerical Examples Conclusions Variance Options • We may typically suppose a model](https://reader034.fdocuments.us/reader034/viewer/2022043005/5f8aed90ea14bb55bc7eb001/html5/thumbnails/61.jpg)
Introduction Free Boundary Problem Optimality Numerical Examples Conclusions
Proof of Submartingale Condition
E [G(Xt , t)|Fs] ≥ G(Xs, s) +∫ t
sE [M(Xu, 0)|Fs] du
−∫ t
sE [M(Xu, 0)|Fs] du +
∫ s
0M(Xs, u) du
−∫ s
0M(Xs, u) du
≥ G(Xs, s).
A somewhat similar computation gives:
E [G(Xt∧τD , t ∧ τD)|Fs] = G(Xs, s)
on {s ≤ τD}.
![Page 62: Robust Pricing and Hedging of Options on Variance · Introduction Free Boundary Problem Optimality Numerical Examples Conclusions Variance Options • We may typically suppose a model](https://reader034.fdocuments.us/reader034/viewer/2022043005/5f8aed90ea14bb55bc7eb001/html5/thumbnails/62.jpg)
Introduction Free Boundary Problem Optimality Numerical Examples Conclusions
Proof of Submartingale Condition
E [G(Xt , t)|Fs] ≥ G(Xs, s) +∫ t
sE [M(Xu, 0)|Fs] du
−∫ t
sE [M(Xu, 0)|Fs] du +
∫ s
0M(Xs, u) du
−∫ s
0M(Xs, u) du
≥ G(Xs, s).
A somewhat similar computation gives:
E [G(Xt∧τD , t ∧ τD)|Fs] = G(Xs, s)
on {s ≤ τD}.