Robust Pricing and Hedging of Options on Variance · Introduction Free Boundary Problem Optimality...

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


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)


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’


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,. . .


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.


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 ).


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?


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)


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)


Variance Call

• Finding lower bound on price of variance call with givencall prices is equivalent to:



• 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).


Variance Call








Variance Call








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.


Questions

Question








Questions

Question








Questions

Question








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.


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.


(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 |.


(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 |.


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.


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.


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).


Optimality


M(x , t) = E(x ,t)f (τD),

and

Z (x) = 2∫ x

0

∫ y

0

M(z, 0)σ2(z)

dz dy ,


G(x , t) =∫ t

0M(x , s) ds − Z (x).


Optimality


M(x , t) = E(x ,t)f (τD),

and

Z (x) = 2∫ x

0

∫ y

0

M(z, 0)σ2(z)

dz dy ,


G(x , t) =∫ t

0M(x , s) ds − Z (x).


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.


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.


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).


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).


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.


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.


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?


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



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



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




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



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



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


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


BarrierAsset Price



• 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




• 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




• 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




• 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




• 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



Fre

quen

cy


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.



• Payoff:(

∫ T0 σ2

t dt − K)


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


BarrierAsset Price



• Payoff:(

∫ T0 σ2

t dt − K)


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




• Payoff:(

∫ T0 σ2

t dt − K)


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



• Payoff:(

∫ T0 σ2

t dt − K)


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




• Payoff:(

∫ T0 σ2

t dt − K)


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




• Payoff:(

∫ T0 σ2

t dt − K)


subhedge = 0.0076.

−0.01 0 0.01 0.02 0.03 0.04 0.050

500

1000

1500

2000

2500

3000

3500



Fre

quen

cy


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.


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).


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).


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).


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).


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



E [G(Xt , t)|Fs] =

∫ t


= G(Xs, s) +∫ t


−∫ s


≥ G(Xs, s) +∫ t−s


−∫ s

0M(Xs, u) du −

∫ t

sE [M(Xu, 0)|Fs] du

+

∫ t




E [G(Xt , t)|Fs] =

∫ t


= G(Xs, s) +∫ t


−∫ s


≥ G(Xs, s) +∫ t−s


−∫ s

0M(Xs, u) du −

∫ t

sE [M(Xu, 0)|Fs] du

+

∫ t




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}.



E [G(Xt , t)|Fs] ≥ G(Xs, s) +∫ t

sE [M(Xu, 0)|Fs] du

−∫ t


∫ s

0M(Xs, u) du

−∫ s

0M(Xs, u) du

≥ G(Xs, s).



on {s ≤ τD}.



E [G(Xt , t)|Fs] ≥ G(Xs, s) +∫ t

sE [M(Xu, 0)|Fs] du

−∫ t


∫ s

0M(Xs, u) du

−∫ s

0M(Xs, u) du

≥ G(Xs, s).



on {s ≤ τD}.

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