Robust Pricing and Hedging of Options on Variance · Introduction Free Boundary Problem Optimality Numerical Examples Conclusions Variance Options • We may typically suppose a model

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}.

Documents

Robust Pricing and Hedging of Options on Variance · Introduction Free Boundary Problem Optimality Numerical Examples Conclusions Variance Options • We may typically suppose a model