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Corridor Variance Swap Spread

Bryan Jianfeng Liang

Quant Research, Bloomberg L.P.

MathFinance Conference, Frankfurt, April 16, 2018

Bryan Jianfeng Liang Quant Research, Bloomberg L.P.



Two assets:

S1: typically SPXS2: Asian or European index, NKY, HSI, SX5E

Payoff at maturity T :

1

T

∫ T

0

[σ22,t − σ2

2

]1L≤S2,t≤Udt−

1

T

∫ T

0

[σ21,t − σ2

1

]1L≤S2,t≤Udt

for 0 ≤ L < U ≤ ∞. The fixed volatilities σ1 and σ2 are typically chosen to makeeach leg of the swap spread have zero value at its inception.

The realized variance is accrued only when the level of the corridor index is withina predefined corridor. Vega exposures in specific range of market levels.

Sell Side: Natural hedge for the vega exposure of bank’s structured productbusiness, especially products with barriers, such as autocallable notes

Buy Side: Construct relative value volatility trade in specific ranges of marketlevels



Autocallable Notes

Popular structured product in Asian markets

The investor receives a higher coupon

Autocalled (the product terminates early) if the underlying hits a settarget (typically 100% - 110% of the spot)

Down-and-in ATM put option with barrier set at 60% - 70% of the spot

The investor is effectively selling the embedded option to the issuer.The option premium provides the additional yield.

The issuer has to short other options to hedge the long vega exposure,which puts downward pressure on implied volatilities in the listedoptions market.

Digital risks of the barriers.



Autocallable Notes

Autocallable Note Payoff Structure

- Autocall level: 100, observed quarterly- Coupon: 5% Quarterly, Coupon Barrier: 90- Conversion Barrier: 75, observed daily



Variance Swap

Payoff at maturity T :1

T

∫ T

0σ2t dt− σ2

The fixed volatility σ is chosen to make the swap have zero value at its inception.

In the absence of jumps, the payoff of a variance swap can be perfectly replicatedby a static portfolio of vanilla options and a dynamic position in the underlying:

dSt

St= µtdt+ σtdWt (diffusion process, no jumps)

d logSt =1

StdSt −

1

2σ2t dt∫ T

0σ2t dt = −2 log

ST

S0+

∫ T

0

2

StdSt (replication formula)

EQ[∫ T

0σ2t dt

]= EQ

[−2 log

ST

S0

]+ 2

∫ T

0(rt − qt) dt = EQ

[−2 log

ST

F0,T

]=

∫ F0,T

0

2

K2Put(K,T )dK +

∫ ∞F0,T

2

K2Call(K,T )dK =

∫ ∞0

OTMF(K,T )dK

where Call(K,T ) = EQ [(ST −K)+], Put(K,T ) = EQ [(K − ST )+

]and

OTMF (K,T ) = Call(K,T ) if K ≥ F0,T else Put(K,T ).



Variance Swap Replication

σ2 =1

T

[∫ F0,T

0

2

K2Put(K,T )dK +

∫ ∞F0,T

2

K2Call(K,T )dK

]

SPX Index, as of 03-20-2018, expiry 04-20-2018, σ = 18.9%



Variance Swap and Smile Extrapolation

Options are only traded in a finite range of strikes on the market

Variance Swap replication is very sensitive to the prices/vols of deepout-of-the-money options, especially puts.

Market quotes of variance swap rates can be used to select theextrapolation of implied volatilities outside of the range of tradedstrikes.

Example I - Mixed Lognormals

For a given maturity T and E [XT ] = F , the risk-neutral density function φof XT is given by

φ(x) =n∑

i=1

wiφi(x), with φi = pdf of Lognormal

(log(µiF )−

1

2σ2i T, σ

2i T

)n∑

i=1

wi = 1 andn∑

i=1

wiµi = 1

Call(K) = E[(XT −K)+

]=

n∑i=1

wiBlackCall(K,T, µiF, σi)




Calibrate (wi, µi, σi)1≤i≤n to vanilla option smiles.

SX5E Index, as of 03-23-2018, expiry 06-15-2018

Var swap rate from mixed lognormal fit 20.75 %Var swap rate mid market quote 21.07 %




Calibrate (wi, µi, σi)1≤i≤n to both vanilla option smiles and var swap rate.

VS2 =

n∑i=1

wi

(σ2i −

2

Tlogµi

)





Example II - Cubic Spline Regression

- The total implied variance function x 7→ σ2(x)T where x = log(K/F ) isrepresented by a cubic spline, best fit to the vanilla smile, and linearlyextrapolated for low and high strikes.- Use var swap rate to find slopes for linear extrapolation at low and high strikes.


Var swap rate from cubic spline regression fit 20.85 %Var swap rate mid market quote 21.73 %



Variance Spreads

Relative value volatility trades

There exists a premium of implied volatility to realized volatility

Buy variance on index with low premium (such as Asian or Europeanindices) and sell variance on index with high premium (typically SPX)

More stable P&L



Variance Spreads



Single-Asset Corridor Variance Swap

Payoff at maturity T :

1

T

∫ T

0

[σ2t − σ2

]1L≤St≤Udt, 0 ≤ L < U ≤ ∞

The fixed volatility σ is chosen to make the swap have zero value at its inception.

Range-limited vega exposure

Excellent alternatives for investors who wish to long volatility and have arange-bound view on the index.

To replicate corridor variance swap, we define

g(S) =

log(L) + 1

L(S − L) if S < L

log(S) if L ≤ S ≤ Ulog(U) + 1

U(S − U) if S > U

g′(S) =

1L

if S < L1S

if L ≤ S ≤ U1U

if S > U

and g′′(S) =

0 if S < L

− 1S2 if L ≤ S ≤ U

0 if S > U




The function g is equal to log function inside the corridor [L,U ] and linearlyextrapolated beyond the corridor.

dg(St) = g′(St)dSt +1

2g′′(St)S

2t σ

2t dt = g′(St)dSt −

1

2σ2t 1L≤St≤Udt




∫ T

0

σ2t 1L≤St≤Udt =

∫ T

0

2g′(St)dSt − 2 [g(ST )− g(S0)]

EQ[∫ T

0

2g′(St)dSt

]= EQ

[∫ T

0

2

(St

L1St<L + 1L≤St≤U +

St

U1St>U

)(rt − qt) dt

]

= EQ[∫ T

0

2

(−

(L− St)+

L+ 1 +

(St − U)+

U

)(rt − qt) dt

]

=

∫ T

0

[−

2

LPut(L, t) +

2

UCall(U, t)

](rt − qt) dt+ 2

∫ T

0

(rt − qt) dt

EQ[∫ T

0

σ2t 1L≤St≤Udt

]=

∫ T

0

[−

2

LPut(L, t) +

2

UCall(U, t)

](rt − qt) dt

+

∫ U

L

2

K2OTMF (K,T ) dK − 2

(g (F0,T )− g (S0)−

∫ T

0

(rt − qt) dt)

Note that the last term in the right-hand side of the equation vanishes if L ≤ S0 ≤ U .

The payoff of a corridor variance swap can be replicated by vanilla options with strikeswithin the corridor at the final maturity, a continuous stream of vanilla options ofintermediate maturities, and a dynamic portfolio in the underlying S.



Two-Asset Corridor Variance Swap

Two legs of a corridor variance swap spread:

Leg 1: 1T

∫ T0

[σ22,t − σ2

2

]1L≤S2,t≤Udt Single-Asset Corridor Var Swap

Leg 2: 1T

∫ T0

[σ21,t − σ2

1

]1L≤S2,t≤Udt Two-Asset Corridor Var Swap

The challenge of valuation/risk management mainly comes from the two-asset leg.

Notations

VS2i (T ) = E

[1

T

∫ T

0σ2i,tdt

]Vanilla var swap on i-th asset

VS2i,j(L,U, T ) = E

[1

T

∫ T

0σ2i,t1L≤Sj,t≤Udt

]Two-asset corridor var swap

VS2i (L,U, T ) = VS2

i,i (L,U, T ) Single-asset corridor var swap

Consistency Conditions

1 As S2 → S1, VS1,2(L,U, T )→ VS1(L,U, T )

2 As L ↓ 0 and U ↑ ∞, VS1,2(L,U, T ) ↑ VS1(T )

3 As U ↓ L, VS1,2(L,U, T ) ↓ 0



Two-Asset Corridor Variance Swap - Cross Local Variance

Question: What is the risk exposure of a two-asset corridor var swap?

E[∫ T

0

σ21,t1L≤S2,t≤Udt

]=

∫ T

0

E[E[σ21,t|S2,t

]1L≤S2,t≤U

]dt

The key is to evaluate the cross local variance:

E[σ21,t

∣∣S2,t

]Two simple cases:

1 S1 and S2 are identical:

E[σ21,t

∣∣S2,t

]= σ2

1,loc (t, S1,t)

2 S1 and S2 are independent:

E[σ21,t

∣∣S2,t

]= E

[σ21,t

]= E

[σ21,loc (t, S1,t)

](const)



Two-Asset Corridor Variance Swap - Cross Local Variance

As difference of two down-variance payoffs (lower bound=0):

E[∫ T

0σ21,t1L≤S2,t≤Udt

]= E

[∫ T

0σ21,t1S2,t≤Udt

]− E

[∫ T

0σ21,t1S2,t≤Ldt

]=: V1,2(U, T )− V1,2(L, T )

where V1,2(K,T ) is the undiscounted value of the down-variance payoff:

V1,2(K,T ) =

∫ T

0E[E[σ21,t|S2,t

]1S2,t≤K

]dt

∂V1,2

∂K=

∫ T

0E[E[σ21,t|S2,t

]δ (S2,t −K)

]dt =

∫ T

0E[σ21,t |S2,t = K

] ∂2C2

∂K2(K, t)dt

E[σ21,T

∣∣∣S2,T = K]

=

∂2V1,2

∂T∂K∂2C2

∂K2

where C2(K,T ) = E[(S2,T −K

)+].

The cross local variance E[σ21,t

∣∣∣S2,t = K]

can be locked in by trading two-asset

corridor variance swaps.

The similar analysis can be applied to single-asset corridor variance swaps wherewe can lock in the local variance σ2

loc (t, St) = E[σ2t |St

].



Two-Asset Corridor Variance Swap LV + LV

Basket Local Volatility Model (LV+LV)

dS1,t

S1,t= (r1,t − q1,t) dt+ σ1,loc (t, S1,t) dW

S1,t

dS2,t


S2,t

dWS1,tdW

S2,t = ρS1S2

t dt

VS21,2 (L,U ; LV+LV) =

1

T

∫ T

0E[E[σ21,loc (t, S1,t)

∣∣∣S2,t

]1L≤S2,t≤U

]dt

Pros

Simple multi-asset model, consistent with vanilla smile of each asset

The correlation ρt can be calibrated to the market prices of basket options orother products with exposure to S1 v.s. S2 correlation.

Cons: What about exotic risks such as volatility-of-volatility risk, (cross)spot/volatility correlation risk?



Two-Asset Corridor Variance Swap SLV + LV

One-Factor SLV Model + Local Vol Model (SLV+LV)

dS1,t

S1,t= (r1,t − q1,t) dt+ σ1,loc (t, S1,t)

√V1,t√

E [V1,t|S1,t]dWS

1,t

dV1,t = a1(t, V1,t)dt+ b1(t, Vt)dWV1,t

dS2,t


S2,t

where

dWS1,tdW

S2,t = ρS1S2

t dt, dWS1,tdW

V1,t = ρS1V1

t dt, dWS2,tdW

V1,t = ρS2V1

t dt.

Three correlations need to be specified and they must satisfy thepositive-definiteness constraint:

1−(ρS1S2t

)2−(ρS1V1t

)2−(ρS2V1t

)2+ 2ρS1V1

t ρS1S2t ρS2V1

t > 0 (D)

��

V1

S1 S2

The SDE for S1,t is a McKean SDE and can be efficiently simulated by particlemethod [GL].



Two-Asset Corridor Variance Swap SLV+LV vs LV+LV

We assume that the two models (SLV+LV and LV+LV) have the same local vol

functions σ1,loc and σ2,loc, and the same spot/spot correlation ρS1S2t .

We shall compare the following three functions:

1 E[σ21,loc (t, S1,t)

V1,t

E[V1,t|S1,t]

∣∣∣∣S2,t

]in SLV + LV

2 E[σ21,loc (t, S1,t)

∣∣∣S2,t

]in SLV + LV

3 E[σ21,loc (t, S1,t)

∣∣∣S2,t

]in LV + LV

Note that (2) and (3) are not the same in general even if S1,t and S2,t have the

same marginals and the same correlation ρS1S2t drives WS

1,t and WS2,t, the joint

distribution of S1,t and S2,t are not the same in SLV+LV and LV+LV due to

stochastic volatility V1,t and cross spot-vol correlation ρS2V1 in SLV+LV.




Example 1.

(SLV+LV)

dS1,t

S1,t=√V1,tdW

S1,t

dV1,t =κ1 (θ1 − V1,t) dt+ ω1

√V1,tdW

V1,t

dS2,t

S2,t=σ2,loc (t, S2,t) dW

S2,t

with

dWS1,tdW

V1,t = ρS1V1dt

dWS1,tdW

S2,t = ρS1S2dt

dWS2,tdW

V1,t = ρS2V1dt

LV+LV

dS1,t


S1,t

dS2,t


S2,t

with dWS1,tdW

S2,t = ρS1S2dt

Numerical values of model parameters:

S1,0 = 100, S2,0 = 100, ρS1S2 = 0.3

V1,0 = 0.02, κ1 = 3, θ1 = 0.05, ω1 = 0.5, ρS1V1 = −0.75

σ1,loc is the local vol surface in Heston model(V0 = V1,0, κ = κ1, θ = θ1, ω = ω1, ρ = ρS1V1

)σ2,loc ≡ 0.2

The positive definiteness constraints (D) gives −0.856 ≤ ρS2V1 ≤ 0.406

ρS2V1∗ = ρS1S2ρS1V1 = −0.225.




t = 0.5




Question: In SLV + LV, when do we have

E[σ21,t

∣∣S2,t

]= E

[σ21,loc (t, S1,t)

∣∣∣S2,t

]?

One sufficient condition that makes the above equality hold is

E[σ21,t

∣∣S1,t, S2,t

]= E

[σ21,t

∣∣S1,t

](A)

Assume (A) holds. Then

E[σ21,t

∣∣S2,t

]= E

[E[σ21,t

∣∣S1,t, S2,t

]∣∣S2,t

]= E

[E[σ21,t

∣∣S1,t

]∣∣S2,t

]= E

[σ21,loc (t, S1,t)

∣∣∣S2,t

]The assumption (A) says that given S1,t, the other variable S2,t provides noadditional information in estimating σ2

1,t.

In other words, the dependence of σ21,t on S2,t is only through the dependence of

σ21,t on S1,t and then the dependence of S1,t on S2,t.




Let(W 1

t ,W2t ,W

3t

)be the 3-dimensional uncorrelated standard Brownian motion.

In the SLV+LV model, if we let

dWS1,t = dW 1

t

dWS2,t = ρS1S2

t dW 1t +

√1−

(ρS1S2t

)2dW 2

t

dWV1,t = ρS1V1

t dW 1t +

√1−

(ρS1V1t

)2dW 3

t

=⇒ dWS

2,tdWV1,t = ρS1S2

t ρS1V1t dt.

If we chooseρS2V1t = ρS1S2

t ρS1V1t (C)

Numerical results show that

E[σ21,loc (t, S1,t)

∣∣∣S2,t

]in SLV + LV ≈ E

[σ21,loc (t, S1,t)

∣∣∣S2,t

]in LV + LV

However, E[σ21,t

∣∣∣S2,t

]in SLV+LV may differ even if (C) holds.




Example 1a.

Same model parameters as in Example 1, except that the correlations now aretime-varying:

ρS1V1t = −0.75, ρ

S1S2t = 0.3 if t ≤ 0.25 else 0, ρ

S2V1t = ρ

S1S2t ρ

S1V1t

E [V1,t|S2,t] in SLV+LV differs markedly from E[σ21,loc (t, S1,t)

∣∣∣S2,t

]in SLV+LV and

LV+LV.




Example 2.

SLV + LV

dS1,t

S1,t

= σ1

√V1,t√

E [V1,t|S1,t]dW

S1,t

dV1,t = κ (θ − V1,t) dt+ ω√V1,t dW

V1,t

dS2,t

S2,t

= σ2 dWS2,t

with

dWS1,tdW

S2,t = ρ

S1S2dt

dWS1,tdW

V1,t = ρ

S1V1dt

dWS2,tdW

V1,t = ρ

S2V1dt

Here σ1 and σ2 are constants. In particular, the corresponding local vol functions areconstants: σ1,loc (t, S1,t) = σ1 and σ2,loc (t, S2,t) = σ2.

LV + LV

dS1,t

S1,t

= σ1 dWS1,t

dS2,t

S2,t

= σ2 dWS2,t

with dWS1,tdW

S2,t = ρ

S1S2t dt.

Since σ1 is constant, the correlation ρS1S2t has no impact on VS1,2(L,U, T ) in LV+LV.




Numerical Values of model parameters:

σ1 = 0.15, σ2 = 0.2, S1,0 = 100, S2,0 = 100, L = 70, U = 110, T = 1

V0 = 0.0165, κ = 3, θ = 0.05, ω = 0.5 and 0.1, ρS1V1 = −0.75, ρ

S1S2 = 0.3

LV+LV is unable to capture the vol-of-vol risk, cross spot/vol correlation risk.



Two-Asset Corridor Variance Swap - Approximation I

Postulate:E[σ21,t

∣∣S2,t

]≈ αt + βtσ

22,loc (t, S2,t) (PI)

Motivation: if σ21,t = αt + βtσ2

2,t + εt and E [εt|S2,t] = 0, then the postulate holds.

VS21,2(L,U, T ) = E

[1

T

∫ T

0σ21,t1L≤S2,t≤Udt

]=

1

T

∫ T

0E[E[σ21,t

∣∣S2,t

]1L≤S2,t≤U

]dt

≈1

T

∫ T

0αtE

[1L≤S2,t≤U

]dt+

1

T

∫ T

0βtE

[σ22,loc(t, S2,t)1L≤S2,t≤U

]dt

If αt ≡ α and βt ≡ β are both constants, then the formula reduces to

VS21,2(L,U, T ) ≈ α RA2(L,U, T ) + β VS2

2 (L,U, T )

The parameters α and β can be estimated from historical data (regressing σ21,t on

σ22,t) or calibrated to the current market prices of relevant instruments.




For a given set of maturities T1 < · · · < Tn, we assume αt and βt take constantvalues between each Ti and Ti+1.

Then we can calibrate αt and βt, if for each Ti, there exists a set of lower andupper bounds (Li, Ui) such that the prices of the following instruments areavailable:

1 Vanilla var swaps, VS1(Ti) and VS2(Ti)

2 Range accural and single-asset corridor var swap on S2, RA2 (Li, Ui, Ti) andVS2 (Li, Ui, T )

3 Two-asset corridor var swap VS1,2(Li, Ui, T )

Note that VS1(Ti), VS2(Ti), RA2 (Li, Ui, Ti) and VS2 (Li, Ui, T ) can be easilycomputed (say, by replication) from an implied vol surface or local volatility modelcalibrated to the market prices of vanilla call and put options.

Now let T = T1 and αt ≡ α and βt ≡ β for 0 ≤ t ≤ T , then we have

VS21,2 (L,U, T ) = αRA2(L,U, T ) + βVS2

2(L,U, T ) (A1)

Letting L = 0 and U =∞ gives another constraint

VS21(T ) = α+ βVS2

2(T ) (A2)

We can solve equations (A1) and (A2) to obtain α and β.The values of α and β at further maturities can be obtained by bootstrap.




Example 3. How good is the approximation?

dS1,t

S1,t=√V1,tdW

S1,t

dV1,t =κ1 (θ1 − V1,t) dt+ ω1

√V1,tdW

V1,t

dS2,t


S2,t

with

dWS1,tdW

V1,t = ρS1V1dt

dWS1,tdW

S2,t = ρS1S2dt

dWS2,tdW

V1,t = ρS2V1dt

Numerical values of model parameters:

S1,0 = 100, S2,0 = 100, ρS1S2 = 0.3, ρS2V1 = −0.5

V1,0 = 0.02, κ1 = 3, θ1 = 0.05, ω1 = 0.5, ρS1V1 = −0.75σ2,loc is the local vol surface in another Heston model with parameters V0 = 0.04,κ = 1, θ = 0.06, ω = 0.25 and ρ = −0.85

For a list of maturities Ti = i12

, i = 1, · · · , 12, and corridor L = 70 and U = 110,

the above model is used to compute the values of VS1(Ti), VS2(Ti),RA2(L,U, Ti), VS2(L,U, Ti) and VS1,2(L,U, Ti). Then we calibrate piece-wiseconstants αt and βt to these values by bootstrapping.




We apply the approximation formula with calibrated αt and βt to other corridors.

VS1,2 (L,U, T ) from simulation

L/U 56/88 63/99 70/110 77/121 84/132T = 0.25 5.67 12.63 16.67 16.88 16.53T = 0.50 8.41 14.00 17.56 17.98 17.29T = 0.75 9.81 14.62 17.82 18.36 17.61T = 1.00 10.61 14.88 17.81 18.43 17.72

VS1,2 (L,U, T ) from approximation

L/U 56/88 63/99 70/110 77/121 84/132T = 0.25 5.66 12.50 16.67 16.89 16.53T = 0.50 8.55 14.03 17.56 17.93 17.22T = 0.75 10.10 14.75 17.82 18.24 17.43

1.00 11.03 15.10 17.81 18.23 17.42

approximation − simulationL/U 56/88 63/99 70/110 77/121 84/132

T = 0.25 -0.01 -0.13 0.0 0.00 0.00T = 0.50 0.14 0.03 0.0 -0.05 -0.07T = 0.75 0.29 0.13 0.0 -0.12 -0.17T = 1.00 0.43 0.21 0.0 -0.19 -0.30




The accuracy can be improved with more market quotes. For example, if there aremarket quotes for two sets of corridors (L,U) and (L, U) (typically (70%, 110%) and(70%,130%) of the spot), we can modify the postulate (PI) as follows:

E[σ21,t

∣∣∣S2,t

]≈ αt +

(βt + γt

S2,t

S2,0

)σ22,loc (t, S2,t) (PIa)

If αt ≡ α, βt ≡ β and γt ≡ γ, for 0 ≤ t ≤ T , then

VS21,2(L,U, T ) = αRA2(L,U, T ) + βVS

22 (L,U, T ) + γGS

22 (L,U, T )

VS21,2(L, U, T ) = αRA2(L, U, T ) + βVS

22

(L, U, T

)+ γGS

22 (L,U, T )

VS21(T ) = α+ βVS

22(T ) + γGS

22(T )

where

GS2i (L,U, T ) = E

[1

T

∫ T

0

Si,t

Si,0σ2i,t1L≤Si,t≤Udt

]is the corridor Gamma swap on Si with corridor (L,U) and GSi(T ) = GSi(0,∞, T ) isthe vanilla Gamma swap.

The single-asset corridor Gamma swap can be perfectly replicated by static positions ofvanilla options (replicating truncated and linearly extrapolated payoff ST logST ) anddynamically rebalanced delta-hedging portfolio. In particular, if rt ≡ qt, then

GS2i (L,U, T ) =

1

T

∫ U

L

2

KOTM(K,T )dK.

For a list of maturities {Ti}, we shall assume αt, βt and γt take on constant valuesbetween Ti and Ti+1 so that their values can be calibrated by bootstrapping.




Simple and fast calibration and pricing, no Monte Carlo simulation needed

Establishes explicit dependence on volatilities and correlations, well-suited forrisk management

Once the values of αt, βt (and γt) are calibrated, the approximation formula(PI) (or (PIa)) for the cross local variance can be used to price two-assetcorridor var swaps with other corridors (for deals entered earlier when thecorridors were set relative to the spot at the time of the trade), or even othergeneral cross weighted variance payoffs, i.e.,

E[

1

T

∫ T

0σ21,tw (S2,t) dt

]=

1

T

∫ T

0E[E[σ21,t

∣∣S2,t

]w (S2,t)

]dt

=E[

1

T

∫ T

0

(αt +

(βt + γt

S2,t

S2,0

)σ22,loc (t, S2,t)

)w (S2,t) dt

]The last expectation can be easily evaluated by replication, PDE or MonteCarlo simulation in the local vol model.



Two-Asset Corridor Variance Swap - Approximation II

Postulate: The variance index S1,t follows a local volatility dynamics.

dS1,t

S1,t= σ1,loc (t, S1,t) dW1,t

Then

VS21,2(L,U) = E

[1

T

∫ T

0σ21,loc (t, S1,t) 1L≤S2,t≤Udt

]=

1

T

∫ T

0E[σ21,loc (t, S1,t)E

[1L≤S2,t≤U

∣∣∣S1,t

]]= E

[1

T

∫ T

0σ21,loc(t, S1,t) (Φt (U |S1,t)− Φt (L|S1,t)) dt

]where

Φt(K|S1,t) = E[

1S2,t≤K

∣∣∣S1,t

]is the conditional CDF of S2,t given S1,t.

In this case we price the two-asset corridor variance swap as a single-assetweighted variance swap on S1 with weight Φt (U |S1,t)− Φt (L|S1,t).

The conditional CDF Φt may be derived from the copula of S1,t and S2,t, whichin turn can be calibrated to the market prices of basket payoffs S1,t and S2,t ifavailable.



References

[B] BNP Paribas U.S. Equities & Derivatives Strategy, Corridor Variance Swaps:A Cheaper Way to Buy Volatility? August 14, 2007

[CL] P. Carr and K. Lewis, Corridor Variance Swaps, Risk, Feb 2004.

[M] JP Morgan European Equity Derivatives Research, Variance Swaps,Investment Strategies: No. 28, Nov 17, 2006

[GL] J. Guyon and P. Henry-Labordere, Nonlinear Option Pricing, CRC Press,2013.



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