Prices and Asymptotics of Variance Swaps Carole Bernard Zhenyu - - PowerPoint PPT Presentation

Prices and Asymptotics of Variance Swaps

Carole Bernard Zhenyu (Rocky) Cui Beirut, May 2013.

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Motivation Convex Order Conjecture Variance Swap Numerics Conclusions

Outline

◮ Motivation ◮ Convex order conjecture ◮ Discrete variance swaps: prices and asymptotics ◮ Conclusion & Future Directions

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Motivation Convex Order Conjecture Variance Swap Numerics Conclusions

Variance Swap

◮ A variance swap is an OTC contract: Notional × 1 T Realized Variance − Strike

• ◮ Realized Variance: RV =

n−1

• i=0
• ln

Sti+1 Sti

2 with 0 = t0 < t1 < ... < tn = T. ◮ Quadratic Variation: QV = lim

n→∞, max

i=0,1,...,n−1(ti+1−ti)→0 RV .

◮ In practice, variance swaps are discretely sampled but it is typically easier to compute the continuously sampled in popular stochastic volatility models. ◮ Question: Finding “fair” strikes so that the initial value of the contract is 0.

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Motivation Convex Order Conjecture Variance Swap Numerics Conclusions

Model Setting (1/2)

◮ Under the risk-neutral probability measure Q, (M)

• dSt

St

= rdt + √VtdW (1)

t

dVt = µ(Vt)dt + σ(Vt)dW (2)

t

where E[dW (1)

t

dW (2)

t

] = ρdt.

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Motivation Convex Order Conjecture Variance Swap Numerics Conclusions

Three Stochastic Volatility Models

Assume E[dW (1)

t

dW (2)

t

] = ρdt. ◮ The correlated Heston model: (H)

• dSt

St

= rdt + √VtdW (1)

t

, dVt = κ(θ − Vt)dt + γ√VtdW (2)

t

◮ The correlated Hull-White model: (HW)

• dSt

St

= rdt + √VtdW (1)

t

, dVt = µVtdt + σVtdW (2)

t

• The correlated Sch¨
• bel-Zhu model:

(SZ)

• dSt

St

= rdt + VtdW (1)

t

dVt = κ(θ − Vt)dt + γdW (2)

t

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Motivation Convex Order Conjecture Variance Swap Numerics Conclusions

Model Setting (2/2)

◮ Under the risk-neutral probability measure Q, (M)

• dSt

St

= rdt + √VtdW (1)

t

dVt = µ(Vt)dt + σ(Vt)dW (2)

t

where E[dW (1)

t

dW (2)

t

] = ρdt. ◮ The fair strike of the “discrete variance swap” is KM

d (n) := 1

T E n−1

• i=0
• ln Sti+1

Sti 2 = 1 T E[RV ] ◮ The fair strike of the “continuous variance swap” is KM

c := 1

T E T Vsds

• = 1

T E[QV ]

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Motivation Convex Order Conjecture Variance Swap Numerics Conclusions

Contributions

◮ A general expression for the fair strike of a discrete variance swap in the time-homogeneous stochastic volatility model: ◮ Application in three popular stochastic volatility models

(1) Heston model: more explicit than Broadie and Jain (2008). (2) Hull-White model: a new closed-form formula. (3) Sch¨

• bel-Zhu model: : a new closed-form formula.

◮ Asymptotic expansion of the fair strike with respect to n, T, vol of vol... ◮ A counter-example to the “Convex Order Conjecture”.

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Motivation Convex Order Conjecture Variance Swap Numerics Conclusions

Convex Order Conjecture

◮ Notations:

(1) RV = n−1

i=0 (log(Sti+1/Sti))2: discrete realized variance for a

partition of [0, T] with n + 1 points; (2) QV = T

0 Vsds: continuous quadratic variation.

◮ Usual practice: approximate E[f (RV )] with E[f (QV )], see Jarrow et al (2012). ◮ B¨ ulher (2006): “while the approximation of realized variance via quadratic variation works very well for variance swaps, it is not sufficient for non-linear payoffs with short maturities”. ◮ Call option on RV: (RV − K)+; Call option on QV: (QV − K)+.

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Motivation Convex Order Conjecture Variance Swap Numerics Conclusions

Convex Order Conjecture (Cont’d)

◮ The convex-order conjecture (Keller-Ressel (2011)): “The price of a call option on realized variance is higher than the price of a call option on quadratic variation” ◮ Equivalently, E[f (RV )] E[f (QV )] where f is convex. ◮ When f (x) = x, our closed-form expression shows that when the correlation between the underlying and its variance is positive, it is possible to observe K M

d (n) < K M c

(Illustrated by examples in Heston, Hull-White and Sch¨

• bel-Zhu models

(M)).

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Motivation Convex Order Conjecture Variance Swap Numerics Conclusions

Conditional Black-Scholes Representation

◮ Recall (M)

• dSt

St

= rdt + √VtdW (1)

t

dVt = µ(Vt)dt + σ(Vt)dW (2)

t

◮ Cholesky decomposition: dW (1)

t

= ρdW (2)

t

+

• 1 − ρ2dW (3)

t

. ◮ Key representation of the log stock price ln(ST) = ln(S0) + rT − 1 2 T Vtdt + ρ

• f (VT) − f (V0) −

T h(Vt)dt

• +
• 1 − ρ2

T

• VtdW (3)

t

where f (v) = v

√z σ(z)dz, h(v) = µ(v)f ′(v) + 1 2σ2(v)f ′′(v).

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Motivation Convex Order Conjecture Variance Swap Numerics Conclusions

Proposition Under some technical conditions, (∆ = T

n ):,

E

• ln St+∆

St 2 = r2∆2 − r∆ t+∆

t

E [Vs] ds + 1 4E t+∆

t

Vsds 2 + (1 − ρ2) t+∆

t

E [Vs] ds + ρ2E

• (f (Vt+∆) − f (Vt))2

+ ρ2E t+∆

t

h(Vs)ds 2 + ρE t+∆

t

h(Vs)ds t+∆

t

Vsds

• − ρE
• (f (Vt+∆) − f (Vt))

t+∆

t

(2ρh(Vs) + Vs)ds

• .

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Motivation Convex Order Conjecture Variance Swap Numerics Conclusions

Sensitivity w.r.t. interest rate

Proposition (Sensitivity to r) The fair strike of the discrete variance swap: K M

d (n) = bM(n) − T

n K M

c r + T

n r2, where bM(n) does not depend on r. dK M

d (n)

dr = T n (2r − K M

c )

K M

d (r) reaches minimum when r∗ = K M

c

2 .

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Motivation Convex Order Conjecture Variance Swap Numerics Conclusions

Three Stochastic Volatility Models

Assume E[dW (1)

t

dW (2)

t

] = ρdt. ◮ The correlated Heston model: (H)

• dSt

St

= rdt + √VtdW (1)

t

, dVt = κ(θ − Vt)dt + γ√VtdW (2)

t

◮ The correlated Hull-White model: (HW)

• dSt

St

= rdt + √VtdW (1)

t

, dVt = µVtdt + σVtdW (2)

t

• The correlated Sch¨
• bel-Zhu model:

(SZ)

• dSt

St

= rdt + VtdW (1)

t

dVt = κ(θ − Vt)dt + γdW (2)

t

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Motivation Convex Order Conjecture Variance Swap Numerics Conclusions

Heston Model

The fair strike of the discrete variance swap is KH

d (n) =

1 8nκ3T

• 2κT
• κ2T (θ − 2r)2 + nθ
• 4κ2 − 4ρκγ + γ2

+n

• γ2 (θ − 2V0) + 2κ (V0 − θ)2

e−2κT − 1 1 − e

κT n

1 + e

κT n

+4 (V0 − θ)

• n
• 2κ2 + γ2 − 2ρκγ
• + κ2T (θ − 2r)

1 − e−κT −2n2θγ (γ − 4ρκ)

• 1 − e− κT

n

• + 4 (V0 − θ) κTγ (γ − 2ρκ) 1 − e−κT

1 − e

κT n

• The fair strike of the continuous variance swap is

KH

c = 1

T E T Vsds

• = θ + (1 − e−κT)V0 − θ

κT .

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Motivation Convex Order Conjecture Variance Swap Numerics Conclusions

Hull-White Model

The fair strike of the discrete variance swap is KHW

d

(n) = r2T n + V0 µT

• 1 − rT

n

• (eµT − 1)

− V 2

• e(2 µ+σ2)T − 1

e

µT n − 1

• 2Tµ(µ + σ2)
• e

(2 µ+σ2)T

n

− 1 + V 2

• e(2 µ+σ2)T − 1
• 2T(2µ + σ2)(µ + σ2)

+ 8ρ

• e

3(4µ+σ2)T 8

− 1

• V03/2σ(e

µT n − 1)

µT (4 µ + 3 σ2)

• e

3(4µ+σ2)T 8n

− 1

• −

64ρ

• e

3(4µ+σ2)T 8

− 1

• V03/2σ

3T(4µ + σ2) (4 µ + 3 σ2) The fair strike of the continuous variance swap is KHW

c

= 1 T E T Vsds

• = V0

Tµ(eµT − 1).

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Motivation Convex Order Conjecture Variance Swap Numerics Conclusions

Sch¨

• bel-Zhu Model

The fair strike of the discrete variance swap is explicit but too complicated to appear on a slide. The fair strike of the continuous variance swap is KSZ

c

= γ2 2κ + θ2 + (V0 − θ)2 2κT − γ2 4κ2T

• (1 − e−2κT)

+ 2θ(V0 − θ) κT (1 − e−κT).

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Motivation Convex Order Conjecture Variance Swap Numerics Conclusions

Heston model: Expansion w.r.t n

KH

d (n)= KH c + aH 1

n + O 1 n2

• .

where aH

1 is a linear and decreasing function of ρ:1

aH

1 0

⇐ ⇒ ρ ρH where ρH

0 =

r2T − rK H

c T +

• θ2

4 + θγ2 8κ

• T + c1

−

• γ(θ−V0)

2κ

(1 − e−κT) − θγT

2

• .

1Explicit expression of aH 1 is in Proposition 5.1, Bernard and Cui (2012). Carole Bernard Lebanese Mathematical Society 19/32

Motivation Convex Order Conjecture Variance Swap Numerics Conclusions

Hull-White model: Expansion w.r.t n

KHW

d

(n) = KHW

c

+ aHW

1

n + O 1 n2

• where aHW

1

is a linear and decreasing function of ρ:2 aHW

1

⇐ ⇒ ρ ρHW where ρHW = 3(4µ + σ2)

• r2T − rK HW

c

T + V 2

4 e(2µ+σ2)T −1 2µ+σ2

• 4σV

3 2

0 (e

3 8 (4µ+σ2)T − 1)

> 0.

2Explicit expression of aHW 1

is in Proposition 5.3, Bernard and Cui (2012).

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Motivation Convex Order Conjecture Variance Swap Numerics Conclusions

Sch¨

• bel-Zhu model: Expansion w.r.t n

The asymptotic behavior of the fair strike of a discrete variance swap in the Sch¨

• bel-Zhu model is given by

KSZ

d (n) = KSZ c

+ aSZ

1

n + O 1 n2

• ,

where aSZ

1

= r2T − rTK SZ

c

+ d1 + d2 γ 2κρ. (1) and where d1 and d2 are explicit.

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Motivation Convex Order Conjecture Variance Swap Numerics Conclusions

Other Expansions

Given that the expressions are explicit, it is straightforward to

• btain expansions for the discrete variance swaps as a function of

the different parameters, and for example with respect to the maturity or to the volatility of volatility.

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Motivation Convex Order Conjecture Variance Swap Numerics Conclusions

Expansion of the fair strike for small maturity T

In the Heston model, an expansion of K H

d (n) when T → 0 is

KH

d (n) = V0 + bH 1 T + bH 2 T2 + O

• T3

where bH

1 = κ(θ − V0)

2 + 1 4n

• (V0 − 2r)2 − 2γV0ρ
• bH

2 = κ2(V0 − θ)

6 + (V0 − θ)κ(γρ + 2r − V0) + γ2V0

2

4n + γρκ(V0 + θ) − γ2V0

2

12n2 . and we have KH

d (n) − KH c = 1

4n

• (V0 − 2r)2 − 2ργV0
• T + O(T2).

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Motivation Convex Order Conjecture Variance Swap Numerics Conclusions

Expansion of the fair strike for small maturity

In the Hull-White model, an expansion of K HW

d

(n) when T → 0 is KHW

d

(n) = V0 + bHW

1

T + bHW

2

T2 + O

• T3

where bHW

1

= V0 µ 2 + 1 4n

• (V0 − 2r)2 − 2ρV03/2σ
• bHW

2

= V0µ2 6 + V0 4n

• σ2V0

2 − 3ρ V01/2σ(σ2 + 4µ) 8 + µ(V0 − 2r)

• + V03/2σ
• ρ(3σ2 − 4µ) − 4σ√V0
• 96n2

Note also KHW

d

(n) − KHW

c

= 1 4n

• (V0 − 2r)2 − 2ρV03/2σ
• T + O(T2).

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Motivation Convex Order Conjecture Variance Swap Numerics Conclusions

Expansion of the fair strike for small maturity

In the Sch¨

• bel-Zhu model, an expansion of K HW

d

(n) when T → 0 is In the Sch¨

• bel-Zhu model, K SZ

d (n) can be expanded

when T → 0 as KSZ

d (n) = V2 0 + bSZ 1 T + O(T2)

(2) where bSZ

1

= κV0(θ − V0) + γ2 2 + 1 n

• r2 − rV 2

0 + V 2 0 (V 2 0 − 4ργ)

4

• Note also

KSZ

d (n) − KSZ c

= 1 4n

• (V2

0 − 2r)2 − 4ρV2 0γ

• T + O(T2).

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Motivation Convex Order Conjecture Variance Swap Numerics Conclusions

Parameters

◮ Heston model: First set of parameters from Broadie and Jain (2008). Second set is when T = 1/12. ◮ Hull-White model: obtain µ by numerically solving K H

c = K HW c

, and determine σ so that the variances of VT in the Heston and Hull-White models match.

(matched) Heston Hull-White T r V0 ρ γ θ κ µ σ Set 1 1 3.19% 0.010201

• 0.7

0.31 0.019 6.21 1.003 0.42 Set 2 1/12 3.19% 0.010201

• 0.7

0.31 0.019 6.21 4.03 1.78

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5 10 15 20 25 30 35 0.0176 0.0178 0.018 0.0182 0.0184 0.0186 0.0188 Discretization step n Kd

H

Heston Model (T=1) Kc

H

ρ = − 0.7 ρ = 0 ρ = 0.7 5 10 15 20 25 30 35 0.0176 0.0178 0.018 0.0182 0.0184 0.0186 Discretization step n Kd

HW

Hull−White Model (T=1) Kc

HW

ρ = − 0.7 ρ = 0 ρ = 0.7 5 10 15 0.0121 0.0121 0.0122 0.0123 0.0123 0.0124 Discretization step n Kd

H

Heston Model (T=1/12) Kc

H

ρ = − 0.7 ρ = 0 ρ = 0.7 5 10 15 0.0121 0.0121 0.0122 0.0123 0.0123 Discretization step n Kd

HW

Hull−White Model (T=1/12) Kc

HW

ρ = − 0.7 ρ = 0 ρ = 0.7

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0.085 0.09 0.095 0.1 γ Kd

H and Kc H

Heston Model 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0.084 0.086 0.088 0.09 0.092 0.094 0.096 γ Kd

HW and Kc HW

Hull−White Model Kd

H

Kd

H

Kd

H

Kc

H

Kc

H

Kd

HW

Kd

HW

Kd

HW

Kc

H

Kc

HW

Kc

HW

Kc

HW

V0=0.072 V0=0.09 V0=0.108 V0=0.108 V0=0.09 V0=0.072

−1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1 −1.5 −1 −0.5 0.5 1 1.5 2 2.5 x 10

−5

Correlation coefficient ρ Kd

H − Kc H

Heston Model r = 0% r = 3.2% r = 6% −1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1 −1 1 2 3 4 5 6 7 x 10

−5

Correlation coefficient ρ Kd

HW − Kc HW

Hull−White Model r = 0% r = 3.2% r = 6% ρH

0=0.04

ρH

0=0.21

ρH

0=0.97

ρ0

HW=0.18

ρ0

HW=1.05

ρ0

HW=5.06

Figure 4: Asymptotic expansion with respect to the correlation coefficient ρ and the risk-free rate r

Parameters correspond to Set 1 in Table 1 except for r that can take three possible values r = 0%, r = 3.2% or r = 6%. Here n = 250, which corresponds to a daily monitoring as T = 1.

−1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1 −6 −4 −2 2 4 6 x 10

−5

Correlation coefficient ρ Kd

SZ − Kc SZ

Schobel−Zhu Model r = 0% r = 3.2% r = 6% ρ0

SZ=0.177

r=6% ρ0

SZ=0.040

r=3.19% ρ0

SZ=0.054

r=0%

Figure 8: Asymptotic expansion with respect to the correlation coefficient ρ and the risk-free rate r.

Parameters are similar to Set 1 in Table 1 for the Heston model except for r that can take three possible values r = 0%, r = 3.2% or r = 6%. Precisely, we use the following parameters for the Sch¨

• bel-Zhu model: κ = 6.21, θ =

√ 0.019, γ = 0.31, ρ = −0.7, T = 1, V0 = √ 0.010201. Here n = 250, which corresponds to a daily monitoring as T = 1.

Motivation Convex Order Conjecture Variance Swap Numerics Conclusions

Conclusions & Future Directions

◮ Explicit expressions and asymptotics for K M

d (n) in any time

homogeneous stochastic volatility model (M). ◮ Allow to better understand the effect of discretization. ◮ Future directions:

1

Extend our study with the 3/2 model (dSt = St √VtdW1(T), dVt = (ωVt − θV 2

t )dt + ξV 3/2 t

dW2(t).

2

Work on expansions valid in a more general setting...

3

Find out whether the first term in the expansion is always linear in the correlation ρ.

4

Generalize the explicit pricing formula to the case of discrete gamma swaps under the Heston model. Notional × 1 T ×

n−1

• i=0

Sti+1 S0

• ln Sti+1

Sti 2

5

Generalize to the mixed exponential jump diffusion model for which it is possible to compute discrete and continuous fair strikes.

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Motivation Convex Order Conjecture Variance Swap Numerics Conclusions

◮ Bernard, C., and Cui, Z. (2011): Pricing timer options, Journal of Computational Finance, 15(1), 69-104. ◮ Broadie, M., and Jain, A. (2008): The Effect of Jumps and Discrete Sampling on Volatility and Variance Swaps, IJTAF, 11, 761-797. ◮ B¨ ulher, H. (2006): Volatility markets: consistent modeling, hedging and practical implementation , Ph.D. dissertation, TU Berlin. ◮ Cai, N., and S. Kou (2011): Option pricing under a mixed-exponential jump diffusion model, Management Science, 57(11), 2067–2081. ◮ Carr, P., Lee, R., and Wu, L. (2012): Variance swaps on Time-changed L´ evy processes, Finance and Stochastics, 16(2), 335-355. ◮ Cui, Z. (2013): PhD thesis at University of Waterloo. ◮ H¨

• rfelt, P. and Torn´

e, O. (2010): The Value of a Variance Swap - a Question of Interest, Risk, June, 82-85. ◮ Jarrow, R., Y. Kchia, M. Larsson, and P. Protter (2013): “Discretely Sampled Variance and Volatility Swaps versus their Continuous Approximations,” Finance and Stochastics, 17(2), 305–324. ◮ Keller-Ressel, M., and C. Griessler (2012): “Convex order of discrete, continuous and predictable quadratic variation and applications to options

• n variance,” ArXiv Working paper.

◮ Keller-Ressel, M., and J. Muhle-Karbe (2012): “Asymptotic and exact pricing of options on variance,” Finance and Stochastics, forthcoming.

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