Prices and Asymptotics of Variance Swaps Carole Bernard Zhenyu (Rocky) Cui
Beirut, May 2013.
Carole Bernard
Lebanese Mathematical Society
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Motivation
Convex Order Conjecture
Variance Swap
Numerics
Conclusions
Outline
I Motivation I Convex order conjecture I Discrete variance swaps: prices and asymptotics I Conclusion & Future Directions
Carole Bernard
Lebanese Mathematical Society
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Motivation
Convex Order Conjecture
Variance Swap
Numerics
Conclusions
Variance Swap I A variance swap is an OTC contract: 1 Realized Variance − Strike Notional × T I Realized Variance: RV =
n−1 P
ln
i=0
Sti+1 Sti
2
with
0 = t0 < t1 < ... < tn = T .
I Quadratic Variation: QV =
lim n→∞,
max
i=0,1,...,n−1
(ti+1 −ti )→0
RV .
I In practice, variance swaps are discretely sampled but it is typically easier to compute the continuously sampled in popular stochastic volatility models. I Question: Finding “fair” strikes so that the initial value of the contract is 0. Carole Bernard
Lebanese Mathematical Society
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Motivation
Convex Order Conjecture
Variance Swap
Numerics
Conclusions
Model Setting (1/2)
I Under the risk-neutral probability measure Q, ( √ (1) dSt = rdt + Vt dWt St (M) (2) dVt = µ(Vt )dt + σ(Vt )dWt (1)
(2)
where E[dWt dWt ] = ρdt.
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Motivation
Convex Order Conjecture
Variance Swap
Numerics
Conclusions
Three Stochastic Volatility Models (1)
(2)
Assume E[dWt dWt ] = ρdt. I The correlated Heston model: ( √ (1) dSt = rdt + Vt dWt , S t (H) √ (2) dVt = κ(θ − Vt )dt + γ Vt dWt I The correlated Hull-White model: ( √ (1) dSt = rdt + Vt dWt , S t (HW) (2) dVt = µVt dt + σVt dWt • The correlated Sch¨ obel-Zhu model:
( (SZ)
Carole Bernard
(1)
dSt St
= rdt + Vt dWt
dVt
= κ(θ − Vt )dt + γdWt
(2)
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Model Setting (2/2) I Under the risk-neutral probability measure Q, ( √ (1) dSt = rdt + Vt dWt S t (M) (2) dVt = µ(Vt )dt + σ(Vt )dWt (1)
(2)
where E[dWt dWt ] = ρdt. I The fair strike of the “discrete variance swap” is "n−1 # X Sti+1 2 1 1 M = E[RV ] ln Kd (n) := E T Sti T i=0
I The fair strike of the “continuous variance swap” is Z T 1 1 M Kc := E Vs ds = E[QV ] T T 0 Carole Bernard
Lebanese Mathematical Society
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Motivation
Convex Order Conjecture
Variance Swap
Numerics
Conclusions
Contributions
I A general expression for the fair strike of a discrete variance swap in the time-homogeneous stochastic volatility model: I 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¨ obel-Zhu model: : a new closed-form formula.
I Asymptotic expansion of the fair strike with respect to n, T , vol of vol... I A counter-example to the “Convex Order Conjecture”.
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Lebanese Mathematical Society
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Motivation
Convex Order Conjecture
Variance Swap
Numerics
Conclusions
Convex Order Conjecture I Notations: Pn−1 (1) RV = i=0 (log(Sti+1 /Sti ))2 : discrete realized variance for a partition of [0, T ] with n + 1 points; RT (2) QV = 0 Vs ds: continuous quadratic variation.
I Usual practice: approximate E[f (RV )] with E[f (QV )], see Jarrow et al (2012). I 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”. I Call option on RV: (RV − K )+ ; Call option on QV: (QV − K )+ . Carole Bernard
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Motivation
Convex Order Conjecture
Variance Swap
Numerics
Conclusions
Convex Order Conjecture (Cont’d)
I 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” I Equivalently, E[f (RV )] > E[f (QV )] where f is convex. I 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 KdM (n) < KcM (Illustrated by examples in Heston, Hull-White and Sch¨ obel-Zhu models (M)).
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Motivation
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Numerics
Conclusions
Conditional Black-Scholes Representation I Recall ( (M)
dSt St
dVt
= rdt +
√
(1)
Vt dWt
(2)
= µ(Vt )dt + σ(Vt )dWt (1)
I Cholesky decomposition: dWt
(2)
= ρdWt
+
p (3) 1 − ρ2 dWt .
I Key representation of the log stock price Z 1 T ln(ST ) = ln(S0 ) + rT − Vt dt 2 0 p Z Z T 2 + ρ f (VT ) − f (V0 ) − h(Vt )dt + 1 − ρ 0
where f (v ) = Carole Bernard
Rv
√
z 0 σ(z) dz,
T
p (3) Vt dWt
0
h(v ) = µ(v )f 0 (v ) + 12 σ 2 (v )f 00 (v ). Lebanese Mathematical Society
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Numerics
Conclusions
Proposition Under some technical conditions, (∆ = Tn ):, " # Z t+∆ St+∆ 2 2 2 = r ∆ − r∆ E ln E [Vs ] ds St t "Z 2 # Z t+∆ t+∆ 1 2 + E Vs ds + (1 − ρ ) E [Vs ] ds 4 t t "Z 2 # h i t+∆ + ρ2 E (f (Vt+∆ ) − f (Vt ))2 + ρ2 E h(Vs )ds t
Z + ρE
t+∆
Z
t+∆
h(Vs )ds t
Vs ds
t
Z
− ρE (f (Vt+∆ ) − f (Vt ))
t+∆
(2ρh(Vs ) + Vs )ds .
t
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Motivation
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Sensitivity w.r.t. interest rate Proposition (Sensitivity to r ) The fair strike of the discrete variance swap: KdM (n) = b M (n) −
T M T Kc r + r 2 , n n
where b M (n) does not depend on r . dKdM (n) T = (2r − KcM ) dr n KdM (r ) reaches minimum when r ∗ =
Carole Bernard
KcM 2 .
Lebanese Mathematical Society
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Motivation
Convex Order Conjecture
Variance Swap
Numerics
Conclusions
Three Stochastic Volatility Models (1)
(2)
Assume E[dWt dWt ] = ρdt. I The correlated Heston model: ( √ (1) dSt = rdt + Vt dWt , S t (H) √ (2) dVt = κ(θ − Vt )dt + γ Vt dWt I The correlated Hull-White model: ( √ (1) dSt = rdt + Vt dWt , S t (HW) (2) dVt = µVt dt + σVt dWt • The correlated Sch¨ obel-Zhu model:
( (SZ)
Carole Bernard
(1)
dSt St
= rdt + Vt dWt
dVt
= κ(θ − Vt )dt + γdWt
(2)
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Heston Model The fair strike of the discrete variance swap is 1 n 2 2 2 2 2κT κ T (θ − 2r ) + nθ 4κ − 4ρκγ + γ 8nκ3 T 1 − e κT n 2 2 −2κT +n γ (θ − 2V0 ) + 2κ (V0 − θ) e −1 κT n 1 + e +4 (V0 − θ) n 2κ2 + γ 2 − 2ρκγ + κ2 T (θ − 2r ) 1 − e −κT 1 − e −κT 2 − κT −2n θγ (γ − 4ρκ) 1 − e n + 4 (V0 − θ) κT γ (γ − 2ρκ) κT 1−e n KH d (n) =
The fair strike of the continuous variance swap is Z T 1 V0 − θ H Kc = E Vs ds = θ + (1 − e −κT ) . T κT 0 Carole Bernard
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Motivation
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Hull-White Model The fair strike of the discrete variance swap is r 2T V0 rT HW Kd (n) = + 1− (e µT − 1) n µT n µT 2 2 V02 e (2 µ+σ )T − 1 e n − 1 V02 e (2 µ+σ )T − 1 + − (2 µ+σ2 )T 2T (2µ + σ 2 )(µ + σ 2 ) 2 n 2T µ(µ + σ ) e −1 3(4µ+σ 2 )T 3(4µ+σ 2 )T µT 3/2 8 8 − 1 V0 σ(e n − 1) 64ρ e − 1 V0 3/2 σ 8ρ e − + 3(4µ+σ 2 )T 3T (4µ + σ 2 ) (4 µ + 3 σ 2 ) 2 8n µT (4 µ + 3 σ ) e −1 The fair strike of the continuous variance swap is Z T V0 µT 1 HW Kc = E Vs ds = (e − 1). T Tµ 0 Carole Bernard
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Motivation
Convex Order Conjecture
Variance Swap
Numerics
Conclusions
Sch¨obel-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 (V0 − θ)2 γ2 γ2 2 = + θ + − (1 − e −2κT ) KSZ c 2κ 2κT 4κ2 T 2θ(V0 − θ) (1 − e −κT ). + κT
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Heston model: Expansion w.r.t n
KH d (n)=
KH c
aH + 1 +O n
1 n2
.
1 where aH 1 is a linear and decreasing function of ρ:
aH 1 >0
⇐⇒
ρ 6 ρH 0
where 2 2 r 2 T − rKcH T + θ4 + θγ 8κ T + c1 . ρH 0 = 0) −κT ) − θγT − γ(θ−V (1 − e 2κ 2 1
Explicit expression of a1H is in Proposition 5.1, Bernard and Cui (2012).
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Motivation
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Numerics
Conclusions
Hull-White model: Expansion w.r.t n
KHW d (n)
=
KHW c
aHW + 1 +O n
1 n2
where aHW is a linear and decreasing function of ρ:2 1 aHW >0 1
⇐⇒
ρ 6 ρHW 0
where ρHW = 0
2
3(4µ + σ 2 ) r 2 T − rKcHW T + 3
3
4σV02 (e 8 (4µ+σ
2 )T
V02 e (2µ+σ2 )T −1 4 2µ+σ 2
> 0.
− 1)
Explicit expression of a1HW is in Proposition 5.3, Bernard and Cui (2012).
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Motivation
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Numerics
Conclusions
Sch¨obel-Zhu model: Expansion w.r.t n
The asymptotic behavior of the fair strike of a discrete variance swap in the Sch¨obel-Zhu model is given by aSZ 1 SZ SZ 1 +O , Kd (n) = Kc + n n2 where a1SZ = r 2 T − rTKcSZ + d1 + d2
γ ρ. 2κ
(1)
and where d1 and d2 are explicit.
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Numerics
Conclusions
Other Expansions
Given that the expressions are explicit, it is straightforward to obtain 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
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Numerics
Conclusions
Expansion of the fair strike for small maturity T In the Heston model, an expansion of KdH (n) when T → 0 is H H 2 3 KH d (n) = V0 + b1 T + b2 T + O T where 1 κ(θ − V0 ) + (V0 − 2r )2 − 2γV0 ρ 2 4n 2 κ (V0 − θ) (V0 − θ)κ(γρ + 2r − V0 ) + b2H = + 6 4n γ 2 V0 γρκ(V0 + θ) − 2 + . 12n2 b1H =
γ 2 V0 2
and we have H KH d (n) − Kc = Carole Bernard
1 (V0 − 2r)2 − 2ργV0 T + O(T2 ). 4n Lebanese Mathematical Society
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Expansion of the fair strike for small maturity In the Hull-White model, an expansion of KdHW (n) when T → 0 is HW HW 2 3 KHW d (n) = V0 + b1 T + b2 T + O T where V0 µ 1 + (V0 − 2r )2 − 2ρV0 3/2 σ 2 4n ! 2 V0 σ 2 V0 3ρ V0 1/2 σ(σ 2 + 4µ) V0 µ HW + − + µ(V0 − 2r ) b2 = 6 4n 2 8 √ V0 3/2 σ ρ(3σ 2 − 4µ) − 4σ V0 + 96n2 Note also 1 HW 2 3/2 KHW (n) − K = (V − 2r) − 2ρV σ T + O(T2 ). 0 0 c d 4n b1HW =
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Expansion of the fair strike for small maturity In the Sch¨ obel-Zhu model, an expansion of KdHW (n) when T → 0 is In the Sch¨obel-Zhu model, KdSZ (n) can be expanded when T → 0 as 2 SZ 2 KSZ d (n) = V0 + b1 T + O(T )
(2)
where b1SZ
γ2 1 = κV0 (θ − V0 ) + + 2 n
V02 (V02 − 4ργ) 2 2 r − rV0 + 4
Note also SZ KSZ d (n) − Kc =
Carole Bernard
1 2 (V0 − 2r)2 − 4ρV02 γ T + O(T2 ). 4n Lebanese Mathematical Society
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Parameters
I Heston model: First set of parameters from Broadie and Jain (2008). Second set is when T = 1/12. I Hull-White model: obtain µ by numerically solving KcH = KcHW , and determine σ so that the variances of VT in the Heston and Hull-White models match.
Set 1 Set 2
Carole Bernard
T 1 1/12
r 3.19% 3.19%
V0 0.010201 0.010201
ρ -0.7 -0.7
Heston γ θ κ 0.31 0.019 6.21 0.31 0.019 6.21
(matched) Hull-White µ σ 1.003 0.42 4.03 1.78
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Heston Model (T=1)
Hull−White Model (T=1)
0.0188
0.0186 KH
0.0186
c
d
KH
0.0182 0.018
0.018 0.0178 0.0176
0.0176 5
10 15 20 25 Discretization step n
30
35
0
5
10 15 20 25 Discretization step n
30
35
Hull−White Model (T=1/12)
Heston Model (T=1/12) 0.0123
0.0124
KHW
KH c
c
ρ = − 0.7 ρ=0 ρ = 0.7
d
KH
0.0123 0.0122
ρ = − 0.7 ρ=0 ρ = 0.7
0.0123 KHW d
0.0123
0.0122 0.0121
0.0121 0.0121 0
c
ρ = − 0.7 ρ=0 ρ = 0.7
0.0182
0.0178
0
KHW
0.0184
KHW d
ρ = − 0.7 ρ=0 ρ = 0.7
0.0184
5 10 Discretization step n
15
0.0121 0
5 10 Discretization step n
15
Heston Model
Hull−White Model
0.1
0.096
KHW d V0=0.108
0.094
KH d
KHW c
V0=0.108 0.095
0.092 KHW and KHW d c
KH and KH d c
KH c KH d
V =0.09 0
0.09
KHW d V0=0.09
0.09 KHW c 0.088
KH c
KHW d
KH d
0.086 V0=0.072
0.085 0
KH c 0.2 0.4 0.6 0.8 1 γ
1.2 1.4 1.6 1.8
KHW c 2
0.084 0
0.2 0.4 0.6 0.8
1 γ
V0=0.072 1.2 1.4 1.6 1.8
2
Heston Model
−5
2.5
x 10
−5
7 r = 0% r = 3.2% r = 6%
2
x 10
Hull−White Model r = 0% r = 3.2% r = 6%
6 HW
1.5
1
=5.06
4
ρH=0.97
c
KHW − KHW
0
0.5
3
d
KHd − KHc
ρ0
5
0
−0.5
−1
2 ρH=0.04
HW
ρ0
1
0
H ρ0 =0.21
=1.05
0 HW
ρ0 −1.5 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 Correlation coefficient ρ
0.8
1
=0.18
−1 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 Correlation coefficient ρ
0.8
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
Schobel−Zhu Model
−5
6
x 10
r = 0% r = 3.2% r = 6% ρSZ =0.054 0
4
r=0%
ρSZ =0.177 0 r=6%
c
KSZ − KSZ
2
d
0
ρSZ=0.040 0
−2
r=3.19%
−4
−6 −1
−0.8
−0.6
−0.4
−0.2
0 0.2 Correlation coefficient ρ
0.4
0.6
0.8
1
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¨ obel-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 I Explicit expressions and asymptotics for KdM (n) in any time homogeneous stochastic volatility model (M). I Allow to better understand the effect of discretization. I Future directions: 1
2 3
4
Extend our study with the 3/2 model √ 3/2 (dSt = St Vt dW1 (T ), dVt = (ωVt − θVt2 )dt + ξVt dW2 (t). Work on expansions valid in a more general setting... Find out whether the first term in the expansion is always linear in the correlation ρ. Generalize the explicit pricing formula to the case of discrete gamma swaps under the Heston model. 2 n−1 St 1 X Sti+1 Notional × × ln i+1 T S0 Sti i=0
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
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I Bernard, C., and Cui, Z. (2011): Pricing timer options, Journal of Computational Finance, 15(1), 69-104. I Broadie, M., and Jain, A. (2008): The Effect of Jumps and Discrete Sampling on Volatility and Variance Swaps, IJTAF, 11, 761-797. I B¨ ulher, H. (2006): Volatility markets: consistent modeling, hedging and practical implementation , Ph.D. dissertation, TU Berlin. I Cai, N., and S. Kou (2011): Option pricing under a mixed-exponential jump diffusion model, Management Science, 57(11), 2067–2081. I Carr, P., Lee, R., and Wu, L. (2012): Variance swaps on Time-changed L´evy processes, Finance and Stochastics, 16(2), 335-355. I Cui, Z. (2013): PhD thesis at University of Waterloo. I H¨ orfelt, P. and Torn´e, O. (2010): The Value of a Variance Swap - a Question of Interest, Risk, June, 82-85. I 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. I Keller-Ressel, M., and C. Griessler (2012): “Convex order of discrete, continuous and predictable quadratic variation and applications to options on variance,” ArXiv Working paper. I Keller-Ressel, M., and J. Muhle-Karbe (2012): “Asymptotic and exact pricing of options on variance,” Finance and Stochastics, forthcoming. Carole Bernard
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