- Email: [email protected]
Stochastic volatility models for the implied correlation index.
Finance Research Letters xxx (xxxx) xxxx
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Stochastic volatility models for the implied correlation index. Evidence, properties and pricing Marcos Escobar ,a, Lin Fang1,b ⁎
a b
Associate Professor, Department of Statistical and Actuarial ScienceS, Western University, Canada Department of Statistical and Actuarial Sciences, Western University, Canada
ARTICLE INFO
ABSTRACT
Keywords: Implied correlation index Stochastic volatility models Mean reversion Correlation derivatives
This paper studies the implied correlation index (CIX), revealing a new stylized fact: heteroscedasticity in correlation. A correlation stochastic volatility (C-SV) model is proposed and a consistent estimation methodology is implemented on CBOE S&P 500 CIX historical data. The impact of the SV parameters is studied for two types of crisis-motivated CIX derivatives, and the empirical study demonstrates that new parameters can have a significant influence of up to 60% on digital option prices.
1. Introduction Since 2009, the Chicago Board Options Exchange (CBOE) has been releasing daily values of the S& P 500 implied correlation index (CIX) as a measure of the market’s expectation on future correlations; see Exchange (2009). The CIX, proposed by Skintzi and Refenes (2005), is defined as the ratio of the sum of the weighted covariance to that of the weighted variance between underlying assets. Stochastic correlation processes (SCPs) are natural ways of modeling the CIX. The study of SCPs has attracted some attention, mostly from the perspective of stochastic covariance models; see Gouriéroux (2006). The works of van Emmerich (2006) and Ma (2009) are two of a number of studies that treat correlation as a stand-alone process in continuous-time2. In van Emmerich (2006) and Teng et al. (2016), the authors constructed SCP models via a convenient function of well-known processes. This general approach benefits from the ease of construction and a better degree of analytical tractability. van Emmerich (2006) also introduced a second approach where the SCP is described directly by Jacobi processes. In this direction, a modified Jacobi process was proposed in Ma (2009) where boundaries for correlation are more flexible than 1 and 1. Stochastic correlation process modelling ensures a critical stylized fact of stochastic correlations: the mean-reverting property. However, the volatility is assumed to be constant. In this paper, we provide empirical evidence of heteroscedasticity in the time series of the CIX; in other words, CIX exhibits timevarying volatility and periods of swings mixed with periods of relative calm (volatility clustering). Furthermore, we propose a correlation stochastic volatility (CSV) model, which can better describe data along the lines of Teng et al. (2016)3. We assume that the variance follows a Cox-Ingersoll-Ross (CIR) process, which is a solid choice in finance (see Heston (1993)). We also study the impact of this model on two types of digital options on the CIX. This is motivated by two facts, first derivatives on correlations have been of interest over the past few years; see Salvi and Swishchuk (2014) and Faria et al. (2018). Secondly, it is recognized that in times of
Corresponding author. E-mail addresses: [email protected] (M. Escobar), [email protected] (L. Fang). 1 Tel.: +1 226-998-1966 2 see Markopoulou et al. (2016) and literature therein for discrete time analyses 3 A second model based on Ma (2009) was explored with similar results ⁎
https://doi.org/10.1016/j.frl.2019.101309 Received 16 January 2019; Received in revised form 13 September 2019; Accepted 6 October 2019 1544-6123/ © 2019 Elsevier Inc. All rights reserved.
Please cite this article as: Marcos Escobar and Lin Fang, Finance Research Letters, https://doi.org/10.1016/j.frl.2019.101309
Finance Research Letters xxx (xxxx) xxxx
M. Escobar and L. Fang
market stress, the correlation increases as stock prices tend to move together; investors can use our digital options to hedge the risk of a financial crisis. The main contributions of our paper are as follows:
• Detect heteroscedasticity on the CBOE S&P 500 CIX, this represents a new stylized fact of financial data. • Introduce a CSV model in which the variance follows a CIR process. • Perform discretization and partial estimations of our new model, numerically demonstrating the consistency of the estimators. • Price two types of crisis-motivated CIX derivatives, detecting a significant influence of the parameters in the stochastic volatility process.
2. About the implied correlation index Let us consider a portfolio of N assets. At time t, the portfolio variance N 2 P, t
2 P, t
can be calculated as follows:
N 1
wi2, t
=
2 i, t
+2
i=1
wi, t wj, t
ij, t i, t j, t ,
0
t
T
(1)
i =1 j>i
where wi, t is the weight of asset i on the portfolio at time t, σi,t is the volatility of asset i at time t, and ρij,t is the pairwise correlation between assets i and j at time t. Definition 1 (CIX formula, see Skintzi and Refenes (2005)). The CIX (ρt) is the correlation value holding for any pair of assets in the portfolio ( t = ij, t ):
CIXt =
t
=
2 P, t N 1 i=1
2
N i=1 j>i
wi2, t
wi, t wj, t
2 i, t
(2)
i, t j , t
Here σP,t represents the implied volatility of the portfolio, while σi,t are implied volatilities from options on each asset i. All implied volatilities are at-the-money options with the same time-to-maturity. The CIX can also be interpreted as a weighted average of the pair wise correlations among the portfolio asset returns. This interpretation allows for the use of correlation as an indicator of market crisis, i.e., during a crisis all correlations increase simultaneously leading to large CIX values. The CBOE employs this methodology to compute the CIX (see Exchange (2009)) using the 50 largest components of the S&P 500 PS index where wi = 50 i i , Pi is the price and Si is the outstanding float-adjusted shares of index component i. i = 1 Pi Si
3. Model and properties We build the SCP as per Teng et al. (2016), using a Vasicek model Xt and a mapping
d
t
2 t )((
= (1
(
artanh( t ))
t
2) dt
+ dWt ), 0
t
t
= f (Xt ) with f (x ) = tanh(x ) .
T
(3)
where 0 ( 1, 1), κ > 0, ς > 0, and Wt represents a Brownian motion. In our CSV model next, we introduce stochastic volatility to the process ρt in Equation (3). Definition 2 (CSV model).
dXt = ( d t= ( d WX, W
Xt ) dt + t ) dt + t
=
t dWt
X
t dWt
1 dt
(4)
Here η is the long-run equilibrium level of the transformed CIX (Xt), similarly ϑ is the reversion rate and t is the instantaneous volatility of Xt. On the other hand, θ is the long-run level of t , while κ is the speed of reversion and ς is the volatility of t (vol of vol). Furthermore, WtX and Wt are correlated Brownian motions. Under Feller condition 2κθ > ς2, the process t is strictly positive. Although the process in (3) displays what is called dynamic ”local” volatility, in Section 4.1, we empirically demonstrates that this is not sufficient to explain the heteroscedastic behaviour of the CIX. Next we rely on the Euler-Maruyama method to perform a discretization of the CSV model. The following is some useful notation first: t0 is the initial time; T is the horizon; N is the number of simulation steps; and t = T /N , Δt > 0 is the partition size with points ti = i t , i = 0, 1, …, N . Here we choose t 0 = 0, T = 2 and N = 500 . Since the correlation between WtX and Wt is ξ1, we can rewrite the Eq. (4) as follows:
dXt = ( d
t
= (
where Wt and
Wt1
Xt ) dt + t ) dt
+
t
( 1 dWt +
1
2 1 1 dWt )
t dWt
(5)
are independent Brownian motions. 2
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Algorithm 1 (Simulation of correlation (4)). For given values of , , , produce = ( 0 , 1, …, T ) via the following equation: ti + 1
=
ti
+ (
max(0,
ti ))
t+
max(0,
ti )
0,
we rely on Lord et al. (2010)’s full truncation method to
Wti , i = 0, 1, ….,N
(6)
where Wti , i =0, 1, ….,N are iid normal random variables with a mean of 0 and variance Δt.Given ϑ, η, X0, we use the EulerMaruyama method to simulate the underlying process over [0, T]:
Xti + 1 = Xti + (
Xti ) t +
max(0,
for i = 0, 1, ….,N
ti ) ( 1
Wti +
1
2 1
Wt1i ),
(7)
i = 0, 1, ….,N are iid normal r.v. with a mean of 0 and variance Δt, and Wt1i is independent of Wti .Next we apply the where mapping f (x ) = tanh (x ) to Xt, leading to: Wt1i ,
ti
= f (Xti ) = tanh (Xti ), i = 0, 1, ….,N
(8)
For exemplary purposes, we choose the value of parameters , , , 0 as in Heston (1993); see Table 1. We keep 0 = for varying θ and X0 = for varying η to identify the mean reversion in our figures. Plots in Fig. 1 illustrate the process of correlation with varying κ, θ and ς respectively. 4. Empirical study In this section, we work with daily CIX data from CBOE. Heteroscedastic behaviour is tested first, and then ordinary least squares is used for partial estimation which is numerically shown to provide consistent estimators. The CIX data is displayed in Fig. 2 and the values are always in (0,1). A higher CIX suggests a type of herd behavior associated with crises, which has been discussed in Dhaene et al. (2012b). Note that the CIX almost exceeded the upper bound of 1 during the2008 s ”Great Recession”. Moreover, it was steadily greater than 0.5 for 8 years after the crisis (December 2016), while falling from 2017 to mid 2019. 4.1. Estimation Next, we estimate the parameters directly related to the observable data (the CIX) using a discretization and least squares. We exclude parameters driving the hidden volatility process as a full estimation exercise would not only be challenging because of the hidden nature of the SV process and the lack of closed-form Fourier transforms, but it would also be prone to inaccuracies given the small sample sizes and the available highly numerical estimation methods (eg. extended Kalman Filter, sequential Monte Carlo for the likelihood, approximate Bayesian). As the variance of the disturbance is heteroscedastic, we rely on the theoretical consistency of the least squares estimator in the presence of ARCH errors; see Pantula (1988). We also confirm the consistency of the estimators via simulations. The autoregressive discretization of Eq. (5) can be written in the following general form:
Y=
0 1N
+
1X
(1)
where Xt = arctanh(CIXt), 0 1
= =1
(9)
+ is a constant volatility, 1N is an N-vector of ones, ϵ is an error vector, and
t (10)
t
Y
= [ Xt 1 , Xt 2 ,
, Xt N ]
X (1)
= [ X t 0 , X t 1,
, X tN 1 ]
Z t 0,
=[
Zt1,
,
ZtN 1 ]
(11)
Table 2 presents the results of an ARCH Lagrange multiplier test with a lag of 12 on the residuals of our model. The ARCH test (pvalues less than 0.05) confirms heteroscedasticity in the time series of the CIX4 The least squares approach above leads to the following estimators:
^ = ^ =
^
1
1
t ^
0
^ t
=
^
0
1
(12)
^
1
4 Fewer lags with Likelihood Ratio Test and Ljung-Box Q-test also confirmed ARCH effect. We also tried fitting an ARMA(p,k) to the raw CIX instead of the AR(1) in eq. (9) with no improvement in the fitting.
3
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Table 1 Default parameters for simulation of ρt on CSV model. Parameter
Value
Initial variance Speed of reversion of volatility parameter Long term mean level of the squared volatility parameter Volatility of the squared volatility parameter Initial value of the underlying process Long term mean level of the underlying process Speed of reversion of the underlying process
= 0.01 =2 = 0.01 = 0.1 X0 = 0.3 = 0.3 = 1.1 1 = 0.7
Correlation of WtX and Wt
0
Fig. 1. Correlation process ρt on CSV model for three varying volatility parameters: a) Speeds of reversion (κ), b) Long term mean level (θ) and c) Volatilities of volatility parameter (ς).
, i.e. the estimation of θ can be derived Since follows a mean-reverting process with mean θ, we can estimate θ via the constant ^ Var( ) by = t . The estimation results are listed in Table 3. Next we numerically demonstrate the consistency of our estimators. The methodology is described in Algorithm 2.
Fig. 2. S& P 500 Implied Correlation Indexes between 2007-01-03 and 2019-09-09 released by CBOE. Symbols KCJ, ICJ, and JCJ are cycled as time elapses. 4
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Table 2 Table of ARCH effects for residuals in Eq. (9). Year
χ2 value
df
2009
313.04
12
2010
52.181
2011 2013 2014
210.04
2016
68.757
2018 2019 2020
< 2.2*10
16
12
12 12
2.052*10 0.01416 2.312*10
5
< 2.2*10
16
12
5.71*10
12 12 12
5.467*10 10 0.0002211 0.0001094 < 2.2*10 16
12
40.602
2017
16
7
12
25.144 42.939
2015
< 2.2*10
12
81.589
12
37.028 38.898 126.13
ARCH effects
5.759*10
12
161.87
2012
P value
5
Y Y Y Y
Y Y Y Y Y
Y Y Y
df degree of freedom, Y means ARCH effects.
Algorithm 2 (Algorithm for consistency of estimators). Fix parameters κ, ς and ξ1, as well as the initial values 0 and X0.5.Assume parameters ϑ, η and θ are known; (i.e. from Table 4); we take = 5.75, = 0.783 and = 0.56.Simulate vectors ( t(1i) , t(2i ) , …, t(Ni ) ) and (Xt(1i) , Xt(2i) , …, Xt(Ni) ), i = 1, …, M .For each path i {1, 2, 3, .,M }, perform a least squares on (Xt(1i) , Xt(2i) , …, Xt(Ni) ) and derive the triple (i ) (i ) (i ) (i ) (i ) (i ) (^ , ^ , ^ ).Repeat the estimation M times: (^ , ^ , ^ ), i = 1, …, M .Calculate the mean of squared errors: (N )
(N )
(N )
(N )
( i) (^ ( N )
MSE ( ^ ) : =
M i=1
MSE ( ^) : =
^ (i) M ( (N ) i=1 M
MSE ( ^) : =
M i=1
(N )
(N )
)2
M
(i ) (^
(N )
)2
(13)
)2
M
for N =500, 1000, 1500, 2000, 3000, 4000, 5000 and 10000.
^ ^ Fig. 3 plots the MSE as a function of N. It can be seen that MSE ( ), MSE ( ^), and MSE ( ) approach zero as N becomes large, confirming the consistency of estimators for (ϑ, η, θ). We also tried other values of κ, ς and ξ1 with similar convergence to zero in N. 4.2. Pricing We design two European digital options on the CIX with the same maturity T, and we price them with our CSV model (4) via Monte Carlo. The correlation between stocks is known to be higher when the market goes down, see Guillaume and Linders (2015). This notion is also in line with previous empirical studies (see, for instance, Dhaene et al. (2012a), Rubbaniy et al. (2014) and Skintzi and Refenes (2005)). Furthermore, this particular phenomenon can be observed directly with the CIX historical data; see Fig. 2. The CIX hit its upper bound 1 with the 2008 financial crisis, and it spiked again during the2010 s ”European sovereign debt crisis” and the fear of contagion from such crisis in 2011. High correlations reduce the benefits of diversification (see Linders and Schoutens (2014)), exposing investors to random market conditions; and hence additional losses during crises. Investors can use this connection to hedge crisis risk. For instance, if the CIX exceeds a critical threshold, then holders can execute the option as an insurance. The critical threshold of the CIX is set to 0.9 as that is a level only achieved during the 2008 financial crisis. In general such a strike value could be lower depending on risk aversion. Our derivatives are chosen in the spirit of credit structural models (see Black and Cox (1976)). Here we use the CIX as a measure of market crisis levels and a high historical value as the indicator of crisis conditions. The first digital CIX option is designed for investors who worry about the financial crisis happening at maturity, while the second digital CIX option works for investors who consider that the crisis could arise over the lifetime of the contract:
1{CIXT
0 . 9} :
=
1 if CIXT 0.9 0 otherwise
(14)
1{CIXT
0 . 9} :
=
1 if CIXT 0.9 0 otherwise
(15)
where CIXT = max 0 5
t T CIXt
is the maximum CIX over the period [0, T]. Based on no arbitrage principles, the price of the options can
They play no role on ensuring consistency of (ϑ, η and θ) 5
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Table 3 Estimates for CSV parameters (daily unit) with T = 2, t = 0.004, r = 2% . SE is standard error. CIX
^
SE( ^0 )
^
SE(
KCJ2009 ICJ2010 JCJ2011 KCJ2012 ICJE2013 JCJ2014 KCJ2015 ICJ2016 JCJ2017 KCJ2018 ICJ2019 JCJ2020
0.311 0.027 0.091 0.086 0.028 0.018 0.025 0.054 0.027 0.003 0.019 0.087
0.026 0.009 0.017 0.017 0.010 0.008 0.009 0.012 0.010 0.004 0.006 0.012
0.448 0.959 0.882 0.898 0.965 0.977 0.961 0.918 0.958 0.992 0.959 0.805
0.041 0.013 0.022 0.020 0.01248 0.010 0.014 0.019 0.015 0.007 0.013 0.027
0
1
^
1)
^
^
^
138 10.25 29.5 25.5 8.75 5.75 9.75 20.5 10.5 2 10.188 48.808
0.563 0.659 0.771 0.843 0.8 0.783 0.641 0.659 0.647 0.375 0.469 0.444
18.496 0.287 0.610 1.26 0.400 0.561 0.245 0.385 0.304 0.195 0.252 0.253
Table 4 Default parameters for pricing of the CSV model. Parameter
Value
Initial squared volatility Speed of reversion of the squared volatility parameter Long term mean level of the squared volatility parameter Volatility of the squared volatility parameter Initial value of the underlying process Long term mean level of the underlying process Speed of reversion of the underlying process
= 0.56 = 0.6 = 0.56 = 0.1 X0 = 0.783 = 0.783 = 5.75 1 = 0.1
Correlation of WtX and Wt Risk free rate Time to maturity Number of simulation steps Number of trajectories Benefit (dollars)
0
r = 0.02 T=2 N = 1000 M = 50000 L = 1000
Fig. 3. The mean squared errors (MSE) of three parameters in the CSV model a the speed of reversion of Xt (ϑ), b Long term mean level of Xt (η), and c long term mean level of variance (θ).
be represented as an expectation under a risk-neutral measure.
Price1 =
[e
rT ·L · 1 {CIXT 0 . 9} ]
(16)
Price2 =
[e
rT ·L · 1 {CIXT 0 . 9} ]
(17)
We set the risk-free rate to r = 2% (average yield for the period 2007–2018), while the benefit/principal of the option is set to L = 1000 and time to maturity T = 2 . The default values of the parameters are estimated from the same data set -JCJ 2014, see Section 4.1. All default parameters are listed in Table 4. Furthermore, we use the Monte Carlo method to perform a sensitive analysis on parameters that were not estimated (i.e., κ, ς). In our model, the level of the CIX is affected by two groups of parameters. One is embedded in the underlying process Xt consisting 6
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Fig. 4. The digital option prices simulated in the CSV model with three varying parameters: the speed of reversion of volatility parameter (κ), the volatility of volatility parameter (ς) and the correlation between two Brownian motions (ξ1). a) κ vs. Price1, b) ς vs. Price1, c) ξ1 vs. Price1, d) κ vs. Price2, e) ς vs. Price2, and f) ξ1 vs. Price2.
Fig. 5. The digital option prices simulated in the CSV model with three varying parameters: long-run variance (θ); speed of reversion of underlying process (ϑ) and mean reversion of underlying process (η). g) θ vs. Price1, h) ϑ vs. Price1, i) η vs. Price1, j) θ vs. Price2, k) ϑ vs. Price2, and l) η vs. Price2.
of the following: ϑ, η, and ξ1. The other group contains the stochastic volatility parameters κ, θ, and ς. To study sensitivities, we vary each parameter to observe its effect on prices. In this exercise we keep 0 = and X0 = . Fig. 4 illustrates Price1 and Price2 in the CSV model with varying κ, ς, and ξ1 respectively. As expected Price2 is more expensive than Price1 because of more opportunities to claim the insurance. From panels a) and d), we observe that both Price1 and Price2 decrease with more frequent reversions to long-term volatilities (increase in κ) by 16% and 3% respectively. On the other hand, higher volatility of volatility (ς) leads to an increase of 37% for Price1 and 7% for Price2, see panels b) and e). Interestingly, panels c) and f) indicate significant increases in price with the correlation parameter (ξ); Price1 goes up 80%, and Price2 increases 70%. Fig. 5 depicts the option prices with regard to θ, ϑ and η. Since θ is the long-run variance and η is the mean reversion level of Xt, both can have significant impacts on option prices, this is confirmed by panels g), i), j), and l. Price1 and Price2 remain around zero for small θ or η; however they could increase rapidly. Moreover, from panels h) and k) we see the substantial impact of ϑ with Price1 and Price2 dropping over 80% as ϑ increases. 5. Conclusions We modeled the CIX by considering a SCP with stochastic volatilities. This was motivated by detecting heteroscedasticity in the time series of the CIX, a new stylized fact. The CSV model (4) is the combination of well-known underlying processes and a proper mapping function. In the empirical study, we analyzed the CIX historical data. The ARCH test revealed that all CIX series have heteroscedasticity. We estimated partial parameters in the CSV model by ordinary least squares. In addition, we numerically demonstrate the estimators to be consistent. We also motivated and priced two types of digital CIX options aiming at protection/insurance from crises. We observed that parameters κ, θ, ξ1 and ς can have a significant influence (i.e., 60%) on option prices. Therefore the use of stochastic volatilities in the modeling of the CIX is strongly recommended. 7
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Supplementary material Supplementary material associated with this article can be found, in the online version, at doi:10.1016/j.frl.2019.101309. References Black, F., Cox, J.C., 1976. Valuing corporate securities: some effects of bond indenture provisions. J Finance 31.2, 351–367. Dhaene, J., Dony, J., Forys, M., Linders, D., Schoutens, W., 2012. FIX the Fear Index: Measuring Market Fear. In: Cummins, M., Murphy, F., Miller, J. (Eds.), Topics in Numerical Methods for Finance. Springer Proceedings in Mathematics & Statistics, New York, NY. Dhaene, J., W., L.D.S., Vyncke, D., 2012. The herd behavior index: a new measure for the implied degree of co-movement in stock markets. Insurance: Math. Econ. 50 (3), 357–370. van Emmerich, C., 2006. Modelling correlation as a stochastic process. Preprint 6 (03). Exchange, C.B.O., 2009. Cboe s&p 500 implied correlation index. Technical report. Working Paper Faria, G. A., Kosowski, R., Wang, T., 2018. he correlation risk premium: International evidence. Available at SSRN 3276601. Gouriéroux, C., 2006. Continuous time wishart process for stochastic risk. Econ. Rev. 25 (2–3), 177–217. Guillaume, F., Linders, D., 2015. Stochastic modelling of herd behaviour indices. Quant. Finance 15 (12), 1963–1977. Heston, S.L., 1993. A closed-form solution for options with stochastic volatility with applications to bond and currency options. Rev. Financ. Stud 6 (2), 327–343. Linders, D., Schoutens, W., 2014. A framework for robust measurement of implied correlation. J. Comput. Appl. Math. 271, 39–52. Lord, R., Koekkoek, R., Dijk, D.V., 2010. A comparison of biased simulation schemes for stochastic volatility models. Quant. Finance 10 (2), 177–194. Ma, J., 2009. A stochastic correlation model with mean reversion for pricing multi-asset options. Asia-Pacific Financ. Markets 16 (2), 97–109. Markopoulou, C., Skintzi, V., Refenes, A., 2016. On the predictability of model-free implied correlation. Int. J. Forecast. 32 (2), 527–547. Pantula, S.G., 1988. Estimation of autoregressive models with arch errors. Sankhy 119–138. Rubbaniy, G., Asmerom, R., Rizvi, S.K.A., Naqvi, B., 2014. Do fear indices help predict stock returns? Quant. Finance 14 (5), 831–847. Salvi, G., Swishchuk, A.V., 2014. Covariance and correlation swaps for financial markets with markov-modulated volatilities. Int. J. Theor. Appl. Finance 17 (01), 1450006. Skintzi, V.D., Refenes, A.P.N., 2005. Implied correlation index: a new measure of diversification. J. Futures Markets 25 (2), 171–197. Teng, L., Van Emmerich, C., Ehrhardt, M., Günther, M., 2016. A versatile approach for stochastic correlation using hyperbolic functions. Int. J. Comput. Math. 93 (3), 524–539.
8
Contents lists available at ScienceDirect
Finance Research Letters journal homepage: www.elsevier.com/locate/frl
Stochastic volatility models for the implied correlation index. Evidence, properties and pricing Marcos Escobar ,a, Lin Fang1,b ⁎
a b
Associate Professor, Department of Statistical and Actuarial ScienceS, Western University, Canada Department of Statistical and Actuarial Sciences, Western University, Canada
ARTICLE INFO
ABSTRACT
Keywords: Implied correlation index Stochastic volatility models Mean reversion Correlation derivatives
This paper studies the implied correlation index (CIX), revealing a new stylized fact: heteroscedasticity in correlation. A correlation stochastic volatility (C-SV) model is proposed and a consistent estimation methodology is implemented on CBOE S&P 500 CIX historical data. The impact of the SV parameters is studied for two types of crisis-motivated CIX derivatives, and the empirical study demonstrates that new parameters can have a significant influence of up to 60% on digital option prices.
1. Introduction Since 2009, the Chicago Board Options Exchange (CBOE) has been releasing daily values of the S& P 500 implied correlation index (CIX) as a measure of the market’s expectation on future correlations; see Exchange (2009). The CIX, proposed by Skintzi and Refenes (2005), is defined as the ratio of the sum of the weighted covariance to that of the weighted variance between underlying assets. Stochastic correlation processes (SCPs) are natural ways of modeling the CIX. The study of SCPs has attracted some attention, mostly from the perspective of stochastic covariance models; see Gouriéroux (2006). The works of van Emmerich (2006) and Ma (2009) are two of a number of studies that treat correlation as a stand-alone process in continuous-time2. In van Emmerich (2006) and Teng et al. (2016), the authors constructed SCP models via a convenient function of well-known processes. This general approach benefits from the ease of construction and a better degree of analytical tractability. van Emmerich (2006) also introduced a second approach where the SCP is described directly by Jacobi processes. In this direction, a modified Jacobi process was proposed in Ma (2009) where boundaries for correlation are more flexible than 1 and 1. Stochastic correlation process modelling ensures a critical stylized fact of stochastic correlations: the mean-reverting property. However, the volatility is assumed to be constant. In this paper, we provide empirical evidence of heteroscedasticity in the time series of the CIX; in other words, CIX exhibits timevarying volatility and periods of swings mixed with periods of relative calm (volatility clustering). Furthermore, we propose a correlation stochastic volatility (CSV) model, which can better describe data along the lines of Teng et al. (2016)3. We assume that the variance follows a Cox-Ingersoll-Ross (CIR) process, which is a solid choice in finance (see Heston (1993)). We also study the impact of this model on two types of digital options on the CIX. This is motivated by two facts, first derivatives on correlations have been of interest over the past few years; see Salvi and Swishchuk (2014) and Faria et al. (2018). Secondly, it is recognized that in times of
Corresponding author. E-mail addresses: [email protected] (M. Escobar), [email protected] (L. Fang). 1 Tel.: +1 226-998-1966 2 see Markopoulou et al. (2016) and literature therein for discrete time analyses 3 A second model based on Ma (2009) was explored with similar results ⁎
https://doi.org/10.1016/j.frl.2019.101309 Received 16 January 2019; Received in revised form 13 September 2019; Accepted 6 October 2019 1544-6123/ © 2019 Elsevier Inc. All rights reserved.
Please cite this article as: Marcos Escobar and Lin Fang, Finance Research Letters, https://doi.org/10.1016/j.frl.2019.101309
Finance Research Letters xxx (xxxx) xxxx
M. Escobar and L. Fang
market stress, the correlation increases as stock prices tend to move together; investors can use our digital options to hedge the risk of a financial crisis. The main contributions of our paper are as follows:
• Detect heteroscedasticity on the CBOE S&P 500 CIX, this represents a new stylized fact of financial data. • Introduce a CSV model in which the variance follows a CIR process. • Perform discretization and partial estimations of our new model, numerically demonstrating the consistency of the estimators. • Price two types of crisis-motivated CIX derivatives, detecting a significant influence of the parameters in the stochastic volatility process.
2. About the implied correlation index Let us consider a portfolio of N assets. At time t, the portfolio variance N 2 P, t
2 P, t
can be calculated as follows:
N 1
wi2, t
=
2 i, t
+2
i=1
wi, t wj, t
ij, t i, t j, t ,
0
t
T
(1)
i =1 j>i
where wi, t is the weight of asset i on the portfolio at time t, σi,t is the volatility of asset i at time t, and ρij,t is the pairwise correlation between assets i and j at time t. Definition 1 (CIX formula, see Skintzi and Refenes (2005)). The CIX (ρt) is the correlation value holding for any pair of assets in the portfolio ( t = ij, t ):
CIXt =
t
=
2 P, t N 1 i=1
2
N i=1 j>i
wi2, t
wi, t wj, t
2 i, t
(2)
i, t j , t
Here σP,t represents the implied volatility of the portfolio, while σi,t are implied volatilities from options on each asset i. All implied volatilities are at-the-money options with the same time-to-maturity. The CIX can also be interpreted as a weighted average of the pair wise correlations among the portfolio asset returns. This interpretation allows for the use of correlation as an indicator of market crisis, i.e., during a crisis all correlations increase simultaneously leading to large CIX values. The CBOE employs this methodology to compute the CIX (see Exchange (2009)) using the 50 largest components of the S&P 500 PS index where wi = 50 i i , Pi is the price and Si is the outstanding float-adjusted shares of index component i. i = 1 Pi Si
3. Model and properties We build the SCP as per Teng et al. (2016), using a Vasicek model Xt and a mapping
d
t
2 t )((
= (1
(
artanh( t ))
t
2) dt
+ dWt ), 0
t
t
= f (Xt ) with f (x ) = tanh(x ) .
T
(3)
where 0 ( 1, 1), κ > 0, ς > 0, and Wt represents a Brownian motion. In our CSV model next, we introduce stochastic volatility to the process ρt in Equation (3). Definition 2 (CSV model).
dXt = ( d t= ( d WX, W
Xt ) dt + t ) dt + t
=
t dWt
X
t dWt
1 dt
(4)
Here η is the long-run equilibrium level of the transformed CIX (Xt), similarly ϑ is the reversion rate and t is the instantaneous volatility of Xt. On the other hand, θ is the long-run level of t , while κ is the speed of reversion and ς is the volatility of t (vol of vol). Furthermore, WtX and Wt are correlated Brownian motions. Under Feller condition 2κθ > ς2, the process t is strictly positive. Although the process in (3) displays what is called dynamic ”local” volatility, in Section 4.1, we empirically demonstrates that this is not sufficient to explain the heteroscedastic behaviour of the CIX. Next we rely on the Euler-Maruyama method to perform a discretization of the CSV model. The following is some useful notation first: t0 is the initial time; T is the horizon; N is the number of simulation steps; and t = T /N , Δt > 0 is the partition size with points ti = i t , i = 0, 1, …, N . Here we choose t 0 = 0, T = 2 and N = 500 . Since the correlation between WtX and Wt is ξ1, we can rewrite the Eq. (4) as follows:
dXt = ( d
t
= (
where Wt and
Wt1
Xt ) dt + t ) dt
+
t
( 1 dWt +
1
2 1 1 dWt )
t dWt
(5)
are independent Brownian motions. 2
Finance Research Letters xxx (xxxx) xxxx
M. Escobar and L. Fang
Algorithm 1 (Simulation of correlation (4)). For given values of , , , produce = ( 0 , 1, …, T ) via the following equation: ti + 1
=
ti
+ (
max(0,
ti ))
t+
max(0,
ti )
0,
we rely on Lord et al. (2010)’s full truncation method to
Wti , i = 0, 1, ….,N
(6)
where Wti , i =0, 1, ….,N are iid normal random variables with a mean of 0 and variance Δt.Given ϑ, η, X0, we use the EulerMaruyama method to simulate the underlying process over [0, T]:
Xti + 1 = Xti + (
Xti ) t +
max(0,
for i = 0, 1, ….,N
ti ) ( 1
Wti +
1
2 1
Wt1i ),
(7)
i = 0, 1, ….,N are iid normal r.v. with a mean of 0 and variance Δt, and Wt1i is independent of Wti .Next we apply the where mapping f (x ) = tanh (x ) to Xt, leading to: Wt1i ,
ti
= f (Xti ) = tanh (Xti ), i = 0, 1, ….,N
(8)
For exemplary purposes, we choose the value of parameters , , , 0 as in Heston (1993); see Table 1. We keep 0 = for varying θ and X0 = for varying η to identify the mean reversion in our figures. Plots in Fig. 1 illustrate the process of correlation with varying κ, θ and ς respectively. 4. Empirical study In this section, we work with daily CIX data from CBOE. Heteroscedastic behaviour is tested first, and then ordinary least squares is used for partial estimation which is numerically shown to provide consistent estimators. The CIX data is displayed in Fig. 2 and the values are always in (0,1). A higher CIX suggests a type of herd behavior associated with crises, which has been discussed in Dhaene et al. (2012b). Note that the CIX almost exceeded the upper bound of 1 during the2008 s ”Great Recession”. Moreover, it was steadily greater than 0.5 for 8 years after the crisis (December 2016), while falling from 2017 to mid 2019. 4.1. Estimation Next, we estimate the parameters directly related to the observable data (the CIX) using a discretization and least squares. We exclude parameters driving the hidden volatility process as a full estimation exercise would not only be challenging because of the hidden nature of the SV process and the lack of closed-form Fourier transforms, but it would also be prone to inaccuracies given the small sample sizes and the available highly numerical estimation methods (eg. extended Kalman Filter, sequential Monte Carlo for the likelihood, approximate Bayesian). As the variance of the disturbance is heteroscedastic, we rely on the theoretical consistency of the least squares estimator in the presence of ARCH errors; see Pantula (1988). We also confirm the consistency of the estimators via simulations. The autoregressive discretization of Eq. (5) can be written in the following general form:
Y=
0 1N
+
1X
(1)
where Xt = arctanh(CIXt), 0 1
= =1
(9)
+ is a constant volatility, 1N is an N-vector of ones, ϵ is an error vector, and
t (10)
t
Y
= [ Xt 1 , Xt 2 ,
, Xt N ]
X (1)
= [ X t 0 , X t 1,
, X tN 1 ]
Z t 0,
=[
Zt1,
,
ZtN 1 ]
(11)
Table 2 presents the results of an ARCH Lagrange multiplier test with a lag of 12 on the residuals of our model. The ARCH test (pvalues less than 0.05) confirms heteroscedasticity in the time series of the CIX4 The least squares approach above leads to the following estimators:
^ = ^ =
^
1
1
t ^
0
^ t
=
^
0
1
(12)
^
1
4 Fewer lags with Likelihood Ratio Test and Ljung-Box Q-test also confirmed ARCH effect. We also tried fitting an ARMA(p,k) to the raw CIX instead of the AR(1) in eq. (9) with no improvement in the fitting.
3
Finance Research Letters xxx (xxxx) xxxx
M. Escobar and L. Fang
Table 1 Default parameters for simulation of ρt on CSV model. Parameter
Value
Initial variance Speed of reversion of volatility parameter Long term mean level of the squared volatility parameter Volatility of the squared volatility parameter Initial value of the underlying process Long term mean level of the underlying process Speed of reversion of the underlying process
= 0.01 =2 = 0.01 = 0.1 X0 = 0.3 = 0.3 = 1.1 1 = 0.7
Correlation of WtX and Wt
0
Fig. 1. Correlation process ρt on CSV model for three varying volatility parameters: a) Speeds of reversion (κ), b) Long term mean level (θ) and c) Volatilities of volatility parameter (ς).
, i.e. the estimation of θ can be derived Since follows a mean-reverting process with mean θ, we can estimate θ via the constant ^ Var( ) by = t . The estimation results are listed in Table 3. Next we numerically demonstrate the consistency of our estimators. The methodology is described in Algorithm 2.
Fig. 2. S& P 500 Implied Correlation Indexes between 2007-01-03 and 2019-09-09 released by CBOE. Symbols KCJ, ICJ, and JCJ are cycled as time elapses. 4
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M. Escobar and L. Fang
Table 2 Table of ARCH effects for residuals in Eq. (9). Year
χ2 value
df
2009
313.04
12
2010
52.181
2011 2013 2014
210.04
2016
68.757
2018 2019 2020
< 2.2*10
16
12
12 12
2.052*10 0.01416 2.312*10
5
< 2.2*10
16
12
5.71*10
12 12 12
5.467*10 10 0.0002211 0.0001094 < 2.2*10 16
12
40.602
2017
16
7
12
25.144 42.939
2015
< 2.2*10
12
81.589
12
37.028 38.898 126.13
ARCH effects
5.759*10
12
161.87
2012
P value
5
Y Y Y Y
Y Y Y Y Y
Y Y Y
df degree of freedom, Y means ARCH effects.
Algorithm 2 (Algorithm for consistency of estimators). Fix parameters κ, ς and ξ1, as well as the initial values 0 and X0.5.Assume parameters ϑ, η and θ are known; (i.e. from Table 4); we take = 5.75, = 0.783 and = 0.56.Simulate vectors ( t(1i) , t(2i ) , …, t(Ni ) ) and (Xt(1i) , Xt(2i) , …, Xt(Ni) ), i = 1, …, M .For each path i {1, 2, 3, .,M }, perform a least squares on (Xt(1i) , Xt(2i) , …, Xt(Ni) ) and derive the triple (i ) (i ) (i ) (i ) (i ) (i ) (^ , ^ , ^ ).Repeat the estimation M times: (^ , ^ , ^ ), i = 1, …, M .Calculate the mean of squared errors: (N )
(N )
(N )
(N )
( i) (^ ( N )
MSE ( ^ ) : =
M i=1
MSE ( ^) : =
^ (i) M ( (N ) i=1 M
MSE ( ^) : =
M i=1
(N )
(N )
)2
M
(i ) (^
(N )
)2
(13)
)2
M
for N =500, 1000, 1500, 2000, 3000, 4000, 5000 and 10000.
^ ^ Fig. 3 plots the MSE as a function of N. It can be seen that MSE ( ), MSE ( ^), and MSE ( ) approach zero as N becomes large, confirming the consistency of estimators for (ϑ, η, θ). We also tried other values of κ, ς and ξ1 with similar convergence to zero in N. 4.2. Pricing We design two European digital options on the CIX with the same maturity T, and we price them with our CSV model (4) via Monte Carlo. The correlation between stocks is known to be higher when the market goes down, see Guillaume and Linders (2015). This notion is also in line with previous empirical studies (see, for instance, Dhaene et al. (2012a), Rubbaniy et al. (2014) and Skintzi and Refenes (2005)). Furthermore, this particular phenomenon can be observed directly with the CIX historical data; see Fig. 2. The CIX hit its upper bound 1 with the 2008 financial crisis, and it spiked again during the2010 s ”European sovereign debt crisis” and the fear of contagion from such crisis in 2011. High correlations reduce the benefits of diversification (see Linders and Schoutens (2014)), exposing investors to random market conditions; and hence additional losses during crises. Investors can use this connection to hedge crisis risk. For instance, if the CIX exceeds a critical threshold, then holders can execute the option as an insurance. The critical threshold of the CIX is set to 0.9 as that is a level only achieved during the 2008 financial crisis. In general such a strike value could be lower depending on risk aversion. Our derivatives are chosen in the spirit of credit structural models (see Black and Cox (1976)). Here we use the CIX as a measure of market crisis levels and a high historical value as the indicator of crisis conditions. The first digital CIX option is designed for investors who worry about the financial crisis happening at maturity, while the second digital CIX option works for investors who consider that the crisis could arise over the lifetime of the contract:
1{CIXT
0 . 9} :
=
1 if CIXT 0.9 0 otherwise
(14)
1{CIXT
0 . 9} :
=
1 if CIXT 0.9 0 otherwise
(15)
where CIXT = max 0 5
t T CIXt
is the maximum CIX over the period [0, T]. Based on no arbitrage principles, the price of the options can
They play no role on ensuring consistency of (ϑ, η and θ) 5
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M. Escobar and L. Fang
Table 3 Estimates for CSV parameters (daily unit) with T = 2, t = 0.004, r = 2% . SE is standard error. CIX
^
SE( ^0 )
^
SE(
KCJ2009 ICJ2010 JCJ2011 KCJ2012 ICJE2013 JCJ2014 KCJ2015 ICJ2016 JCJ2017 KCJ2018 ICJ2019 JCJ2020
0.311 0.027 0.091 0.086 0.028 0.018 0.025 0.054 0.027 0.003 0.019 0.087
0.026 0.009 0.017 0.017 0.010 0.008 0.009 0.012 0.010 0.004 0.006 0.012
0.448 0.959 0.882 0.898 0.965 0.977 0.961 0.918 0.958 0.992 0.959 0.805
0.041 0.013 0.022 0.020 0.01248 0.010 0.014 0.019 0.015 0.007 0.013 0.027
0
1
^
1)
^
^
^
138 10.25 29.5 25.5 8.75 5.75 9.75 20.5 10.5 2 10.188 48.808
0.563 0.659 0.771 0.843 0.8 0.783 0.641 0.659 0.647 0.375 0.469 0.444
18.496 0.287 0.610 1.26 0.400 0.561 0.245 0.385 0.304 0.195 0.252 0.253
Table 4 Default parameters for pricing of the CSV model. Parameter
Value
Initial squared volatility Speed of reversion of the squared volatility parameter Long term mean level of the squared volatility parameter Volatility of the squared volatility parameter Initial value of the underlying process Long term mean level of the underlying process Speed of reversion of the underlying process
= 0.56 = 0.6 = 0.56 = 0.1 X0 = 0.783 = 0.783 = 5.75 1 = 0.1
Correlation of WtX and Wt Risk free rate Time to maturity Number of simulation steps Number of trajectories Benefit (dollars)
0
r = 0.02 T=2 N = 1000 M = 50000 L = 1000
Fig. 3. The mean squared errors (MSE) of three parameters in the CSV model a the speed of reversion of Xt (ϑ), b Long term mean level of Xt (η), and c long term mean level of variance (θ).
be represented as an expectation under a risk-neutral measure.
Price1 =
[e
rT ·L · 1 {CIXT 0 . 9} ]
(16)
Price2 =
[e
rT ·L · 1 {CIXT 0 . 9} ]
(17)
We set the risk-free rate to r = 2% (average yield for the period 2007–2018), while the benefit/principal of the option is set to L = 1000 and time to maturity T = 2 . The default values of the parameters are estimated from the same data set -JCJ 2014, see Section 4.1. All default parameters are listed in Table 4. Furthermore, we use the Monte Carlo method to perform a sensitive analysis on parameters that were not estimated (i.e., κ, ς). In our model, the level of the CIX is affected by two groups of parameters. One is embedded in the underlying process Xt consisting 6
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M. Escobar and L. Fang
Fig. 4. The digital option prices simulated in the CSV model with three varying parameters: the speed of reversion of volatility parameter (κ), the volatility of volatility parameter (ς) and the correlation between two Brownian motions (ξ1). a) κ vs. Price1, b) ς vs. Price1, c) ξ1 vs. Price1, d) κ vs. Price2, e) ς vs. Price2, and f) ξ1 vs. Price2.
Fig. 5. The digital option prices simulated in the CSV model with three varying parameters: long-run variance (θ); speed of reversion of underlying process (ϑ) and mean reversion of underlying process (η). g) θ vs. Price1, h) ϑ vs. Price1, i) η vs. Price1, j) θ vs. Price2, k) ϑ vs. Price2, and l) η vs. Price2.
of the following: ϑ, η, and ξ1. The other group contains the stochastic volatility parameters κ, θ, and ς. To study sensitivities, we vary each parameter to observe its effect on prices. In this exercise we keep 0 = and X0 = . Fig. 4 illustrates Price1 and Price2 in the CSV model with varying κ, ς, and ξ1 respectively. As expected Price2 is more expensive than Price1 because of more opportunities to claim the insurance. From panels a) and d), we observe that both Price1 and Price2 decrease with more frequent reversions to long-term volatilities (increase in κ) by 16% and 3% respectively. On the other hand, higher volatility of volatility (ς) leads to an increase of 37% for Price1 and 7% for Price2, see panels b) and e). Interestingly, panels c) and f) indicate significant increases in price with the correlation parameter (ξ); Price1 goes up 80%, and Price2 increases 70%. Fig. 5 depicts the option prices with regard to θ, ϑ and η. Since θ is the long-run variance and η is the mean reversion level of Xt, both can have significant impacts on option prices, this is confirmed by panels g), i), j), and l. Price1 and Price2 remain around zero for small θ or η; however they could increase rapidly. Moreover, from panels h) and k) we see the substantial impact of ϑ with Price1 and Price2 dropping over 80% as ϑ increases. 5. Conclusions We modeled the CIX by considering a SCP with stochastic volatilities. This was motivated by detecting heteroscedasticity in the time series of the CIX, a new stylized fact. The CSV model (4) is the combination of well-known underlying processes and a proper mapping function. In the empirical study, we analyzed the CIX historical data. The ARCH test revealed that all CIX series have heteroscedasticity. We estimated partial parameters in the CSV model by ordinary least squares. In addition, we numerically demonstrate the estimators to be consistent. We also motivated and priced two types of digital CIX options aiming at protection/insurance from crises. We observed that parameters κ, θ, ξ1 and ς can have a significant influence (i.e., 60%) on option prices. Therefore the use of stochastic volatilities in the modeling of the CIX is strongly recommended. 7
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Supplementary material Supplementary material associated with this article can be found, in the online version, at doi:10.1016/j.frl.2019.101309. References Black, F., Cox, J.C., 1976. Valuing corporate securities: some effects of bond indenture provisions. J Finance 31.2, 351–367. Dhaene, J., Dony, J., Forys, M., Linders, D., Schoutens, W., 2012. FIX the Fear Index: Measuring Market Fear. In: Cummins, M., Murphy, F., Miller, J. (Eds.), Topics in Numerical Methods for Finance. Springer Proceedings in Mathematics & Statistics, New York, NY. Dhaene, J., W., L.D.S., Vyncke, D., 2012. The herd behavior index: a new measure for the implied degree of co-movement in stock markets. Insurance: Math. Econ. 50 (3), 357–370. van Emmerich, C., 2006. Modelling correlation as a stochastic process. Preprint 6 (03). Exchange, C.B.O., 2009. Cboe s&p 500 implied correlation index. Technical report. Working Paper Faria, G. A., Kosowski, R., Wang, T., 2018. he correlation risk premium: International evidence. Available at SSRN 3276601. Gouriéroux, C., 2006. Continuous time wishart process for stochastic risk. Econ. Rev. 25 (2–3), 177–217. Guillaume, F., Linders, D., 2015. Stochastic modelling of herd behaviour indices. Quant. Finance 15 (12), 1963–1977. Heston, S.L., 1993. A closed-form solution for options with stochastic volatility with applications to bond and currency options. Rev. Financ. Stud 6 (2), 327–343. Linders, D., Schoutens, W., 2014. A framework for robust measurement of implied correlation. J. Comput. Appl. Math. 271, 39–52. Lord, R., Koekkoek, R., Dijk, D.V., 2010. A comparison of biased simulation schemes for stochastic volatility models. Quant. Finance 10 (2), 177–194. Ma, J., 2009. A stochastic correlation model with mean reversion for pricing multi-asset options. Asia-Pacific Financ. Markets 16 (2), 97–109. Markopoulou, C., Skintzi, V., Refenes, A., 2016. On the predictability of model-free implied correlation. Int. J. Forecast. 32 (2), 527–547. Pantula, S.G., 1988. Estimation of autoregressive models with arch errors. Sankhy 119–138. Rubbaniy, G., Asmerom, R., Rizvi, S.K.A., Naqvi, B., 2014. Do fear indices help predict stock returns? Quant. Finance 14 (5), 831–847. Salvi, G., Swishchuk, A.V., 2014. Covariance and correlation swaps for financial markets with markov-modulated volatilities. Int. J. Theor. Appl. Finance 17 (01), 1450006. Skintzi, V.D., Refenes, A.P.N., 2005. Implied correlation index: a new measure of diversification. J. Futures Markets 25 (2), 171–197. Teng, L., Van Emmerich, C., Ehrhardt, M., Günther, M., 2016. A versatile approach for stochastic correlation using hyperbolic functions. Int. J. Comput. Math. 93 (3), 524–539.
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