Implied basket correlation dynamics

Wolfgang Karl Härdle; Elena Silyakova

doi:10.1515/strm-2014-1176

Article

Implied basket correlation dynamics

Wolfgang Karl Härdle and Elena Silyakova

Published/Copyright: July 28, 2016

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From the journal Statistics & Risk Modeling Volume 33 Issue 1-2

Abstract

Equity basket correlation can be estimated both using the physical measure from stock prices, and also using the risk neutral measure from option prices. The difference between the two estimates motivates a so-called “dispersion strategy”. We study the performance of this strategy on the German market and propose several profitability improvement schemes based on implied correlation (IC) forecasts. Modelling IC conceals several challenges. Firstly the number of correlation coefficients would grow with the size of the basket. Secondly, IC is not constant over maturities and strikes. Finally, IC changes over time. We reduce the dimensionality of the problem by assuming equicorrelation. The IC surface (ICS) is then approximated from the implied volatilities of stocks and the implied volatility of the basket. To analyze the dynamics of the ICS we employ a dynamic semiparametric factor model.

Keywords: Correlation risk; dimension reduction; dispersion strategy; dynamic factor models

MSC 2010: 62H25; 62H15; 62H20

Funding source: Deutsche Forschungsgemeinschaft

Award Identifier / Grant number: CRC 649 “Economic Risk”

Funding statement: The authors gratefully acknowledge financial support from the Deutsche Forschungsgemeinschaft through CRC 649 “Economic Risk”.

A Switch point selection for correlation regimes

The dependence of ρ and σB observed in RV and RC is not pronounced in case of ATM IV and IC, see Figures 5 and 6. Therefore we propose a market regime correction scheme for the IC dataset. The idea is to find a breakpoint between two segments of a piecewise linear regression of ρt+τ on σB,t+τ. Using the procedure described in [34], we fit a segmented linear regression with one break point. Based on results summarized in Table 9 we make the following state-dependent correction: if σ^B,t⁢(1,τ)>21 (high volatility regime), then ρ^t⁢(κ,τ)=0.0091⁢σ^B,t⁢(κ,τ).

Table 9

Segmented linear regression of ρt+τ on σB,t+τ with one break point,τ=0083,0.25,0.5 for t+τ, from 20100104 till 20120801.

τ	σB,t+τ	ρt+τ	Slope 1	Slope 2
0.083	20.24	0.5917	0.0361	0.0085
0.25	20.34	0.5728	0.0336	0.0093
0.5	22.42	0.6008	0.0286	0.0094
Average	21.00	0.5884	0.0328	0.0091

B Smoothing parameters selection

For both (3.5) and (3.6) kernel bandwidths hμ=(hμ,1,hμ,2)⊤ and hϕ=(hϕ,1,hϕ,2)⊤ are to be selected. As suggested in [28] we use the penalizing function approach to select optimal hμopt, minimizing the mean integrated squared error (MISE):

(B.1)1T⁢∑t=1T1Jt⁢∑j=1Jt{Yt,j-∑l=1LZ^t,l⁢m^l⁢(Xt,j)}2⁢wh*,t⁢(Xt,j)⁢ΞAIC⁢{Wh*,t,j⁢(Xt,j)T⁢Jt},

with the Akaike (1970) information criterion (AIC) as penalizing function ΞAIC⁢(q)=exp⁡(2⁢q) and Wh*,t,j⁢(Xt,j) defined by

Wh*,t,j⁢(Xt,j)=𝒦h⁢(0)Jt-1⁢∑k=1Jt𝒦h⁢(Xt,k-Xt,j)

for every Xt,j, 1≤t≤T, 1≤j≤Jt.

Since the distribution of the observations is very uneven, we use the weighted version of the criterion with weights wh*,t⁢(u¯):=ph*,t-1⁢(u¯), where ph*,t⁢(u¯) is the average design density. For every Xt,j, 1≤t≤T, 1≤j≤Jt it is defined by

ph*,t⁢(Xt,j)=Jt-1⁢∑k=1Jt𝒦h⁢(Xt,k-Xt,j).

The bandwidth hμ⁢AICopt=(hμ⁢1,hμ⁢2)⊤ corresponding to the minimal criterion (B.1) is taken as optimal. The bandwidth h* of the weighting function is constant and does not depend on the choice of hμ.

Acknowledgements

Thanks go to L. Udvarhelyi for editorial assistance.

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Received: 2014-12-25

Revised: 2016-6-14

Accepted: 2016-6-30

Published Online: 2016-7-28

Published in Print: 2016-6-1

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https://doi.org/10.1515/strm-2014-1176

Keywords for this article

Correlation risk; dimension reduction; dispersion strategy; dynamic factor models