The Implied Volatility Index Risk Management Institute National University of Singapore First version: October 6, 8, this version: October 8, 8 Introduction This document describes the formulation and implementation of the 3-day Chinese 5 ETF implied volatility index VI in Risk Management Institute RMI. The Cboe volatility index VIX is colloquially deemed to be the fear gauge in the US market, as it measures the fluctuations of the composite stock index. It was first published in 993, originally determined as the implied volatilities of the at-the-money S&P index options. On September, 3, the methodology was revised to accommodate the out-ofmoney S&P5 option data as introduced in [] and []. This novel approach was also applied to evaluate SSE 5 ETF volatility index in China, first released on June 6, 5, but suspended in February, 8. RMI keeps updating the Chinese 5 ETF volatility index for academic purposes. In the following section, we first investigate the realized variance of the asset returns and its connection with the variance swap, and then provide the formulation of the implied volatility index through the decomposition of the log contract. The implementation in Chinese 5 ETF is outlined subsequently. The last section discusses the implementation issue in Chinese markets. The Realized Variance To formulate VI, we first examine the realized variance of the underlying asset. Suppose under the physical measure P, the price process of the underlying asset {S t : t > } is described as follows: ds t = µ t S t dt + σ t S t dw t,. where µ t and σ t are the drift and the volatility terms, be it deterministic or stochastic; W t is the standard Brownian motion. Equivalently, we have d ln S t = µ t σ t dt + σt dw t. It is easy to verify that the quadratic variation of ln S t over [, T ] is σ t dt. In fact, if µ t and σ t are deterministic, the variance of the log-return R T := ln S T S same as the quadratic variation of ln S t : E RT Var R T = E RT = µ t σ t dt + = σt dt µ t σ t dt will be the σ t dt.. In a more generic sense, without entailing the deterministicity of µ t and σ t, the authors in [7] have shown that under some mild conditions, the realized variance can be asymptotically This documentation is authored by ZHU Xianhao from NUS-RMI Project Team.
approximated by the expectation of the quadratic variation of the log-price process. Define the realized variance of series S ti = t <... < t M = T as Π M T = M i= ln S t i. S ti For M, its expectation satisfies: E ΠM T VarR T E σt dt..3 This motivates us to evaluate the variance swap with the strike Kvol in terms of the quadratic variation, because it would be easier to express the swap value explicitly. The payoff of a variance swap equally partitioned over [, T ] at its maturity is as in [3] and [6] V T = A M Π M T K vol, where A is the annual factor representing the number of days in a year usually 5, 36 or 365 in regard to daily observations and M is the number of days counted for the payoff of the variance swap. From.3, under risk-neutral measure P, the value of swap at t = is approximately determined as V = e r tdt E T σ t dt K vol where r t is the deterministic risk-free interest rate and E is the expectation under P. 3 The Volatility Index The volatility index is the square root of the par strike value rendering V =. At time, it is expressed as T VI; T = E σt T dt 3. Before evaluating VI, we introduce the following property of the log contract. Remark Decomposition of log contracts. The log contract, whose payoff at the maturity is ln S T K, is constructed by buying forward contracts and selling calls and puts see [4], [5] and [6]. More specifically, its payoff can be decomposed to ln S T = S T K K K K where K is an arbitrary positive number. Proof. As for S T K, the right-hand side of 3. is reduced to. K K S T + dk K K S T K + dk 3. S T K K ST K K S T KdK = ln S T K A similar proof is available in the case of S T < K.
Suppose K i K j is an ascending strike sequence with increments K i = K i+ K i. The decomposition in 3. can be discretized as: ln S T S T K K K i K i K i S T + {K i K } K i j K j S T K j + {K j > K } K j. In particular, if we choose K as the at-the-money strike level, the log contract becomes a portfolio consisting of forward contracts and out-of-money OTM calls and puts. e rtdt E ln S T K e rtdt F, T K i P OTM ; K i Ki K i j C OTM ; K j Kj K j 3.3 where F, T is the arbitrage-free forward price at time with maturity T ; P OTM ; K i and C OTM ; K j are OTM call and put options identified by K. With dividend yield q t, 3. can be specified as VI; T = = T = T E ln S T + S ln S e rt qtdt K r t q t dt + σ t dw t E ln S T K ln F, T E T K ln S T, 3.4 K Note the expectation inside 3.4 is the undiscounted log contract value. Therefore, the volatility index is further characterized in terms of options and forward contracts: VI; T T F,T r T K + e t dt P OTM ;K i T i K Ki i + C OTM ;K j j K Kj j 3.5 The first term under the square root in 3.5 comes from Taylor expansion. 4 The Implementation RMI provides 3-day implied volatility index for Chinese 5 ETF on the basis of the end-of-day price. We refer to the methodologies in [] and [8] to process the data and evaluate 3.5. With respect to the data processing, there are nine criteria in [8]. We deal with the market data according to the first eight rules to determine the option premium values, based on the trading volumes and bid/ask/close prices. The ninth rule is currently unexecutable due to technical difficulties in data filtering. The rules employed in our computation are If trading volume is greater than zero, and there are bid/ask prices, and if the trade price is between the last bid and ask quotes, then the option value is the last execution price; otherwise, if the trade price is beyond the bid and ask quotes, the last mid-price is selected as the option premium. If trading volume is greater than zero, but only the last bid ask price is available: the option price is the maximum minimum of the last execution price and the bid ask quote. It can be the spot value S, the forward price F, T, or even a value near them. These criteria are directly translated from [8]. 3
If trading volume is greater than zero, but there are no bid/ ask prices: the last execution price is identified as the option premium. If there is no trading for the day, but there are bid/ask quotes, the option premium is the mid-price. If there is no trading for the day and only the last bid ask is available, then the option premium is the maximum minimum of the last execution price of the previous day and the bid ask price. If there is no trading for the day and both bid/ask quotes are unavailable, the option premium is quantified as the last execution price of the previous day. As for the computation of 3.5, F, T is implied from the put-call parity at a strike level with the minimum absolute difference between the call and put prices see []. K is the largest market strike satisfying K F, T. The interest rate is captured from RMI Shibor curve. The market observations generally do not have the exactly 3-day maturities. In Shanghai Stock Exchange, the existing options have four tenors: current month, next month and the following two consecutive quarters. The options with the nearest maturities denoted as T longer than seven days are selected as the near-term options. Those have the second nearest tenors T are the next-term options. Interpolation typically needs to be applied between the near-term and the next-term VIs in order to obtain the 3-day result. Specifically, with T < T, 7 365 < T < 3 365 and T = 3 365, the 3-day VI referred to as VI earlier at time zero is reported as follows: VI = T VI T T, T T + VI T, T T T. 4. T T T T Note if T 3 365, VI is set to the value of VI, T, following the criteria in [8]. In this case, no interpolation is adopted and the data of the next-term options are not used. The results produced by RMI can be accessed online. The 3-day implied volatility index for 5 ETF is available from December, 6 onwards. Apart from this, the -day, 3- day, 5-day and 9-day moving averages can be displayed optionally. The users are able to download the historical data in Excel format. Figure illustrates RMI online Volatility Index online dashboard. 5 Further Discussion Data sparsity in SSE 5 ETF options may affect the accuracy of the result. Recall the discretization in 3.3 and the interpolation in 4.. It appears that a growing number of OTM options can better approximate 3.; a closer T to T will produce preciser VI in 4.. Concerning the S&P5 index options, there are more than 4 near-term next-term observations taken into VIX calculation. The gap between near- and next-terms are only seven days. In contrast, there are merely up to around sixteen strikes per tenor in Chinese market in 8; the difference between tenors is typically as large as one month. The author proposes two potential solutions to this issue. First, one may fit an arbitragefree implied volatility surface, and evaluate additional option data points based on this surface. Alternatively, assume the asset price follows some specific stochastic process and calibrate the parameters of the model, after which one can generate extra data. These methods can significantly increase the number of options and to some extent improve the implementation. However, fitting two-dimensional no-arbitrage implied volatility surface with both interpolation and extrapolation is challenging; calibration can be time-consuming and not stable. Further research remains appealing in this topic. 4
Figure : RMI Volatility Index Dashboard: the lines from top to bottom in the graph area display the historical volatility indices VI, the moving averages and the close prices of Chinese 5 ETF. The input/output area allows the user to select the calendar date and the moving average type, after which the VI and moving average will be retrieved from RMI database. Clicking the button beneath will download the entire historical results. References [] Adhikari, B. K., & Hilliard, J. E. 4. The VIX, VXO and realised volatility: a test of lagged and contemporaneous relationships. International Journal of Financial Markets and Derivatives, 33, -4. [] Cboe Exchange, Inc 8. The Cboe Volatility Index VIX. Retrieved from: https://www.cboe.com/micro/vix/vixwhite.pdf. [3] Carr, P., & Lee, R. 7. Realized volatility and variance: Options via swaps. Risk, 5, 76-83. [4] Dai, M. 5. Chapter 5: Variance and Volatility Product lecture note in QF 5 Structured Product in NUS MSc. in Quantitative Finance program. [5] Demeterfi, K., Derman, E., Kamal, M., & Zou, J. 999. More than you ever wanted to know about volatility swaps.. Goldman Sachs quantitative strategies research notes, 4, -56. [6] Gatheral, J.. Lecture 6: Volatility and Variance Swaps. Case Studies in Financial Modelling Course Notes. [7] Barndorff-Nielsen, O. E., & Shephard, N.. Estimating quadratic variation using realized variance. Journal of Applied econometrics, 75, 457-477. [8] Shanghai Stock Exchange 6. Shanghai Stock Exchange 5 ETF Volatility Index Methodology. [In Chinese]. 5
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