# VXJ: Volatility Index Japan

A volatility index proposed by the VXJ Research Group

### What is implied volatility?

Implied volatility is the options market participants’ expectation of the price variability of an asset, implied by the market price of an option written on the asset. When high price variability (high volatility) is foreseen, the prices of the call/put options written on the asset usually become high. In other words, the market prices of options reflect information on the risk of future price variability. The volatility parameter value backed out from the price of an option using the Black-Scholes pricing formula is called the Black-Scholes implied volatility and is particularly well-known, but usually takes different values depending on the maturity and the strike price of the option used for its calculation.

### A model-free volatility index

The volatility index that aggregates information contained in the market prices of options with various maturities and strikes into a single value is a useful reference for quantitatively grasping the future volatility expected at a given day. One aggregation method targets the expectation of the cumulative volatility (cumulative variance) over a given future period starting from today. It can be shown that this target quantity can be approximated using the market prices of call/put options without assuming that a particular model such as the Black-Scholes model holds. The Chicago Board Options Exchange (CBOE) has developed one particular approximation method, and calculates and disseminates the values of the S&P 500 volatility index, VIX, and other volatility indices for various indices and asset prices.

### Approximation errors in implied volatilities

A volatility index based on the CBOE’s VIX calculation method is just one of many possible approximations to the theoretical implied volatility value derived in a model-free fashion from the expected cumulative volatility. Since it approximates an integral over an infinite range, which appears in a particular representation of the expected volatility, by a simple Riemann sum, the CBOE's VIX formula gives rise to errors due to discretization and cutting off of portions of the integration range into a finite range. During the recent financial crisis, still fresh in our memory, the index based on the CBOE formula severely underestimated volatility due to cut-off errors. During this period of market turbulence, implied volatilities were very high reflecting the market's fear, but the volatility indices based on the VIX formula did not completely capture the degree of this fear.

### A more reliable model-free volatility

The CSFI-VXJ Research Group has developed a new approximation method for the expected cumulative volatility, and started publicly releasing the values of the volatility index, VXJ, for the Nikkei Stock Average based on our new method. By applying a nonlinear transformation specific to the problem, option prices are more naturally interpolated and extrapolated, and numerical integration is avoided. Our simulation study verifies that this new method substantially reduces approximation errors relative to the CBOE's VIX method. Also, our empirical study indicates that the new index has an improved ability to forecast the realized volatility. For more details, please see the discussion paper.

### Revision history

2011/8/12
We adopted a more precise approximation for the normal distribution function and modified the CSFI-VXJ values.

2013/9/2
In light of the potential effects of the new option trading rules, the CSFI recommends the use of CSFI-VXJ as a more reliable benchmark of future volatility in the Japanese market. Starting from September 2, 2013, the weekly updates will be confined to the CSFI-VXJ simply referred to as VXJ. To avoid possible confusion, the formerly VXJ is relabeled as VJO and will remain available for download from the historical database with monthly updates.

2014/1
The VJO index is discontinued as of December 31, 2013.

VXJ(formerly labeled as MMDS-VXJ): VXJ-Guide
VXJ(formerly labeled as MMDS-VXJ): Technical Report (IJTAF(2011), errors corrected)
VJO(formerly labeled as VXJ): VJO-Guide (This document explains VJO with former label VXJ.)

VXJ Research Group Members :
Kazuhiko Nishina, Kosuke Oya, Nabil Maghrebi, Masato Ubukata, Masaaki Fukasawa, Kazutoshi Yamazaki, Isao Ishida, Yuta Kurose, Makoto Takahashi and Seiya Kuno