Price Impact in the VIX Futures Market and Mean-Field Games in two Order Books

Santawisook, Patchara

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Price Impact in the VIX Futures Market and Mean-Field Games in two Order Books

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This Ph.D. thesis deals with the price impact in the VIX futures market from a statistical and mathematical perspective. The CBOE volatility index, VIX, is known by investors as the fear index. It was introduced to measure the investors' view on the future expected volatility of the S&P 500 stock index. Investors cannot trade the VIX index directly; however, one can trade VIX futures, which gauge the market's expectation of the 30-day implied volatility. Market volatility spiked on February 8, 2018, drawing wide attention to volatility-based products. On that day, the VIX went up more than 100% in intraday trading. The XIV, one of the VIX-based exchange-traded products (ETPs), dropped more than 80%, triggering an "acceleration event." As a consequence, the XIV issuer had to terminate this product. One of the factors contributing to this event was the architecture of the ETPs written on VIX: a daily contracts rolling where the short-term (mid-term) ETPs roll every day to maintain a weighted average of one month (five months) to expiration. Therefore, a large number of shares is expected to be acquired and liquidated every day before the market closes. We study the effect of VIX ETPs on the price of VIX futures by investigating the impact curves at different times of the trading day. We find that the impact curve corresponding to the time before market close is the lowest. Our empirical results show that impact curves exhibit a power-law. This is theoretically justified by using dimensional analysis to show that if the immediate price impact is a function of the trade size, it is given by a power function. We propose a mean-field game framework for the VIX futures market to complement our empirical study, where traders can trade in a regular limit order book (RLOB) and a trade-at-settlement order book (TASLOB). We assume that there are many high-frequency traders (HFTs) in the market, and they trade in both order books. We investigate the case where the number of HFTs tends to infinity. While transactions in RLOB suffer from a temporary price impact, transactions in TASLOB do not, but they trade at an unknown price, the daily settlement price that is only determined at the end of the trading day. We use a extended mean-field games approach where interactions between agents happen through controls instead of states, to solve an optimal liquidation problem in two order books. We solve the optimal liquidation problem for three cases. In the first case, there are only two types of traders: HFTs and noise traders. We introduce the exchange-traded funds' issuer (ETFs) into the market in the second case. They provide a large amount of additional liquidity on one side of the TAS limit order book (TASLOB). In the third case, we solve the optimal liquidation problem of HFT, where ETFs buy a limited number of shares in TASLOB.

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