By Tim Leung, Marco Santoli
This booklet offers an research, lower than either discrete-time and continuous-time frameworks, at the expense dynamics of leveraged exchange-traded money (LETFs), with emphasis at the roles of leverage ratio, discovered volatility, funding horizon, and monitoring blunders. This research presents new insights at the hazards linked to LETFs. It additionally results in the dialogue of recent probability administration options, comparable to admissible leverage ratios and admissible chance horizon, in addition to the mathematical and empirical analyses of a number of buying and selling options, together with static portfolios, pairs buying and selling, and stop-loss suggestions concerning ETFs and LETFs. the ultimate a part of the booklet addresses the pricing of suggestions written on LETFs. due to the fact that assorted LETFs are designed to trace an identical reference index, those money and their linked strategies percentage very related resources of randomness. The authors offer a no-arbitrage pricing method that regularly price ideas on LETFs with assorted leverage ratios with stochastic volatility and jumps within the reference index. Their effects are priceless for industry making of those techniques, and for choosing rate discrepancies around the LETF innovations markets. because the marketplace of leveraged exchange-traded items develop into a huge attached a part of the monetary marketplace, it is important to raised comprehend its suggestions impression and broader industry effect. this can be vital not just for person and institutional traders, but in addition for regulators.
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Additional resources for Leveraged Exchange-Traded Funds: Price Dynamics and Options Valuation
11: Table of leverage ratios pairing (β+ , β− ), the static Δ-neutral portfolio weight ω ∗ (short position for the β+ -LETF), and realized variance coeﬃcient −β−2 β+ . Note that the (short) weight for the β− -LETF is (1 − ω ∗ ). We now backtest the Δ-neutral strategy. 5 of the β− LETF with β+ = −β− = 2 and hold the position for 10 days. The return RT depends on the relative weights on the long/short-LETFs but not the absolute cash amounts. Dividing the price data from the trading days during 2013–2015 into 10-day rolling periods, we calculate the returns from the strategy over each period.
Chapter 4 Options on Leveraged ETFs The popularity of ETFs has also led to increased trading of options written on ETFs. 06 billion contracts, of which 282 million were ETF options while 473 million were stock options. As we have seen, LETFs that are referenced to the same underlying index share the same source of randomness. This leads to an important question of consistent pricing of options written on LETFs with the same reference index. In other words, we will study the price relationships among LETF options, not only across strikes and maturities but also for various leverage ratios.
As for the calls on the short LETFs (−1, −2, −3 calls in the ﬁgure) that have a low adjusted moneyness, they correspond to the original OTM calls and are thus more likely to expire worthless. We remark that aligning the average option returns by adjusting moneyness does not mean that the option payoﬀs have the same distribution. Indeed, LETFs with diﬀerent leverage ratios do not share the same dependence on the reference index. In particular, they have diﬀerent exposures to volatility decay. 3 shows the empirical distributions of the returns of the 1-month bullish ATM options.