Leveraged Exchange-Traded Funds: Price Dynamics and Options by Tim Leung, Marco Santoli

By Tim Leung, Marco Santoli

This e-book offers an research, below either discrete-time and continuous-time frameworks, at the expense dynamics of leveraged exchange-traded cash (LETFs), with emphasis at the roles of leverage ratio, discovered volatility, funding horizon, and monitoring mistakes. This research presents new insights at the dangers linked to LETFs. It additionally ends up in the dialogue of latest possibility administration ideas, equivalent to admissible leverage ratios and admissible probability horizon, in addition to the mathematical and empirical analyses of a number of buying and selling concepts, together with static portfolios, pairs buying and selling, and stop-loss ideas related to ETFs and LETFs. the ultimate a part of the ebook addresses the pricing of suggestions written on LETFs. considering the fact that diversified LETFs are designed to trace an identical reference index, those money and their linked innovations percentage very related assets of randomness. The authors supply a no-arbitrage pricing process that regularly price thoughts on LETFs with assorted leverage ratios with stochastic volatility and jumps within the reference index. Their effects are precious for industry making of those innovations, and for selecting cost discrepancies around the LETF innovations markets. because the industry of leveraged exchange-traded items develop into a huge hooked up a part of the monetary marketplace, it is important to higher comprehend its suggestions impact and broader marketplace influence. this can be very important not just for person and institutional traders, but in addition for regulators.

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Extra info for Leveraged Exchange-Traded Funds: Price Dynamics and Options Valuation

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It is common to consider option prices as a function of © The Author(s) 2016 T. Leung, M. 1007/978-3-319-29094-2 4 51 52 4 Options on Leveraged ETFs moneyness, which is defined by the ratio of the strike over the spot LETF price. As such, at-the-money options have moneyness of value 1. The prices of options on LETFs with different leverage ratios are generally not comparable at the same strike or moneyness. The reason is that LETFs can differ greatly in their dynamics and dependence on the reference index.

20) that the derivative would take the same sign as ∂p(z,β) ∂β , unless the drift term ψ(β) is so negative that the denominator also becomes negative. The intra-horizon risk measure motivates a stop-loss exit strategy in order to limit downside risk during the investment horizon. Incorporating a stopL loss level L0 , we denote RT = TL∧τ and express the expected relative 0 value as E {RT } = P{τ < T } + E LT 1{τ L0 >T } . 3 Intra-Horizon Risk and Stop-Loss Exit 49 The first term is given by P{τ < T } = log( ) − ψ(β)T √ |β|σ T Φ + 2ψ(β)/(βσ)2 Φ log( ) + ψ(β)T √ |β|σ T .

We plot their average returns over their adjusted moneyness. In other words, each data point corresponds to a single adjusted moneyness for the corresponding LETF call/put. As we can see, the returns match very well across leverage ratios and option types. Here, the OTM calls (with high adjusted moneyness) for the long LETFs tend to expire worthless in one month and yield a return of −1, while ITM calls with low adjusted moneyness are more likely to have a positive return; see +1, +2, +3 calls in the figure.

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