By Tim Leung, Marco Santoli
This publication offers an research, less than either discrete-time and continuous-time frameworks, at the fee dynamics of leveraged exchange-traded money (LETFs), with emphasis at the roles of leverage ratio, learned volatility, funding horizon, and monitoring mistakes. This examine offers new insights at the dangers linked to LETFs. It additionally ends up in the dialogue of latest possibility administration innovations, resembling admissible leverage ratios and admissible chance horizon, in addition to the mathematical and empirical analyses of a number of buying and selling techniques, together with static portfolios, pairs buying and selling, and stop-loss ideas regarding ETFs and LETFs. the ultimate a part of the e-book addresses the pricing of thoughts written on LETFs. considering that various LETFs are designed to trace an analogous reference index, those cash and their linked strategies proportion very related assets of randomness. The authors supply a no-arbitrage pricing technique that continuously price thoughts on LETFs with diverse leverage ratios with stochastic volatility and jumps within the reference index. Their effects are invaluable for industry making of those ideas, and for selecting fee discrepancies around the LETF suggestions markets. because the industry of leveraged exchange-traded items turn into a tremendous hooked up a part of the monetary marketplace, it is necessary to higher comprehend its suggestions impact and broader industry impression. this can be vital not just for person and institutional traders, but additionally for regulators.
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Additional info for Leveraged Exchange-Traded Funds: Price Dynamics and Options Valuation
Example text
2 Admissible Risk Horizon 43 Given any investment horizon T and leverage ratio β, the conditional value-at-risk for the LETF is given by CV aRα (β, T ) = 1 − e (β(μ−r)+r−f )T √ Φ(Φ−1 (α) − |β|σ T ) . 12) The CV aRα (β) is decreasing in β for β ≤ β ∗∗ and increasing for β ≥ β ∗∗ , with the critical leverage β ∗∗ satisfying √ √ μ−r √ Φ(Φ−1 (α) − |β ∗∗ |σ T ) = sign(β ∗∗ )φ(Φ−1 (α) − |β ∗∗ |σ T ). 14) log(1 − z) − ψ(β)T √ . 12). Next, we compute the derivative ∂CV aRα e(β(μ−r)+r−f )T = × ∂β α √ √ √ (r − μ)Φ(Φ−1 (α) − |β|σ T ) + sign(β)σ T φ(Φ−1 (α) − |β|σ T ) .
1: The mean-variance frontier (solid), where each point corresponds to a different leverage ratio within [0, 3]. The mark ’o’ (resp. 23 (resp. 25). 1 Admissible Leverage Ratio Given a risk budget based on a risk measure, one can determine the leverage ratios that are deemed acceptable. To this end, we now compute analytically the value-at-risk (VaR) and conditional VaR associated with a long position in a leveraged ETF, and derive the associated admissible leverage ratios. 1 Admissible Leverage Ratio 39 where Φ(·) is the normal cumulative distribution function and ψ(β) = β(μ − r) + r − f − β 2 σ2 .
4, we see similar return distributions among the 54 4 Options on Leveraged ETFs 1-month bearish ATM options. Here, we see that the SPXU (β = +3) call has about a 50% chance to expire worthless yielding a return of −1, and this probability is significantly higher than other bearish ATM options. 5 2 Fig. 1: Empirical average returns for the 1-month LETF options (calls on the long LETFs and puts on the short LETFs) that are bullish with respect to S&P 500, plotted against adjusted moneyness. 5 2 Fig.
