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In the book Leveraged Exchange-Traded Funds: Price Dynamics and Options Valuation, it describes a static delta-hedged long volatility position by simultaneously shorting regular/inverse leveraged ETFs pairs. This is due to the fact that the leveraged ETF price, L(t), can be associated with the underlying unleveraged ETF price, S(t), by

$$ L(t) = L(0) \left( \frac{S(t)}{S(0)}\right)^{\beta}e^{\frac{(\beta - \beta^2)\sigma^2 t}{2}}, $$ where ${\beta}$ is the leveraged factor. This formula is obtained by solving the Ito's integral for GBM with the leveraged factor. However, when I tried to formulate the price change by Tylor expansion, it ends up with $$ \begin{align*} dL&=\frac{\partial L}{\partial S} dS + \frac{1}{2} \frac{\partial^2 L}{\partial S^2} dS^2 \\ &=\beta L \frac{dS}{S} + \frac{1}{2} \left[\beta (\beta - 1)\right] L \left( \frac{dS}{S} \right)^2 \end{align*} $$ If the position (short leveraged ETF + long ETF) is delta-hedged, the PnL difference becomes $$ dP \approx -\frac{1}{2} \left[\beta (\beta - 1)\right] L \left( \frac{dS}{S} \right)^2, $$ which implies that the position is shorting the underlying volatility.

How come the same position is both longing/shorting (realized) volatility at the same time? I strongly believe that I misunderstood the concepts of short gamma and volatility decay, I am wondering is there anyone who can help me bridge the knowledge gap? Any comments are highly appreciated!!

Update: should I add ${\frac{\partial L}{\partial \sigma}}$ and ${\frac{\partial L}{\partial t}}$ terms into analysis? which terms are responsible for the volatility decay (bleeding)?

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