Trading gamma has traditionally been left to the “experts” on Wall Street. With the proliferation of options trading knowledge and tools in the retail market, that no longer needs to be the case. For a primer on Gamma trading, I would suggest reading Scalping Gamma and Long Gamma, Short Vega.
There are two positions that you can take by buying options (long gamma) or selling options (short gamma) while delta hedging the equity exposure:
- Long Gamma – Profit when realized volatility is greater than the implied volatility of the purchased option
- Short Gamma – Profit when realized volatility is less than the implied volatility of the sold option
Instead of talking about gamma trading, let us go through an example for better clarification.
Since short term implied volatility is trading relatively cheaply compared to the last 4 years of historical volatility, we could consider this an opportune time to purchase options. If you believe the market is going to go up very quickly or down very quickly, then you might just purchase a straddle on the SPDR S&P 500 (SPY). If we look at the April options, let us suggest that you would buy a call at a strike of $140 and a put at a strike of $140:
If you bought these options near the mid-point, you might be able to purchase this straddle for a total of $4.26, or 3.04%. If you simply purchased this straddle and held it to maturity, you would make money if the SPY closed above $144.26 or below $135.74 on April 21st.
Instead of making an explicit directional call, let us instead say that we just want to make a trade that suggest volatility will be a lot higher than the 13.3% annual figure that is built into these option prices. In order to do that, I would buy the straddle and then delta hedge the position daily:
As shown above, the total delta of the position is $0, the Gamma is +$149, the Vega is +$318.98 and the theta is -$72.08. This means that if SPY goes up by one point, the delta moves to $149. If implied volatility moves up 1%, you make $320 and if one day passes without anything else happening you lose $72. If we look at instantaneous shocks on SPY, we can see what the P&L of this aggregate position is all else equal:
You can estimate what you will make or lose with a gamma long/short position with the following formula:
If we put a point change of $2 in this equation, we would calculate .5*149*2^2 = $298 which is fairly close to the +317/$310 in the above scenario table. The slippage is due to the change in delta over that range in value.
On the flip side, we lose $72 per day of holding the option. If we use this as our break-even starting point, we can calculate an approximate point move that gives us a break-even for the day:
The next question is to look at how frequently you would delta hedge this position, but we will leave that for another day.
This in no way is a trade recommendation, just an educational example.
Praise for The Volatility Surface
“I’m thrilled by the appearance of Jim Gatheral’s new book The Volatility Surface. The literature on stochastic volatility is vast, but difficult to penetrate and use. Gatheral’s book, by contrast, is accessible and practical. It successfully charts a middle ground between specific examples and general models–achieving remarkable clarity without giving up sophistication, depth, or breadth.”
–Robert V. Kohn, Professor of Mathematics and Chair, Mathematical Finance Committee, Courant Institute of Mathematical Sciences, New York University
“Concise yet comprehensive, equally attentive to both theory and phenomena, this book provides an unsurpassed account of the peculiarities of the implied volatility surface, its consequences for pricing and hedging, and the theories that struggle to explain it.”
–Emanuel Derman, author of My Life as a Quant
“Jim Gatheral is the wiliest practitioner in the business. This very fine book is an outgrowth of the lecture notes prepared for one of the most popular classes at NYU’s esteemed Courant Institute. The topics covered are at the forefront of research in mathematical finance and the author’s treatment of them is simply the best available in this form.”
–Peter Carr, PhD, head of Quantitative Financial Research, Bloomberg LP Director of the Masters Program in Mathematical Finance, New York University
“Jim Gatheral is an acknowledged master of advanced modeling for derivatives. In The Volatility Surface he reveals the secrets of dealing with the most important but most elusive of financial quantities, volatility.”
–Paul Wilmott, author and mathematician
“As a teacher in the field of mathematical finance, I welcome Jim Gatheral’s book as a significant development. Written by a Wall Street practitioner with extensive market and teaching experience, The Volatility Surface gives students access to a level of knowledge on derivatives which was not previously available. I strongly recommend it.”
–Marco Avellaneda, Director, Division of Mathematical Finance Courant Institute, New York University
“Jim Gatheral could not have written a better book.”
–Bruno Dupire, winner of the 2006 Wilmott Cutting Edge Research Award Quantitative Research, Bloomberg LP