Short-End of the Vol Cone

We use Gamma Scalping Realized Volatility (rVol), see below, to value short-dated rVol for various trailing windows, measured in number of minutes. The long gamma and short gamma last values are entered into the Vol cone graph for various trailing windows. The size of the re-hedge steps used (again, see below for an explanation of how Gamma Scalping rVol works) is chosen to make sure we have about 80 re-hedge executions during the window, to ensure a low variance result (for a static distribution).

This process is similar to the usual vol cone, with various trailing windows for vol calculations, but instead of only considering daily returns, we consider various rehedge-step sizes to ensure the average contains about 80 executions. This would be analogous to considering various return durations, say daily returns for a 50 day moving rVol estimator using the usual 1-standard-deviation formula, and 2 day returns for a 100 day moving rVol, with the objective of obtaining similar accuracy.

Gamma Scalping Realized Volatility

The idea is to have a measure of intra-day realized volatility (rVol) that is not critically dependent on assumptions about the security price change process, such as normal returns (lognormal price changes) and even allows for jumps.

Method:

1. The above graphs are essentially P&L generation rates for different gamma scalping strategies; let's say different traders.

2. P&L generation rates are not model or even distribution dependent: it's a robust powerful concept for examining market dynamics and micro-structure.

3. The long case (third graph) is selling at the bid, when underlying price moves up, and buying at the ask, when underlying price moves down, with limit orders, entered at various constant (bps, 0.01% of price) step separations and indicated in the graph legend with different colors for each step-size/strategy; when a trade gets executed the last-executed limit-order is then re-entered. This maintains a ladder of limit orders. This is the (dynamic-replication) delta-hedging strategy typically executed for a long (constant) Gamma position. Repeatedly selling high and buying low generates a profit when the stock stays within a range. This profit counters the time decay on the option position.

4. The short case (second graph) is selling at the bid, when underlying price moves down, and buying at the ask, when underlying prices moves up, as soon as the price hits the ladder rung levels. This is not quite, but is very similar to, market stop-loss orders (which execute market orders when a trade is registered at the rung level) which are often used to dynamically hedge short Gamma positions. Repeatedly buying high and selling low generates a loss when the stock stays within a range. This loss is countered by the (negative) time decay, i.e., profit on the short gamma option position.

5. Each color line is a different strategy or “trader”. For each strategy, for example, the fastest trader represented by the red line, we count the number of times the re-hedge trade occurs over a trailing period of N minutes, as indicated in the legend, and hence calculate an average rate of re-hedges, i.e., rate of P&L generation. Note that the slowest trader, the purple line, has the widest re-hedge ladder steps and requires the longest averaging period (N minutes) to ensure requisite accuracy/stability.

6. The last step of the analysis is to translate these P&L generation rates into any model variance input, via a decay term. For the lognormal case of the Black-Scholes option model, this translation is

P&L rate = 1/2 * rVol^2 * S^2 * Gamma = 1/2 * (% S)^2 * Gamma * rate of re-hedges

i.e., rVol = % / sqrt (average re-hedge time)

The formula rVol = % / sqrt (expected re-hedge time) is derived in these papers[1,2,3]. These rVols are used for the y-axis (instead of P&L generation rate). We are expressing P&L generation rate in the same "basis" as iVol making them directly comparable.

7. Note finally, any model can be used, thus any security can be analyzed, such as interest rates and spreads, which are commonly understood to be not well modeled by lognormal price change distributions, simply by using a different translation from P&L rate to model input parameter(s).

Interpretation:

1. Generally speaking, for a perfectly stable lognormal-distribution (without jumps), the long side shows the rainbow with spaced flat lines from red (fastest trader with tightest re-hedge steps) upwards to purple (slowest trader with largest re-hedge steps). The short side shows higher vols and the opposite; flat lines with from red down to purple. The limit for large re-hedge steps (compared to bid/ask spread size) is that the purple lines will show the same values for long and short. The observed "slippage", i.e., differences from the limiting purple case, is due to "paying" the bid/ask spread. See a “pure” 20 vol case below, i.e., using theoretically generated stochastic bid/ask markets to illustrate the pure lognormal returns case.