# Delta Hedging made simple (sort of…)

As any options trader knows, dynamically hedging option book’s delta exposure is an art as much as it is a science. As assets dynamic follow some kind of Brownian motion (stochastic/random process), we cannot know a priory the terminal dynamic of the underlying. Therefore, we cannot determine the optimal delta hedging frequency (meaning — how often we should offset the delta risk of the option book).

There have been dozens of papers and theoretical works published trying to offer numerical/closed-form solutions to this question, but at the end of the day, the Delta Hedging puzzle is rather subjective and highly dependent on the trader’s utility function (or target function). Some traders will target minimum P&L variance, and therefore, will look to delta-hedge almost continuously (highly unlikely, given non-zero Transaction Cost), while others will try to maximize the option book return. The suggested framework targets Sharpe Ratio (i.e. volatility-adjusted return).

Our framework starts characterizing the underlying dynamic of the asset. Any financial asset (whether it’s currency, interest rate, equity index, etc.…) following some kind of stochastic process. A stochastic process (without going into the nitty-gritty of the math) is a process where the underlying dynamic evolves with a degree of drift (slope), but the individual returns are random and exhibit noisy behavior

So we know that the underlying dynamic is governed by two main factors:

1. Noise (Volatility)
2. Drift (Trend)

Now that we acknowledged that these two factors are dictating the underlying dynamic we can reduce our model to target these two factors. For simplicity reasons, we want to assume that there are two regimes for each factor: Low (vol/trend) or High (vol/trend). let’s visualize that by a 2x2 matrix:

Obviously assuming that there are only two states (low/high) is rather oversimplifying the true dynamic, but as we will see later on, it actually makes perfect sense for our purpose.

After laying out the ground for the possible regimes we will run a very simple regime-switch model (one that is not mathematical intensive and can be easily run on an excel spreadsheet, with minimum computation involved). As we have already acknowledged that the dynamic is governed by volatility and trend, we will make two thresholds (one for each factor) to determine in which state we are in for each factor (i.e. which regime). Let’s review possible decision rules to determine our current state:

Volatility: when analyzing the volatility factor we are NOT actually looking to forecast the forward volatility, but rather have a sense of whether we are in a high or low volatility regime. We can do so by looking at the realized (historical volatility) and compare that to the short-dated (say 1–2d) realized volatility. Let’s look at the following example: S&P500 historical 1-month volatility is currently 20% (in annualized term), with a standard deviation of +-1vol (meaning that it has a 67% chance of realizing between 19–21% vol). Now let’s create a rule that says: “if my realized 2day volatility is greater than 1-month vol + stdev.*x (our threshold), we assume we are in high vol state, otherwise, we are in low vol state”. obviously, our signal will be highly sensitive to the selection of x…

Trend: when analyzing the persistence of the trend we are basically measuring the autocorrelation of the underlying returns. while returns are assumed to be i.i.d (independent and identically distributed) we know that that is not the case for most assets that exhibit non-zero autocorrelation (i.e., either mean-reversion- negative autocorrelation, or trend-positive autocorrelation). There are many tests we can run to measure the persistence of trends in time series, but we will take one which is rather quite intuitive — “Variance Ratio”. Variance Ratio is the ratio between two Realized Variance measures (one measured on a high-frequency basis and one measured on a close-close basis). To measure the High Frequency realized variance we can either use 1h fixings or we can use some kind of high/low/open/close realized variance measure (Garman Klass/Parkinson volatility comes to mind. if you are interested in learning more please see a link in the appendix)

Once we have both the HF variance and the close-close variance we divide the HF by the c-c variance. A ratio larger than 1 means that the asset has some degree of mean reversion, as the asset tends to have lower volatility on a c-c basis compared to intraday sampling. a ratio smaller than 1 will suggest some degree of trend.

Our decision rule will be “ If VR (var. ratio) < y, we assume high trend, otherwise, we are in a mean-reverting dynamic”. Again our signal will be sensitive to the selection of y

Ok, so let’s recap what we have done so far: We acknowledged that the underlying asset is some kind of random process. We also assumed that there are two factors that affect the dynamic, and each has two possible states. Lastly, we created two decision rules to determine the regime we are currently in (with a high degree of probability)… What we need now to execute it is a hedging strategy. Let’s review three possible hedging strategies:

1. Hedging as a function of time — as traders we are costumed to look at assets dynamic as a function of time (sometimes without even realizing that). Just think about the common way of measuring volatility…

As we can see, by dividing our total variance by the number of observations, we are making this time-based realized volatility measure. This strategy makes some sense, as some practitioners tend to offset their delta exposure once a day, but it doesn’t take into account the intraday dynamic, and actually independent of the dynamic.

2. Hedging as a function of % change of the underlying asset — This strategy actually takes into account the underlying dynamic, as we set our hedging threshold to be a function of the underlying movement (rather than an arbitrary selection of points in time), that said, as our gamma profile varies as we move away (and toward strikes) the \$delta (the total amount hedged) can vary significantly under the same % move (for example delta accumulated around the strike with 1% move might be twice as much as 1% move when the underlying spot is slightly away from the strike).

3. Hedging as a function of accumulated delta — This strategy is the only strategy among the three strategies that takes into account our options book gamma/delta profile. This strategy basically sets the threshold as the delta accumulated since our last delta offset. The difference between this strategy and the % change is that this strategy will hedge more frequently as we trade around the strike and closer to expiry while hedging less frequently as we drift away from the strike and have relatively low gamma.

So now that we are familiar with the regimes, the thresholds, and strategies, let’s build a simple hedging scheme (this strategy is from a long gamma perspective, and should be viewed as a mirror image for a short gamma):

1. We will start by determining our volatility/drift thresholds:

our volatility threshold (that we multiply by the stdev of the 1-month volatility) is going to be 1.5 (that’s 1.5 times the 1-stdev, or in other words the threshold that covers about 87% of our distribution, so relatively conservative threshold).

Our drift threshold is going to be 1.2 (so the HF variance is 20% higher than the close-close variance).

2. After determining the thresholds we need to determine our strategies and hedging scheme for each regime:

a . low volatility/Low drift — hedge frequent using low volatility assumption (so ideally we want to hedge frequently, so we will either choose %change/delta based strategies, depends on the proximity to strike and gamma profile)

b. Low volatility/High drift — as the underlying asset is presumed to be trending and doesn’t exhibit high mean reversion, we would like to capture the trend, so our choice should ideally alternate between time-based (less frequent hedging) or %change (with some sufficient threshold to capture the trend). one would probably see a case for “trailing stop” strategy (but this seems a topic for another discussion)

c. High Volatility/ Low drift — this dynamic is basically long gamma “wet dream”, characterized by choppy price action around the mean, which is highly profitable for a strategy that takes profit frequently. an ideal choice would be delta based (with high volatility assumption, meaning that we set our options book volatility slightly higher to reflect that into our greeks)

d. High Volatility/High drift — this dynamic is somewhat similar to the lv/ld dynamic, but here we set our volatility parameter of the option book to high volatility, which, as a result, lowers our options book’s delta and allows us to “let the delta run” (meaning that we don’t offset a large portion of our options’ delta throughout the move).

Now that we have both the thresholds and hedging scheme we can review some hypothetical example:

This is Thursday, and we have just initiated a 1-month long SPX straddle after yesterday some massive volatility selling that took place. we bought the ATM volatility @ 24vol, with 1-month SPX realizing about 20 (on a close-close basis). As we open our spreadsheet we see that the overnight move in the SPX was around 23 (as some hedge fund was called on margin during the Asia session). Our volatility indicator screams high vol state, so we move on to the drift indicator, and see that our HF volatility is actually 28vol while the close-close is 23vol. Based on these two factors (and the thresholds we set) we conclude that we are in mean reverting/high vol regime, and act accordingly (setting our vol to high and use delta-based hedging). After few hours we sample again, as we see that we continue to buy delta on the move down, without much correction of the underlying spot, and find that the underlying spot dynamic exhibits a trend-like behavior (our variance ratio signal falls to 1), so we switch our hedging scheme to high vol/less frequent hedging (as the vol is moving higher our %change threshold widens accordingly)

So this is the general framework of our hedging scheme. Now we are left with determining our risk-aversion/appetite to make decisions with regard to the actual delta/ %change/ time we want to use as hedging thresholds. Unfortunately, there is no closed-form solution for that question… Throughout the years I’ve tested a lot of ways to optimize that, but I think everyone should have his/her own input (or secret sauce) into the model.

Feel free to share your thought, comment, ideas… 