Every now and then I get DMs from novice trades asking me for advice on the best way to go about learning options trading. My answer is always to Start with introductory-level books like Natenberg’s “Option Volatility Strategies”, Euan Sinclair’s “Options Trading”, or John Hull’s “Option, Futures, and Other Derivatives”. These books provide the foundations for anyone who wants to dive into the rabbit hole of derivatives pricing and trading, yet not overwhelming with math/stats/probability theory (apparently some are intimidated by quantitative language).

The level of quantitative knowledge tends to be the 2nd most asked question that I get. People want to engage in options trading, yet fear that their level of quantitative knowledge is insufficient. While I LOVE all aspects of quantitative research/analysis, I don’t think that novice traders should have an MS/Ph.D. in physics/Financial Engineering/Computer Science to trade derivatives (namely options). Obviously, as we get more comfortable with options trading, and want to enhance our strategies (or trading style), we should attempt to expand our quantitative knowledge (mostly to know what mistakes to avoid), but for novice traders who want to trade “vanilla” strategies a basic understanding of mathematical concepts (i.e., just the intuition ex. the actual proof) is sufficient in my view.

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As I love to explain option trading and quantitative concepts for non-professionals (I’m already making some nice progress with my daughters, who are 12yrs and 9yrs old), I thought of putting together a blog post that explains the very basics of volatility trading. I will try to keep the quant knowledge to the bare minimum, and the only prerequisite with regards to options trading is that the reader knows what Put/Call options are. Ready? so let’s dive in…

# Volatility is the price of an option

When we price an option we will use, in most cases, the Black-Scholes option pricing model. While there are few underlying flaws in the assumptions of the model (you can check out my blog post about volatility model for more), this model serves us well as an accounting standard (meaning that it produces are robust price given our parameter). To be able to price an option (any option), we need to provide the following parameter:

• The spot price of the underlying asset
• Strike price
• Time to expiry
• Option type (Call/Put)
• Risk-free rate and dividend yield/Domestic rate/Cost-of-carry
• Volatility

Five out of the 6 above parameters are relatively known and observed in the market (one could argue that for long-dated options, the dividend yield is not given, but that’s a topic for another occasion). Still, the one that drives the entire options market volatility, is unknown/unobservable and therefore is the underlying parameter that we, as option traders, trade.

To understand the relation between volatility and the price of an option, we need to understand the calculation of the Black-Scholes model.

According to the Black-Scholes formula, the price of European Call option (an option that can be exercised only at expiry) is:

So now you are probably saying, “But he said we are keeping the quant level low, and he brings these equations like WTF is N(d1)?”, bear with me…

Now… Let’s break down this complex formula into the different parameters (and keep in mind that I saved you 99% of the actual proof, including the super fun Ito’s Lemma, which you can google if you are curious):

F= Forward price of the underlying asset (taking into account the rate differential)

K = Strike Price

T-t = Time to maturity

σ = Volatility

N(·) = Cumulative distribution function (of standard normal distribution).

If you are still feeling a bit puzzled and confused, we can make that even simpler:

The price of any option can be broken into two parts:

1. The intrinsic value of the option — How much “inside the money” the option is. If I had exercised the option now, what would have been the value of the option:

2. The time value of the option — The time value of the option represents the part of the option price governed by the effect of time to maturity and volatility. The more time we have, the more expensive the option(because it has more time to go deeper into ITM). The higher the volatility, the more expensive the option.

The latter is probably less intuitive than the former, as volatility can also take our option OTM. The thing about volatility is that the lower bound of an option is 0 (as in the worst case, we will not exercise the option), but the upper bound is infinite (the spot can go to the moon..). The higher the volatility, the more valuable the time value is (as our option can go further deeper into ITM, hypothetically speaking).

So we can comfortably say that all else being equal (and that we cannot turn back time), the only real unknown in options is volatility.

# A quick and dirty option pricing

After we established that our primary motivation in options trading is trading volatility, we want to put that into use and see how volatility translates into the price of an option and how we can turn that into a trading strategy.

Our starting point as volatility traders is that we are directional indifferent (meaning that we don’t care whether the market goes up or down). As we take no view on the direction we will trade ATM Straddle (i.e. Buy/Sell Call + Put with a strike = Forward(t))

The cost of our ATM Straddle can be approximated in the following way:

In plain English, The price of the ATM straddle equals the price of Call + Put, which equals (approximately) 0.8× Spot ×Vol ×Time (in a 1-year term). If you are puzzled about where this “0.8” all of a sudden came from, it’s something to do with approximation to the normal distribution, so let’s leave it like that…

So now we can translate volatility into ATM straddle price and calculate the “ATM straddle breakeven” (and it’s important to note that we didn’t specify which kind of volatility we are talking about, so we can use any type of volatility to approximate our straddle price).

The “ATM straddle breakeven” shows us how much we need the market to move to breakeven against the cost of the strategy.

Let’s look at the following example: Our market-maker/dealer is kind enough to price the implied volatility term structure (curve) flat at 16% between 1-day and infinity, and Forward=Spot price=100. This is how the “ATM straddle breakeven” chart will look like:

As we can clearly see, our straddle’s value doesn’t grow linearly but rather at the square root of time. The explanation as to why option price grows at the rate of the square root of time is related to the fact that although we talk (and trade) volatility, we actually trade variance (as volatility is not additive). The problem with variance is that it’s not intuitive to investors, so it’s easier to use the square root of the variance (i.e., volatility).

# Talking Greeks

Now that we know how to price options (kind of), how to approximate our ATM straddle breakeven (roughly), and how volatility and time affect our option price, let’s understand how each parameter affects our option price. The easiest way to test our option sensitivity to the underlying parameters is to use nonparametric approximation (remember there are four changing variables in the B&S formula: Spot, Rates, Time, and Volatility). While this might sound intimidating, it’s simply shifting the different parameters up/down and testing the option’s sensitivity to each underlying parameter separately.

Obviously, shifting parameters up/down is time-consuming, so luckily, Black-Scholes came up with an easier way to measure the option’s derivatives — “The Greeks.” Greeks are closed-form formulas representing the option's sensitivity to the different underlying parameters. Let’s review the most widely used Greeks (I left the sensitivity to rates outside the scope):

1. Delta (Δ) — The sensitivity of the option price to a move of the underlying spot price
2. Vega(ν) — The sensitivity of the option price to the change of implied volatility.
3. Theta( Θ) — The sensitivity of the option price to the passage of time (aka, “time decay”)

These three Greeks are known as 1st order derivatives, as the option's value is directly derived from a change in these parameters. However, there is one 2nd order derivative that is highly important for us as novice options traders:

Gamma(Γ) — The sensitivity of the option’s delta to the change in the underlying spot price

Some might mention Vanna/Charm/Volga, but they are less relevant to us now…

From now on, we move away from the option price and the underlying asset and enter the matrix. The only thing that matters to us from now on is the magical triangle of “Volatility-Gamma-Theta.”

# The Volatility-Gamma-Theta Triangle

By now, we’ve come a long way from an introduction to options (and options trading) to understanding that we trade volatility and the different derivatives.

Theta (time decay) and Gamma (our delta sensitivity to the underlying spot change). These two are probably the main drivers of our options portfolio (assuming you don’t trade far OTM long-dated options).

Both Theta and Gamma are highly sensitive, you guessed right, to volatility. The relation between the three can make (or break) your option strategy, so it’s important to understand how the Volatility-Gamma-Theta dynamic works.

Volatility-Gamma — Gamma, as we know by now, is “how much our delta changes with respect to the change of the underlying spot.” As volatility traders, this tells us at what PACE our delta will accumulate. Why is this important to us? If we BUY volatility, we want our gamma to be HIGH (so we will accumulate delta faster, and our option price will increase faster). If we SELL volatility, we want the exact opposite. Volatility has an INVERSE effect on gamma. The higher the volatility, the lower the gamma (and vice versa).

Volatility-Theta — Theta represents the rate at which our option price decays (loses value) as a function of time passage. As higher volatility translates into higher option prices, volatility will have a DIRECT effect on our theta value. The higher the volatility, the higher our theta (i.e., the option will decay faster), and vice versa. This should not come as a surprise if we recall the “ATM Straddle breakeven,” which means that for us to break even against the cost of ATM straddle, our move should be higher than the implied volatility.

So we know that volatility has a DIRECT effect on theta and an INVERSE effect on gamma; therefore, we can think of the three as vertices of a triangle.

The following chart shows the effect of volatility (x-axis) on both gamma and theta for a long ATM straddle (note that theta should be negative as our option value decays, but I made it an absolute value)

Now that we understand the effect of volatility on gamma and theta, we will see how they affect our options trading profitability.

Let’s think about the following example: We buy a 1-week ATM straddle on SPY, with a strike price of 390, and implied volatility of 16%. Using our “quick-and-dirty” option pricing formula we can see that the straddle cost equals to (roughly) 1.77% ( or ~6.9pts in SPY terms). This cost means that our options’ portfolio will break even at 383.10 (lower bound) or 396.90 (upper bound). However, we can run the same calculation for different time intervals (say 1-day), and get that our 1-day breakeven rates are 387.40/392.60. This transformation shows that we can treat the straddle strategy (or any strategy for that matter) as a series of T-interval straddle breakeven. If we bought the straddle, we need the underlying spot to move MORE than the breakeven rate, while if we sold the straddle we need the underlying spot to move LESS than the breakeven rate (Later we will see how dynamic hedging makes our trading far more interesting).

Now let’s say that we bought the option, but the market decided not to move at all the first two days (so was stagnated at 390). Our option value decayed the first two days (as nothing can stop time from moving forward), and now we need the market to move more each day for the remainder of the options’ life to breakeven (the terminal breakeven rates are still 383.1/396.9, as this was determined the moment we bought the options). After two days of a stale market, our 1-day breakeven rates will be 386.90/393.10 (compared to 387.40/392.60), so we see that we need a bigger move to compensate for the fact that the market didn’t move (or didn’t realize) close enough to the implied volatility (which is the “expected” move priced by the market).

## Volatility and Dynamic Hedging

Now that we know how to price options, Greeks, and the magical triangle of Volatility-Gamma-Theta, we can move on to the fun part of options — Volatility Trading. As we trade volatility, we should maintain our options portfolio delta-neutral (or, in other words, indifferent to the market direction), which means we don’t care whether the market goes up or down as long as our view on volatility is correct (either volatility will realize above the implied volatility we bought, or below the implied volatility we sold). The way we “realize” volatility is by using “Dynamic Hedging”. This supercool term is essentially just buying/selling the underlying asset (based on our option’s delta to maintain the portfolio delta-neutral).

To understand how dynamic hedging works, let’s go back to our SPY example:

After we happily bought 1-week 390 straddles (thinking that the market would relive the glory days of March 2020), the market started going sideways… 1.5% down, only to recoup losses and close unchanged. The following day market opened strong, up 2%, only to erase gains and close flat. At the end of the first two days of the option, the market was unchanged (on a close-close basis), but in fact, it moved on AVERAGE ~1.77%. This is where dynamic hedging comes to play…

Let’s make a trading rule: each day, we will buy/sell the accumulated options portfolio’s delta once the market reaches (or goes beyond) our daily breakeven move (in case of 16%, that would be 1%), so in our example, we will buy at 386.1 (and sell back at 390) on the first day, and then sell at 393.9 (and buy back at 389.95).

As we buy low and sell high, we profit from the market volatility via dynamic delta hedging (aka, gamma scalping) and offset the theta (time decay). The more volatile the market, the greater our hedging profits (in other words, we realize above the implied volatility we paid). The below chart represents a theoretical (randomly simulated) path:

Obviously, our P&L at expiry will be highly path-dependent, as the underlying spot path dynamic is unknown, and there could be infinite realized spot paths between the initiation of the trade and expiry, but ON AVERAGE if the realized volatility is higher than the implied volatility a long gamma exposure will be profitable (and vice versa).

# Final Thoughts

Although I tried making this write-up an introduction to option (volatility) trading, I barely scratched the surface of the world of options and volatility. That reminds me of the early stage of my boxing experience. When I started boxing my coach told me: “You can’t learn boxing from online videos and tutorials. Nothing beats the actual experience inside the ring when you face danger”. This resembles trading a lot, as any book (no matter how good it is), cannot teach you how to manage and run the day-to-day risk of option trading volatility. Options trading is a journey that can take many shapes and forms. It depends on our scope of interest, ability to learn new concepts, and adapts to ever-changing markets.