As an individual investor, should you really trade options ?

We’ve seen too many people around us using options only to take views on the underlying direction. They are often buyers (premium payers) and justify their decision with accessing leverage and above all limiting risk. They in fact delude their own loss-aversion into thinking the market is offering a free-lunch with almost no downside. But markets are not that inefficient and they end up losing money. It is the same situation when the investment decision is not proactively made, but passively consented to an unscrupulous banker who is selling a structured product. All you hear is “riskless performance booster”, you give in, he gets his commissions. You’ve actually just “bought” a bunch of options. Far from riskless.

The point with options is that you’re not really trading the underlying, nor you’re really trading its volatility (or variance), you’re trading a complex mix of those factors.

Assume you buy a 6 month at-the-money call, with an underlying at $100, and an implied volatility of 30%, and 3 months later, the underlying has rallied up to $105, with the implied volatility going down to 25%. You’re now 5% in the money! Have you actually made money? Indeed no: the “loss” on the implied vol and time decay have eaten up your “profit” on the underlying.

The additional layer of complexity an option brings should be carefully weighed before going ahead,
and compared to an implementation of the trade idea with more straightforward instruments.
With a view on the underlying direction, consider using the spot itself, or if you want leverage,
a futures, a forward, a CFD (check out the costs on the latter though).
With a view on realized volatility, you might consider using a variance swap.
Note we’re not addressing here the issue of using options and option strategies as *hedging* instruments, but as
*investments* in their own.

There is a fundamental formula that, in our view, explains a lot about options, but is not
enough taught. Education sites and day trader books too quickly jump from basic definitions onto
describing strategies like “condors” and “victory spreads”, avoiding any formula (of course: with each
equation in any book, you divide by 10 the number of potential readers). Quantitative books often put their
efforts into describing, *ad nauseam*, the concepts of “no-arbitrage pricing”, failing to convey any
intuition as to why any one should trade options.

While most individual option investors know about delta risk, time decay and vega risk, this formula, explained below, sheds some light on the less-known gamma risk. But before laying out the nasty math details, let’s make a short detour on the option market changing forces.

The equity option industry is healthy, to say the least, with a 14% annual growth rate since 2000. And this is not just because of asset managers, hedge funds and commercial hedgers, but also individuals, whose trading activity, according to some estimates, account for about 20% of daily traded volume (institutions account for 30% and market makers for the remaining 50%). One striking fact is that the ETF option market is quietly heading to become the dominant one in the “equity” option space in the US: out of the 16 million equity option contracts traded each day (for a notional of more than $100 billion), about 45% flows into ETF options (versus 35% in 2015). And half of it goes into a single name: the SPY options (SPY is an ETF tracking the S&P 500).

Although FX and commodity options are available to individuals, equity options get all their attention. And the growing liquidity on listed ETF options now unlocks access for the public to options on all asset classes. ETF options on high yield corporate bonds (HYG) and WTI Crude Oil (USO) are among the most liquid ones.

There are already about 1000 ETFs in the US that have listed options tied to them. Europe is lagging behind with less than 50 ETF options listed on Euronext and Eurex. One complication in Europe is that the ETF market itself is still mostly OTC (estimates indicate a 70%-30% split for off- versus on- exchange volumes in Europe, while it is the reverse in the US). This might be explained in part by different regulations: in the US, ETFs are covered by the REG NMS regulation system, making best order-routing and public reporting of transactions mandatory, while the European analogue MIFID did not include ETFs in their ruling (MIFID II will change this).

With all this new investing power just a few clicks away, it is important retail investors keep a cool head and do not rush into frantic careless trading. A good way to keep you hair on is to read the next section.

We’ll assume here some familiarity with the Black-Scholes model. If you’re an investor on the buy side, your option value is either marked by the market or by your (prime) broker, and your P&L is . Easy, but it does little in helping to understand the drivers underpinning this P&L. One way to elicit the factors and risks in the trade is to use the option delta-hedge , write and look more closely at the hedged option part . Let’s boldly do a “Taylor expansion” on , as a function of the underlying price and time (this is not rigorous, but it works). We’ll assume interest rates are zero, and the underlying does not yield any dividend. The profit (or loss) from one day to the next is where we used the usual Greeks notation for partial derivatives, except for (lower case), to avoid confusion with the upper cased which denotes here a change in variable. Now, since the portfolio is delta-neutral (), we have Recall the Black-Scholes equation: where is the “hedging” volatility, i.e. the value we choose to use for the parameter appearing in the Black-Scholes formula and the Greeks. With our assumptions (), we have: Then rearranging a bit: Now turning back to , we get where we used .

This is the **gamma trading equation**, or **volatility trading equation**.
is the realized variance;
while is our assumed “hedging” variance (the variance implied by our choice of hedging
volatility).
This key formula shows that the P&L of an option trade is linked to the difference between realized variance and hedging
variance, but in a non trivial way: the factor will, depending on its
value ( is a bell-shaped function of ), magnify or squeeze this difference.
A period of high realized volatility will not necessarily
builds up profit if it occurs when your gamma is low.

One issue with the above derivation is that we “differentiated” without strict consideration to the stochastic calculus rules (actually a bit like in the original Black-Scholes paper). We can make this more rigorous by resorting to the concept of self-financing portfolios: our option hedge is not just made up of some quantity of the underlying, but also of some cash invested in a risk-less bond , with a continuous rebalancing between the two. We stick here to the friction-less Black-Scholes world (continuous trading and lending/borrowing are feasible at no cost). We’ll only relax the hypothesis that the “true” volatility of the underlying process is known and constant. We’ll hedge using the Black-Scholes formulae with a volatility that may be random but only through dependence on the current underlying. With this assumption, our assumed option price is a function of only and : The bond price dynamics are defined by Denote the true underlying process as: (this is a general positive diffusion, and may be stochastic).

The trader on the other hand assumes: Applying Itô’s lemma to , the true dynamics of are

Let be the self-financing hedge portfolio intended to replicate the option value. It is made up of a quantity of and a quantity of (both stochastic processes): Following Black-Scholes, the trader uses The self financing property is by definition: which gives us (writing in terms of and ):

Under the trader’s assumptions, the function verifies the Black-Scholes *equation* with :
(note may not be the usual Black-Scholes *formula*, as we did not assume to be
a constant)

From expanding and with the above and simplifying, we find which solves to which shows our approximate derivation above yields a correct result.

For those who didn’t read the optional section, we’re now using to denote the true volatility of the underlying (so that what we called “realized variance” is now ). Let’s assume and are constant. Assume again interest rates are zero.

At maturity , as shown above, with , we have: With and constant, and using the approximation we can write Taking (risk-neutral) expectations: Since is the Black-Scholes formula under our assumptions (true volatility and hedging volatility constant) we then have

Let’s introduce the vega, noted , the (illegitimate, out-of-model) sensitivity of to :

Identifying the equation above with a Taylor expansion of as a function
of yields
This gives an interpretation of vega as proportional to the **average gamma trading costs**.

The gamma is more of a short-term matter, while the vega is more of a long-term one. Relating to this, the figure below shows how the vega declines with time, while the gamma spikes around the strike when approaching maturity (note vega and gamma are the same for puts and calls).

Note there is a connection between and derived directly from the Black-Scholes formulae for the Greeks: but this local relationship does not provide as much insight, in our view, as the equation above.

Next time trading an option itches you, think twice: are you prepared to take a view on short term realized vol (“trade” the gamma), or to take a view on longer term implied vol (“trade” the vega), then you’re ready to move on to the next step (it is not the end of the journey yet). Now you need to understand the all-important impact of dividends on options, and the crucial differences between American and European styles, in particular with respect to dividends (most stock and ETF options are American).

In line with our long term investment approach of capturing risk premia, we do think some option strategies can add value to an investment portfolio. It takes time and effort to implement properly. If you’re interested in the details, get in touch.

- If you want to know more about the difficulties arising when managing a book of options as opposed to toying with equations on a single option as we have done here, then have a look at Dynamic Hedging by Nassim Taleb.
- If you’re curious about the technicalities involved in building volatility models to help manage a book’s risks, read The volatility Surface by Jim Gatheral, heavy on maths, but short and to-the-point.
- Source of data about ETF option markets: The OCC, the single equity option clearer in the US.