NEW YORK (TheStreet) -- It can be a winning strategy to sell uncovered or unhedged at-the-money short-term puts, then wait until maturity and repeat the trade. Investors comfortable with the risk of equities can beat the index in the long run by exploiting a flaw in the classical option theory. *Note that with the levered strategy below you can theoretically lose more than the capital invested -- *think about it carefully.

The strategy is to sell unhedged at-the-money short-term puts on the **S&P 500**I:GSPC, wait until maturity, and repeat.

So why tell you this? I can hardly think of anything simpler. That's why I don't feel I'm revealing any substantial trade secret.

This strategy is different from the core strategy of the start-up hedge fund I'm running, and the current market regime might not be ideal to begin implementing this strategy. Or perhaps I'm just telling you this because I don't want too many people to bash front-end option premiums.

What makes this strategy interesting in my opinion is that:

- It outperforms the S&P 500 in the long run, in terms of return and risk.
- It exploits a flaw in classical option pricing theory.

Figures 1 and 2 compare the strategy's evolution since March 1994 vs. the S&P 500, rebased at 100 using monthly and weekly maturities. To make our results easy to reproduce, we ignored interest rates and dividends. Over the 20-year period, the S&P 500 registered a 0.41 Sharpe ratio. The monthly strategy is slightly better at 0.55, but the weekly strategy is much better at 0.76.

You start with $100 of capital, sell naked at-the-money puts with $100 notional, and roll on every listed expiry with a variable notional equal to the amount of capital. (With the S&P at 2,000, you would actually use $200,000 of capital and sell one CBOE contract with multiplier 100. **S&P 500 ETF** (**SPY**) - Get Report options may provide an alternative for smaller notional or for fine-tuning.)

Up next: the finer points.

The recursive formula for the strategy's capitalization simplifies to:

where *t* is the trade date, *t* + 1 is the expiry date, *K* is the accumulated capital, *p* is the at-the-money put premium collected and *S* is the S&P 500 index level. For example with $100 starting capital, a $10 weekly put premium and the S&P 500 at 2,000, the capital would grow to $100.50 if the S&P 500 closed at 2,000 or above after one week, and it would shrink to $99 if the S&P 500 dropped to 1,970.

Figure 3 shows the weekly strategy with an option leverage factor of 2.

*Note that with the levered strategy you can theoretically lose more than the capital invested.*

**Figure 3. 2x-leveraged weekly strategy vs. S&P 500 index**

We have a trading strategy that back-tests well -- and for many people that's good enough. At Ogee, however, we like to know *why* a particular strategy works and have some assurance that it will continue to work.

As every good option trader knows, the price of a short-term at-the-money call or put is approximately:

0.4σ*

*S*

√

*T,*

(Eq. 1)

where σ* is the option's implied volatility, *S* is the index spot level and *T* is the time to expiry measured in years. For short maturities σ* tends to match the current realized volatility regime of the S&P 500. This proxy is classically derived using a first-order approximation of the Black-Scholes formulas. (See my book on options for more information: Bossu and Henrotte, *An Introduction to Equity Derivatives: Theory & Practice* 2nd Ed., from John Wiley & Sons.)

What intrigued me is the root-time order of the formula: as *T* goes to zero the at-the-money option price goes to zero as required, but far more reluctantly than *T*.

Why should this at all be intriguing? Because continuous capital compounding is of order *T*, not *√T*. Suppose for a moment that the S&P 500 remains flat and that you can continuously sell instantaneous at-the-money-options for 0.4σ**S*√*dt*. The differential equation for capital appreciation is then *dK* = *K*/*S* 0.4σ**S*√*dt* = 0.4σ**K*√*dt*, i.e. *dK/dt* is of order 1/√*dt* which goes to infinity and *K* thus solves to infinity! In contrast if you receive interest *rdt* your capital equation is *dK* = *Kr**dt* which solves finitely to exp(*r**t*) over a time period *t*.

So the root-time behavior of Equation (1) suggests that repetitively selling naked at-the-money options is more lucrative with short-term expiries than long-term expiries, as previously empirically verified. But of course the S&P 500 does not remain flat over an entire week or month, and the true equation governing the strategy's capital appreciation in discrete time is:

Statisticians may now analyze this type of process at will using the model of their choice for the S&P 500 spot price evolution in the "real world" (as opposed to the "risk-neutral world" used for option pricing.) Here it is worth emphasizing that if we use the classic geometric Brownian motion *dS**t*/*S**t* = *µ**dt* + *σdW**t* then (*K**t*) only admits a continuous-time limit when σ = σ*. The implication is that classical theory fails to properly price very short term at-the-money options.

The good news as far as our put-selling strategy is concerned is that this flaw isn't really news: professional option traders have long known about the "pin risk" that occurs when the index level is close to the strike price near the option's expiry. As such it can be assumed that historical market prices of short-term at-the-money options reflect true supply and demand rather than market-makers using a "wrong" pricing formula waiting to be corrected.

*This article is written for educational purposes and does not constitute investment advice or a solicitation to invest in any security offered by Ogee.*

*This article represents the opinion of a contributor and not necessarily that of TheStreet or its editorial staff.*

Sébastien Bossu currently runs a start-up hedge fund,

Ogee

, and is Adjunct Professor of Finance at Pace University. He has almost 10 years' experience in banking and the financial industry and has published several papers and textbooks in the field. His latest textbook

*Advanced Equity Derivatives: Volatility & Correlation*

was published in May by John Wiley & Sons.