This article is a re-post of a blog entry I accidentally deleted. Read on if you haven't read it before. You will never cover a call the same way again...

- Covered calls allow you to generate income on a price-neutral investment.
- Selecting the price and strike of a covered call can be difficult.
- This article presents a quantitative method for optimally pricing covered calls.

Writing covered calls is a well-known strategy for generating income from an open stock position. This author has found it difficult to find straightforward advice on how best to price a covered call. When is the premium too low to justify the cap on one's profit inherent in writing a covered call? And what is the relationship between premium and strike price that optimally balances reward and risk?

This article proposes a principled method for covered call pricing based on the gambling concept of expected value. This approach can assist the covered call writer in determining whether the premium for a call option is too low to justify a short call position. By reviewing the options chain, the trader can rule out poorly priced options and select the call that provides the best risk to reward ratio.

Overview of covered calls.

The owner of a long stock position that has appreciated substantially may decide that the stock will not appreciate much further. The owner can still generate income from this position by writing (selling) out-of-the-money call options with the expectation that the price will not rise to the strike before expiry. If the price stays below the strike, then the owner keeps the premium from the call sale and decides whether to repeat the strategy for the next option term. If the price rises above the strike, the loss on the option is offset by the gains in the underlying stock. The stock may be called away, but the owner has still reaped the benefit of the stock appreciation (up to the strike price) as well as the premium earned from the option sale.

The following is an example. Your author purchased SSO at $95.17 on February 3 and by February 14 it had risen to $101.33. The price was not expected to rise much further, so a February 28 $103 call was sold for $0.78 each. This effectively lowered the cost basis to $94.39. The trade off for this was a cap on profits at $103. Since SSO closed above $103 on February 28, the author lost out on the profits above the strike. The benefit received in return for the cap on profits was the $0.78 premium from selling the options.

Why did the author choose the $103 call for $0.78? Why not the $102 call for more premium or the $104 call for less? The method below provides a basis for selecting the best option.

Know when to hold 'em.

Sometimes, but not always, playing poker is as easy as Kenny Rogers suggests. If you know you have a 50% chance to win a hand, and the pot of money in front of you is more than the required bet, you will place the bet every time. No need to think about it. In the long run you will make money, because the times you win outpay the times you lose.

Even if you are likely to lose, a bet can still be profitable in the long run. If you know you have only a 33% chance to win a poker hand, but the pot has been built up to three times as much as the required bet, then you always make that bet. Two times out of three you will lose one bet (-2). But one time out of three you will win __three__ bets (+3). In the long run, you come out ahead.

As a further example, consider the roulette wheel with 36 numbers, the "0", and the "00". There are 38 separate outcomes. If the casino paid you 37x the bet to guess correctly, then you and the casino would break even in the long run because, on average, for every 37 losing bets you make, you will win 37 times your bet. If you are offered 38 times your bet, you will always make money if you play long enough, even though your chance of winning any one spin is a paltry 1/38. If you are offered only 36 times your bet (and this is what casinos usually offer) then you can see why the house always wins.

The above examples illustrate the gambling concept of expected value, which refers to the relationship between what you stand to win, what you stand to lose, and the probability of each. As you can see, whether a bet is profitable does not depend solely on whether it is likely that the bet will win. If the prize for winning is large enough, it might be profitable to play the game, even if your chance of winning any single spin of the wheel is low.

A mathematical representation of expected value.

Very simply, the expected value ("EV") of any bet is the sum of the amount you expect to win and the amount you expect to lose. The amount you expect to win is the prize times the likelihood of winning. The amount you expect to lose is the price of playing time the likelihood of losing. In mathematical terms:

EV = P[win] * $[win] + P[loss] * $[loss]

In the equation above, P is the probability and $ is the dollar amount of a win or a loss. You multiply each outcome by its probability and add these together to get the total expected value of your bet.

For example, say the casino has a game where you win 35% of the time. It costs $5 to play and you stand to win $15. Your expected value is $0.25:

EV = 35% * $15 - 100% * $5 = 5.25 - 5 = $0.25

You would happily mortgage your home and play this game at the high roller table.

If, however, the odds were slightly altered to a 32% win rate, then you would never play this game:

EV = 32% * $15 - 100% * $5 = 4.8 - 5 = -$0.20

The above examples are based on a simple win-loss scenario. Sometimes, there can be more than two possible outcomes, and each needs to be accounted for. For instance, say there is a game that costs $2 to play. Also assume that 20% of the time you will win $5, 20% of the time you will win $3, 15% of the time you will win $2, and 45% of the time you will win nothing. You calculate EV by adding up all of these scenarios:

EV = 20% * $5 + 20% * $3 + 15% * $2 - 100% * $2 = $1 + $0.60 + $0.30 - 2 = -0.10.

Because you lose a dime of value every time you play, in the long run you will lose at this game.

Generalizing expected value to investing.

In the above examples, we knew the chances of winning and losing. With roulette, we know we have a 1/38 chance of hitting our number. The odds of every casino game is known. By contrast, the behavior of the stock market is not necessarily probabilistic. Indeed, the stock market is doubly ambiguous -- not only do we not know what the price will be in the future, we do not even know the __probability__ of what the price will be in the future. Thus, before we apply these concepts to investing, we must first derive a model of the "odds" of the stock market. In other words, we need a basis for estimating the probability that stock XYZ will hit price $ at a certain time in the future. For purposes of this article, we will use a one-month time horizon.

Let P($) be a mathematical function that represents our model of future price performance. In particular, assume that P($) provides the probability that the stock will hit price $ one month in the future. With this in hand, we can now apply the concept of expected value to stock trades.

In our one-month trade, your gain or loss on the trade is the difference between the current price and the price (called "$") at month's end. You must multiply each gain or loss by the value of P($) and add all of that up to determine the total expected value. This is where an integral comes in handy to express the general form of the expected return of your investment:

__Equation 1__

In equation 1, each value of P($) at the price $ is multiplied by the gain or loss at that price and added together to provide a total expected value EV.

Probability models and the stock market: how normal is the market?

There are as many ways as there are investors to derive a probability function of future stock performance. For purposes of simplicity, we will assume that the market behaves stochastically over short time periods, i.e., it behaves like a system of random events like the spins on a roulette wheel. Following this assumption, we will assume more specifically that prices one month in the future are normally distributed according to the Bell curve based on the statistics (mean and standard deviation) of recent price performance.

These seem like big assumptions, and they are. But a back-of-the-envelope analysis shows they are not too far from reality. From 1/1/1994 to 12/31/2013, for SPY, I calculated the 63-day (3-month) standard deviation of price performance over time and called this "s". I also calculated the average daily gain over the previous 63 days and called this "d". Assuming a normal distribution, we can predict that the prices in 21 days (one month) will generally follow a Bell curve having a peak and standard deviation as follows.

Peak of Bell curve in 21 days = 21/2 * d + current price

Standard deviation of Bell curve in 21 days = sqrt(21) * s

To check the validity of our assumption that the market followed a Bell curve on our one-month time frame, I next determined the percentage of time that the price moved a certain distance from the center of the Bell curve and compared these percentages to what we would expect from a true normal distribution as shown below:

SPY 21-day estimated price distribution compared to normal distribution (n=5185) | |||

Standard deviations from peak | Normal distribution | SPY 63/21-day distribution | Delta |

0.5 | 38.20% | 39.10% | 0.90% |

1 | 68.20% | 70.70% | 2.50% |

1.5 | 86.60% | 89.30% | 2.70% |

2 | 95.40% | 96.20% | 0.80% |

2.5 | 98.80% | 98.90% | 0.10% |

While not perfect, these numbers show that our assumption is close enough to a normal distribution to apply the principles of expected value on a one-month time scale.

Expected value and covered calls.

Now we can begin to apply these concepts to the pricing of covered call options. Writing a covered call is like placing a bet except in reverse. When you sell the call, you immediately win some money, i.e., the premium. You then have to wait to see if the call costs you money when the stock rises above the strike price. What we want to know is, what is the expected value of entering the covered call position? Using the tools discussed above, we can calculate this value based on [i] the current price of the underlying stock, [ii] the current premiums and strike prices in the option chain, and [iii] an estimate of future price performance. For the latter, we use the normal distribution Bell curve based on the average and standard deviation of daily price changes over the last three months, as discussed above.

To determine expected value, we need to know the gains, losses, and probabilities of each gain and loss. The gain from a covered call is easy: it is the amount of the option premium, which you receive 100% of the time you sell a call option.

The loss from a covered call is more difficult to estimate. Any movement above the strike price of a covered call is considered a loss because you give up a chance to earn those profits. In the example of SSO above, the $103 short call puts a limit on the trade at $103. If SSO climbs to $104.50 when the option expires, we would not receive any of the returns above $103. The net loss from selling this call would be $104.50 minus the $103 strike minus the $0.78 premium, or $0.72. In other words, if you hadn't opened the short call, you would be $0.72 richer.

Caution: math ahead.

As in equation 1 above, to find the expected loss of selling a call option before we enter the trade, we need to [i] multiply the loss at each price point by the probability of that price point and [ii] add all of those together for every price, from the strike price (where losses begin) to infinity. Using a little calculus, we can represent the total expected value with the integral equation below:

__Equation 2__

The value "strike" is the option strike price and P($) is a function representing the probability that the stock will hit price $ at expiration in one month (21 days).

As discussed above, the function P($) is assumed to be a normal distribution Bell curve with a peak "U" and a standard deviation "S." These values are, in turn, derived from prior price performance. Thus, as we showed above, if d is the average daily price change over the past 3 months and s is the standard deviation of daily price changes over the same period, then we know that:

__Equations 3.1 and 3.2:__

With these values in mind, we can then present the equation for the probability of prices in one month:

__Equation 4:__

When equation 4 is substituted into equation 2, we get a final expression for the expected value based on the premium, the strike, the current price, and the predicted mean U and standard deviation S:

__Equation 5:__

Finding the best premium.

To decide whether to sell a call option, one desires a positive expected value. We set EV > 0 in the above equation and, via algebraic manipulation and a portion of the author's weekend, we can solve for the premium:

__Equation 6:__

The author used Excel to generate exemplary results of this equation, as depicted graphically below assuming U=100 and S=15 for strike prices from 101 to 130:

Using the expected value equation in an example.

Now let's apply this method to a covered call decision by looking at SPY. As I type this portion of the article, SPY was trading at about 186.7. Using Excel and data from Yahoo! Finance, I was able to calculate the average daily gain of SPY prices over the past 63 days (3-months) as 0.0979 and the daily standard deviation of price over that time period was 1.312.

Using equations 3.1 and 3.2 above, we estimate the peak of the price distribution a month from now ("U") to be 187.73. The standard deviation of that price distribution ("S") we estimate to be 6.01.

We can then make a table using equation 6 above that gives us the minimum premium we would pay at a given strike price. In this table, I have included the bid prices of the SPY March 28 calls as of the time of this writing. I have also included the difference between those prices and the minimum premium suggested by equation 6:

Strike | Minimum premium per equation 7 | March 28 SPY calls ((bid)) | Difference |

188.00 | 2.27 | 1.43 | -0.84 |

188.50 | 2.03 | 1.21 | -0.82 |

189.00 | 1.82 | 1.02 | -0.80 |

189.50 | 1.62 | 0.85 | -0.77 |

190.00 | 1.43 | 0.7 | -0.73 |

190.50 | 1.26 | 0.57 | -0.69 |

191.00 | 1.11 | 0.46 | -0.65 |

191.50 | 0.97 | 0.38 | -0.59 |

192.00 | 0.84 | 0.31 | -0.53 |

192.50 | 0.73 | 0.25 | -0.48 |

193.00 | 0.63 | 0.2 | -0.43 |

193.50 | 0.54 | 0.18 | -0.36 |

194.00 | 0.46 | 0.16 | -0.30 |

194.50 | 0.39 | 0.14 | -0.25 |

195.00 | 0.33 | 0.11 | -0.22 |

195.50 | 0.28 | 0.1 | -0.18 |

196.00 | 0.23 | 0.08 | -0.15 |

As you can see, given the recent upward trajectory of SPY over the past several months, it is not surprising that this method does not identify any calls that would provide positive expected value. The calls are all too cheap, i.e., the premium you get from the sale of the short call would not cover the risk of loss you would suffer due the cap on your expected profits.

It must be remembered that equation 6 is a special case that assumes a normal distribution of prices in the future. An investor may choose a different probability model of future price behavior. Any probability distribution that you wish to use may be substituted into equation 2 and solved for the premium according to the method above.

Whither commissions?

Commissions are generally incurred with every covered call, just like the premium. To account for these, reduce your premium by the per-share commissions and use this "effective premium" in the calculations described above.

Conclusion.

Using equation 6, which can be implemented in an Excel spreadsheet, the minimum recommended premiums can be estimated and used as a guide for covered call writing. Although equation 6 assumes a normal distribution of prices in the near future, the investor is not limited to a normal distribution pricing model, and can use any probability distribution model with equation 2, which presents the general form of the principles in this article.

Accordingly, the gambling concept of expected value can be applied to the pricing of covered calls to determine whether the option chain is offering sufficient premiums to cover the risk taken on when a call is sold.

**Disclosure: **I am long EDV.