An option pricing model can be the first step in finding profitable equity options trading opportunities. Using pricing models effectively, we can uncover discrepancies that we may be able to exploit with speculative positions. Pricing models may be applied to call options, as shown in “Wells Fargo calls” (below), or to put options, which will be described later.

On Aug. 9, 2013, Wells Fargo shares were priced at $43.23. For strike prices ranging from $50 to $43, market prices for calls extended from 0.250 to 2.240 with premiums from $25 to $224. The model shows call prices predicted by regression analysis based on the market prices for nine strike prices.

Price variations between predicted and market values are small, with the largest difference at $3.51 at $135 market premium. The column for price variations is the first place to look for potential trades: Buying undervalued options and hedging over short time periods with concurrent sales of overpriced options on the same equity. Buying the $46 strike at $95 and selling the $45 strike at $135 is an example that ended on Aug. 16 with the $46 strike at $83 and the $45 strike at $130, for a net gain of $100.

Because this trade is hedged against movements in the underlying stock, it is possible to hold out for a profit, although the chance for larger gain is stronger close to the original buy and sell date.

A long straddle also is possible when there are underpriced options on both the puts and calls. Buy an undervalued call and put with the intention of profiting when the prices return to normal no matter the direction of movement in the underlying stock. Regression analysis may be used to value puts as well as calls; however, arbitrage pricing usually keeps puts and calls close to parity, as we explain later.

A straddle trade in Starbucks options was possible on Sept. 6, 2013, with the stock at $71.57. The January 2015 $75 call and $70 put were underpriced by $9.28 and $13.08 according to regression analysis. By Sept. 12, Starbucks’ stock had increased to $75.67. Changes in the call and put prices were plus $212 and minus $134 respectively, for a $78 net profit before trading cost.

**Volatility’s role**

Expected, or implied, volatility is the primary factor underlying the price of options for equities, futures and any contract in which the holder has the right to buy or sell at a fixed price over a specified time period an underlying asset or liability whose price is free to vary. Traders in the options market determine implied volatility based on the fluctuation of the underlying price — including historical price movements and expected future variations.

Based on the assessment of volatility, the options market sets an upper and lower price range for the underlying through the expiration date. On “Wells Fargo calls,” an option price is computed for each of nine strike prices, with the underlying stock at $43.23. Upper and lower prices are breakeven stock prices at the January 2014 expiation date that result in zero gain or loss from a delta-neutral trade on Aug. 9 (hedging the number of options sold short determined by the inverse of the slope at a specific strike against a long position in the underlying stock).

“Wells Fargo price curve” (below) shows the dynamics of pricing a call for which the strike price is equal to the stock price. A line tangent to the option’s price curve has a slope equal to 0.575. The upper breakeven price occurs where the sloped line intersects the horizontal axis at $48.207. The lower breakeven price is $39.558, directly beneath the intersection of the sloped line and the intrinsic value line, on which each point equals the stock price less the $43.23 strike price. The sloped line on the chart is lowered slightly to make it easier to see the details.

The height of the option price curve where the stock equals the strike price can be used as a measure of relative volatility. “Four equities” (below) shows volatilities compared for Facebook, Google, Starbucks and Deere.

The heights along the curves equal the ratio of call price-to-strike price, with the primary comparison made where the ratio of stock price-to-strike price equals 1.0. It is easy to see that Facebook is far above the other three equities in terms of implied volatility, roughly twice the height of either Google or Deere. The volatility measure for Facebook January 2014 calls is approximately the same as that for Deere’s January 2015 options, equalizing the time value of 16 months for Deere with four months for Facebook.

**Delta neutral**

The column of delta values, or slopes, in the option price curve at each strike price may be used in the design of delta-neutral trades — the sale of calls hedged by a long position in the stock.

The number of calls in the trade is determined by the slope, equal to 1/delta. For Wells Fargo, using the $44 strike relates to a slope of 0.509 so that 1.96 calls are sold for each share held long. As indicated by “Wells Fargo calls” and “Wells Fargo delta neutral” (below), we know that the breakeven stock prices are $48.30 and $39.85 with the peak profit $338 at the strike, $44.

The regression analysis used on “Wells Fargo calls” can be used to compute the heights of put and call option price curves. Applied to equities in an industry such as banking, the results help us understand how the stocks are related. The chart on “Four banks” (below) shows movements in the cumulative percentage price changes over a five-month period beginning April 2, 2013. Similar short-term price changes occur each day at the same time the spread between prices becomes larger or smaller. Because all large banks are controlled by the same economic, regulatory and competitive factors, it isn’t surprising that they move more or less as a unit.

On “Four banks puts and calls” (below) differences in volatility are seen as calls increase in price when the stock price increases relative to strike price while the put price curves fall with stock prices that are larger than the strike. Citicorp and Bank of America are the strongest banks as measured by implied volatility, while Wells Fargo is represented by the lowest price curve for put and call options. Reflecting the similarity in stock price movements on “Four banks,” the price curves all are in the same area of volatility — no Facebook level of volatility is shown by the banks at the present time.

**Parity pricing**

Based on the market price of the call option at a specific strike, a put option can be created by the following process: 1) buy the call at its market price, 2) sell the underlying stock at market price and 3) buy a bond at a discounted price that will equal the value of the strike price at the expiration date.

Arbitrage ensures that put prices will be related to calls at or near parity. For example, on Sept. 16, 2013, Citigroup stock is $51.00. One strike price is $45 with the corresponding call priced by the market at 7.05.

The current discount rate is one-half of 1% for one-third of a year to expiration, so the formula used to compute the put’s value is (-7.05 + 51 – 45 / (1.005)^0.333) x –1. The computed parity price, 0.9753, is close to the put’s market value of 0.9500.

The options market usually calculates the price of puts at or near the parity price with call options. “Citigroup January 2014” (below) is an excellent example of parity pricing. When calls are mispriced, the corresponding puts may be overvalued or undervalued at the same time. This pricing relationship supports the long straddle trades described above, with the possibility of originating a trade based on an undervalued call.

*Paul Cretien is an investment analyst and financial case writer. His e-mail is PaulDCretien@aol.com. *