The principles discussed primarily apply to traded options on listed stock but many of the heuristics can be applied to non-traded options or options on non-traded stock. Options, which come in the form of calls and puts, grant a right, but not an obligation to a buyer. As a result, plain vanilla options can be worth something or nothing at expiry; they cannot be worth a negative value to a buyer since there are no net cash outflows after purchase.
A seller of plain vanilla options is on the opposite side of the trade and can only lose as much as the buyer gains.
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It is a zero-sum game when this is the only transaction. A call on a stock grants a right, but not an obligation to purchase the underlying at the strike price. If the spot price is above the strike, the holder of a call will exercise it at maturity. The payoff not profit at maturity can be modeled using the following formula and plotted in a chart.
Based on the strike price and stock price at any point of time, the option pricing may be in, at, or out of the money:. The distinction of moneyness is relevant since options trading exchanges have rules on automatic exercise at expiration based on whether an option is in-the-money or not. The Options Clearing Corporation has provisions for the automatic exercise of certain in-the-money options at expiration, a procedure also referred to as exercise by exception.
The option pricing will hence depend on whether the spot price at expiry is above or below the strike price. Intuitively, the value of an option prior to expiry will be based on some measure of the probability of it being in-the-money with the cash flow discounted at an appropriate interest rate.
Though options have been in use since the historical period of Greek, Roman and Phoenician civilizations, Fisher Black originally came up with this option pricing model in , extensively used now, linking it to the derivation of heat-transfer formula in physics.
The formula looks as follows:. For calls, their value before maturity will depend on the spot price of the underlying stock and its discounted value, then the strike price and its discounted value and finally, some measure of probability. The components of this break down as follows:. The remainder of the calculation is all about discounting the cash outflow at a continuously compounded discount rate, adjusting for any dividends, or cash flows before maturity and, for probability using a normal distribution. The BSM model assumes a normal distribution bell-curve distribution or Gaussian distribution of continuously compounded returns.
The model also implies that as the ratio of current stock price to exercise price increases, the probability of exercising the call option increases, taking N d factors closer to 1, and implying that the uncertainty of not exercising the option decreases. As the N d factors get closer to 1, the result of the formula gets closer to the value of the intrinsic value of the call option. N D2 is the probability that stock price is above the strike price at maturity. N D1 is a conditional probability. The term D1 combines these two into a conditional probability that if the spot at maturity is above strike, what will be its expected value in relation to current spot price.
The following model is what I use in Excel for BSM calculations the shaded cells are calculations linked to other cells :. A call lets the buyer enjoy the upside of a stock without really holding it for a period till expiry. Intuitively, if the upside is paid out during the period of holding, then the calls should be less valuable since the right to that upside is not being derived by the option holder. Of course, the reverse applies in the case of puts. The model assumes that dividends are also paid out at a continuously compounded rate.
Now that special dividends are being discussed due to changes in the US tax code, it is worth mentioning that you will see an adjustment factor to traded options for one-time dividends above a certain percentage of the stock price. One-time special dividends have a big impact on option pricing. Options Industry Council OIC has a free calculator which will display the traded option values and the greeks.
The last traded price of calls and puts are clearly correlated to the strike price and form this hockey stick-esque graph.
Should I Offer Stock Options to My Employees?
What happens when the spot price changes for AAPL? Intuitively, and based on the BSM model the option pricing also should change too. This is measured by Delta, which is the approximation of how the value of an option changes for a change in spot price. Delta is used as a hedging ratio. If you are looking to hedge an underlying position with an option that has a delta of 0. Delta is an approximation, though. It works well for a small movement in price and for short periods of time. We see the relationship of the call to changes in stock price below as well as the change in delta over the same range of stock prices.
This becomes more noticeable nearer to the strike price. Gamma is always a positive value and Delta is positive for a call and negative for a put for the buyer. Gamma or the rate of change in delta approaches zero as the strike price moves away from the spot price for deep out-of-the-money or in-the-money option positions. Intuitively, the longer the time to expiry, the higher the likelihood that it will end up in-the-money.
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Hence, longer dated options tend to have higher values, regardless of whether they are puts or calls. The time value subsequently decays to 0 as it nears expiry. The rate of decay is not a straight line. It is easier to think of it using the analogy of a ball rolling down a slope. The speed picks up as the ball rolls further down the slope—slowest being at the top and fastest at the bottom at expiry. The rate of decay is represented by Theta and is positive for calls and puts.
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Interest rates have an impact on option value through the use as a discount rate. Intuitively, calls imply getting the upside of holding the underlying shares without dishing out the full price. The sensitivity to interest rates is measured by Rho, with higher interest rates increasing the value of calls and vice-versa for puts.
Vega, though not actually in the Greek alphabet, is used to denote the sensitivity of option value to volatility.
What You Need to Know About Stock Options
Volatility refers to the possible magnitude of price moves up or down. The higher the volatility from a spot price, the higher the likelihood that the price may reach the strike. Hence, the higher the volatility, the higher the price of options. Implied volatility is calculated with the BSM Model, using the traded prices of options. IV has become a traded asset class by itself in through VIX options. If you buy an option in a very calm market and there is a sudden uptick and downtick in the price of the underlying, with the price ending back where it was before, you may see that option pricing has increased in value.
For qualifying positions , your cost basis is simply the acquisition cost allowing the discount received to be reported as a capital gain instead of ordinary income. These stock units are awarded to an employee as a form of compensation. The employee does not receive the stock at the time of the award, but has a specific vesting plan outlining when the employee will receive the stock. At the time the stock vests, the employee receives the units and the fair market value FMV of the stock received on that date is considered income. The amount reported to you as income on Form W-2 by your employer at the time the stock vests will then be your adjusted cost basis in these stock units.
ISO units must be held for at least one year after the options are exercised. In addition, you cannot sell the shares until at least two years after the options are awarded to you. For these reasons, any discount you receive by purchasing these options are taxed as a long-term capital gain , which yields a lower tax rate than ordinary income.
When these options are granted, they are granted at a predetermined price. When option is exercised, the employee has ordinary income for the difference between the price they pay grant price and the fair market value FMV on the date they purchased the stock exercise price.
Any compensation income received from your employer in the current year is included on Form W-2 in Box 1. Review Boxes 12 and 14 as they list any income included on Form W-2 related to your employee stock options. You will receive a Form B in the year you sell the stock units. The form reports any capital gain or loss resulting from the transaction on your tax return.