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# Option Pricing Tutorial

by Jerry White

It is important to distinguish between an option premium and the theoretical or fair value of an option. An option premium is the price which an option buyer and seller agree to when they enter into an option contract; it is a function of supply and demand. The fair value of an option is a price derived from a mathematical model. Subject to the accuracy of the data and the assumptions and constraints of the model, fair value is the price at which both the buyer and the seller of an option should expect to break even. Therefore, fair value is an estimate of where an option should sell in an efficient market.

### Black-Scholes

The Black-Scholes model, for European options (options exercisable only at maturity) on non-dividend paying stocks, was published in 1973. Black and Scholes determined that, under certain assumptions, it is possible to set up a perfectly hedged or riskless position consisting of a long position in an underlying stock and a short position in options on that stock (or vice versa) such that any profit resulting from an instantaneous increase in the stock price would be exactly offset by a loss on the option position, or vice versa. The option premium at which this equilibrium condition obtains is the fair value of the option.

The fair value of an option has two components: intrinsic value and time value. Intrinsic value is the sum of money that would be realized if the option were exercised today. Time value reflects the potential risky changes in the price of the underlying instrument from today until expiration. Fair value is a function of five variables: price of the underlying instrument, time to expiration, strike price, interest rate, and volatility.

For a call option, intrinsic value is the amount by which the price of the underlying instrument exceeds the option striking price; for a put option, it is the amount by which the price of the underlying instrument is less than the striking price. For a fixed striking price, the intrinsic value and fair value of a call option increase, dollar for dollar, with increases in the underlying instrument price; conversely, the intrinsic value and fair value of a put option increase as the underlying instrument price declines. Intrinsic value may not be less than zero, as the option buyer is under no obligation to exercise at a loss. An option that has some intrinsic value is said to be in-the-money; an option that has no intrinsic value is said to be out-of-the-money. An option whose strike price is equal to the market price is said to be at-market.

Time value is determined by the interrelationship of all the key option variables:

Time to expiration. All other things being equal, the more life an option has, the greater its time value. Thus, fair values of options with far-out expirations are greater than those with nearby expirations. Time value diminishes to zero on expiration day. It is this characteristic which makes an option a "wasting asset".

Underlying instrument price vs. strike price. Time value decreases for strike prices that are increasingly far from the current underlying commodity price. The reason is that there is a decreasing likelihood that price fluctuations will cause the underlying price to reach the striking price. Thus, option fair values for increasingly out-of-the-money options approach zero as time value approaches zero. For increasingly in-the-money options, option fair values approach intrinsic value as time value approaches zero.

Interest rates. When other variables are held constant, time value for call options rises when the domestic interest rate increases and falls when the foreign interest rate increases. For puts, the opposite relationship is true. Thus, when domestic interest rates increase, call option values increase and put option values decrease; when foreign interest rates increase, call option values decrease and put option values increase.

Volatility. Volatility is an annualized statistical measure of the degree of day-to-day fluctuation of the underlying instrument price. Increases in volatility increase option fair value. Decreases in volatility lower option fair value.

Volatility has the largest impact on time value and fair value of any of the variables mentioned. Yet, it is the one variable which cannot be stipulated from contract terms or inherent market prices. It is relatively easy to calculate a volatility based on a history of underlying prices, but history, while a good reference, may not always be an adequate predictor of the future. The ideal volatility is that which will correctly predict the magnitude of price fluctuations during the life of an option. Ultimately, this is a matter of judgment.

One of the preferred methods of determining volatility is to look to the option market itself. For an actively traded option with a strike price at or near the market, arbitrage opportunities will keep the market price generally in line. It is reasonable to assume that the market price for such a liquid option is "fair." We can then determine, by iterative solution, the value of volatility that will cause an option model to yield a fair value equal to the observed market price. This volatility is called the implied volatility. If the implied volatility is then used to calculate fair values of all options, the resultant fair values will be consistent with the originally selected liquid option's fair value, and market opportunities relative to the selected option will be apparent.

The hedge equilibrium condition which is the basis for the Black-Scholes model leads to two additional significant relationships: the hedge ratio or delta, and put-call parity.

Delta The delta is the first derivative of option value with respect to underlying price and describes how much the option value will move for each unit move in the price of the underlying instrument. A delta of 0.5 means that for each one cent move of the underlying instrument, the option will move a half cent in the same direction. Thus, a delta-neutral position appropriate for a 0.5 delta consisting, say, of two short IMM DM options and one long IMM DM futures contract will change very little in value if the futures contract goes up or down one cent. Large moves of the underlying commodity change the delta, however; so, to maintain a neutral hedge, it is necessary periodically to re-balance options and underlying contracts. The delta may also be regarded as a surrogate for the probability that the option will be exercised at expiration. The option seller should use the delta to manage his short option position so that he is covered if he receives an unexpected assignment.

### Put/Call Parity

The hedge equilibrium condition also leads to the put/call parity relationship. Put/call parity provides that a put option can be converted to a call option having the same strike and expiration if it is combined with a long underlying instrument position (for the same quantity). Conversely a call option can be converted to a put option if combined with a short underlying instrument position.

For example, a put option on 1 million Swiss francs with a strike of 55 cents and expiration date of January 1st plus a sale of 1 million Swiss francs for January 1st at a price of 55 cents, is equivalent to a call option on 1 million Swiss francs with a strike of 55 cents and expiration date of January 1st.

The reciprocal nature of this relationship means that for European cash options, the deltas for a put and call option for the same strike and maturity will always add up to exactly one.

It should also be noted that the deltas for an at-market European cash call and put option pair, while summing to 1.0, nevertheless are not each 0.5. That is, they are not equally likely to be exercised. There are two reasons for this: first, the lognormal price distribution assumed in the model assumes an upward bias; that is, prices can move up infinitely, but cannot move below zero. Second, any non-zero interest rate has the effect, over time, of pushing underlying prices up because of the costs of financing the underlying instrument.

For European options on futures, the deltas add up to less than one. This is so because the expected value of a futures contract is today's futures price and, in taking the first derivative of the option value formula, a present value term is applied to the delta, discounting the value from maturity to the present time. (At maturity, then, the deltas will add up to one.)

For American options, which may be exercised early, and for which both the call and put for a given strike and maturity may yield a profit and be exercised at different times, the deltas may add up to more or less than one. Strictly speaking, because of early and independent exercise, put/call parity does not apply to American options.

If there are payouts against the underlying instrument, such as stock dividends, bond coupon yields or commodity interest rate differential (domestic vs. foreign), fair value, delta calculations and parity relationships are even more complex. With currencies, for example, both the spot and forward price must be taken into consideration; so long as the interest rate differential is not zero, these prices will be different. If forward prices for a commodity are at a premium to spot, calls will be based on the forward prices while puts will be based on spot. Conversely, if forward prices for a commodity are at a discount to spot, calls will be based on spot and puts will be based on the forward prices.

### Other European Models

The Black-Scholes model has been the foundation for most other option valuation models that have been developed subsequently. The model has been adjusted for stocks with dividends and physical commodities with carrying costs. Black adapted the model to European options on futures on the basis of the spot/forward price relationship in a full carry market. More recently, Garman and Kohlhagen described an adaptation of the Black-Scholes model for options on foreign currencies, wherein the foreign interest rate, domestic interest rate, and the interest rate differential are appropriately used in the calculation. Thus, with appropriate modifications to account for dividends, interest rates, or carrying costs, the Black-Scholes model has proven to be an appropriate basis for valuing European-type options on physical commodities, financial instruments, and currencies, as well as stocks.

With regard to futures options, the futures prices contain the statistical information about foreign interest rates (or dividends, or coupon, or physical carrying cost) as well as subjective and intuitive factors that do not need to be modeled. Therefore, the appropriate underlying futures price can be used directly, and no correction for dividends, coupon, or foreign interest rate needs to be provided for. Black's 1976 futures option model is appropriate for valuation of European options on futures.

### American Options

In the commodity markets, over-the-counter options are usually European. But nearly all exchange-traded options are American-type options. An American-type option allows the option holder to exercise the option at any time prior to expiration. This characteristic produces additional risks for the option writer, and additional opportunities for the option buyer.

For American put options on physical instruments, and for both American calls and puts on futures, early exercise may be advantageous. It may be worth more to the option holder to take a certain profit early and invest the proceeds than to forego the use of the funds until expiration.

Whereas the Black-Scholes European option model and most of its derivatives can easily be programmed, American-type option valuation is much more complex. In fact, there has been no accepted closed-form solution to the general case of valuing American options. The mathematical approaches generally used, such as the binomial method, involve extensive iteration and require substantial computational power.

But American call options on physical (cash) instruments are rarely exercised early. It has been shown that the value of an American physical call option is always at least as great as the value it would have if exercised immediately. This has given rise to the expression that an American physical call option is "worth more alive than dead." Thus, a rational investor will not exercise an American physical call option early.