Options

E - Options and other derivatives

Derivatives are useful financial instruments to increase return or to hedge risk (as illustrated in Chapter 4 Section E and in Chapter 13 Section 13G). A complete discussion of all forms of derivatives is beyond the purpose of this text. But, a few guidelines on how some typical derivatives are priced should convince readers that valuation techniques for derivatives follow the same philosophy as the valuation of all other financial assets.

There are essentially three types of derivatives: forward contracts, futures and options. A forward contract is to buy or sell some asset at specified price and date in the future to an identified counter-party. Foreign currency forward contracts for 30, 60 or 90 days can be arranged with any major bank. Selling stocks short (i.e. selling stocks that one does not own now), is also a forward contract. The problem with forward contracts with individual investors as counter-party is that an individual may renege on a promise. Futures avoid this problem by having individuals deal with institutions (e.g. an exchange, or a clearing corporation) without knowing who is the counter-party, and by having standardized contracts. There are futures to buy commodities, major currencies, government securities (called interest rate futures) and stock indexes (called index futures). Options are also handled through an exchange, but they are different from futures in that a holder of an option does not have to exercise it if it is not beneficial (but the writer does). There are options to buy called call option and options to sell called put option. Futures and options are often not exercised but canceled with a reverse contract. For instance, a promise to buy is canceled by a promise to sell the same item.

Warrants and shareholder preemptive rights (as well as stock options given to key employees) are similar to call options since they give an owner the right to buy shares of corporate stock. Options differ from rights and warrants in that options are written (i.e. issued and sold) by any investor in an organized options market through a broker, whereas rights and warrants are issued by the corporation and distributed directly to existing shareholders or bondholders. Another difference is that all call and put options have an expiration date, but not all warrants, rights and employee stock options. As indicated above, futures and options are traded on exchanges, and their price is determined by supply and demand. That price is affected by elements of value discussed below.

1) - Valuation of options

The price of an option (including the initial price paid which is called a premium), stems from two elements:
- 1 intrinsic value, or difference between market price of the underlying stock and strike price (which is also known as exercise price because it is the price at which an option holder can buy from or sell to the option writer the underlying stock through the options exchange), and
- 2 time value, which represents the possibility that prices may go up or down in the future.

There is a difference between American options which are exercisable at any time, and European options which are only exercisable during a designated period just prior to expiration. The difference is not material because American style call options have time value up until expiration, and it would not make sense to exercise an option prior to expiration because of this time value: one would usually sell it instead.

Graph G-3.1 below represents the pattern of a call option value with a strike or exercise price of $40.

Graph G-3.1

The maximum value Vmax of the call option is the stock price itself (i.e. one would not pay more for the right to buy a stock than the current price of that stock): it is represented by a Vmax line at a 45 degree angle. An option cannot have a negative value (because one remembers that an option does not have to be exercised). Therefore, the minimum value Vmin of the call option is zero as long as the stock price is below the strike price of $40. If the stock price is above the strike price, Vmin is equal to the excess of the stock price over strike price (i.e. the intrinsic value). If the price of the stock is below the strike price, the option is said to be out-of-the-money, and its value V stems only from time value. If the stock price is above the strike price, the option is said to be in-the-money, and its value V is the sum of intrinsic plus time value. (If the strike price equals the market price, the option is said to be at-the-money.) The further away is the expiration date, the larger is the time value: as time approaches expiration, the value curve V shifts ever closer and closer to the intrinsic value.

Graph G-3.2 presents the pattern of a put option value V with an exercise price of $40.

Graph G-3.2

The maximum value Vmax of the put option is the strike price (i.e. one would not pay more for the right to sell a stock than the price at which the stock can be sold in the market). Since, an option cannot have a negative value, the minimum value Vmin of the put option is zero if the stock price is above the strike price of $40, and if the stock price is below $40, it is equal to the excess of the strike price over the stock price (i.e. the intrinsic value). If the price of the stock is above the strike price, the option is said to be out-of-the-money, and its value V stems only from time value. If the stock price is below the strike price, the option is said to be in-the-money, and its value V is the sum of intrinsic value plus time value. The further away is the expiration date, the larger is the time value: as time approaches expiration, the value curve V shifts ever closer and closer to the intrinsic value.

Most finance textbooks give a mathematical model developed by Fischer Black and Myron Scholes for pricing options. The Black-Scholes option value V_c is

V_c = P₀N(d₁) - Se^-rt N(d₂)

where P₀ = current stock price: S = strike price; r = risk free rate; t = time remaining to expiration in years or fraction of year; e = natural logarithm base 2.7183...; N(d₁) and N(d₂) are cumulative probabilities from standard normal distribution with; d₁ = (ln(P₀/S) + (r + 0.5sigma²)t) / sigma t^.5; d₂ = d₁ - sigma t ^.5
where ln(P/S) = natural logarithm of P/S: sigma = standard deviation of stock's annualized continuously compounded rate of return
The formula shows that the value of a call option
- increases with a higher stock price
- decreases with a higher strike price
- decreases with time closer to expiration (i.e. in the present value of strike price)
- increases with higher risk free rate (i.e. also in the present value calculation of strike price)
- increases with volatility of stock (measured by the standard deviation sigma) because it gives more opportunity for positive time value from possible higher stock prices, without any detriment from potential lower stock prices
Although the call option formula appears complex, one can observe that as t gets closer to zero, N(d₁) and N(d₂) tend to 1 and the value of the option tends to its intrinsic value P-S, if positive.

The value of a put option V_p can be derived from the call option value because of the put-call parity which is discussed in Chapter 4 Section E

V_p = V_c - P₀ + Se^-rt

where V_c = value of call option: P₀ = stock price; S = strike price; r = risk free rate; t = time remaining to expiration in years or fraction of year; e = natural logarithm base 2.7183...; Se^-rt = present value of strike price S

Substituting for V_c, gives

V_p = P₀N(d₁) - Se^-rt N(d₂) - P₀ + Se^-rt

= P₀(N(d₁) - 1) - Se^-rt (N(d₂) - 1)

Empirical results show that the call option formula is reasonably accurate, with the exceptions of i) stocks that pay dividends, ii) options that are out-of-the-money, and iii) stocks with unusually high or low volatility.

2)- Valuation of warrants

Warrants are essentially call options. As explained in Chapter 13, warrants have an exercise price usually set above the current market price of the stock. This makes the warrant a call option that is out-of-the-money. Consequently, the value of a warrant is initially only its time value.

3)- Valuation of preemptive rights

A preemptive right is given to each existing common shareholders for each common share owned when a corporation decides to raise new capital with a seasoned stock issue that is first offered to its own shareholders. The corporation stipulates the number of rights N needed to subscribe to one new share and the additional amount S that needs to be paid to the corporation to exercise the rights. Thus, to buy a new share, an investor must have N rights to subscribe to one new share at a subscription price S.

If one right is issued for one share, the value of a right must be equal to the difference between the stock price P1 of a share with the right and stock price P2 without the right, regardless of the behavior of stock prices in the market. Thus

R = P1 - P2

An investor could acquire a share in the stock market by either paying stock price P2 for a share without right, or purchase a share from the corporation by paying the subscription price S and redeeming the required N rights. Thus, the equality between P2 and S plus N rights R

P2 = S + NR

The value of a right is therefore in terms of stock price P2 (i.e.a share without right)

R = (P2-S) / N

Substituting for P2 in the difference between P1 and P2 above, gives

R = P1 - (S + NR)

which solves for R

R= (P1 - S) / (N + 1)

See Chapter 13 for further application and discussion of preemptive rights.

See review questions .

See research assignments R-3.15, R-3.16 and R-3.17.

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Last modified: Jun/01/01 Next: Companies