Chapter 3 Principles of Option Pricing 1 Basic

Chapter 3 Principles of Option Pricing 1

Basic Notation And Terminology • S₀ = Stock price today (time 0 = today) • X = exercise price • T = time to expiration is expressed as a decimal fraction of a year. For example, if the current date is April 9 and the option’s expiration date is July 18, we simply count the number of days between these two dates. That would be 21 days remaining in April, 31 in May, 30 in June, and 18 in July for a total of 100 days. The time to expiration therefore would be 100/365=0. 274 2

• r = risk free rate, is the rate earned on a riskless investment. • Sᴛ =stock price at option expiration; that is, after the passage of a period of time of length T. • C (S₀, T, X) = price of a call option in which the stock price is S₀, the time to expiration is T, and the exercise price is X. • P (S₀, T, X) = price of a put option in which the stock price is S₀, the time to expiration is T, and the exercise price is X. 3

Principles of Call Option Pricing Minimum Value of a Call If the call holder sees that it is advantageous to exercise it the call will be exercised. If the exercising it will decrease the call holder's wealth, the holder will not exercise it. The option cannot have negative value, because the holder cannot be forced to exercise it. Therefore, C (S₀, T, X) ≥ 0 4

• For an American call, the statement that a call option has a minimum value of zero is dominated by a much stronger statement: Ca (S₀, T, X) ≥ Max(0, S₀ - X) • The expression Max (0, S₀ - X) means: “Take the maximum value of the two arguments, zero or S₀ - X. ” • The minimum value of an option is called its intrinsic value. 5

• The intrinsic value of an American call is the greater of zero or the difference between the stock price and the exercise price. • Intrinsic value which is positive for in-the-money calls and zero for out-of-the-money calls, is the value the call holder receives from exercising the option and the value the call writer gives up when the option is exercised. 6

Example: • Consider the DCRB June 120 call. The stock price is $125. 94 and the exercise price is $120. • Evaluating the expression gives Max(0, 125, 94 – 120) = 5. 94 call option price. 7

1 - If the call were priced at $3. • An option trader could buy the call for $3, exercise itwhich would entail purchasing the stock for $120 -and then sell the stock for$125. 94 • This arbitrage transaction would net an immediate riskless profit of $2. 94 on each share. • All investors would do this, which would drive up the option price. 8

• When the option price reached $5. 94, the transaction would no longer be profitable. Thus, $5. 94 is the minimum price of the call or intrinsic value. 2 - If the exercise price $125 and the stock price $125. 94 ? Max(0, 125, 94 – 125) = 0. 94 and are priced at no less than 0. 94. 3 - If the exercise price $130 exceeds the stock price $125. 94 ? zero 9

• All those options obviously have nonnegative values. • The intrinsic value concept applies only to an American calls, because a European call can be exercised only on the expiration day. If the price of an European call were less than Max (0, S₀ - X), the inability to exercise it would prevent traders from engaging in the aforementioned arbitrage that would drive up the call’s price. 10

• The option price (premium) of an American call normally exceeds its intrinsic value. • The difference between the call price and the intrinsic value is called the time value or speculative value, which defined as Ca (S₀, T, X) - Max(0, S₀ - X). • The time value reflects what traders are willing to pay for the uncertainty of the underlying stock. 11

INTRINSIC VALUES AND TIME VALUES OF DCRB CALLS Time Value Exercise Price Intrinsic Value May June July 120 5. 94 2. 81 9. 46 14. 96 125 0. 94 4. 81 12. 56 17. 66 130 0. 00 3. 60 11. 35 16. 40 • The time values increase with the time to expiration. 12

Maximum Value of a Call A call option also has a maximum value C (S₀, T, X) ≤ S₀ • The maximum value of a call is the price of the stock. 13

Value of a Call at Expiration • The price of a call at expiration is given as C (ST, 0, X) = Max(0, ST - X) Because no time remains in the option’s life, the call price contains no time value. The prospect of future stock price increases is irrelevant to the price of the expiring option, which will be simply its intrinsic value. • At expiration, an American option and a European option are identical instruments. 14

Effect of Time to Expiration • Consider two American calls that differ only in their times to expiration. One has a time to expiration of T 1 and the price of Ca (S₀, T 1, X); the other has a time to expiration of T 2 and a price of Ca (S₀, T 2, X). Remember that T 2 is greater than T 1. Which of these two option s will have the greater value? 15

• Suppose that today is the expiration day of the shorter -lived option. The value of the expiring option is Max(0, ST 1 -X). The second option has a time to expiration of T 2 -T 1. Its minimum value is Max(0, ST 1 X). Therefore, Ca(S₀, T 2, X) ≥ Ca(S₀, T 1, X) • A longer-lived American call must always be worth at least as much as a shorter-lived American call with the same terms. 16

• The longer the time to expiration, the grater the call’s value. • The time value of a call option varies with the time to expiration and the proximity of the stock price to the exercise price. • Investors pay for the time value of the call based on the uncertainty of the future stock price. 17

• If the stock price is very high, the call is said to be deep-in-the-money and the time value will be low. • If the stock price is very low, the call is said to be deep-out-the-money and the time value likewise will be low. 18

Principles of Put Option Pricing Minimum Value of a Put • A put is an option to sell a stock. A put holder is not obligated to exercise it and will not do so if exercising will decrease wealth. Thus a put can never have a negative value. P (S₀, T, X) ≥ 0 • Because a put option need not be exercised, its minimum value is zero. 19

• An American put can be exercised early. Therefore, Pa(S₀, T, X) ≥ Max(0, X - S₀) • The value, Max(0, X- S₀) is called the put’s intrinsic value. • An in-the-money put has a positive value, while an out-of-the-money put has an intrinsic value of zero. • The intrinsic value of an American put is the greater of zero or the difference between the exercise price and the stock price. 20

• The difference between the put price and intrinsic value is the time value or speculative value. • Time value is defined as Pa(S₀, T, X) – Max(0, X - S₀). As with calls, the time value reflects what an investor is willing to pay for the uncertainty of the final outcome. 21

• Maximum Value of a Put: The maximum value of an American put is the exercise price. Pa (S₀, T, X) ≤ X 22

Value of Put at Expiration On the put’s date, no time value will remain. Expiring American puts therefore are the same as European puts. The value of either type of put must be the intrinsic value. Thus, P (ST, 0, X) = Max(0, X - ST) 23

• If X > ST and the put price is less than X-ST, investors can buy the put and exercise the put for immediate risk-free profit. • If the put expires out-of-the-money ( X < ST ), it will be worthless. 24

Effect of Time to Expiration • A longer-lived American put must always be worth at least as much as a shorter-lived American put with the same terms. 25