# Options - Introduction

Options – Introduction, page 1 of 7

**OPTIONS - INTRODUCTION**

**SOME DEFINITIONS**

**Underlying asset: **The asset on which the option is written.

**Current value of underlying asset (V):** The current market price of the underlying asset.

**Expiration date value of underlying asset (V*): **The market price of the underlying asset at the moment the option expires.

**Option contract:** A contract providing the buyer the right (but not the obligation) to buy or sell a given quantity of a particular asset at a fixed price over some stated time period. An option to buy is a **call; an option to sell is a put.**

**Exercise price (X)**: Price paid for asset by a call option holder when the option is exercised; price received by a put option holder when the option is exercised.

**Expiration date (T)**: When the option expires.

**Premium (CPrem for call option, and PPrem for put option): The market price** that the buyer of the option pays to the seller of the option in purchasing the option. When you sell or write a call option, you receive CPrem; when you buy a call option you pay CPrem . When you sell or write a put option, you receive PPrem; when you buy a put option, you pay PPrem.

American option: An option that can be exercised at any time up to the date of expiration.

European option: An option that can be exercised only on the date of expiration.

**Intrinsic value: **The greater of zero and the profit from immediately exercising the option. For an American or European call option, intrinsic value equals Max[0, V X]. For an American or European put, intrinsic value equals Max[0, X V].

**In-the-money option: **An option with a positive intrinsic value.

**At-the-money option: **An option with X = V (zero intrinsic value).

**Out-of-money option: **An option not at-the-money but with a zero intrinsic value (that is, a call option for which V < X; or a put option for which X < V).

**Time value of an option: **The difference between the option’s premium and its intrinsic value. Thus, for an American or European option, Premium = Intrinsic Value + Time Value; thus, Time Value = Premium Intrinsic Value. For American options, time value must be equal to or greater than zero because the option can be exercised at any time.

Delta: The change in the option premium with respect to a small change in the price of the underlying asset (and ). Also called “riskless hedge ratio.”

Time decay: The decline in the value of the option as its time to expiration diminishes.

ANALYSIS: Assume the following definitions.

V: the current price of the underlying asset

V*: the price of the underlying asset at the option expiration date

X: exercise price

CPrem: the call’s premium

PPrem: the put’s premium

The term “payoff” below refers to what is received at the expiration date T if the option is held to expiration (which we will assume). It is not the profit on buying or selling the option. The profit is the net gain taking into account both the time T payoff and the time 0 cash flow associated with the transaction. So, if you pay CPrem = $5 at time 0 for a call option having an exercise price E = $40, and at expiration date T the share price V* = $65, your payoff from exercising the option at time T is [V* E] = $25, but your profit (ignoring the time value of money) is $20 ($25 payoff minus $5 price paid for the option).

Expiration date payoff on **a share of stock** = V* (1)

Expiration date payoff on **stock short sale** = V* (2)

Expiration date payoff from buying a call option = max[0, V* X] (3a)

= min[0, X V*] (3b)

Expiration date payoff from selling a call option = max[0, V* X] (4a)

= min[0, X V*] (4b)

Expiration date payoff from buying a put option = max[0, X V*] (5a)

= min[0, V* X] (5b)

Expiration date payoff from selling a put option = max[0, X V*] (6a)

= min[0, V* X] (6b)

Eqtn. # / Formula / V* = $60 and X = $20 / V* = $60 and X = $85Buy a call:

(3a) / max[0, V* X] / max[0, $60 $20] = $40 / max[0, $60 $85] = 0

(3b) / min[0, X V*] / min[0, $20 $60] = $40 / min[0, $85 $60] = 0

Sell a call:

(4a) / max[0, V* X] / max[0, $60 $20] = $40 / max[0, $60 $85] = 0

(4b) / min[0, X V*] / min[0, $20 $60] = $40 / min[0, $85 $60] = 0

Buy a put:

(5a) / max[0, X V*] / max[0, $20 $60] = 0 / max[0, $85 $60] = $25

(5b) / min[0, V* X] / min[0, $60 $20] = 0 / min[0, $60 $85] = $25

Sell a put:

(6a) / max[0, X V*] / max[0, $20 $60] = 0 / max[0, $85 $60] = $25

(6b) / min[0, V* X] / min[0, $60 $20] = 0 / min[0, $60 $85] = $25

Min[ ] and Max[ ] Rules:

max[A, B] = min[ A, B] (7a)

C + max[A, B] = max[C + A, C + B] (7b)

C + min[A, B] = min [C + A, C + B] (7c)

For Z > 0:

Z max[A, B] = max[ZA, ZB] (7d)

Z min[A, B] = min[ZA, ZB] (7e)

For Z < 0:

Z max[A, B] = min[ZA, ZB] (7f)

Z min[A, B] = max[ZA, ZB] (7g)

Examples: To illustrate the above relationships, suppose that A = 5, B = 3, C = 4 (the relationships also hold if one or more of A, B and C are negative).

(7a): max[5, 3] = min[ 5, 3] = ( 5) = 5

(7b): 4 + max[5, 3] = max[4 + 5, 4 + 3] = 9

(7c): 4 + min[5, 3] = min [4 + 5, 4 + 3] = 7

For Z = 2:

(7d): 2 max[5, 3] = max[25, 23] = 10

(7e): 2 min[5, 3] = min[25, 23] = 6

For Z = 2:

(7f): (2) max[5, 3] = min[( 2) 5, ( 2) 3] = 10

(7g): (2) min[5, 3] = max[(2) 5, (2) 3] = 6

Let’s use the above equations to analyze the “financial alchemy” in Brealey, Myers and Allen (BM&A) Figure 20.5 on page 571. In this illustration, exercise price X = $60.

Figure 20.5: Using (1), (4b) and (7c), we have:

Share of stock + Sell a call = V* + min[0, X V*]

= min[V*, X]

= r.h.s. of equality in Figure 20.5 since:

- For V* X, min[V*, X] = V*

- For V* > X, min[V*, X] = X = $55

Figure 20.6, Row 1: Using (1), (5a) and (7b), we have:

Share of stock + Buy a put = V* + max[0, X V*]

= max[V*, X]

= r.h.s. of equality in Row 1 of Figure 20.6 since:

- For V* X, max[V*, X] = X = $55

- For V* > X, max[V*, X] = V*

Figure 20.6, Row 2: Using (3a) and (7b), we have:

Bank deposit X + Buy a call = X + max[0, V* X]

= max[X, V*]

= r.h.s. of equality in Row 2 of Figure 20.6 since:

- For V* X, max[V*, X] = X = $55

- For V* > X, max[V*, X] = V*

PUT-CALL PARITY:

The Put-Call Parity relationship (which applies only to European puts and calls) is:

Value of Call + PV of Exercise Price = Value of Put + Value of a Share (8)

“PV of Exercise Price” is the present value (using the riskless interest rate as the discount rate) of amount X to be received at time T. The reason that (8) holds in perfect (competitive and frictionless) capital markets is that buying a call and investing [PV of Exercise Price] in a riskless asset will produce exactly the same time T payoff as buying a put and buying a share. In perfect markets, two portfolios that generate identical payoffs must have the same market price. Therefore, to establish (8), we will show that the left-hand side of (8) produces the same payoff at time T (expiration date) as does the right-hand side of (8). Let’s take a look.

- Left-hand side of (8) produces the following time T payoff:

oBy (3a), buying a call produces max[0, V* X] (9a)

oInvesting [PV of Exercise Price] in a riskless asset produces exercise price X (9b)

- Right-hand side of (8) produces the following time T payoff:

oBy (5a), buying a put produces max[0, X V*] (9c)

oBy (1), buying a share of stock produces V* (9d)

So, using (9a) and (9b), the left-hand side of (8) produces max[0, V* X] + X: but, using (7b):

Left-hand side of (8) produces max[0, V* X] + X= max[X, V*] (10a)

Using (9c) and (9d), the right-hand side of (8) produces max[0, X V*] + V*; but using (7b):

Right-hand side of (8) produces max[0, X V*] + V*= max[V*, X] (10b)

Since max[X, V*] = max[V*, X] it follows that the left-hand side of (8) produces the same time T payoff as the right-hand side of (8) and so they must have the same value.

Rearranging (8) we get:

Value of Put = Value of Call + [PV of Exercise Price] Value of a Share (11)

Equation (11) says that a put produces the same time T payoff as the combination of call, a riskless asset having current price X, and a short sale position of the firm’s stock.

We can also rearrange (8) as follows:

Value of Call = Value of Put + Value of a Share [PV of Exercise Price] (12)

Equation (12) says that buying a call produces the same payoff as buying a put, buying a share and borrowing that amount such that X is owed at time T.

OPTION VALUE – CALL OPTION

At expiration date T, the value of a European call option (CPrem) is either $0 or [V* X]. What about CPrem before expiration date T? Take a look at BM&A Figure 20.10. The figure illustrates the potential range for CPrem before expiration date T.

Here are some properties of CPrem before expiration date T:

- Upper bound for CPrem = V (value of a share). The reason for the upper bound is that no one would pay more for the option than for the share itself.

- Lower bound for CPrem = intrinsic value = max[0, V X]. Note that the option cannot sell for less than zero. Second, option price CPrem cannot fall below (V X) because, if it did, there would be an immediately arbitrage profit. Here’s why.

Assume that now (time 0): CPrem < V X (13a)

To show that (13a) implies a guaranteed arbitrage profit, assume (13a). At time 0, you sell a share short (and receive V from the short-sale), buy a call option for CPrem, and invest X in a riskless asset that will pay you X at time T, where r is the risk-free interest rate per period. The resulting time 0 net cash payoff to you is (where the inequality in (13b) follows from (13a)):

Time 0 payoff = V CPrem X > 0 (13b)

So, the time 0 payoff is positive. Now consider the time T payoff. At date T, you receive X from the riskless asset you bought at time 0; you also exercise the call option (i.e., buy a share for exercise price X) and use the share purchased with the call option to cover the short position you established at time 0. Your time T payoff is:

Time T payoff = X X > 0 (13c)

So both the time 0 payoff and the time T payoff are positive from this arbitrage transaction (that is, the arbitrage nets positive cash flow at time 0 and time T).

- As the share price V decreases, CPrem approaches zero. The reason for this is that, as the share price decreases, the probability of exercise at time T approaches zero.

- If share price V = X, the option intrinsic value (V X) = 0, but CPremis positive if there is a positive probability of V* > X. CPrem > 0 because, if date T share value V* > X, the option holder will at date T exercise the option and gain (V* X).

- As the share price V increases,CPrem approaches {V [PV of Exercise Price X]}. As V increases, the probability of exercise at time T approaches 1, in which case the option-holder in effect now owns the share
*minus the PV of X (computed using the risk-free rate)*. The higher is the interest rate and the further into the future is date T, the lower is [PV of X] and the higher is the value of the call option. Notice that limit {V [PV of X]} = (V X)+ (X [PV of X]) = call option intrinsic value + (X [PV of X]).

Risk and Value: The higher is the risk (variance) of the stock price change per time period, the more valuable is the call option. The intuition is this. As the variance of the stock price change per unit time goes up, the probability distribution of V* spreads out. The right-hand tail of the distribution gets bigger (high values for V* become more probable) and the probability of V* = 0 also goes up. But notice that, whereas the downside potential stays at zero (the value of the share and the value of the option cannot fall below zero), the upside potential for the stock price grows because there is no limit on the upside. So, greater variance implies a greater chance for the owner of the call option to make very large gains; whereas the downside potential remains at zero. The result is that greater stock price variance means higher call option value.

Also, as the time to expiration (T) increases, the variance of V* increases (for any given variance of V per unit time) because there is more time for stock price V to reach very high territory or decline to a point near zero. If is the variance of the daily change in stock price V, the variance of V* is T, where T is the number of days to expiration (see BM&A, footnote 15).

OPTION VALUE – EUROPEAN PUT OPTION

At expiration date T, the value of a European put option (PPrem) is either $0 or [X V*]. Here are some properties of PPrem before expiration date T:

- Upper bound for PPrem = X (option exercise price). The reason for this upper bound is that no one would pay more for the option than the maximum possible payoff that the option might produce; this maximum payoff is exercise price X (where this maximum would be achieved only if the stock price V* were zero).

- Lower bound for PPrem = intrinsic value = max[0, X V]. The reason for the lower bound is this. First, the option cannot sell for less than zero. Second, if PPrem were less than (X V), anyone could earn an arbitrage profit on the option (using an argument similar to that for the call option).

- As the share price decreases, PPrem approaches [PV of Exercise Price] V. The reason for this is that, as the share price decreases, the probability of exercise at time T approaches 1. If the probability of exercise is approximately 1, the owner of the option in effect owns the present value of the exercise price X (computed using the risk-free rate) and a short position in a share of stock.

- As the share price increases, PPrem approaches zero. The reason for this is that, as the share price increases, the probability of exercise at time T approaches zero.

Risk and Value: The higher is the risk (variance) of the stock price change per time period, the more valuable is the put option. The intuition is similar to the call option. As the variance of the stock price change per unit time goes up, the probability distribution of V* spreads out. The left-hand tail of the distribution gets bigger (low values for V* become more probable) and therefore there is an increasing likelihood of a substantial gain on the put option (where the maximium gain occur if V* = 0). Expansion of probability of very high values for V* does not negatively impact the value of the option since any V* > X means that the option will not be exercised.

7/5/05