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Subject 8. Factors that Affect the Value of an Option PDF Download
The previous discussion tells us that the price is somewhere between zero and maximum, which is either the underlying price, the exercise price, or the present value of the exercise price - a fairly wide range of possibilities. The range will be tightened up a little on the low side by establishing a lower bound on the option price.

For American options, which are exercisable immediately:
C0 >= Max (0, S0 - X)
P0 >= Max (0, X - S0)
If the option is in-the-money and is selling for less than its intrinsic value, it can be bought and exercised to net an immediate risk-free profit.

However, European options cannot be exercised early; thus, there is no way for market participants to exercise an option selling for too little with respect to its intrinsic value. Investors have to determine the lower bound of a European call by constructing a portfolio consisting of a long call and risk-free bond and a short position in the underlying asset.

First the investor needs the ability to buy and sell a risk-free bond with a face value equal to the exercise price and current value equal to the present value of the exercise price. The investor buys the European call and the risk-free bond and sells short (borrows the asset and sells it) the underlying asset. At expiration the investor shall buy back the asset.

This combination produces a non-negative value at expiration, so its current value must be non-negative. For this situation to occur, the call price has to be worth at least the underlying price minus the present value of the exercise price:

The lower bound of a European put is established by constructing a portfolio consisting of a long put, a long position in the underlying, and the issuance of a zero-coupon bond. This combination produces a non-negative value at expiration so its current value must be non-negative. For this situation to occur, the put price has to be at least as much as the present value of the exercise price minus the underlying price.

For both calls and puts, if this lower bound is negative, we invoke the rule that an option price can be no lower than zero.

Example

  • All options expire in 60 days, have the same exercise price (X) of $60 and the same underlying asset.
  • The current price of the underlying (S0) is $50.
  • The risk-free rate (r) is 5%.
  • Find the lower bounds of American and European calls and puts.

Solution

  • Time to expiration (T) = 60/365 = 0.1644
  • European Call (c0): MAX[0, 50 - 60/(1 + 5%)0.1644] = MAX[0, -9.52] = 0
  • American Call (C0): MAX[0, 50 - 60/(1 + 5%)0.1644] = MAX[0, -9.52] = 0
  • European Put (p0): MAX[0, 60/(1 + 5%)0.1644 - 50] = MAX[0, 9.52) = 9.52
  • American Put (P0): MAX[0, 60 - 50) = 10
  • Note that the lower bound of the American put is above the lower bound of the European put.

Factors that Affect the Value of an Option

There are primarily six factors that determine the value of an option.

Value of the Underlying

The call option can be viewed as buying the underlying and the put option can be viewed as selling the underlying. So, the value of call option increases with an increase in the value of the underlying and the value of put option decreases with an increase in the value of the underlying.

Exercise Price

The lower the exercise price, the higher the value of the call option because we would be able to buy the underlying at a lower price. The opposite is true for the put option i.e. the higher the exercise price; the higher the value of the put option as we would be able to sell the underlying at a higher price. Hence, the value of a European call option is inversely proportional to the exercise price and the value of a European put option is directly proportional to the exercise price.

Time to Expiration

The more the time to expiration, the greater the value of the option. The logic is that the underlying has more potential for movement and thus the option will have a higher value. With the same logic, even the put option will increase with an increase in the time to expiration. But there is some exception to the European put options. If the risk-free rate is high, the volatility is lower, and the European put option is deep-in-the-money, then the value of put option can decrease with increase in the time to expiration.

You can easily remember it with the example of a bankrupt company. Suppose you buy a European put option with one year to expiry for the exercise price of $100. Just after the option purchase, the company gets caught in a scandal and goes bankrupt. The price of the stock falls to zero and is never going to recover and is going to remain at the price of zero. Since the option is European, we cannot exercise the option before the expiration. So, the value of the option will be simply the present value of $100. The value will keep on increasing as the time to expiration decreases and we move closer to the expiry.

The long-dated European put option having 10-years, 15-years or 20-years to expiration almost always decrease in value with increase in the time to expiration because the negative impact of the discount factor of the risk-free rate dwarfs the positive impact of the movement of the underlying due to longer time to expiration. Because the lower movement is limited because the underlying price cannot fall below zero.

Risk-Free Interest Rate

The value of call option increases in the value with an increase in the risk-free rate and the value of put option decreases with an increase in the risk-free rate. As per put-call parity, c0 + X*(1+r)-T = p0 + S0. If we increase the risk-free rate, then the value of factor X*(1+r)-T falls and the value of call option has to increase for the parity of the equation.

Volatility

Both call options and put options increase in value with an increase in volatility. The call option increases in value because the underlying price can increase to a higher price because of high volatility. Similarly, the put option increases in value because the underlying price can fall to a lower price due to higher volatility. The volatility factor and time to expiration factor are combined to get the time value of an option. The volatility can have more impact if the time to expiration is longer. The option prices generally decrease as the options approach expiration date and this is referred to as time value decay.

Cost of Carry

The call option is equivalent to the long position in the underlying and the put option is equivalent to the short position in the underlying. The value of the underlying decreases with benefits and increases with the cost of carry. So, the value of European call option is inversely proportional to the benefits and directly proportional to the cost incurred in holding the underlying. The opposite is true for the European put option i.e. the value of European put option increases with more benefits and decreases with more cost of carry.

Learning Outcome Statements

k. identify the factors that determine the value of an option and explain how each factor affects the value of an option;

CFA® 2022 Level I Curriculum, Volume 5, Module 46

User Contributed Comments 23

User Comment
6162 the reason behind this is that the buyer can use his part of money for some other investment purposes when the interest rates r high in the market but paying only a margin amount of the option whereas the writer or the seller cannnot sell the goods until the buyer wishes so he is stuck with the stock and cannot convert it into cash and benefit the high rate of interest in the market.
mtcfa Am I the only one that thinks this section is somewhat off base? If I have an equity call that goes in my favor (ie deep in the money), you can bet I'm going to excercise that option and take my profits. Why would I sit around and wait for the market to turn againat me and lose what ever profits I already made?
cwrolfe I'd just short the call option and lock-in my gains
antarctica Here it's assumed that price rises continuously. So if you exercise now you'll be worse off than keeping it until expiration, assuming that stock price either goes up or stay the same.
Winner Euro Call Max I get 9.92 rather than 9.52 as detailed above. Does anyone else get this answer?
Winner Sorry, Now I see how they get the 9.52, its
clearer if you see it like this
50 - [(60/(1.05^.1644)] = -9.52
uberstyle kind of a strong assumption to assume prices continue to rise - I am not disagreeing but wonder why it is not noted?
Tom81 Solution

* Time to expiration (T) = 60/365 = 0.1644.
* European Call (c0): MAX[0, 50 - 60/(1 + 5%)0.1644] = MAX[0, -9.52] = 0.
* American Call (C0): MAX[0, 50 - 60/(1 + 5%)0.1644] = MAX[0, -9.52] = 0.
* European Put (p0): MAX[0, 60/(1 + 5%)0.1644 - 50] = MAX[0, 9.52) = 9.52.
* American Put (P0): MAX[0, 60 - 50) = 10.
* Note that the lower bound of the American put is above the lower bound of the European put.


Should that not just be for an American Max(0,50-60), i.e. not discounted.
olagbami Why is the American call option discounted?
tschorsch You would just sell the option to lock in profits.
It is still worth more than the intrinsic value because the underlying could still go more in your favor.
Also, exercising an option would give exposure to the underlying, and you would also usually incure more transaction costs by exercising.
tschorsch Also, suppose you are long a call. When the underlying price hits your target, you want to exit the position. When you exercise, you are still net long the position and must sell the underlying to remove your exposure. Exercising is not instantaneous and you will have some time exposure when you cannot exit (unless you sell the underlying short).
The moral of the story is, unless you want to hold the underlying for the longer term (or for puts you already have the underlying and want to unload it) it is really not worth exercising just to take profits.
Shammel @ olagbami: I think the American call option is discounted as we are finding the lower bound.
cardinal08 The call is using leverage, "borrowing," to buy the underlying later. The put is holding the premium. In a high interest rate environment, This is beneficial to the call holder because he is achieving leverage without needing to finance through high rates, and this disadvantages the put holder because he is holding a premium that would otherwise be paying a high rate.
Juhee American call can be exercised before expire date not like European call. so that is why we need to discount to get the present value
anova The lower bounds of American and European Calls are the same else the American Call lower bound would be lower the European Call lower bound.
co>= Max[0,So-X/(1+r)^T]
Co>= Max[0,So-X/(1+r)^T]
papajeff They are referring only to exercising the option, not selling the calls. if you sell the call you lock in the time value.
moneyguy At least this stuff is not at all confusing.
Emily1119 Why a long term european put can be worth more or less than a short term european?
Oksanata in this case they should have discounted American put as well, shouldn't they?
Oksanata Oh, now I see..All the explanations are in No.9 of this subject..
Oksanata If Volatility, Time to Expiration is UP - Call and Put values are UP
All other factors,if UP, affecting option prices in opposite directions as follows:
Dividents, Strike Price UP: Call Down, Put Up
Intr.Rate, Underlying UP: Call UP, Put Down
gill15 Why would it be confusing to you? You're the Moneyguy.
SKIA This section should have a couple of questions to hammer on the relationships.
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Your review questions and global ranking system were so helpful.
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