Chapter 10: Properties of Stock Options Flashcards
What is the most important relationship between options and stock prices?
Put-Call Parity
What is Put-Call parity (definition)
A relationship between the price of a European call option, the price of a European put option, and the underlying stock price.
Is it ever optimal to exercise an American call option on a non-dividend paying stock prior to that options expiration?
No
Could it be optimal to exercise an American put early?
Yes, only under certain circumstances
S0
Current stock price
K
Strike price
Ó
Volatility of the stock price
T
Time to expiration
r
risk free rate
Are dividends expected to be paid a factor for option prices?
Yes
If the current stock price increases, what will happen to a European call’s value as a result? (all else held constant)
Increase
If the current stock price increases, what will happen to a European put’s value as a result? (all else held constant)
Decrease
If the current stock price increases, what will happen to an American call’s value as a result? (all else held constant)
Increase
If the current stock price increases, what will happen to an American put’s value as a result? (all else held constant)
Decrease
If the strike price of a European call increases, what will happen to the call option?
Decrease
If the strike price of a European put increases, what will happen to the put option?
Increase
If the strike price of an American call increases, what will happen to the call option?
Decrease
If the strike price of an American put increases, what will happen to the call option?
Increase
If time to expiration increases, what will happen to a European call value?
Unknown (either)
If time to expiration increases, what will happen to a European put value?
Unknown (either)
If time to expiration increases, what will happen to an American Call?
Increase in value
If time to expiration increases, what will happen to an American Put?
Increase in value
If volatility increases, what will happen to all stock options (call/put/American/European)?
Increase in value
Which types of options increase in value as the risk free rate increases?
American and European calls
Which types of options decrease in value as the risk free rate increases?
American and European puts
How do dividends affect European calls?
Decrease value of the option
How do dividends affect European puts?
Increase the value of the option
How do dividends affect American calls?
Decrease the value of the option
How do dividends affect American puts?
Increase the value of the option
How does a call option gain value?
Stock > Strike
How does a call option gain value?
So>K
How does a put option gain value?
K>So
How does a put option gain value? (which variable must be greater than another?)
Strike > Stock
As time to expiration increases, American options either increase in value or at least do not decrease in value (T/F)?
True
Why does the time to expiration leave uncertainty to the pricing of European put and call options?
If there is a dividend expected in a longer life option (six weeks, 2 month option) but there is no dividend for the shorter life (one month), then the pricing for these options will be affected: the longer life one (dividend one) will increase the value of the puts, but the shorter one without the dividend may increase the value of the call.
Why is higher volatility always driving up prices in these options?
With calls and puts, the potential to gain value is far higher as well as them both experiencing limited loss (the most they can lose is the value of the option).
In general, when interest rates rise, stock prices tend to…
Fall
In general, when interest rates fall, stock prices tend to…
Rise
What do dividends do to the stock price on the ex-dividend date?
Reduce the stock price
St
Stock price on the expiration date
r’
Continuously compounded risk-free rate of interest for an investment maturing in time T
C (capital)
Value of American call option to buy one share
P (capital)
Value of American put option to buy one share
*It should be noted that r is the nominal risk-free rate of interest, not the real risk-free rate of interest
**The real rate of interest is the rate of interest earned after adjustment for the effects of inflation. For example, if the nominal rate of interest is 3% and inflation is 2%, the real rate of interest is approximately 1%
c (lowercase)
Value of European call option to buy one share
p (lowercase)
Value of Euorpean put option to sell one share
If an option price is above the upper bound or below the lower bound, can one perform profitable arbitrage?
Yes
(T/F) An American or European call option gives the holder the right to buy one share of a stock for a certain price. No matter what happens, the option can never be worth more than the stock. Hence, the stock price is an upper bound to the option price.
True (If false, arbitrage is possible)
c <= S0 and C<=S0
Upper bounds of European and American options
An American put option gives the holder the right to sell one share of a stock for K. No matter how low the stock price becomes, the option can never be worth more than K, therefore:
P <=K
For European (put) options, we know that at maturity the option cannot be worth more than K. It follows that it cannot be worth more than the present value of K today.
p<=Ke^(-rt)
If P<=Ke^(-rt) were not true…
arbitrageur could make a riskless profit by writing the option and investing the proceeds of the sale at the risk-free interest rate.
Lower bound for the price of a European call option on a NON-DIVIDEND paying stock is (theoretical minimum)
So-Ke^(-rt)
Can a call option’s value become negative?
No
Worst that can happen to a call:
c>= max( So-Ke^-rt, 0)
Lower bound equation (Euro Call):
So-Ke^-rt
How do you formulate the lower bound for puts on non-dividend paying stocks
Ke^(-rt) -S0
Worst that can happen to a put:
p=>max(Ke^-rt - S0, 0)
put call parity (formula)
c+Ke^(-rt) = p +S0
put call parity meaning
Shows that the value of a European call with a certain exercise price and exercise date can be deduced from the value of a European put with the same exercise price and exercise date and vice versa
According to put-call parity, which ever option is overpriced, you would
short the overpriced and stock and buy the one that is underpriced
3 Names for the BSOPM
Fischer Black, Myron Scholes, and Robert Merton
Why should an American call never be exercised early (non-dividend)?
A call option being held instead of the stock itself insures the holder against the stock price falling below the strike price. 2) Time value: from the perspective of the option holder, the later the strike price is paid out, the better.
*Because American call options are never exercised early when there are no dividends, they are equivalent to…
European call options
It can be optimal to exercise which option on a non-dividend paying stock early?
American put option
(T/F) A put option that is deep in the money should always be exercised early?
True
*In general the early exercise of a put option becomes more attractive as S0 _________, as r __________, and as volatility _______.
** SO decreases, r increases, and volatility decreases
What are the European put option’s lower and upper bounds:
Max( K^e-rt - S0, 0) <=p <= Ke^-rt
An American put option is ALWAYS worth more than a European put options
True (243)
When dividends are expected, we can no longer assert that an American call option will not be exercised early (T/F)
True
It is sometimes optimal to exercise an American call immediately prior to an ex-dividend date
(True, 244)
Put-call parity equation with Dividends:
c+D+Ke^-rt = p + S0
(T/F) Put call parity HOLDS for American options
False. It does not hold
(T/F) Call < Put for at the money options
False
(T/F) Call > Put for at the money options
True
(T/F) Put > Call for at the money options
False
(T/F) Put < Call for at the money options
True
If you do not use the c+Ke^(-rt) = p + S0 formula for the put-call parity, what is another one you can use (with minuses)?
c-p=S0-Ke^(-rt)
