Week 1: Financial Options
Understanding Financial Options
- A financial option is a contract that gives its owner the right, but not the obligation, to buy or
sell an underlying asset at a fixed price, on or before a specified date in the future.
- Call (Put) option: A call (put) option gives the holder the right to buy (sell) an asset; it is not
an obligation to buy (sell)
Three different options：
1. In-the-money: if the payoff from exercising an option immediately is positive, the option is
said to be in-the-money. （the most expensive one）
◼ Call options with strike prices below the current stock price are in-the-money, as are put
options with strike prices above the current stock price.
2. Out-of-the-money: if the payoff from exercising the option immediately is negative, the option
is out-of-the-money. (the cheapest one)
◼ Cal options with strike prices above the current stock price are out-of-the-money, as are
put options with strike prices below the current stock price.
3. At-the-money: when the exercise price of an option is equal to the current price of the stock,
the option is said to be at-the-money. (highest level of trading volume when close to at-the-
money)
Factors Affecting Option Prices
1. Strike price:
- The value of a call option increase (decreases) as the strike price decreases (increases); all other
things held constant.
The value of a put option increase (decreases) as the strike price increases (decreases); all other
things held constant.
2. Stock price：
- The value of a call option increases (decreases) as the stock price increases (decreases); all other
things held constant.
The value of a put option increases (decreases) as the stock price decreases (increases); all other
things held constant.
3. Exercise date:
- For American options, the longer the time to the exercise date, the more valuable the option
- A European option with a later exercise date may potentially trade for less than an otherwise
identical with an earlier exercise date. (Dividend Situation)
An American option carries all the same rights and privileges as an otherwise equivalent European
option, it cannot be worth less than a European option.
- The intrinsic value of an option is the value it would have if it expired immediately.
---- the intrinsic value is the amount by which the option is currently in-the-money, or zero if
the option is out-of-money.
---- if an American option is worth less than its intrinsic value, you could make arbitrage profits
by purchasing the option and immediately exercising it. Thus, an American option cannot be
worth less than its intrinsic value. (European option 不存在这种情况，因为他只能在规定
的那一个时间 exercise， 所以不会造成 arbitrage profits，它的 value 可以小于 intrinsic value)
- The time value of an option is the difference between the current option price and its intrinsic
, value.
---- Because an American option cannot be worth less than its intrinsic value, it cannot have a
negative time value.
4. Volatility: the value of an option generally increases with the volatility of the stock.
Exercising Options Early: non-dividend paying stock
- Although an American option cannot be worth less than its European counterpart, they may
have equal value.
◼ For a non-dividend paying stock, the put-call parity formula for the value of the call
option is:
C = P +S – PV(K) [European option]
We can write the price of the zero-coupon bond as PV(K) = K – dis(K)
Where dis(K) is the amount of the discount from face value of the zero-coupon bond,
C = S – K + dis(K) + P
Intrinsic value + Time value
- Because dis(K) and P are positive before the expiration date, a European call always has a
positive time value.
---- The price of any call option on a non-dividend-paying stock always exceeds its intrinsic
value prior to expiration if the interest rate is positive.
- It is never optimal to exercise a call option on a non-dividend-paying stock early- you are
always better off just selling the option.
---- when you exercise an option, you get its intrinsic value. But the price of a call option on
a non-dividend-paying stock always exceeds its intrinsic value.
---- if you want to liquidate your position in a call on a non-dividend-paying stock, you will get
a higher price if you sell it rather than exercise it.
---- an American call on a non-dividend-paying stock has the same price as its European
counterpart
◼ However, for a non-dividend paying stock, the put-call parity formula for the value of the
put option:
P = K – S + C - dis(K)
Intrinsic value + Time value
When the strike price is high and a put option is sufficiently deep in-the-money, dis(K) will be
large relative to the value of the call, and the time value of a European put option will be
negative.
An American put option can be worth more than an otherwise identical European option.
Exercising Options Early: dividend paying stock
- For a dividend paying stock, the put-call parity formula can be written as:
S + P = PV(K) + PV(Div) + C
PV(Div) is the present value of the future dividends paid by the stock during the life of the
options.
- When stocks pay dividends, the right to exercise an option on them early is generally valuable
for both calls and puts.
C = S – K + dis(K) + P – PV (Div)
, ---- if PV(Div) is large enough, the time value of a European call option can be negative, and
the price of the American option can exceed the price of a European option
---- when a company pays a dividend, the price drop hurts the owner of a call option because
the option holder does not get the dividend as compensation.
---- a call should only be exercised early to capture the dividend; it will only be optimal to do
so just before the stock’s ex-dividend date.
- Dividends have the opposite effect on the time value of a put option. Again, from the put-call
parity relation, we can write the put value as:
P = K – S + C – dis(K) + PV(Div)
---- because the time value of a put includes the present value of the expected dividends paid
during the life of the option, dividends reduce the likelihood of early excercise
---- the likelihood of early exercise increases whenever the stock goes ex dividend (due to the
expected drop European)
---- for the deep in-the-money calls, the present value of the dividends is large than then interest
earned on the low strike prices, making it costly to wait to exercise the option
---- for the deep in-the-money puts, the interest on the high strike prices exceeds the dividends
earned, again making it costly to wait
Week 2: Option Valuation
◼ Risk-Neutral Probabilities and Option Pricing
---- For Multiperiod
, First, we compute the risk-neutral probability that the stock price will increase. At time 0, we
have:
Because the stock has the same returns (up 20% or down 10%) at each date, we can check that
the risk-neutral probability is the same at each date as well.
For the call option with a strike price of $1.2S, this call pays $30.24S if the stock goes up
twice, and zero otherwise. The risk-neutral probability that the stock will go up twice is 0.433
* 0.433, so the call option has an expected payoff of 0.433 * 0.433 * $0.24S = $0.045S
We compute the current price of the call option by discounting this expected payoff at the risk-
free rate:
◼ The Binomial Pricing Formula: Replicating Strategy
--- A Multiperiod Model: Binomial Lattice
Understanding Financial Options
- A financial option is a contract that gives its owner the right, but not the obligation, to buy or
sell an underlying asset at a fixed price, on or before a specified date in the future.
- Call (Put) option: A call (put) option gives the holder the right to buy (sell) an asset; it is not
an obligation to buy (sell)
Three different options：
1. In-the-money: if the payoff from exercising an option immediately is positive, the option is
said to be in-the-money. （the most expensive one）
◼ Call options with strike prices below the current stock price are in-the-money, as are put
options with strike prices above the current stock price.
2. Out-of-the-money: if the payoff from exercising the option immediately is negative, the option
is out-of-the-money. (the cheapest one)
◼ Cal options with strike prices above the current stock price are out-of-the-money, as are
put options with strike prices below the current stock price.
3. At-the-money: when the exercise price of an option is equal to the current price of the stock,
the option is said to be at-the-money. (highest level of trading volume when close to at-the-
money)
Factors Affecting Option Prices
1. Strike price:
- The value of a call option increase (decreases) as the strike price decreases (increases); all other
things held constant.
The value of a put option increase (decreases) as the strike price increases (decreases); all other
things held constant.
2. Stock price：
- The value of a call option increases (decreases) as the stock price increases (decreases); all other
things held constant.
The value of a put option increases (decreases) as the stock price decreases (increases); all other
things held constant.
3. Exercise date:
- For American options, the longer the time to the exercise date, the more valuable the option
- A European option with a later exercise date may potentially trade for less than an otherwise
identical with an earlier exercise date. (Dividend Situation)
An American option carries all the same rights and privileges as an otherwise equivalent European
option, it cannot be worth less than a European option.
- The intrinsic value of an option is the value it would have if it expired immediately.
---- the intrinsic value is the amount by which the option is currently in-the-money, or zero if
the option is out-of-money.
---- if an American option is worth less than its intrinsic value, you could make arbitrage profits
by purchasing the option and immediately exercising it. Thus, an American option cannot be
worth less than its intrinsic value. (European option 不存在这种情况，因为他只能在规定
的那一个时间 exercise， 所以不会造成 arbitrage profits，它的 value 可以小于 intrinsic value)
- The time value of an option is the difference between the current option price and its intrinsic
, value.
---- Because an American option cannot be worth less than its intrinsic value, it cannot have a
negative time value.
4. Volatility: the value of an option generally increases with the volatility of the stock.
Exercising Options Early: non-dividend paying stock
- Although an American option cannot be worth less than its European counterpart, they may
have equal value.
◼ For a non-dividend paying stock, the put-call parity formula for the value of the call
option is:
C = P +S – PV(K) [European option]
We can write the price of the zero-coupon bond as PV(K) = K – dis(K)
Where dis(K) is the amount of the discount from face value of the zero-coupon bond,
C = S – K + dis(K) + P
Intrinsic value + Time value
- Because dis(K) and P are positive before the expiration date, a European call always has a
positive time value.
---- The price of any call option on a non-dividend-paying stock always exceeds its intrinsic
value prior to expiration if the interest rate is positive.
- It is never optimal to exercise a call option on a non-dividend-paying stock early- you are
always better off just selling the option.
---- when you exercise an option, you get its intrinsic value. But the price of a call option on
a non-dividend-paying stock always exceeds its intrinsic value.
---- if you want to liquidate your position in a call on a non-dividend-paying stock, you will get
a higher price if you sell it rather than exercise it.
---- an American call on a non-dividend-paying stock has the same price as its European
counterpart
◼ However, for a non-dividend paying stock, the put-call parity formula for the value of the
put option:
P = K – S + C - dis(K)
Intrinsic value + Time value
When the strike price is high and a put option is sufficiently deep in-the-money, dis(K) will be
large relative to the value of the call, and the time value of a European put option will be
negative.
An American put option can be worth more than an otherwise identical European option.
Exercising Options Early: dividend paying stock
- For a dividend paying stock, the put-call parity formula can be written as:
S + P = PV(K) + PV(Div) + C
PV(Div) is the present value of the future dividends paid by the stock during the life of the
options.
- When stocks pay dividends, the right to exercise an option on them early is generally valuable
for both calls and puts.
C = S – K + dis(K) + P – PV (Div)
, ---- if PV(Div) is large enough, the time value of a European call option can be negative, and
the price of the American option can exceed the price of a European option
---- when a company pays a dividend, the price drop hurts the owner of a call option because
the option holder does not get the dividend as compensation.
---- a call should only be exercised early to capture the dividend; it will only be optimal to do
so just before the stock’s ex-dividend date.
- Dividends have the opposite effect on the time value of a put option. Again, from the put-call
parity relation, we can write the put value as:
P = K – S + C – dis(K) + PV(Div)
---- because the time value of a put includes the present value of the expected dividends paid
during the life of the option, dividends reduce the likelihood of early excercise
---- the likelihood of early exercise increases whenever the stock goes ex dividend (due to the
expected drop European)
---- for the deep in-the-money calls, the present value of the dividends is large than then interest
earned on the low strike prices, making it costly to wait to exercise the option
---- for the deep in-the-money puts, the interest on the high strike prices exceeds the dividends
earned, again making it costly to wait
Week 2: Option Valuation
◼ Risk-Neutral Probabilities and Option Pricing
---- For Multiperiod
, First, we compute the risk-neutral probability that the stock price will increase. At time 0, we
have:
Because the stock has the same returns (up 20% or down 10%) at each date, we can check that
the risk-neutral probability is the same at each date as well.
For the call option with a strike price of $1.2S, this call pays $30.24S if the stock goes up
twice, and zero otherwise. The risk-neutral probability that the stock will go up twice is 0.433
* 0.433, so the call option has an expected payoff of 0.433 * 0.433 * $0.24S = $0.045S
We compute the current price of the call option by discounting this expected payoff at the risk-
free rate:
◼ The Binomial Pricing Formula: Replicating Strategy
--- A Multiperiod Model: Binomial Lattice
