Options: valuation
Intro: Individual Equilibrium
• Option valuation: What is the equilibrium price for an option?
• In essence, we are interested in market equilibrium prices. It
is, however, easier to understand from an individual investor’s
point of view first.
Note: the timing of buying/selling an option contract!!!
• Example: Suppose you plan to long/short a European call
option on IBM.
Payoff
Buyer pays C
IBM price at T
x
t=0
t=T
Payoff
Time(t)
Seller gets C
IBM price at T
x
Options?
Terminology
Arbitrage
Binomial
Black-Scholes
Intro: Individual Equilibrium
• You buy/sell in order to acquire a future payoff structure. Your
future payoff after you have engaged yourself in a long/short
position of the IBM option is now “contingent” on the future
IBM’s share price.
• Depending on your belief, your portfolio, interest rate, etc.,
you will have in your mind a price you would be willing to
pay/get in order to engage in such a future payoff structure.
Payoff
Buyer pays C
IBM price at T
x
t=0
t=T
Payoff
Time(t)
Seller gets C
IBM price at T
x
Options?
Terminology
Arbitrage
Binomial
Black-Scholes
Individual => Market Eqm.
• We learnt from CAPM that a market equilibrium
must satisfy the following: Every individual
investment position is in his equilibrium.
• Therefore, market equilibrium essentially involve
all individual equilibria.
• In option pricing, we carry on the same idea. But
we use a short cut to formulate the equilibrium
option prices. We employ the concept called, “no
arbitrage”
Options?
Terminology
Arbitrage
Binomial
Black-Scholes
Arbitrage
In the last lecture, we have studied the Put-Call Parity.
In fact, it also uses the concept of no arbitrage.
What is arbitrage?
Definition:
“An arbitrage opportunity arises when an investor can construct a zero
investment portfolio that will yield a sure profit.”
•
Options?
Effectively, if the law of one price is violated, arbitrage opportunity
emerges. If a product is trading at different prices in two very close
locations, you take advantage by buying from the lower-priced location
and immediately selling at the higher-priced location. Your profit is equal
to the price differential. Again, arbitrage appears because the law of one
price is violated.
Terminology
Arbitrage
Binomial
Black-Scholes
Arbitrage
•
Imagine the above scenario would induce not only you but other people to
jump into it to take advantage of the price differential. It is the fact that so
many people are ready to jump on to an arbitrage opportunity that
essentially keeps the law of one price holds. Because the increased
demand at the lower-priced location will quickly jack up the price, while
the increased supply at the higher-priced location will push down the
price. This adjustment process goes on until the two prices equalize.
But how are we going to apply the concept to risky assets?
•
Imagine there are two portfolios each composed of totally different assets.
If their future payoffs across EVERY possible future state are EXACTLY
the same, the two portfolios should have the same present value. (e.g., if
IBM or Bombardier shares offer the exact same payoff structure to
investor, their share prices should be the same, regardless of them being
different companies. The bottom line: payoff structure, not assets)
•
What if their prices do differ? There is an arbitrage opportunity. Anyone
can construct the lower-cost portfolio. And sell it at a higher price and
earn immediate profit. Such forces of trying to take advantage of the misprice will eliminate the arbitrage opportunity.
Options?
Terminology
Arbitrage
Binomial
Black-Scholes
No Arbitrage: An Example
•
Similarly put-call parity employs the concept of no arbitrage. A risk-less
portfolio should be priced as a risk-less asset.
Payoffs of 3 different assets in each of the 3 possible states
Risky
assets
•
•
•
Options?
x
y
z
Possible states
Good Normal Bad
100
80
70
15
25
30
70
30
10
It may not be that obvious, but imagine a portfolio with (2y + 1z) would
have a payoff structure exactly the same as if you hold 1x alone.
No arbitrage means, Px = 2Py + Pz
Payoff structure being the same = payoffs at EVERY possible state are
the same
Terminology
Arbitrage
Binomial
Black-Scholes
•
No Arbitrage: Put-Call Parity
We set up a similar table as the previous slide. Payoffs at expiration date
(i.e., Time = T) are listed in the table cells.
Possible states
ST >X
ST ≤X
X
X
ST
ST
Short 1 Call option
-(ST - X)
0
Long 1 put option
0
(X-ST)
Investments Risk-free Investment with an
amount equal to X/(1+R)T
Long a share of Stock
•
•
•
Same idea here. A portfolio consisting of the bottom 3 items would have a
payoff exactly the same as if you hold the top risk-free investment alone.
No arbitrage means, P2 - P3 + P4 = P1
Thus,
S0 + P – C = X/(1+Rf)T
Options?
Terminology
Arbitrage
Binomial
Black-Scholes
No Arbitrage: Put-Call Parity
$
•
<= Long 1 put
ST
x
ST
The graph of combining
different options and
assets is such that the
payoffs of all assets are
added up vertically.
<= Long 1 stock
x
ST
<= Short 1 call
x
x
Total Payoff
ST
<= Total payoffs
x
Options?
Terminology
Arbitrage
Binomial
Black-Scholes
Financial Engineering
•
One of the many attractions of options is the ability they provide to create
investment positions with the resulting payoff structure dependent on a variety
of ways on the underlying securities’ prices.
$
•
<= Long 1 put
ST
x
ST
<= Long 1 call
•
x
•
ST
<= Short 1 put
ST
<= Short 1 call
x
Imagine the 4 different
payoffs patterns:
• Long Put
• Long Call
• Short Put
• Short Call
And imagine options with
different exercise prices
and expiration dates.
Wisely and creatively
combines options and
you can build up different
types of payoff structure
tailored towards your
investment needs.
x
Options?
Terminology
Arbitrage
Binomial
Black-Scholes
Option strategies
•
There are unlimited number of ways for how you combine different options to
form a specific payoff structure that you want.
•
To appreciate the power of using options, you need to be very familiar with the
payoff structures of options.
•
To be a successful financial controller, fund manager, pension fund manager,
investment banker, etc., or purely to get the most out of your personal
investments, you have to be creative in using options.
Options?
Terminology
Arbitrage
Binomial
Black-Scholes
Strategy: Protective Put
•
•
You would like to invest in Google, or you have already invested in Google. Since
recently, its share price has hit the 5-month low, you are unwilling to bear potential
loss beyond a given level. What you can do is the following:
• Invest in the Google stock
• Buy one put per share of Google stock
Such an option strategy is called protective put.
• The final payoff structure is such
that no matter how much Google’s
share drops in price, your overall
<= Long 1 stock
loss is limited to a fixed amount,
ST
whereas if Google’s share
x
increases in price, you will still
x
gain from it.
• The precise exercise price you
<= Long 1 put
ST
choose will dictate the maximum
x
Total Payoff
loss you are willing to bear.
• Again, it is a protective way of
x
holding a stock, that’s why it’s
<= Total Payoffs
called Protective Put.
ST
x
Options?
Terminology
Arbitrage
Binomial
Black-Scholes
Strategy: Covered Call
•
•
What if you're neutral on Google’s performance? (i.e., you think its stock price
will remain relatively unchanged) To potentially profit from such expectation:
• Invest in the Google stock
• Sell one call per share of Google stock
Such an option strategy is called covered call.
• The final payoff structure is such
that no matter how much Google’s
share price drops, your overall
loss is limited to the price you pay
<= Long 1 stock
today. And you still have the
ST
amount you acquired from selling
x
a call.
x
• If share price increases, and the
call holder exercises its right to
<= Short 1 call
ST
buy from you, you have a stock to
x
fulfill your obligation.
• If share price does not change
x
Total Payoff
much, for example, it remains at X
<= Total Payoffs
on the expiration date, then you’ve
ST
gained C, the sales price of the
x
call you sold.
Options?
Terminology
Arbitrage
Binomial
Black-Scholes
Strategy: Straddle
•
•
Imagine another scenario. A pharmaceutical company just release a drug which is
soon to be approved or disapproved by the FDA. You anticipate either a big jump of
its share price if FDA approves, or a big drop otherwise. To profit from it:
• Buy one call of that company’s stock.
• Buy one put of that company’s stock
Such an option strategy is called Straddle.
•
<= Long 1 call
ST
x
•
x
ST
<= Long 1 put
x
Total Payoff
The final payoff structure is such
that if that company’s stock price
varies a lot, you will benefit the
most.
If instead, the company’s stock
price doesn’t vary a lot because of
the news, you will likely make a
loss.
x
<= Total Payoffs
ST
x
Options?
Terminology
Arbitrage
Binomial
Black-Scholes
Valuation: Option definitions revisited
•
There are 2 basic types of options: CALLs & PUTs
•
A CALL option gives the holder the right, but not the obligation
•
•
To buy an asset
•
By a certain date
•
For a certain price
A PUT option gives the holder the right, but not the obligation
•
To sell an asset
•
By a certain date
•
For a certain price
•
an asset – underlying asset
•
Certain date – Maturity date/Expiration date
•
Certain price – strike price/exercise price
Options?
Terminology
Arbitrage
Binomial
Black-Scholes
Valuation: No arbitrage
•
We have mentioned that if the law of one price is violated, people will
jump into the opportunity and make pure profit out of nothing.
•
In equilibrium, such opportunity should have been eliminated.
•
The no arbitrage condition serves as one of the most basic unifying
principles in the study of financial markets
•
An application of that is given out in the previous slides to illustrate the
put-call parity.
•
And we’ll keep on using the no arbitrage condition in order to derive the
equilibrium option prices.
Options?
Terminology
Arbitrage
Binomial
Black-Scholes
Range of possible call option values
•
Let us first look at the boundary for a call option. Assuming the underlying
stock doesn’t payout dividend before the call option expires.
First, its value cannot be negative. Because the holder of a call option need
not be obligated to exercise it if it is not profitable to do so.
C≥0
[1 – lower bound]
Second, its value cannot be higher than the present stock price. Because
Stock price – exercise price is the payoff of the call.
C≤S0
[2 – Upper bound]
Third, its value cannot be lower than the present stock price minus the
present value of the exercise price.
C≥S0 - Present value of X
or
C≥S0 – X/(1+R)T
[3 – lower bound]
•
Reason for [3]: if you compare 2 different portfolios:
{a} buy a stock now at S0 and borrow X/(1+R)T
{b} buy a call option with exercise price X.
Options?
Terminology
Arbitrage
Binomial
Black-Scholes
Range of possible call option values
C≥S0 – X/(1+R)T
•
[3 – lower bound]
Reason for [3]: if you compare 2 different portfolios:
{a} buy a stock now at S0 and borrow X/(1+R)T
{b} buy a call option with exercise price X.
•
Payoff of {a} at maturity is ST – X (i.e, the stock price at time T - the
amount that you have to repay to your lender)
NOTE: This payoff can be +ve or –ve!
•
Payoff of {b} at maturity is either 0 if you don’t exercise, or ST – X if you
choose to exercise.
•
What we see is {b} has a more favorable payoff structure than that of {a},
if constructing {a} requires S0 – X/(1+R)T amount of money, than to
construct {b}, you need at least more than that amount.
•
Thus we have the lower bound of the value of call as C≥S0 – X/(1+R)T
Options?
Terminology
Arbitrage
Binomial
Black-Scholes
Range of possible call option values
•
•
C≥0
[1 – lower bound]
•
C≤S0
[2 – Upper bound]
•
C≥S0 – X/(1+R)T[3 – lower bound]
With all 3 boundary conditions, we get the following graph:
Call Value (C)
S0
X/(1+R)T
Options?
Terminology
Arbitrage
Binomial
Black-Scholes
Call option value as a function of stock
price
•
The value of call as a function of the current stock price is given in the
following red line.
Call Value (C)
S0
X/(1+R)T
Options?
Terminology
Arbitrage
Binomial
Black-Scholes
Factors affecting the call option value
•
We identify 5 factors that affect an option’s value
1) Stock price (S)
2) Exercise Price (X)
3) Volatility of the underlying stock price (σ)
4) Time to Maturity/expiration (T)
5) Interest rate (Rf)
•
You should familiarize yourself with the following table:
Factor
Stock price
Exercise price
Volatility of stock price
Time to expiration
Interest rate
Options?
Terminology
Effect on Call value
increases
decreases
increases
increases
increases
Arbitrage
Effect on Put value
decreases
increases
increases
increases
decreases
Binomial
Black-Scholes
Factors affecting the call option value
•
Stock price
•
Recall the payoff for call and put. Call: max{0,S-X}, Put: max{0, X-S}
•
The higher the stock price, the more likely that a call option will be
exercised in-the-money to get profit. Thus C ↑ if S0 ↑
•
The higher the stock price, the less likely that a put option will be
exercised in-the-money to get profit. Thus P ↓ if S0 ↑
Factor
Stock price
Exercise price
Volatility of stock price
Time to expiration
Interest rate
Options?
Terminology
Effect on Call value
increases
decreases
increases
increases
increases
Arbitrage
Effect on Put value
decreases
increases
increases
increases
decreases
Binomial
Black-Scholes
Factors affecting the call option value
•
Exercise price
•
Recall the payoff for call and put. Call: max{0,S-X}, Put: max{0, X-S}
•
The higher the exercise price, the less likely that a call option will be
exercised in-the-money to get profit. Thus C ↓ if X ↑
•
The higher the exercise price, the more likely that a put option will be
exercised in-the-money to get profit. Thus P ↑ if X ↑
Factor
Stock price
Exercise price
Volatility of stock price
Time to expiration
Interest rate
Options?
Terminology
Effect on Call value
increases
decreases
increases
increases
increases
Arbitrage
Effect on Put value
decreases
increases
increases
increases
decreases
Binomial
Black-Scholes
Factors affecting the call option value
•
Volatility of stock price
•
Recall the payoff for call and put. Call: max{0,S-X}, Put: max{0, X-S}
•
The higher the volatility of stock price , the higher the probability of S
being higher than X and thus the more likely the call will be exercised inthe-money to get profit. Thus C ↑ if σ ↑
•
Surprisingly, it is also true for put.
The higher the volatility of stock price , the higher the probability of S
being lower than X and thus the more likely the put will be exercised inthe-money to get profit. Thus P ↑ if σ ↑
Factor
Stock price
Exercise price
Volatility of stock price
Time to expiration
Interest rate
Options?
Terminology
Effect on Call value
increases
decreases
increases
increases
increases
Arbitrage
Effect on Put value
decreases
increases
increases
increases
decreases
Binomial
Black-Scholes
Factors affecting the call option value
•
Time to expiration
•
Recall the payoff for call and put. Call: max{0,S-X}, Put: max{0, X-S}
•
The longer the time to expiration, the more time allowed for the stock
price to climb above the exercise price and thus the more likely the call
will be exercised in-the-money to get profit. Thus C ↑ if T ↑
•
Surprisingly, it is also true for put.
The longer the time to expiration, the more time allowed for the stock
price to fall below the exercise price and thus the more likely the put will
be exercised in-the-money to get profit. Thus P ↑ if T ↑
Factor
Stock price
Exercise price
Volatility of stock price
Time to expiration
Interest rate
Options?
Terminology
Effect on Call value
increases
decreases
increases
increases
increases
Arbitrage
Effect on Put value
decreases
increases
increases
increases
decreases
Binomial
Black-Scholes
Factors affecting the call option value
•
Interest rate (risk-free)
•
Recall the put-call parity. S0 + P – C = X/(1+Rf)T
•
Keeping every other variables fixed, the higher the interest rate, the
smaller the RHS, and thus C has to increase to lower the LHS too. Thus
C ↑ if Rf ↑
•
Keeping every other variables fixed, the higher the interest rate, the
smaller the RHS, and thus P has to decrease to lower the LHS too. Thus
P ↓ if Rf ↑
Factor
Stock price
Exercise price
Volatility of stock price
Time to expiration
Interest rate
Options?
Terminology
- the least intuitive
Effect on Call value
increases
decreases
increases
increases
increases
Arbitrage
Effect on Put value
decreases
increases
increases
increases
decreases
Binomial
Black-Scholes
Binomial option pricing
•
With all the insights you have acquired. Let’s go to the first formal option pricing
model.
•
Assumption: The stock price can take only 2 possible values on the date the option
expires, no transaction cost and imperfections, frictionless market.
An example to illustrate, Binomial option pricing concerns about call options. Let’s now
consider a call, with exercise price = $125. Stock price is now $100. At expiration, it
will either go up to $200 or down to $50. (Note: NO probability is given)
$200
$100
$200 - $125 = $75
C
$50
Stock price
$0
Call option value
•
Consider a portfolio that consists of short 1 option and long m shares of
this stock.
•
Payoff of this portfolio is:
Options?
Either [Good state]
$200m - $75 if the stock price rises to $200
or
$50m if the stock price drops to $50.
[Bad state]
Terminology
Arbitrage
Binomial
Black-Scholes
Binomial option pricing
$200m
$100m
-$75
-C
$100m-C
$50m
$0
Long m Stocks + Short 1 Call
•
$200m-$75
=
$50m
The combined portfolio
Choose a specific m* to make the combined portfolio risk-less. (i.e.,
payoffs are the same in both states)
Set
$200m - $75 = $50m, solving, we have m* = 75/150 = 0.5
The ratio is what we needed. That means, if a portfolio consists of longing
1/2 share of the stock and shorting 1 call option, or if a portfolio consists
of longing 1 shares of the stock and shorting 2 call options, the portfolio is
risk-less.
$200m*-$75 = $25
$100m*-C = $50 - C
$50m*
= $25
The combined portfolio with m*
Options?
Terminology
Arbitrage
Binomial
Black-Scholes
Binomial option pricing
$200m*-$75 = $25
$100m*-C=$50 - C
$50m*
= $25
The combined portfolio with m*
•
So, the combined portfolio gives me $25 no matter which state is realized;
i.e, the portfolio is risk-less. The present value of this $25 at maturity
should be equal to the value of the combined portfolio that you pay now
(i.e., no arbitrage condition). Thus:
100m* - C = $50 – C = 25/(1+Rf)T
•
If time to expiration = 1 year, annual risk-free interest rate = 8%, then the
Call option should have a value equal to:
C = $50- 25 /(1+8%)1 = $26.85 (round up 2 significant decimal places)
•
Options?
Using the put-call parity, we can find the put option value with the same
exercise price and expiration date. [DO IT YOURSELF!!!]
Terminology
Arbitrage
Binomial
Black-Scholes
Black-Scholes option pricing formula
•
Generalizing the binomial option pricing, we have the Black-Scholes
formula, which is the Nobel prize winner Prof. Scholes’ main contribution
leading to his 1997 Nobel prize.
•
Black-Scholes formula:
C = S0N(d1) – X•e-RfT•N(d2)
Options?
Where
d1 = [ln(S0/X) + (Rf + σ2/2)T] / σ√T
And
d2 = d1 - σ√T
C
= Call Option Price
S0
= Current Stock Price
N(d1)
= Cumulative normal density function of (d1)
X
= Strike or Exercise price
N(d2)
= Cumulative normal density function of (d2)
Rf
= discount rate (risk free rate)
T
= time to maturity of option (as % of year)
σ
= volatility or annualized standard deviation of daily stock returns
Terminology
Arbitrage
Binomial
Black-Scholes
Black-Scholes option pricing forumla
C = S0N(d1) – X•e-RfT•N(d2)
Where
d1 = [ln(S0/X) + (Rf + σ2/2)T] / σ√T
And
d2 = d1 - σ√T
N(d1)= cumulative area
below d1 for a standard
normal distribution.
Standard Normal
Density Function
~ N(0,1)
-0.5
–0.2
0
If d1 = 0, N(d1) = 0.50
Options?
Terminology
0.2
0.5
If d1 = 0.5, N(d1) = 0.69
Arbitrage
Binomial
Black-Scholes
Black-Scholes option pricing forumla
Some of the important assumptions are as follows:
•
1) The stock will pay no dividends until after the option expiration date.
•
2) Both the interest rate and the standard deviation of daily return on the
stock are constant.
•
3) Stock prices are continuous, meaning that sudden extreme jumps such
as those in the aftermath of an announcement of a take-over attempt are
ruled out.
C = S0N(d1) – X•e-RfT•N(d2)
Options?
Where
d1 = [ln(S0/X) + (Rf + σ2/2)T] / σ√T
And
d2 = d1 - σ√T
Terminology
Arbitrage
Binomial
Black-Scholes
Black-Scholes: An example
C = S0N(d1) – X•e-RfT•N(d2)
Where
d1 = [ln(S0/X) + (Rf + σ2/2)T] / σ√T
And
d2 = d1 - σ√T
Example
What is the price of a call option given the
following?
S0 = 30, Rf = 5%, σ2 = 0.0305, X = $30, T = 1 year
d1 = 0.37362
N(d1) = 0.645657
d2 = 0.198978
N(d2) = 0.57886
C = S0[N(d1)] – Xe-rt[N(d2)]
C = $ 2.85, using put-call parity, we can calculate
the corresponding put option price.
Options?
Terminology
Arbitrage
Binomial
Black-Scholes
Some more insights on options
• American Options can be exercised at anytime
before maturity
• European Options can be exercised at maturity
• It is never optimal to exercise an American call option
early:
Thus, American and European calls should have the
same price
• But it may be optimal to exercise an American put
option earlier than maturity
• Empirical evidence:
– Black-Scholes option pricing model does well at
pricing options that are at the money, but do much
worse as the options go deeper into or out of the
money
Options?
Terminology
Arbitrage
Binomial
Black-Scholes
For Final
• You will not need to remember the Black-Scholes formula.
• You have to try the Black-Scholes formula before the exam
because the final exam will for sure have a question concerning
the Black-Scholes. That means you have to know how to use a
Cumulative normal distribution table.
• You have to be familiar with the put-call parity and no arbitrage
condition.
• You have to know the Binomial option pricing too. Work it out at
least once.
• You should try to get yourself familiar with how to quote an
option price from CBOE. And you should be able to understand
the meaning of a table you see from a CBOE option quote.
• I strongly encourage you to do the exercises on options posted
on the course webpage. Try them before you look into the
solutions.
Options?
Terminology
Arbitrage
Binomial
Black-Scholes