Options: valuation
Arbitrage
We have introduced Put-Call Parity.
In fact, Put-call parity 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 identical product is selling at different prices in two very
close locations, you will buy it from the lower-priced location and
immediately sell it at the higher-priced location and realize a gain. Again,
arbitrage appears because the law of one price is violated.
Terminology
Arbitrage
Binomial
Black-Scholes
Arbitrage
•
It is easy to think of a physical item selling at different prices in close
locations would stimulate huge amount of people taking advantage of it. It
is the fact that so many people would jump on it that essentially keeps the
law of one price holds because the huge demand for a lower cost product
drives up its price until it equalizes to the higher price.
But how are we going to apply the concept to risky assets?
•
Imagine you are holding two portfolios. They have different assets inside
each of them. But ultimately, their payoffs across different possible states
are EXACTLY the same. Under the law of one price, the two portfolios
should have the same value. (e.g., if IBM or Hilton stocks offer the same
payoff structure to you, you should expect them to have the same stock
price, irrespective of them being different companies. The bottom line –
payoff structure, not company name)
•
If they are not valued the same, 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 another risk-less portfolio.
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 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 slides. Payoffs at maturity 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, P1 = P2 - P3 + P4
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 payoffs depend in a variety of ways on the values of
other securities.
$
•
<= Long 1 put
ST
x
ST
<= Long 1 call
•
x
•
ST
<= Short 1 put
x
Total Payoff
ST
<= Short 1 call
Imagine the 4 different
payoffs patterns:
• Long Put
• Long Call
• Short Put
• Short Call
And imagine options with
many different levels of
exercise prices.
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
Example: Protective Put
•
•
As an example. You would like to invest in Google but unwilling to bear potential
losses beyond some given levels. 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.
•
<= Long 1 stock
ST
x
x
<= Long 1 put
ST
x
x
Total Payoff
Profit
ST
X-(S0+P)
Options?
•
•
<= Total Payoffs
x
Terminology
Arbitrage
The final payoff structure is such
that no matter how much Google’s
share drops in price, your overall
loss is limited to a fixed amount,
whereas if Google’s share
increases in price, you will still
gain from it.
The precise exercise price you
choose will dictate how much the
maximum loss you are willing to
bear.
Again, it is a protective way of
holding a stock, that’s why it’s
called Protective Put.
Binomial
Black-Scholes
Example: 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.
•
<= Long 1 stock
ST
x
•
x
ST
<= Short 1 call
x
•
Profit
x
Total Payoff
<= Total Payoffs
C
ST
x
Options?
Terminology
Arbitrage
The final payoff structure is such
that no matter how much Google’s
share drops in price, your overall
loss is limited to its price today,
plus the amount you acquired
from selling a call.
If Google’s price increases, and
the call holder exercises its right to
buy from you, you have a stock to
cover your obligtion.
If Google’s stock does not vary
much in price, for example, it
remains at X on the maturity date,
then you’ve gained C, the sales
price of the call you sold.
Binomial
Black-Scholes
Example: 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 stock 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.
Profit
x
<= Total Payoffs
ST
-(P+C)
Options?
x
Terminology
Arbitrage
Binomial
Black-Scholes
Option strategies
•
Of course, 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 wisely.
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 be violated, people will
jump into the opportunity of making 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 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 first. Assuming the stock
itself doesn’t payout dividend during before the call option matures.
First, its value cannot be negative. Because the holder of a call option need
not be obligated to exercise it if its value of exercise is less than zero.
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]
•
Options?
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.
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 payoff structure actually more favorable than
{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 a formal option pricing
model.
•
Assumption: The stock price can take only 2 possible values at option
expiration.
•
An example to illustrate, let’s consider a call, exercise price = $125. Stock
price is now $100. 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:
[1] $200m - $75 if the stock price rises to $200 or
[2] $50m if the stock price drops to $50.
Options?
Terminology
Arbitrage
Binomial
Black-Scholes
Binomial option pricing
$200m
$100m
-$75
-C
$100m-C
$50m
0
Long m Stocks + Short 1 Call
•
=
$50m
The combined portfolio
Choose a specific m* to make the combined portfolio risk-less. (i.e.,
payoffs are the same for the two different states)
Set
•
$200m-$75
$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 of the two
states is realized. That means, 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 put-call parity, we can find the put option value with the same
exercise price.
Terminology
Arbitrage
Binomial
Black-Scholes
Black-Scholes option pricing forumla
•
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
On the ECMC49F Exam
• You will not need to remember the Black-Scholes formula.
• You will need to try using 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, like
the one posted on lecture note 06.
• I strongly encourage you to do the exercises on options posted
on the course webpage. Solutions will be provided next week.
But try it out before you look into the solutions.
Options?
Terminology
Arbitrage
Binomial
Black-Scholes