Options: valuation

1
Options valuation
2
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 investors 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
tT
Time(t)
t0
Payoff
Seller gets C
IBM price at T
x
Options? Terminology Arbitrage Binomial Black-
Scholes
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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
  IBMs 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
tT
Time(t)
t0
Payoff
Seller gets C
IBM price at T
x
Options? Terminology Arbitrage Binomial Black-
Scholes
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Individual gt 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
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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.
• 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.

Options? Terminology Arbitrage Binomial Black-
Scholes
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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 mis-price will
  eliminate the arbitrage opportunity.

Options? Terminology Arbitrage Binomial Black-
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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

Possible states Possible states Possible states
Good Normal Bad
Risky assets x 100 80 70
Risky assets y 15 25 30
Risky assets z 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

Options? Terminology Arbitrage Binomial Black-
Scholes
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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 Possible states
ST gtX ST X
Investments Risk-free Investment with an amount equal to X/(1R)T X X
Investments Long a share of Stock ST ST
Investments Short 1 Call option -(ST - X) 0
Investments Long 1 put option 0 (X-ST)

• 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/(1Rf)T

Options? Terminology Arbitrage Binomial Black-
Scholes
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No Arbitrage Put-Call Parity

• The graph of combining different options and
  assets is such that the payoffs of all assets are
  added up vertically.

lt Long 1 put lt Long 1 stock lt Short 1
call lt Total payoffs
ST
x
ST
x
ST
x
Total Payoff
x
ST
x
Options? Terminology Arbitrage Binomial Black-
Scholes
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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.

• 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.

lt Long 1 put lt Long 1 call lt Short 1
put lt Short 1 call
ST
x
ST
x
ST
x
ST
x
Options? Terminology Arbitrage Binomial Black-
Scholes
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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-
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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 Googles share drops in price, your
  overall loss is limited to a fixed amount,
  whereas if Googles share increases in price, you
  will still gain from it.
• The precise exercise price you choose will
  dictate the maximum loss you are willing to bear.
• Again, it is a protective way of holding a stock,
  thats why its called Protective Put.

lt Long 1 stock lt Long 1 put lt Total
Payoffs
ST
x
x
ST
x
Total Payoff
x
ST
x
Options? Terminology Arbitrage Binomial Black-
Scholes
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Strategy Covered Call

• What if you're neutral on Googles 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 Googles share price drops, your overall
  loss is limited to the price you pay today. And
  you still have the amount you acquired from
  selling a call.
• If share price increases, and the call holder
  exercises its right to buy from you, you have a
  stock to fulfill your obligation.
• If share price does not change much, for example,
  it remains at X on the expiration date, then
  youve gained C, the sales price of the call you
  sold.

lt Long 1 stock lt Short 1 call lt Total
Payoffs
ST
x
x
ST
x
x
Total Payoff
ST
x
Options? Terminology Arbitrage Binomial Black-
Scholes
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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 companys stock.
• Buy one put of that companys stock
• Such an option strategy is called Straddle.

• The final payoff structure is such that if that
  companys stock price varies a lot, you will
  benefit the most.
• If instead, the companys stock price doesnt
  vary a lot because of the news, you will likely
  make a loss.

lt Long 1 call lt Long 1 put lt Total Payoffs
ST
x
x
ST
x
Total Payoff
x
ST
x
Options? Terminology Arbitrage Binomial Black-
Scholes
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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
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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 well keep on using the no arbitrage
  condition in order to derive the equilibrium
  option prices.

Options? Terminology Arbitrage Binomial Black-
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Range of possible call option values

• Let us first look at the boundary for a call
  option. Assuming the underlying stock doesnt
  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.
  C0 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.
  CS0 2 Upper bound
• Third, its value cannot be lower than the present
  stock price minus the present value of the
  exercise price. CS0 - Present value of
  X or CS0 X/(1R)T 3 lower bound
• Reason for 3 if you compare 2 different
  portfolios
• a buy a stock now at S0 and borrow X/(1R)T
• b buy a call option with exercise price X.

Options? Terminology Arbitrage Binomial Black-
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Range of possible call option values

• CS0 X/(1R)T 3 lower bound
• Reason for 3 if you compare 2 different
  portfolios
• a buy a stock now at S0 and borrow X/(1R)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
  dont 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/(1R)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 CS0 X/(1R)T

Options? Terminology Arbitrage Binomial Black-
Scholes
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Range of possible call option values

• C0 1 lower bound
• CS0 2 Upper bound
• CS0 X/(1R)T 3 lower bound
• With all 3 boundary conditions, we get the
  following graph

Call Value (C)
Upper bound S0
Lower Bound S0 - X/(1R)T
S0
X/(1R)T
Options? Terminology Arbitrage Binomial Black-
Scholes
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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)
Upper bound S0
Lower Bound S0 - X/(1R)T
S0
X/(1R)T
Options? Terminology Arbitrage Binomial Black-
Scholes
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Factors affecting the call option value

• We identify 5 factors that affect an options
  value
• 1) Stock price (S)
• 2) Exercise Price (X)
• 3) Volatility of the underlying stock price (s)
• 4) Time to Maturity/expiration (T)
• 5) Interest rate (Rf)
• You should familiarize yourself with the
  following table

Factor Effect on Call value Effect on Put
value Stock price increases decreases Exerci
se price decreases increases Volatility of
stock price increases increases Time to
expiration increases increases Interest rate
increases decreases
Options? Terminology Arbitrage Binomial Black-
Scholes
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Factors affecting the call option value

• Stock price
• Recall the payoff for call and put. Call
  max0,S-X, Put max0, 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 Effect on Call value Effect on Put
value Stock price increases decreases Exerci
se price decreases increases Volatility of
stock price increases increases Time to
expiration increases increases Interest rate
increases decreases
Options? Terminology Arbitrage Binomial Black-
Scholes
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Factors affecting the call option value

• Exercise price
• Recall the payoff for call and put. Call
  max0,S-X, Put max0, 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 Effect on Call value Effect on Put
value Stock price increases decreases Exerci
se price decreases increases Volatility of
stock price increases increases Time to
expiration increases increases Interest rate
increases decreases
Options? Terminology Arbitrage Binomial Black-
Scholes
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Factors affecting the call option value

• Volatility of stock price
• Recall the payoff for call and put. Call
  max0,S-X, Put max0, 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 in-the-money to get profit. Thus C ? if
  s ?
• 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
  in-the-money to get profit. Thus P ? if s ?

Factor Effect on Call value Effect on Put
value Stock price increases decreases Exerci
se price decreases increases Volatility of
stock price increases increases Time to
expiration increases increases Interest rate
increases decreases
Options? Terminology Arbitrage Binomial Black-
Scholes
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Factors affecting the call option value

• Time to expiration
• Recall the payoff for call and put. Call
  max0,S-X, Put max0, 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 Effect on Call value Effect on Put
value Stock price increases decreases Exerci
se price decreases increases Volatility of
stock price increases increases Time to
expiration increases increases Interest rate
increases decreases
Options? Terminology Arbitrage Binomial Black-
Scholes
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Factors affecting the call option value

• Interest rate (risk-free) - the least
  intuitive
• Recall the put-call parity. S0 P C
  X/(1Rf)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 Effect on Call value Effect on Put
value Stock price increases decreases Exerci
se price decreases increases Volatility of
stock price increases increases Time to
expiration increases increases Interest rate
increases decreases
Options? Terminology Arbitrage Binomial Black-
Scholes
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Binomial option pricing

• With all the insights you have acquired. Lets 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. Lets 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
200 - 125 75
100
C
50
0
Stock price
Call option value

• Consider a portfolio that consists of short 1
  option and long m shares of this stock.
• Payoff of this portfolio is
• Either Good state 200m - 75 if the stock
  price rises to 200
• or Bad state 50m if the stock price drops
  to 50.

Options? Terminology Arbitrage Binomial Black-
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Binomial option pricing
200m
-75
200m-75
100m
-C
100m-C
50m
0
50m
Long m Stocks
Short 1 Call
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-
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Binomial option pricing
200m-75 25
100m-C50 - 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/(1Rf)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 /(18)1 26.85 (round up 2
  significant decimal places)
• Using the put-call parity, we can find the put
  option value with the same exercise price and
  expiration date. DO IT YOURSELF!!!

Options? Terminology Arbitrage Binomial Black-
Scholes
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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) Xe-RfTN(d2)
• Where d1 ln(S0/X) (Rf s2/2)T / svT
• And d2 d1 - svT

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) s
volatility or annualized standard deviation of
daily stock returns
Options? Terminology Arbitrage Binomial Black-
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Black-Scholes option pricing forumla
C S0N(d1) Xe-RfTN(d2) Where d1
ln(S0/X) (Rf s2/2)T / svT And d2 d1 -
svT
N(d1) cumulative area below d1 for a standard
normal distribution.
Standard Normal Density Function N(0,1)
-0.5 0.2 0 0.2 0.5
If d1 0, N(d1) 0.50 If d1 0.5, N(d1) 0.69
Options? Terminology Arbitrage Binomial Black-
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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) Xe-RfTN(d2)
• Where d1 ln(S0/X) (Rf s2/2)T / svT
• And d2 d1 - svT

Options? Terminology Arbitrage Binomial Black-
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Black-Scholes An example
C S0N(d1) Xe-RfTN(d2) Where d1
ln(S0/X) (Rf s2/2)T / svT And d2 d1 -
svT
Example What is the price of a call option given
the following? S0 30, Rf 5, s2 0.0305, X
30, T 1 year
d1 0.37362
N(d1) 0.645657
d2 0.198978
N(d2) 0.57886
C S0N(d1) Xe-rtN(d2)
C 2.85, using put-call parity, we can
calculate the corresponding put option price.
Options? Terminology Arbitrage Binomial Black-
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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-
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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