Chapter 12: Basic option theory

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Chapter 12 Basic option theory

• Investment Science
• D.G. Luenberger

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Before we talk about options

• This course so far has dealt with deterministic
  cash flows and single-period random cash flows.
• Now wed like to deal with random flows at each
  of several time points, i.e., multiple random
  cash flows.
• Multiple random cash flow theories are generally
  very difficult. Here, wed like to focus on a
  special case derivatives, i.e., those assets
  whose cash flows are functionally related to
  other assets whose price characteristics are
  assumed to be known.
• Options are an important category of derivatives.
  

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Option, I

• An option is the right, but not the obligation,
  to buy (or sell) an underlying asset under
  specified terms. Usually there are a specified
  price, called strike price or exercise price (K),
  and a specified period of time, called maturity
  (T) or expiration date, over which the option is
  valid.
• An option is a derivative because whose cash
  flows are related to the cash flows of the
  underlying asset.
• If the option holder actually does buy and sell
  the underlying asset, the option holder is said
  to exercise the option.
• The market price of an option is called premium.

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Option, II

• An option that gives the right to purchase (sell)
  the underlying asset is called a call (put)
  option.
• An American option allows exercise at any time
  before and including the expiration date.
• An European option allows exercise only on the
  expiration date. In this course, we will focus
  on European options because their pricing is
  easier.
• The underlying assets of options can be financial
  securities, such as IBM shares or SP 100 Index,
  or physical assets, such as wheat or corn.

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Option, III

• Many options are traded on open markets. Thus,
  their premiums are established in the market and
  observable.
• Options are wonderful instruments for managing
  business and investment risk, i.e., hedging.
• Options can be particularly speculative because
  of their built-in leverages. For example, if you
  are very sure that IBMs share price will go up,
  betting 1 on IBM call options may generate a
  much higher return than betting 1 on IBM shares.

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Notation

• S the price of the underlying asset.
• C the value of the call option.
• P the value of the put option.

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Value of call option at expiration
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Value of put option at expiration
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Value of option at expiration

• The value of the call at expiration C max (0,
  S K).
• The call option is in the money, at the money, or
  out of money, depending on whether S gt K, S K,
  or S lt K, respectively.
• The value of the put at expiration P max (0, K
  S).
• The put option is in the money, at the money, or
  out of money, depending on whether S lt K, S K,
  or S gt K, respectively.

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11.1 Binomial model, PP. 297-299

• A binomial model can be a single-period model or
  a multiple-period model.
• A basic period length can be a week, a month, a
  year, etc.
• If the price of an asset is known at the
  beginning of a period, say S, the price of the
  asst at the end of the period is one of only two
  possible values, Su and Sd, where u gt 1 gt d gt
  0. Su (Sd) is expected to happen with
  probability p (1 p).
• That is, we have uncertainty, but in the form of
  two possible values.

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Binomial model
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Calibration, I

• Binomial models provide an uncertain structure
  for us to model the underlying assets price
  dynamics.
• This modeling is necessary because option value
  is a function of the underlying assets price
  dynamics.
• Thus, to obtain accurate valuation for an option,
  we need to do a good job on modeling the
  underlying assets price dynamics.
• That is, we need to choose binomial parameters,
  i.e., p, u, and d, carefully such that the
  binomial-based price dynamics is consistent with
  the observable (historical) price
  characteristics, e.g., average return and
  standard deviation, of the underlying asset.

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Calibration, II

• Let v be the expected (average) annual return of
  the underlying asset, v E ln(ST /S0 ), say
  12.
• Let ? be the yearly standard deviation of the
  underlying asset, ?2 var ln(ST /S0 ), say
  15.
• Note that 12 return and 15 standard deviation
  are about the kind of numbers that you would
  expect from a typical SP 500 stock.
• Now we need to define the period length relative
  to a year. If we define a period as a week, a
  period length of ?t is 1/52.

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Calibration, III

• Then, the binomial parameters can be selected as
• p ½ ½ (v / ?) (?t)1/2.
• u e? (?t)1/2.
• d 1/u.
• e is exponential and has a value of 2.7183.
• With these choices, the binomial model will
  closely match the values of v and ? (see pp.
  313-315 for the proof).

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Calibration, IV
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1-period binomial option theory, I

• We assume that it is possible to borrow or lend
  at the risk-free rate, r.
• Let R 1 r, and u gt R gt d.
• Suppose that there is a call option on the
  underlying asset with strike price K and
  expiration at the end of the single period.
• Let Cu (Cd) be the value of the call at
  expiration.

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3 related lattices
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No-arbitrage

• The key to price the call option at time 0 is to
  form a portfolio at time 0 (1) the portfolio
  consists of the underlying asset and the
  risk-free asset, (2) the portfolios value at
  time 1 is equal to the value of the call at time
  1, regardless whether it is up or down.
• This portfolio is called a replicating portfolio
  x dollar worth of the underlying asset and b
  dollar worth of the risk-free asset.
• No-arbitrage because the replicating portfolio
  and the call yield the value at time 1 regardless
  what might happen, the value of the replicating
  portfolio and the call at time 0 must be the
  same.
• That is, x b C.

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Outcome matching

• The value of the replicating portfolio equals to
  the value of the call at time 1 when it is up u
  x R b Cu.
• The value of the replicating portfolio equals to
  the value of the call at time 1 when it is down
  d x R b Cd.
• Solve for x and b from the two equations.

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1-period binomial Solution

• x (Cu Cd) / (u d).
• b (u Cd d Cu) / (R (u d)).
• Based on no-arbitrage, we know that C x b.
• C (Cu Cd) / (u d) (u Cd d Cu) / (R
  (u d)).
• After some algebras, we have C (1/R) q Cu
  (1 q) Cd, where q (R d) / (u d).
• Note that p is not in the pricing equation
  because no trade-off among probabilistic events
  is made.

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1-month IBM call option, I

• Consider IBM with a volatility of its logarithm
  of ? 20. The current price of IBM is 62. A
  call option on IBM has an expiration date 1 month
  from now and a strike price of 60. The current
  interest rate is 10, compounded monthly.
  Suppose that IBM will not pay dividends.

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1-month IBM call option, II
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When no information about v and ?

• If we have a primitive binomial problem in which
  we have no information on expected return and
  standard deviation, we then must know the two
  possible outcomes.
• Example suppose the market price of a stock is
  50. The two possible outcomes for the stock
  price is either 60 or 40 in a year.
• A call option with a one-year expiration and a
  50 exercise price.
• Interest rate is 5.

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Duplicating portfolio

• We need to duplicate the call with the strategy
  of buying stocks and borrowing monies.
• The duplicating strategy is to buy ½ share of the
  stock and borrow 19.05. Why this particular
  combination? We will talk about this later.

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When the stock price is up

• When the stock price is up, the payoff of buying
  a call is 10 (60 - 50).
• When the stock price is up, the payoff the
  duplicating strategy is also 10. The sum of the
  following two positions is 10 (30 - 20) (1)
  buying ½ share ½ 60 30, and (2) borrowing
  19.05 at 5 -19.05 1.05 -20.

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When the stock price is down

• When the stock price is down, the payoff of
  buying a call is 0.
• When the stock price is down, the payoff the
  duplicating strategy is also 0. The sum of the
  following two positions is 0 (20 - 20) (1)
  buying ½ share ½ 40 20, and (2) borrowing
  19.05 at 5 -19.05 1.05 -20.

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No arbitrage

• The call and the duplicating strategy generate
  identical payoffs at the end of the year.
• No arbitrage principle implies that the current
  market price of the call equals to the current
  market price of the duplicating position.
• The market price of the duplicating position is
  5.95 (25 - 19.05). Buying ½ share costs 25
  (1/2 50).
• The call price is 5.95.

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Why ½ share?

• The duplicating portfolio was given to be buying
  ½ share and borrowing 19.05 earlier.
• Why ½ share? This amount is called the delta of
  the call.
• Delta swing of the call / swing of the stock
  (10 - 0) / (60 - 40) ½.

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Why borrowing 19.05?

• Buying ½ share gives us either 30 or 20 at
  expiration, which is exactly 20 higher than the
  payoffs of the call, 10 and 0, respectively.
• To duplicate the position, we thus need to borrow
  a dollar amount such that we will need to pay
  back exactly 20.
• Given the future value is 20, the interest rate
  is 5, and the number of the time period is 1, we
  have the present value to be 19.05 (use your
  financial calculator).

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Multiple-period pricing

• The usefulness of single-period binomial pricing
  is that it can be applied to multiple-period
  problems in a straightforward manner.
• That is, the single period pricing, C (1/R)
  q Cu (1 q) Cd, is repeated at every
  node of the lattice, starting from the final time
  period and working backward toward the initial
  time.

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5-month IBM call option, I
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5-month IBM call option, II
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1-month vs. 5-month

• The call price for 1-month IBM call is 3.14.
  The call price for 5-month IBM call is 5.85.
• Holding other factors constant, the longer the
  maturity, the higher the call premium.
• The reason for this is that additional time
  allows for a greater chance for the stock to rise
  in value, increasing the final payoff of the call
  option.
• See Figure 12.3, p. 324.
• Along the same line of seasoning, the higher the
  standard deviation, the higher the call premium.
  You should verify this numerically.

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How about put option pricing?

• So far, we have focused on call pricing.
• The reason for this is that for European options
  one can calculate the value of a put option, P,
  based on value the call option, C, when they have
  the same strike price and maturity.
• P C S df K, where df is risk-free
  discount factor.
• This relationship is called the put-call parity.

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Put option pricing

• Consider a GM call option and a GM put option,
  both have 3 months to expiration and the same
  strike price, 35. The current price of GM
  shares is 37.78. The call premium is 4.25.
  The interest rate is 5.5, so over 3 months, the
  discount factor is 0.986 ( 1 / (1 0.055/4).
• P C S df K 4.25 37.78 0.986 35
  1.00.

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Options are interesting and important

• A combination of options can lead to a unique
  payoff structure that otherwise would not be
  possible.
• Options make it happen!
• Example a butterfly spread, p. 325.
• Question who would hold a butterfly spread?