Binomial Option Pricing: II

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Binomial Option Pricing II

• Eugene
  Tan

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Binomial Option Pricing II

• Understanding Early Exercise
• Understanding Risk-Neutral Pricing
• The Binomial Tree and Lognormality
• Estimating Volatility
• Stocks Paying Discrete Dividends

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Understanding Early Exercise of a Call

• Receives the stock and therefore receives future
  dividends
• Pays the strike price prior to expiration (this
  has an interest cost)
• Loses the insurance implicit in the call

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Early Exercise for an American Call
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Early exercise for an American Put
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Understanding Risk-Neutral pricing

• The Risk-Neutral Probability
• Pricing an option using real probabilities

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Risk-Neutral Probability

• Risk-averse
• Risk-neutral

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Risk-Neutral Probability II

• Scenario 1
• - Dr. Hill is offered 1,000
• Scenario 2
• - Dr. Hill is offered 2,000 with probability
  0.5 and 0 with probability 0.5

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Risk-Neutral Probability III
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Pricing an Option using Real Probability

• 1-period example
• Multi-period example

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1 period example
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Multi-period example
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The Binomial Tree and Lognormality

• The random walk model
• Modeling stock prices as a random walk
• Continuously compounded returns (CCR)
• Standard deviation of returns
• The Binomial Model
• Lognormality and the Binomial Model
• Alternative Binomial Trees

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The Random Walk Model

• Understanding a random walk

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Modeling stock prices as a random walk
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Problems with modeling stock prices as a random
walk I

• If by chance, we get enough cumulative down
  movements, the stock price will become negative.
• Because stockholders have limited liability (they
  can walk away from a bankrupt firm), a stock
  price will never be negative.

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Problems with modeling stock prices as a random
walk II

• The magnitude of the move (1) should depend upon
  how quickly the coin flips occur and the level of
  the stock price.

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Problems with modeling stock prices as a random
walk III

• The stock on average should have a positive
  return.
• The random walk model taken literally does not
  permit this.

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Review of Continuously Compounded Returns (CCR)

• The logarithmic function computes returns from
  prices.
• The exponential function computes prices from
  returns.
• CCRs are additive.
• CCRs can be less than -100
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  

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Logarithmic function and returns from prices

• Let St and Sth be stock prices at times t and
  th.The CCR between t and th is rt,th and is
  defined by

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Exponential function and prices from returns

• If we know the CCR, we can obtain Sth by
  exponentiating both sides of the previous
  equation, giving

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CCRs are additive

• Suppose we have CCRs over a number of periods
  for example, rt,th, rth,t2h, etc. the CCR over
  a long period is the sum of the CCRs over the
  shorter periods, i.e.,

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CCRs can be less than 100

• A CCR that is a large negative number still gives
  a positive stock price.
• Thus if the log of the stock price follows a
  random walk, the stock price cannot become
  negative.

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The standard deviation of returns

• From

we get the annual return
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Variance of the annual return I
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Variance of the annual return II

• Suppose the returns are uncorrelated over time
  and identically distributed, with this
  assumption,

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The standard deviation of returns

• The standard deviation therefore scales with the
  square root of time. This is why

appears in the binomial pricing model
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The Binomial Model I
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The Binomial model II

• The stock price cannot become negative.
• As stock price moves occur more frequently, h
  gets smaller.
• We can guarantee that the expected change in the
  stock price is positive.

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Lognormality and the Binomial Model

• The binomial tree approximates a lognormal
  distribution.
• It is commonly used to model stock prices.

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What is the lognormal distribution?

• The lognormal distribution is the probability
  distribution that arises from the assumption that
  CCRs on the stock are normally distributed.

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A binomial tree
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Construction of a Binomial tree

• Number of ways to reach ith node

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Construction of a Binomial tree II

• Probability of reaching ith node

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Construction of a Binomial Tree III
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Comparison between lognormal and binomial
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Comparison between lognormal and binomial II
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Alternative Binomial Trees

• The Cox-Ross-Rubinstein binomial tree
• The lognormal tree

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The Cox-Ross-Rubinstein binomial tree

• It is the best known way to construct a binomial
  tree, and as such

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The Cox-Ross-Rubinstein binomial tree II

• This approach is often used in practice.
• However, if h is large or s is small, it is
  possible that

which violates
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The Cox-Ross-Rubinstein binomial tree III

• In real applications, h would be small, so the
  previously mentioned problem would not occur.

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The lognormal tree

• This is another alternative to construct the
  tree, using

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The lognormal tree II

• This procedure for generating a tree was proposed
  by Jarrow and Rudd (1983) and is sometimes called
  the Jarrow-Rudd binomial model.

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The lognormal tree III

• In computing the following equation, you will
  find that the risk-neutral probability of an
  up-move is generally close to 0.5.

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Binomial Models

• All three methods of constructing a binomial tree
  yield different option prices for finite n. but
  approach the same price as n approaches infinity.

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Binomial Models II
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Binomial Models II

• While different binomial trees all have different
  up or down movements, all have the same ratio of
  u to d.

or
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Is the Binomial model realistic?

• The binomial model is a form of the random walk
  model, adapted to modeling stock prices.

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Is the Binomial model realistic? II

• The lognormal random walk model, assumes among
  other things, that i) volatility is
  constant ii) large stock price
  movements dont occur
• iii) returns are independent over time

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Is the Binomial model realistic? III

• The random walk model is a useful starting point
  for thinking about stock price behavior.
• Widely used because of its elegant simplicity
• However, it is not sacrosanct (inviolable).

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Estimating volatility

• The most important decision is the value we
  assign to s, which we cannot observe directly.

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Estimating volatility II

• One way of measuring s is by computing the
  standard deviation of CCRs.

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Estimating volatility III

• Volatility computed from historical stock returns
  is historical volatility.

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Estimating volatility IV

• Extra care is required with volatility if the
  random walk model is not a plausible economic
  model of the assets price behavior

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Stocks paying discrete dividends

• Modeling discrete dividends
• Problems with the discrete dividend tree
• A binomial tree using the prepaid forward

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Modeling discrete dividends

• Suppose that a dividend will be paid in time t
  and th, and that its future value at time th is
  D. The time t forward price for delivery at th
  is then

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Modeling discrete dividends II

• Since the stock price at time th will be
  ex-dividend, we create the up and down moves
  based on the ex-dividend stock price

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Modeling discrete dividends III

• How does option replication work when a dividend
  is imminent? When a dividend is paid, we have to
  account for the fact that the stock earns the
  dividend. Thus we have,

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Modeling discrete dividends IV

• Solving the equations, you get

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Problems with the discrete dividend tree

• The practical problem with this procedure is that
  the tree does not completely recombine after a
  discrete dividend.

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Problems with the discrete dividend tree II
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A binomial tree using the prepaid forward

• The key insight for this method is that if we
  know for certain that a stock will pay a fixed
  dividend, then we can view the stock price as
  being the sum of two components the dividend,
  and the present value of the ex-dividend value of
  the stock (prepaid forward price).

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A binomial tree using the prepaid forward II

• Since the dividend is known, all volatility is
  attributed to the prepaid forward component of
  the stock price.

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A binomial tree using the prepaid forward III

• Risk-neutral probabilities for the tree are
  obtained in the same way as in the absence of
  dividends.

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A binomial tree using the prepaid forward IV

• We are constructing the binomial tree for the
  prepaid forward, which pays no dividends.
• The probabilistic equation for an up-move for a
  prepaid forward is the same as a nondividend
  paying stock

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A binomial tree using the prepaid forward V
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Summary

• Risk-neutral option valuation is consistent with
  valuation using more traditional discounted cash
  flow methods.
• The binomial model is a random walk model adapted
  to modeling stock prices.
• Binomial model can also be adapted to price
  options on a stock that pays discrete dividends.

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• End
• -Eugene Tan