BLACK-SCHOLES OPTION PRICING MODEL

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BLACK-SCHOLESOPTION PRICING MODEL

• Chapters 7 and 8

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BOPM and the B-S OPM

• The BOPM for large n is a practical, realistic
  model.
• As n gets large, the BOPM converges to the B-S
  OPM.
• That is, for large n the equilibrium value of a
  call derived from the BOPM is approximately the
  same as that obtained by the B-S OPM.
• The math used in the B-S OPM is complex but the
  model is simpler to use than the BOPM.

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B-S OPM Formula

• B-S Equation

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Terms

• T time to expiration, expressed as a proportion
  of the year.
• R continuously compounded annual RF rate.
• R ln(1Rs), Rs simple annual rate.
• annualized standard deviation of the
• logarithmic return.
• N(d) cumulative normal probabilities.

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N(d) term

• N(d) is the probability that deviations less than
  d will occur in the standard normal distribution.
  The probability can be looked up in standard
  normal probability table (see JG, p.217) or by
  using the following

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N(d) term
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B-S Features

• Model specifies the correct relations between the
  call price and the explanatory variables

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Arbitrage Portfolio

• The B-S equation is equal to the value of the
  replicating portfolio

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Arbitrage Portfolio

• The replicating portfolio in our example consist
  of buying .4066 shares of stock, partially
  financed by borrowing 15.42

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Arbitrage Portfolio

• If the price of the call were 3.00, then an
  arbitrageur should go short in the overpriced
  call and long in the replicating portfolio,
  buying .4066 shares of stock at 45 and borrowing
  15.42.
• Since the B-S is a continuous model, the
  arbitrageur would need to adjust the position
  frequently (every day) until it was profitable to
  close. For an example, see JG 222-223.

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Dividend Adjustments Pseudo-American Model

• The B-S model can be adjusted for dividends using
  the pseudo-American model. The model selects the
  maximum of two B-S-determined values

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Dividend Adjustments Continuous
Dividend-Adjustment Model

• The B-S model can be adjusted for dividends using
  the continuous dividend-adjustment model.
• In this model, you substitute the following
  dividend-adjusted stock price for the current
  stock price in the B-S formula

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Black-Scholes Put Model
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B-S Put Models Features

• The model specifies the correct relations between
  the put price and the explanatory variables
• Note Unlike the call model, the put model is
  unbound.

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Arbitrage Portfolio

• The B-S put equation is equal to the value of the
  replicating portfolio

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Arbitrage Portfolio

• The replicating portfolio in our example consist
  of selling .5934 shares of stock short at 45 and
  investing 33.83 in a RF security

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Dividend Adjustments

• B-S put model can be adjusted for dividends by
  using the continuous dividend-adjustment model
  where is substituted for So. A
  pseudo-American model can also be used. This
  model for puts is similar to calls, selecting
  the maximum of two B-S-determined values

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Barone-Adesi and Whaley Model

• The pseudo-American model estimates the value of
  an American put in reference to an ex-dividend
  date. When dividends are not paid (and as a
  result, we do not have a specific reference date)
  the model cannot be applied.
• This is not a problem with applying the pseudo
  model to calls, since the advantage of early
  exercise applies only when an ex-dividend date
  exist.
• As we saw with the BOPM for puts, early exercise
  can sometimes be profitable, even when there is
  not a dividend.
• A model that addresses this problem and can be
  used to price American puts, as well as calls, is
  the Barone-Adesi Whaley (BAW) model. See JG
  246-248.

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Estimating the B-S Model Implied Variance

• The only variable to estimate in the B-S OPM (or
  equivalently, the BOPM with large n) is the
  variance. This can be estimated using historical
  averages or an implied variance technique.
• The implied variance is the variance which makes
  the OPM call value equal to the market value. The
  software program provided each student calculates
  the implied variance.

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Estimating the B-S Model Implied Variance

• For at-the-money options, the implied variance
  can be estimated using the following formula

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B-S Empirical Study

• Black-Scholes Study (1972) Black and Scholes
  conducted an efficient market study in which they
  simulated arbitrage positions formed when calls
  were mispriced (C not to Cm).
• They found some abnormal returns before
  commission costs, but found they disappeared
  after commission costs.
• Galai found similar results.

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MacBeth-Merville Studies

• MacBeth and Merville compared the prices obtained
  from the B-S OPM to observed market prices. They
  found
• the B-S model tended to underprice in-the-money
  calls and overprice out-of-the money calls.
• the B-S model was good at pricing on-the-money
  calls with some time to expiration.

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Bhattacharya Studies

• Bhattacharya (1980) examined arbitrage portfolios
  formed when calls were mispriced, but assumed the
  positions were closed at the OPM values and not
  market prices.
• Found B-S OPM was correctly specified.

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General Conclusion

• Empirical studies provide general support for the
  B-S OPM as a valid pricing model, especially for
  near-the-money options.
• The overall consensus is that the B-S OPM is a
  useful model.
• Today, the OPM may be the most widely used model
  in the field of finance.

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Uses of the B-S Model

• Identification of mispriced options
• Generating profit tables and graphs for different
  time periods, not just expiration.
• Evaluation of time spreads.
• Estimating option characteristics
• Expected Return, Variance, and Beta
• Options Price sensitivity to changes in S, T, R,
  and variability.

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Expected Return and Risk

• Recall, the value of a call is equal to the value
  of the RP. The expected return, standard
  deviation, and beta on a call can therefore be
  defined as the expected return, standard
  deviation, and beta on a portfolio consisting of
  the stock and risk-free security (short)

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Expected Return and Risk

• In term of the OPM, the total investment in the
  RP is equal to the call price, the investment in
  the stock is equal HoSo, and the investment in
  the RF security is -B. Thus

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Expected Return and Riskfor Puts
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Delta, Gamma, and Theta

• Delta is a measure of an options price
  sensitivity to a small change in the stock price.
• Delta is N(d1) for calls and ranges from 0 to 1.
• Delta is N(d1) - 1 for puts and ranges from -1 to
  0.
• Delta for the call in the example is .4066
• Delta for the put in the example is .5934.
• Delta changes with time and stock prices changes.

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Delta, Gamma, and Theta

• Theta is the change in the price of an option
  with respect to a change in the time to
  expiration.
• Theta is a measure of the options time decay.
• Theta is usually defined as the negative of the
  partial of the option price with respect to T.
• Interpretation An option with a theta of 7 would
  find for a 1 decrease in the time to expiration
  (2.5 days), the option would lose 7 in value.
• For formulas for estimating theta, see JG
  258-259.

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Delta, Gamma, and Theta

• Gamma measures the change in the options delta
  for a small change in the price of the stock. It
  is the second derivative of the option with
  respect to a change in the stock price.
• For formulas for estimating gamma, see JG
  258-259.

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Position Delta, Gamma, and Theta

• The description of call and put options in terms
  of their delta, gamma, and theta values can be
  extended to option positions.
• For example, consider an investor who purchases
  n1 calls at C1 and n2 calls on another call
  option on the same stock at a price of C2.
• The value of the portfolio (V) is

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Position Delta, Gamma, and Theta

• The call prices are a function of S, T,
  variability, and Rf. Taking the partial
  derivative of V with respect to S yields the
  position delta

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Position Delta, Gamma, and Theta

• The position delta measure the change in the
  positions value in response to a small change in
  the stock price.
• By setting the position delta equal to zero and
  solving for n1 in terms of n2 a neutral position
  delta can be constructed with a value invariant
  to small changes in the stock price.

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Position Delta, Gamma, and Theta

• The position theta is obtained by taking the
  partial derivative of V with respect to T

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Position Delta, Gamma, and Theta

• The position gamma is obtained by taking the
  derivative of the position delta respect to S
• Strategy For a neutral position delta with a
  positive position gamma, the value of the
  position will decrease for small changes in the
  stock price and increase for large increases or
  decreases in the stock price.
• StrategyFor a neutral position delta with a
  negative position gamma, the value of the
  position will increase for small changes in the
  stock price and decrease for large increases or
  decreases in the stock price.