Black-Scholes Pricing

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Black-Scholes Pricing Related Models
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Option Valuation

• Black and Scholes
• Call Pricing
• Put-Call Parity
• Variations

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Option Pricing Calls

• Black-Scholes Model

C Call S Stock Price N Cumulative Normal
Distrib. Operator X Exercise Price e
2.71..... r risk-free rate T time to expiry
Volatility
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Call Option Pricing Example

• IBM is trading for 75. Historically, the
  volatility is 20 (s). A call is available with
  an exercise of 70, an expiry of 6 months, and
  the risk free rate is 4.

ln(75/70) (.04 (.2)2/2)(6/12) d1
--------------------------------------------
.70, N(d1) .7580 .2 (6/12)1/2 d2 .70 -
.2 (6/12)1/2 .56, N(d2) .7123 C 75
(.7580) - 70 e -.04(6/12) (.7123)
7.98 Intrinsic Value 5, Time Value 2.98
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Put Option Pricing

• Put priced through Put-Call Parity

Put Price Call Price X e-rT - S
(or
)
From Last Example of IBM Call Put 7.98
70 e -.04(6/12) - 75 1.59 Intrinsic Value
0, Time Value 1.59
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Black-Scholes Variants

• Options on Stocks with Dividends
• Futures Options (Option that delivers
  a maturing futures)
• Blacks Call Model (Black (1976))
• Put/Call Parity
• Options on Foreign Currency
• In text (Pg. 375-376, but not reqd)
• Delivers spot exchange, not forward!

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The Stock Pays no Dividends During the Options
Life

• If you apply the BSOPM to two securities, one
  with no dividends and the other with a dividend
  yield, the model will predict the same call
  premium
• Robert Merton developed a simple extension to the
  BSOPM to account for the payment of dividends

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The Stock Pays Dividends During the Options Life
(contd)
Adjust the BSOPM by following (?continuous
dividend yield)
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Futures Option Pricing Model

• Blacks futures option pricing model for European
  call options

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Futures Option Pricing Model (contd)

• Blacks futures option pricing model for European
  put options
• Alternatively, value the put option using
  put/call parity

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Assumptions of the Black-Scholes Model

• European exercise style
• Markets are efficient
• No transaction costs
• The stock pays no dividends during the options
  life (Merton model)
• Interest rates and volatility remain constant,
  but are unknown

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Interest Rates Remain Constant

• There is no real riskfree interest rate
• Often use the closest T-bill rate to expiry

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Calculating Volatility Estimates

• from Historical Data S, R, T that just was,
  and ? as standard deviation of historical returns
  from some arbitrary past period
• from Actual Data S, R, T that just
  was, and ? implied from pricing of nearest
  at-the-money option (termed implied
  volatility).

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Intro to Implied Volatility

• Instead of solving for the call premium, assume
  the market-determined call premium is correct
• Then solve for the volatility that makes the
  equation hold
• This value is called the implied volatility

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Calculating Implied Volatility

• Setup spreadsheet for pricing at-the-money call
  option.
• Input actual price.
• Run SOLVER to equate actual and calculated price
  by varying ?.

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Volatility Smiles

• Volatility smiles are in contradiction to the
  BSOPM, which assumes constant volatility across
  all strike prices
• When you plot implied volatility against striking
  prices, the resulting graph often looks like a
  smile

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Volatility Smiles (contd)
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Problems Using the Black-Scholes Model

• Does not work well with options that are
  deep-in-the-money or substantially
  out-of-the-money
• Produces biased values for very low or very high
  volatility stocks
• Increases as the time until expiration increases
• May yield unreasonable values when an option has
  only a few days of life remaining