Black-Scholes Model Assumptions

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Black-Scholes Model Assumptions

• How to Improve the BS assumptions
• Constant volatility
• price changes smoothly
• constant short-term interest rate
• No trading cost
• No taxes
• No dividends
• Option be exercised at maturity
• No takeover events over the option life

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Volatility changes

• One of the important important factor in the B-S
  model
• Standard procedure in deriving volatility measure
  (e.g., use of log relative returns)
• Implied volatility from other options to derive
  the inputs for pricing
• Generalized autoregressive conditional
  heteroscedasticity model (GARCH model)

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Jumps

• Big news (bad or good) will have a temporary
  large increase in volatility
• Up jumps and down jumps have different effects on
  option values than symmetric jumps
• Cox, Merton and Ross (JFE, Jan/March, 1976)
  developed a formula for symmetric jumps. (1)
  Compared to the BS formula, their model gives
  high values for both in-the-money and out-of-the
  money options.(2) short-term options are
  particularly subject to jumps

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Interest Rate Changes

• Typically the interest rate should be used in
  continuous compounding format
• When interest rate changes are uncertain, it is
  better to use the yield on the zero coupon bond
  that matches the maturity of the option
• In general, the impact of interest rate is less
  than that of the volatility effect.

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Dividends

• Merton derives a dividend paying option model
  asc Se-dT N(d1) - Ee-rT N(d2)
• d2 d1-sT0.5d is the dividend yield.

ln(S/E)(r-d 0.5s2)T sT0.5
where d1
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Taxes

• Taxes affect option values
• For example c Se-d(1-tax)T N(d1) - Ee-rT
  N(d2)where ln(S/E)r-d(1-tax)0.5s2T
  sT0.5
• d2 d1-sT0.5
• d is the dividend yield.

d1
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Take-over case

• If firm A takes over firm B through an exchange
  of stock, options on firm Bs stock will normally
  become options on firm As stock. We will use
  As volatility instead of Bs in valuing the Bs
  option
• If firm A takes over firm B through a cash tender
  offer, these are two effects.(1) Outstanding B
  options will expire early, reducing the values of
  puts and call

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• Take-over continued(2) Stock Bs share price
  will rise the tender offer. This will increase
  call values but decrease put values.
• Uncertain takeover will affect option values (up
  or down).
• For short-term out-of-the-money call options,
  the chance of take-over will increase the option
  value
• For short-term out-of-the-money put options,
  chance of the take-over will decrease their
  values.