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
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)
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 outof-the money options.
(2) short-term options are particularly subject to
jumps
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.
Dividends
• Merton derives a dividend paying option model
as:
c = Se-dT N(d1) - Ee-rT N(d2)
2)T
ln(S/E)+(r-d
+0.5s
where d1 =
sT0.5
d2 = d1-sT0.5
d is the dividend yield.
Taxes
• Taxes affect option values
• For example:
c = Se-d(1-tax)T N(d1) - Ee-rT N(d2)
where:
2]T
ln(S/E)+[r-d(1-tax)+0.5s
d1 =
sT0.5
d2 = d1-sT0.5
d is the dividend yield.
Take-over case
• If firm A takes over firm B through an exchange of
stock, options on firm B’s stock will normally
become options on firm A’s stock. We will use A’s
volatility instead of B’s in valuing the B’s 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
Take-over continued
(2) Stock B’s 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.