# Black-Scholes Model Assumptions 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 where d1 = ln(S/E)+(r-d +0.5s )T 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: ln(S/E)+[r-d(1-tax)+0.5s2]T 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 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 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.