Chapter 12 BlackScholes Flashcards
Black-Scholes is the binomial model with x steps. x is…
x -> infinity
Black-Scholes doesn’t use Average return on a stock because
the larger average return is offset by the larger discount rate.
Black-Scholes makes a bunch of assumptions:
- Continuously compounded returns on stock are normally distributed and independent
- volatility is known and constant
- future dividends are known
- risk free rate is known and constant
- no transaction costs or taxes
- possible to short sell and borrow at risk free rate
Black-Scholes D1 Formula
[ln(S/K) + (r-δ+(1/2)σ^2)T] / σsqrt(T)
ln(prepaidForward on S/ Prepaid Forward on K) + (1/2)σ^2)T / σsqrt(T)
Black-Scholes D2 Formula
d1 - σ*sqrt(T)
Black-Scholes European Call Formula
Se^(-δt) * N(d1) - Ke^(-rt) * N(d2)
Prepaid Forward on S * N(d1) - Prepaid Forward on K * N(d2)
Black-Scholes European Put Formula
- Se(-δt) * N(-d1) + Ke(-rt) * N(-d2)
- Prepaid Forward on S * N(-d1) + Prepaid Forward on K * N(-d2)
Black-Scholes can be adjusted to other underlying assets by…
Change δ dividend yield
Prepaid Forward Price for a stock with discrete dividends
S - PV(Dividends)
Black-Scholes for futures
set delta = r, risk free rate aka Black Formula
Option greeks measure what
Change in option price given a change in only one of the inputs
Delta (Call vs. Put, In-the-money vs. Out-of-the-money, time)
Option Price Change when stock increases by $1.
Calls -> + and Puts -> -, Higher for in the money, time dilutes the effects of in the money.
formula e^(-rt) * N(d1)
Gamma (Call vs. Put, In-the-money vs. Out-of-the-money)
Derivative of Delta, identical call and put means gamma is the same, if a call is deep in the money or deep out of the money, gamma is zero as delta doesn’t change much
Vega (Call vs. Put, In-the-money vs. Out-of-the-money, time)
Option Change Price when volatility changes by set amount usually 1 percent point (0.01). PCP-> Same for call and put, higher for at the money options, moderate time higher
Theta
Option Price Change when time changes by 1 day or 10 days.
Rho (Call vs. Put, In-the-money vs. Out-of-the-money, time)
Option Price Change with respect to interest. Positive for a call and negative for a put. As time increases or options become more in the money, rho’s effect increases
Psi (Call vs. Put, In-the-money vs. Out-of-the-money, time)
Option Price Change with respect to dividend yield. + for a put and - for a call. Time increase or options more in the money means Psi’s effect increases
Greek measure of a portfolio is
the sum of the individual stock greeks
Elasticity of an option
percent change in option/percent change in stock
e(delta)/optionprice / (e/S) where e is dollar change in stock
volatility of an option equals the volatility of a stock times what factor?
absolute value of the elasticity
Beta or risk premium of an option equals beta of a stock times what?
value of elasticity
Sharpe Ratio for an option is the sharpe ratio of a stock times what?
1 because they are the same. SR is risk premium/volatility, and hence the elasticities cancel out.
Elasticity of a portfolio is
the weighted average of the individual elasticities where weights are percent of portfolio in one option
B of portfolio is
Elasticity of a portfolio times B of the stock.
A Calandar spread is
buying options with same strike but different expirations
implied volatility
volatility found by using black scholes model and market price
risk reversals
implied volatility difference between out of money calls and puts with same delta
