Chapter 12 BlackScholes Flashcards
(27 cards)
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.
