Black-Scholes Merton Model Flashcards
List the assumptions of the Black-Scholes Model.
– The original model assumes a European call option on a non-dividend paying stock with a current price of S, strike price of K, and maturity of T years.
– Stock prices are assumed to be lognormal distributed. Volatility is constant.
– The markets are frictionless (no taxes, transaction costs, restrictions on short selling, and securities are perfectly divisible).
– The continuously compounded risk-free interest rate (r) per annum is constant.
– Investors can borrow and lend at the same risk-free interest rate.
– There are no arbitrage opportunities.
– Trading is continuous.
What are the Call Option pricing formulas for the Black-Scholes Model?
The current value of a call option is:
c = SN(d1 ) - Ke^(-rT) N(d2)
Where:
d1 = (ln(S/K) + (r+σ^2/2)T) / (σ√T)
d2 = d1 - σ√T
N(z) is the area under the standard normal distribution
N(d2) is the probability that the option will be exercised in a risk-neutral world.
N(d1) is not easy to interpret but can be considered as the factor by which the present value of the contingent receipt of the stock exceeds the current stock price. The value of the option does not depend on the expected return of the stock.
When using the S.N. table to calculate N(d1) and N(d2), calculate d1 and d2 to the nearest two decimal places. Do not use the interpolation technique in Hull in the exam.
What are the Put Option pricing formulas for the Black-Scholes Model?
The current value of an otherwise equivalent put option is:
p=Ke^(-rT)(N-d2) - SN(-d1 ), where
d1 = ln((S/K) + (r + σ^2/2)T) / σ√T
d2 = d1 - σ√T
We can derive the put formula using the put-call parity.
N(-d1) = ?
N(-d1) = 1 - N(d1)
Does the assumption that investors are risk neutral means that investors risk preferences affect the value of an option?
Risk Neutrality means investors’ risk preferences do not affect the value of an option when the value is expressed as a function of the price of the underlying instrument.
Finding an option’s value in a risk-neutral world, we should Assume x, Calculate y, and Discount z. What are x, y and z?
x: Assume that the expected return is the rfr.
y: Calculate the expected payoff.
z: Discount the expected payoff at the rfr.
