Ch 13 - Black-Scholes-Merton Model Flashcards
Valuing Stock Options
The first three of the key underlying assumptions of Black-Scholes-Merton
- Price of underlying asset follows lognormal distribution
- Risk-free rate known and constant
- Volatility of underlying asset known and constant
Price of a call (calculation)
c = S0 N(d1) - Ke^(-rT) N(d2)
Price of a put (calculation)
p = Ke^(-rT) N(-d2) - S0 N(-d1)
d1 =
[ln(S0 / K) + (r + sigma^2 /2) T ] / sigma*sqrt(T)
d2 =
[ln(S0 / K) + (r - sigma^2 /2) T ] / sigma*sqrt(T)
d1 - sigma*sqrt(T)
S0 N(d1) represents..
The expected stock price at time T in a risk neutral world where stock price less strike price is zero
N(d2) represents..
Probability that a call option will be exercised in a risk neutral world
When is the assumption that the return on a stock price at any future time has a lognormal probability distribution true?
where the RETURN on the stock (not the stock price itself) follows a random walk (upward move is as likely as downward move)
Volatility is
the standard deviation of the year continuously compounded rate of return in 1 year
= sigma* sqrt(delta_t)
Risk-neutral valuation premise
Any security dependent on other traded securities can be valued on the assumption that investors are risk neutral
In a risk-neutral world
- The E[R] from all investment assets = RFR
2. RFR = appropriate discount rate to any expected future cash flow
Why can a riskless portfolio consisting of a position in the option and a position in the underlying stock be set up?
The stock price and the option price affected by the same source of underlying uncertainty; stock price movements
What is the implied volatility of an option?
The volatility that makes the Black-Scholes-Merton price of an option equal to its market price
How do you calculate the implied volatility of an option?
Through trial and error; systematically test different volatilities until a value gives the European put option price when substituted into the Black-Scholes-Merton formula.
Explain how risk-neutral valuation could be used to derive the Black-Scholes-Merton formulas.
Assuming that the expected return from the stock is the risk-free rate, we calculate the expected payoff from a call option. We then discount this payoff from the end of the life of the option to the beginning at the risk-free rate.
