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.
The last three assumptions of Black-Scholes-Merton
- No taxes or transaction costs
- No cash flows on underlying (no divs)
- European options
How are European options on dividend-paying stocks valued using BSM?
By substituting the stock prices less the PV of divs into the BSM formula
What divs are included when pricing an option using BSM on a div paying stock? What do they represent?
Only divs with ex-div dates during the life of the option
They represent the expected reduction in the stock price on the ex-dividend date
What are the limitations of the BSM model?
Markets often move in ways not consistent with random walk assumption
Volatility is not constant
What is used instead of BSM that overcomes its limitations?
ARCH = autoregressive conditional heteroskedasticity
It replaces constant volatility with random volatility
