Chapter 13 - The Black Scholes Option Pricing Formula

Question 1

What is the purpose of the Black-Scholes model?

Answer 1

To derive the price of a European call or put option

Question 2

What is the stochastic differential equation (SDE) governing the share price in the Black-Scholes model?

Answer 2

dS_t = S_t (μ dt + σ dz_t)

Question 3

What are the constants μ and σ referred to in the Black-Scholes model?

Answer 3

Drift and volatility parameters respectively

Question 4

True or False: Investors can only invest positive amounts in shares according to the Black-Scholes model.

Answer 4

False

Question 5

What type of financial instrument can investors hold alongside shares in the Black-Scholes model?

Answer 5

Risk-free cash bond

Question 6

Fill in the blank: The share price process in the Black-Scholes model is characterized by a _______ distribution.

Answer 6

Continuous

Question 7

What type of amounts can investors invest in shares?

Answer 7

Positive or negative amounts

Question 8

What is the ordinary differential equation for the risk-free cash bond?

Answer 8

dB_t= rB_t dt

Question 9

In the equation dB_t= rB_tdt, what does ‘r’ represent?

Answer 9

The constant risk-free Interest rate

Question 10

What is the Black-Scholes formula for a call option on a non-dividend paying underlying share?

Answer 10

f(t, S) = S * N(d1) - K * e^(-r(T-t)) * N(d2)

Where N(z) is the cumulative distribution function of the standard normal distribution.

Question 11

What is the formula for d1 in the Black-Scholes model?

Answer 11

d1 = (ln(S/K) + (r + (σ^2)/2)(T-t)) / (σ√(T-t))

Where S is the current stock price, K is the strike price, r is the risk-free interest rate, σ is the volatility, and T-t is the time to expiration.

Question 12

What is the formula for d2 in the Black-Scholes model?

Answer 12

d2 = d1 - σ√(T-t)

This is derived from the definition of d1.

Question 13

What is the Black-Scholes formula for a put option?

Answer 13

f(t, S) = K * e^(-r(T-t)) * N(-d2) - S * N(-d1)

This is the corresponding formula for put options.

Question 14

When is the Black-Scholes formula equal when using the Garman-Kohlhagen formula?

Answer 14

q = 0

This indicates that the dividend rate is assumed to be zero.

GK - page 47 of the tables.

Question 15

What are the assumptions of the black Scholes model?

Answer 15

1. The price of the underlying share follows a geometric Brownian motion
   2.There are no risk-free arbitrage opportunities
   3.The risk-free rate of interest is constant, the same for all maturities and the same for borrowing or lending.
   4.Unlimited short selling (that is, negative holdings) is allowed
   5.There are no taxes or transaction costs.
   6.The underlying asset can be traded continuously and in infinitesimally small numbers of units.

Question 16

What is the Black-Scholes PDE?

Answer 16

rf = df/dt + rStdf/ds + 1/2sigma^2St^2*d^2f/ds^2