The Black-Scholes Partial Differential Equation: Practice Problems and Solutions - Version 3.0

The Actuary's Free Study Guide for Exam 3F / Exam MFE - Section 49 (Version 3.0)

This section of sample problems and solutions is a part of The Actuary's Free Study Guide for Exam 3F / Exam MFE, authored by Mr. Stolyarov.

This is Section 49 of the Study Guide. See Section 1 here. See Section 2 here. See Section 3 here. See Section 4 here. See Section 5 here. See Section 6 here. See Section 7 here. See Section 8 here. See Section 9 here. See Section 10

here. See Section 11 here. See Section 12 here. See Section 13 here. See Section 14 here. See Section 15 here. See Section 16 here. See Section 17 here. See Section 18 here. See Section 19 here. See Section 20 here. See Section 21 here. See Section 22 here. See Section 23 here. See Section 24 here. See Section 25 here. See Section 26 here. See Section 27 here. See Section 28 here. See Section 29 here. See Section 30 here. See Section 31 here. See Section 32 here. See Section 33 here. See Section 34 here. See Section 35 here. See Section 36 here. See Section 37 here. See Section 38 here. See Section 39 here. See Section 40 here. See Section 41 here. See Section 42 here. See Section 43 here. See Section 44 here. See Section 45 here. See Section 46 here. See Section 47 here. See Section 48 here.

The Black-Scholes partial differential equation or Black-Scholes equation (as opposed to the Black-Scholes formula) is as follows:
rC(S_t) = (1/2)σ²S_t²Γ_t + rS_t∆_t + θ

This equation holds for American and European call s and puts, but not at times when it is optimal to exercise the options early.

The Black-Scholes equation has the following assumptions:

1. The underlying asset does not pay any dividends.

2. The option does not pay any dividends.

3. The interest rate and volatility are constant.

4. The stock moves one standard deviation over a small time interval.

Source: McDonald, R.L., Derivatives Markets (Second Edition), Addison Wesley, 2006, Ch. 13, pp. 429-430.

Meaning of variables:

S_t = stock price at time t.

C = call option price.

∆ = option delta.

Γ = option gamma.

θ = option theta

r = annual continuously compounded risk-free interest rate

σ = annual standard deviation of the stock price movement.