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

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_t² + rS_t∆_t + θ

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

The Black-Scholes equation entails 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.

Original Practice Problems and Solutions from the Actuary's Free Study Guide:

Problem BSPDF1. The stock of Company Ξ has a current price of $56 per share, and the standard deviation of its price is 0.34. A certain call option on this stock has a delta of 0.42, a gamma of 0.001, and a theta of -0.05. The annual continuously compounded risk-free interest rate is 0.06. What is the price of this call option, as found using the Black-Scholes equation?

Solution BSPDF1. We use the formula rC(S_t) = (1/2)σ²S_t²Γ_t² + rS_t∆_t + θ, which we rearrange thus: C(S_t) = [(1/2)σ²S_t²Γ_t² + rS_t∆_t + θ]/r = [(1/2)0.34256²0.001² + 0.06*56*0.42 - 0.05]/0.06 = C(56) = $22.68968768

Problem BSPDF2. Assume that the Black-Scholes framework holds. The stock of Vicious Co. has a price of $93 per share, and the price has a standard deviation of 0.53. A certain call option on Vicious Co. stock has a price of $4, a delta of 0.53, and a gamma of 0.01. The annual continuously compounded risk-free interest rate is 0.02. What is the theta for this call option?

Solution BSPDF2. We use the formula rC(S_t) = (1/2)σ²S_t²Γ_t² + rS_t∆_t + θ, which we rearrange thus: θ = rC(S_t) - (1/2)σ²S_t²Γ_t² - rS_t∆_t. Thus, θ = 0.02*4 - (1/2)0.53²93²0.01² - 0.02*93*0.53 =

θ = -1.027275205

Problem BSPDF3. Assume that the Black-Scholes framework holds. The price of the stock of Devious Co. has a standard deviation of 0.74. A certain call option on this stock has a price of $25, a delta of 0.29, a gamma of 0.02, and a theta of -0.04. The annual continuously compounded risk-free interest rate is 0.08. What is the current price of one share of Devious Co. stock?

Solution BSPDF3. We use the formula rC(S_t) = (1/2)σ²S_t²Γ_t² + rS_t∆_t + θ, which we rearrange thus: 0 = (1/2)σ²S_t²Γ_t² + rS_t∆_t - (rC(S_t) - θ). We find that

rC(S_t) - θ = 0.08*25 + 0.04 = 2.04, (1/2)σ²Γ_t² = (1/2)0.74²0.02² = 0.00010952, and r∆_t = 0.08*0.29 = 0.0232. We thus have the following equation:

0.00010952S_t² + 0.0232S_t - 2.04 = 0. By the quadratic formula, the relevant positive value of

S_t is S_t = $66.84058205.

Problem BSPDF4. Assume that the Black-Scholes framework holds. The price of the stock of Imperious LLC is $516 per share and has a standard deviation of 0.22. A certain call option on this stock has a price of $67, a delta of 0.03, and a theta of -0.05. The annual continuously compounded risk-free interest rate is 0.1. What is the gamma of this option?

Solution BSPDF4. We use the formula rC(S_t) = (1/2)σ²S_t²Γ_t² + rS_t∆_t + θ, which we rearrange thus: rC(S_t) - θ - rS_t∆_t = (1/2)σ²S_t²Γ_t² , so Γ_t = √[(rC(S_t) - θ - rS_t∆_t)/((1/2)σ²S_t²)] =

√[(0.1*67 + 0.05 - 0.1*516*0.03)/((1/2)0.22²516²)] = 5.202/6443.3952 = Γ_t = 0.0008073383424

Problem BSPDF5. The stock of Obsequious Co. has a price of $40, and this price has a standard deviation of 0.43. A certain call option in this stock has a price of $5, a gamma of 0.02, and a theta of -0.02. The annual continuously compounded risk-free interest rate is 0.056. Find the delta of this call option.

Solution BSPDF5. We use the formula rC(S_t) = (1/2)σ²S_t²Γ_t² + rS_t∆_t + θ, which we rearrange thus: (rC(S_t) - θ - (1/2)σ²S_t²Γ_t² )/rS_t = ∆_t

Thus, (rC(S_t) - θ - (1/2)σ²S_t²Γ_t² )/(rS_t) = (0.056*5 + 0.02 - (1/2)0.43²40²0.02² )/(0.056*40) =

∆_t = 0.1075142857.

See other sections of The Actuary's Free Study Guide for Exam 3F / Exam MFE.