Review of Put-Call Parity and Binomial Option Pricing: Revised Practice Problems and Solutions

This section will review all the concepts we have worked with thus far. It applies the ideas of put-call parity and binomial option pricing discussed in Sections 1 through 31.

The problems in this section were designed to be similar to problems from past versions of Exam 3F / Exam MFE. They use original exam questions as their inspiration - and the specific inspiration for each problem is cited so as to give students a chance to see the original. All of the original problems are publicly available, and students are encouraged to refer to them. But all of the values, names, conditions, and calculations in the problems here are the original work of Mr. Stolyarov.

Problem RPCPBOP1.

Similar to Question 1 from the Society of Actuaries' May 2007 Exam MFE:

On October 1, 3451, the stock of Respectable Co. has a price of $300/share. Two equal dividends will be paid on May 1, 3452 and July 1, 3452. A European put option on Respectable Co. stock with strike price of $290 expiring in one year sells for $30, while a European call option on Respectable Co. stock with strike price of $290 expiring in one year sells for $53. The annual continuously-compounded risk-free interest rate is 12%. Find the amount of each dividend.

Solution RPCPBOP1. We use the formula for put-call parity on stock option:

C(K, T) - P(K, T) = [S₀ - PV_0,T(Div)] - e^-rTK, where T = 1 year and the dividends will be paid in 7/12 years and in 9/12 = ¾ years. K = 290, C(K, T) = 53, P(K, T) = 30, S₀ = 300, r = 0.12.

Let D be the amount of each dividend.

Thus,

PV_0,T(Div) = S₀ - e^-rTK - C(K, T) + P(K, T) = 300 - e^-0.12290 - 53 + 30 = PV_0,T(Div) = 19.79307335 = D(e^-0.12*7/12 + e^-0.12*9/12) and D = 19.79307335/(e^-0.12*7/12 + e^-0.12*9/12) = D = $10.72025418.

Problem RPCPBOP2.

Similar to Question 2 from the Society of Actuaries' May 2007 Exam MFE:

You are aware of the following information in the one-period binomial model for the price of the stock of The Firm Firm. The time period in the binomial model is two years, and the stock does not pay any dividends. In two years, the stock price will change by a factor of 0.333 or by a factor of 4.36. The annual continuously compounded expected return on the stock is 0.23. Find the true probability of the stock price going up in two years.

Solution RPCPBOP2. We use the formula p = (e^αh - d)/(u - d) for the true probability of the stock price's increase. We are given u = 4.36, d = 0.333, h = 2, α = 0.23. Thus,

p = (e^2*0.23 - 0.333)/(4.36 - 0.333) = p = 0.3106714639

Problem RPCPBOP3.

Similar to Question 4 from the Society of Actuaries' May 2007 Exam MFE:

For the stock of Cautious Co., the annual continuously compounded dividend yield is 0.11, and the annual continuously compounded risk-free interest rate is 0.04. The stock currently trades for $367 per share. One-year European call options on Cautious Co. stock have the following prices (C) corresponding to various strike prices (K):

K = $350, C = $14

K = $370, C = $5

K = $450, C = $0.23

K = $1000, C = $0

K = $1100, C = $0

You own five put options with each of the above strike prices:
Put A: K = $350

Put B: K = $370

Put C: K = $450

Put D: K = $1000

Put E: K = $1100

Each of the options can only be exercised either immediately or in two years. Which of these options is it optimal to exercise immediately? More than one answer may be correct.

Solution RPCPBOP3. First, we can use put-call parity to determine optimality of exercise by comparing K-S for each put with the value of P(K, T). If K - S > P(K, T), immediate exercise is optimal.

We can eliminate Put A right away, because K - S = -17 < 0.

For Put B, K - S = 370 - 367 = 3.

P(K, T) = C(K, T) + Ke^-rT - S₀e^-∂T, where T = 3, K = 370, C(K, T) = 14, r = 0.04, T = 2.

Thus, P(370, 2) = 5 + 370e^-0.04*2 - 367e^-0.11*2 = P(370, 2) = 52.02864931 > 3, so it is not optimal to exercise Put B.

For Put C, K - S = 450 - 367 = 83

P(450, 2) = 0.23 + 450e^-0.04*2 - 367e^-0.11*2 = P(450, 2) = 121.107957 > 83, so it is not optimal to exercise Put C.

For Put D, K - S = 1000 - 367 = 633

P(1000, 2) = 1000e^-0.04*2 - 367e^-0.11*2 = P(1000, 2) = 628.5919475 < 633, so it is optimal to exercise Put D.

For Put E, K - S = 1100 - 367 = 733

P(1100, 2) = 1100e^-0.04*2 - 367e^-0.11*2 = P(1100, 2) = 720.9035822 < 733, so it is optimal to exercise Put E.

Thus, it is optimal to immediately exercise Puts D and E.

Problem RPCPBOP4.

Similar to Question 11 from the Society of Actuaries' May 2007 Exam MFE:

The price movements of Jubilant Co. stock follow a two-period binomial model, where every period the stock price can change by a factor of 0.78 or 2. The stock pays no dividends, and the current stock price is $100. The annual continuously-compounded risk-free interest rate is 0.03. Each period in the binomial model is two years. Find the price of a four-year American put option on Jubilant Co. stock with a strike price of $300.

Solution RPCPBOP4.

We first find the stock prices in the binomial tree:
S = 100, u = 2, d = 0.78.

So S_u = 200 and S_uu = 400

S_d = 78, S_dd = 60.84, and S_du = 156.

Thus, at the final nodes of the binomial tree, we have

P_uu = 0, P_du = 300 - 156 = 144, and P_dd = 300 - 60.84 = 239.16

The risk-neutral probability of a stock price increase is (for r = 0.03, h = 2, and ∂ =0)

p* = (e^(r-∂)h - d)/(u - d) = (e^0.03*2 - 0.78)/(2 - 0.78) = p* = 0.2310135627

P_u = e^-rh[p*P_uu + (1 - p*)P_du] = e^-0.03*2[0 + (1 - 0.2310135627)144] = P_u = 104.2853981 > 300 - S_u = 300 - 200 = 100. Thus, P_u = 104.2853981.

P_d = e^-rh[p*P_du + (1 - p*)P_dd] = e^-0.03*2[0.2310135627*144 + (1 - 0.2310135627)239.16] = P_d = 204.5293601 (or so it seems at first). However, 300 - S_d = 300 - 78 = 222 > 204.5293601. Thus, because the put is American, the true value of P_d is 222.

P = e^-rh[p*P_u + (1 - p*)P_d] = e^-0.03*2[0.2310135627*104.2853981 + (1 - 0.2310135627)222] = P = 183.4616929. However, the value of exercising the put immediately is K - S = 300 - 100 = 200 > 183.4616929. Thus, the price of this put is $200.

Problem RPCPBOP5.

Similar to Question 14 from the Society of Actuaries' May 2007 Exam MFE:

Payoff on a straddle option is the absolute value of (K - S) on the expiration date, where K is the strike price of the option and S is the stock price at expiration. Stock Ψ, which pays no dividends, currently sells for $444 per share. In 65 years, the stock will sell for $567 or $230. The annual continuously-compounded risk-free interest rate is 0.0004. A straddle option on Stock Ψ with strike price of $500 can only be exercised in 65 years. Find the straddle's current price.

Solution RPCPBOP5. This problem is a one-period binomial option pricing problem in disguise. In 65 years, the payoff to the straddle will be either 567 - 500 = 67 (in the event of an up move) or 500 - 230 = 270 (in the event of a down move).

We find p* = (e^(r-∂)h - d)/(u - d) = (e^0.0004*65 - 230/444)/(567/444 - 230/444) =

p* = 0.6697192318

The straddle price A = e^-rh[p*A_u + (1 - p*)A_d] =

e^-0.0004*65[0.6697192318*67 + (1 - 0.6697192318)270] = A = $130.6066918

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