Exam-Style Questions on Binomial Option Pricing for Actuaries

This is Section 23 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.

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 ESQBOP1.

Similar to Question 14 from the Casualty Actuarial Society's Spring 2007 Exam 3:

A European call option on Opaque LLC stock has the following specifications:
Strike price = $578, current stock price = $581, time to expiration = 4 years, annual continuously compounded interest rate = 0.12, dividend yield = 0. You can use a two-period binomial model to calculate the option's price.

This is the binomial tree for the stock price movements:

----------------------981.89

----------755.3 -----------

581 -------------- 604.24

---------464.8 ------------

--------------------371.84

Find the price of one European call option on Opaque LLC stock.

Solution ESQBOP1.

First, it is useful to find u and d. Here, u = 755.3/581 = u = 1.3 and d = 464.8/581 = d = 0.8.

Since this is a two-period model, we know that h = 2. Furthermore, r = 0.12, and ∂ = 0.

We calculate p* = (e^(r-∂)h - d)/(u - d) = (e^(0.12)2 - 0.8)/(1.3 - 0.8) = p* = 0.9424983006

We note that C_uu = 981.89 - 578 = C_uu = 403.89 and C_du = 604.24 - 578 = C_du = 26.24

C_dd = 0, since the stock price in that case is lower than the strike price.

Now we use the formula C = e^-2rh[(p*)²C_uu + 2(p*)(1-p*)C_ud + (1 - p*)²C_dd] =

e^-2*0.12*2[(0.9424983006)²403.89 + 2(0.9424983006)(1-0.9424983006)26.24 + 0] =

C = $223.7649974

Problem ESQBOP2.

Similar to Question 15 from the Casualty Actuarial Society's Spring 2007 Exam 3:

Within the framework of a binomial pricing model, you have the following data:

The price of one share of Transparent Co. is $200; the option strike price is $187. The annual continuously compounded risk-free interest rate is 0.07, and the stock's annual continuously-compounded dividend yield is 0.03. The volatility of the stock's price movements in 0.4. The options' time to expiration is 2 years, and the length of one time period within the binomial model is 6 months. Find the risk-neutral probability that the stock price will increase over one time period.

Solution ESQBOP2. First, we need to compute u = e^{(r-∂)h + σ√(h)} and d = e^{(r-∂)h - σ√(h)} , given that r = 0.07, ∂ = 0.03, h = 0.5, so (r - ∂)h = 0.02. σ = 0.4, so u = e^{(r-∂)h + σ√(h)} = e^{0.02 + 0.4√(0.5)} = u = 1.353701527

d = e^{0.02 - 0.4√(0.5)} = d = 0.7688628203.

Thus, p* = (e^(r-∂)h - d)/(u - d) = (e^0.02 - 0.7688628203)/(1.353701527 - 0.7688628203) = p* = 0.4297569854

Solution ESQBOP3.

Similar to Question 16 from the Casualty Actuarial Society's Spring 2007 Exam 3:

The following binomial tree models the possible price movements of a stock, Q, that does not pay dividends:

----------544.5

363 -----------

-----------72.6

t = 0----- t =1

A European call option on Q has a strike price of $400 and expires at t =1. In the replicating portfolio for this option, what is the number of shares of stock Q present?

Solution ESQBOP3. We seek to find ∆. First, we note that C_u = 544.5 - 400 = 144.5, whereas C_d = 0, since S_d< 400. u = 544.5/363 = u = 1.5 and d = 72.6/363 = d = 0.2. S is given as 363.

We use the formula ∆ = e^-∂h(C_u - C_d)/[S(u-d)], which, since the stock pays no dividends, simplifies to ∆ =(C_u - C_d)/[S(u-d)] = 144.5/[363(1.5-0.2)] = ∆ = 0.3062089426

Problem ESQBOP4.

Similar to Question 17 from the Casualty Actuarial Society's Spring 2007 Exam 3:

A binomial tree with valuations made every 4 years is used to calculate the price of one 8-year European put option on the stock of Inscrutable Co. The stock price volatility is 0.5, and annual continuously compounded risk-free interest rate is 0.09. The stock pays no dividends. The put option's strike price is $56, and the stock price is $70. Find the price of such a put option.

Solution ESQBOP4. Here, h = 4, σ = 0.5, ∂ = 0, and r = 0.09.

First, we need to compute u = e^{(r-∂)h + σ√(h)} and d = e^{(r-∂)h - σ√(h)}.

u = e^{(0.09)4 + 0.5√(4)} = u = 3.896193302

d = e^{(0.09)4 - 0.5√(4)} = d = 0.527292424

We also note that p* = (e^(0.09)4 - 0.527292424)/(3.896193302 - 0.527292424) =

p* = 0.2689414214

Now we find S_uu = 3.896193302²*70 = 1062.622557, so P_uu = 0

S_du = 3.896193302*0.527292424*70 = 143.8103247, so P_du = 0

S_dd = 0.527292424²*70 = 19.46261103, so P_dd = 56 - 19.46261103 = P_dd = 36.53738897

Now we use the formula P = e^-2rh[(p*)²P_uu + 2(p*)(1-p*)P_ud + (1 - p*)²P_dd] =

e^-2*0.09*4[0 + 0 + (1 - 0.2689414214)²36.53738897] = P = $9.50495001

Problem ESQBOP5.

Similar to Question 17 from the Casualty Actuarial Society's Fall 2007 Exam 3:

In one year, the stock of Timorous, Inc., will be worth either $230 or $185. The stock is currently worth $200 and pays no dividends. The annual risk-free interest rate is 0.033, compounded continuously. Using the one-period binomial option pricing model, calculate the delta for a call option on Timorous, Inc., stock that expires in 1 year and has a strike price of $175.

Solution ESQBOP5. We use the formula ∆ = e^-∂h(C_u - C_d)/[S(u-d)], which, since the stock pays no dividends, simplifies to ∆ =(C_u - C_d)/[S(u-d)].

C_u = 230 - 175 = 55, and C_d = 185 - 175 = 10. Furthermore,

S = 200, u = 230/200 = 1.15 and d = 185/200 = 0.925. Thus,

∆ = (55 - 10)/[200(1.15-0.925)] = ∆ = 1.

(Yes, this is correct. It means that you would need to hold one share of stock in the replicating portfolio for the call option. Also as part of the portfolio, you would need to borrow some amount of money B, so the B term will be negative and the option price will be less than the stock price, as expected.)

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