More Exam-Style Questions on Binomial Option Pricing for Actuaries

This is Section 24 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.

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 MESQBOP1.

Similar to Question 18 from the Casualty Actuarial Society's Fall 2007 Exam 3:

The stock of Devious Co. currently trades for $65 per share. The annual continuously compounded risk-free interest rate is 0.10. Every 2 years, the stock price either increases by 30% or decreases by 20%. The stock pays no dividends. Using a two-period binomial model, calculate the price of a 4-year American put option on Devious Co. stock with a strike price of $74.

Solution MESQBOP1.

We cannot bypass the intermediate values of put prices here, because for an American put, at each node in the binomial tree, we must choose the higher of two values, K - S or the otherwise equivalent European option price.

We are given that u = 1.3 and d = 0.8. Also, S = 65, r = 0.10, ∂ = 0, h = 2, and K = 74.

We first find p* = (e^(r-∂)h - d)/(u - d) = (e^(0.1)2 - 0.8)/(1.3 - 0.8) = p* = 0.8428055163

S_uu = 1.3²*65 = 109.85, implying that P_uu = 0

S_du = 1.3*0.8*65 = 67.6, implying that P_du = 74 - 67.6 = P_du = 6.4

S_dd = 0.8²*65 = 41.6, implying that P_dd = 74 - 41.6 = P_dd = 32.4

We can calculate the equivalent European option prices two years from now:

P_u = e^-rh[p*P_uu + (1 - p*)P_du] = e^-0.1*2[0 + (1 - 0.8428055163)6.4] = P_u = 0.8236797313

If the stock price increases in two years, S_u = 1.3*65 = 84.5, so K - S < 0 and thus

P_u = 0.8236797313

P_d = e^-rh[p*P_du + (1 - p*)P_dd] = e^-0.1*2[0.8428055163*6.4 + (1 - 0.8428055163)32.4] =

P_d = 8.586075728. If the stock price declines in two years, S_d = 0.8*65 = 52, so K - S = 74 - 52 = 22. Thus, it will be optimal to exercise the option and so P_d =22.

Thus, P = e^-rh[p*P_u + (1 - p*)P_d] = e^-0.1*2[0.8428055163*0.8236797313 + (1 - 0.8428055163)22] = P = $3.399763456. But we compare this to the gain from exercising the option immediately, which is 74 - 65 = 9 > 3.399763456. Thus, the option today is worth $9.

Problem MESQBOP2.

Similar to Question 19 from the Casualty Actuarial Society's Fall 2007 Exam 3:

The price of a European call option on Impeccable LLC stock can be found using a one-period binomial model. The option expires in 5 months, and the stock pays no dividends. The option's strike price is $400, while the stock's current price is $385. The annual continuously compounded risk-free interest rate is 0.06. The annual stock price volatility is 0.5. Find the current value of the call option.

Solution MESQBOP2. Here, h = 5/12, σ = 0.5, ∂ = 0, and r = 0.06.

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

u = e^{(0.06)5/12 + 0.5√(5/12)} = u = 1.415876271

d = e^{(0.06)5/12 - 0.5√(5/12)} = d = 0.7424879687

p* = (e^(r-∂)h - d)/(u - d) = (e^(0.06)5/12 - 0.7424879687)/(1.415876271 - 0.7424879687) =

p* = 0.4200060364.

It is evident that the call option will only have value in 5 months if the stock price increases to uS = 1.415876271*385 = 545.1123643. In that case, C_u = 545.1123643 - 400 = C_u = 145.1123643

We use the formula C = e^-rh[p*C_u + (1 - p*)C_d] = e^-0.06(5/12)[0.4200060364*145.1123643 + 0] =

C = $59.44325579

Problem MESQBOP3.

Similar to Question 4 from the Society of Actuaries' Sample MFE Questions and Solutions:

The stock of Hospitable Co. currently trades for $45 per share. The stock pays no dividends. The annual continuously compounded risk-free interest rate is 0.13. u = 1.4, where uis one plus the rate of capital gain on the stock per period if the stock price goes up. d = 0.7, where dis one plus the rate of capital lose on the stock per period if the stock price goes down. Using a two-period binomial model, calculate the price of a 3-year American call option on Hospitable Co. stock with a strike price of $49.

Solution MESQBOP3. Here, h = 1.5, ∂ = 0, r = 0.13, u = 1.4, d = 0.7.

Thus, p* = (e^(r-∂)h - d)/(u - d) = (e^(0.13)1.5 - 0.7)/(1.4 - 0.7) = p* = 0.7361585521

S_u = 1.4*45 = 63

S_uu = 1.4*63 = 88.2, so C_uu = 88.2-49 = C_uu = 39.2

S_ud = 0.7*63 = 44.1, so C_du = 0

S_d = 0.7*45 = 31.5

S_dd = 0.7*31.5 = 22.05, so C_dd = 0

Thus, since C_du = 0 and C_dd = 0, C_d = 0 as well.

C_u = e^-rh[p*C_uu + (1 - p*)C_ud] = e^-0.13*1.5[0.7361585521*39.2 + 0] = C_u = 23.7448814.

This is greater than S_u - K = 14.

Thus, C = e^-0.13*1.5[0.7361585521*23.7448814 + 0] = C = 14.38314778.

This is greater than S - K = 4. Thus, C = $14.38314778.

Problem MESQBOP4.

Similar to Question 5 from the Society of Actuaries' Sample MFE Questions and Solutions:

One Golden Hexagon (GH) currently trades for 998 Wooden Circles (WC). The volatility of this exchange rate is 0.2. The annual GH-denominated continuously compounded interest rate is 0.04, while the annual WC-denominated continuously compounded interest rate is 0.05. Using a three-period binomial model, calculate the price of a 3-year American put option on Golden Hexagons, denominated in WC. The put option has a strike price of 1023 WC.

Solution MESQBOP4.

We first find u = e^{(r-f)h + σ√(h)}. Here, r = 0.05, f = 0.04, h = 1, and σ = 0.2. S

Thus,

u = e^{0.01 + 0.2√(1)} = u = 1.23367806

d = e^{(r-f)h - σ√(h)} = e^{0.01 - 0.2√(1)} = d = 0.8269591339

Currently, the exchange rate x = 998.

x_duu = 1.23367806²*0.8269591339*998 = 1256.08281, so P_duu and P_uuu are both 0, since the exchange rate will exceed the strike price if the stock goes up consistently or up twice and down once.

x_ddu = 1.23367806*0.8269591339²*998 = 841.977487, so P_ddu = 1023 - 841.977487 =

P_ddu = 181.022513

x_ddd = 0.8269591339³*998 = 564.3943878, so P_ddd = 1023 - 564.3943878 = P_ddd = 458.6056122

p* = (e^(r-f)h - d)/(u - d) =(e^0.01 - 0.8269591339)/(1.23367806 - 0.8269591339) =

p* = 0.4501660026

Thus, it seems at first that P_dd = e^-0.05[0.4501660026*181.022513 + (1 - 0.4501660026)458.6056122] = 317.3749751

But x_dd = 0.8269591339²*998 = 682.4936864, so K - x_dd = 1023 - 682.4936864 = 340.5063136 > 317.3749751, so P_dd = 340.5063136

Since P_duu and P_uuu are both 0, P_uu = 0.

Since P_du = e^-0.05[0.4501660026* P_duu + (1 - 0.4501660026)*P_ddu] =

e^-0.05[0 + (1 - 0.4501660026)*181.022513] = P_du = 94.67808283 (European equivalent)

x_ud = 1.23367806*0.8269591339*998 = 1018.160937, so (1023 - x_ud ) < 94.67808283 and so

P_du = 94.67808283.

Now we find

P_u = e^-rh[p*P_uu + (1 - p*)P_ud] = e^-0.05[0 + (1 - 0.4501660026)94.67808283] = P_u = 49.51836774, which is greater than K - x_u < 0.

P_d = e^-rh[p*P_du + (1 - p*)P_dd] =

e^-0.05[0.4501660026*94.67808283+ (1 - 0.4501660026)340.5063136] = P_d = 218.6332359.

x_d = 0.8269591339*998 = 825.3052157, so K - x_d = 1023 - 825.3052157 = 197.6947843 < 218.6332359. Thus, P_d = 218.6332359.

So P = e^-rh[p*P_u + (1 - p*)P_d] =

e^-0.05[0.4501660026*49.51836774 + (1 - 0.4501660026)218.6332359] = P = 135.5534954, which is greater than K - x = 25. So P = 135.5534954 WC

This problem had enough work for two or three problems of a more typical (and reasonable) size, so I will conclude Section 24 with only 4 problems.

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