Errata for Section 19 of the Actuary's Free Study Guide for Exam 3F / Exam MFE

In the original Section 19 of The Actuary's Free Study Guide for Exam 3F / Exam MFE, there was a typo regarding the formula for directly calculating option prices using a two-period binomial model. The formula was given as

P = e^-2rh[(p*)²P_uu + (p*)(1-p*)P_ud + (1 - p*)²P_dd]

whereas the correct formula is

P = e^-2rh[(p*)²P_uu + 2(p*)(1-p*)P_ud + (1 - p*)²P_dd]. Not the "2" coefficient of the middle term.

As a result, the work for Solutions BOPWP4-5 needs to be revised. These are the correct solutions.

Problem BOPWP4. Gregarious, Inc., stock is currently worth $56. Every year, it can change by a factor of 0.9 or 1.3. The stock pays no dividends, and the annual continuously-compounded risk-free interest rate is 0.04. Using a two-period binomial option pricing model, find the price today of one two-year European put option on Gregarious, Inc., stock with a strike price of $120.

Solution BOPWP4. In one year, the stock will either be worth S_u = 1.3*56 = 72.8, or it will be worth S_d = 0.9*56 = 50.4. In two years, the stock will either be worth

S_uu = 1.3²*56 = 94.64 or S_ud = S_du = 1.3*0.9*56 = 65.52 or S_dd = 0.9*0.9*56 = 45.36.

At S_uu = 94.64, the put is worth P_uu = 120 - 94.64 = P_uu = 25.36

At S_du = 65.52, the put is worth P_du = 120 - 65.52 = P_du = 54.48

At S_dd = 45.36, the call is worth P_dd = 120 - 45.36 = P_dd = 74.64

Now we calculate p* = (e^(r-∂)h - d)/(u - d) = (e^0.04 - 0.9)/(1.3 - 0.9) = 0.3520269355.

We note that there can be a direct calculation of the put price today using the three possible put prices two periods from now using the binomial model. The formula might make intuitive sense to you if you consider the way a binomial probability distribution works:

P = e^-2rh[(p*)²P_uu + 2(p*)(1-p*)P_ud + (1 - p*)²P_dd]

P = e^-2*0.04[(0.3520269355)²25.36 + 2(0.3520269355)(1- 0.3520269355)54.48 + (1 - 0.3520269355)²74.64] = P = $54.77396157.

This approach is much faster than finding the intermediate put prices. The identical kind of formula can be applied to call pricing using the two-period binomial model as well.

Problem BOPWP5. Complicated, Inc., pays dividends on its stock at an annual continuously compounded yield of 0.06. The annual effective interest rate is 0.09. Complicated, Inc., stock is currently worth $100. Every two years, it can change by a factor of 0.7 or 1.5. Using a two-period binomial option pricing model, find the price today of one four-year European put option on Gregarious, Inc., stock with a strike price of $130.

Solution BOPWP5. We are given that r = 0.09, ∂ = 0.06, and h = 2. Thus,

(r-∂)h = (0.09 - 0.06)*2 = 0.06.

p* = (e^(r-∂)h - d)/(u - d). Here, for every time period, p* = (e^0.06 - 0.7)/(1.5 - 0.7) = p* = 0.4522956832.

We find

S_uu = 1.5²*100 = 225, which implies that P_uu = 0

S_ud = S_du = 1.5*0.7*100 = 105, which implies that P_ud = 130 - 105 = P_ud = 25

S_dd = 0.7²*100 = 49, which implies that P_dd = 130 - 49 = P_dd = 81.

Now we use the great time-saving formula

P = e^-2rh[(p*)²P_uu + 2(p*)(1-p*)P_ud + (1 - p*)²P_dd] =

e^-2*0.09*2[0 + 2(0.4522956832)(1-0.4522956832)25 + (1 - 0.4522956832)²81] =

P = $25.59397445

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