Standard Deviation of Returns and Multi-Period Probabilities in the Binomial Model: Practice Problems and Solutions

This is Section 29 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.

The standard deviation of returns on an asset over time period of length h can be expressed as

σ_h = σ√(h), where σ is the standard deviation over a time period of length 1.

Using the binomial model, stock prices can be modeled as follows:

S_t+h = S_te^{(r- δ)h ± σ√(h)}.

When we take the natural logs of both sides, we get ln(S_t+h/S_t) = (r- δ)h ± σ√(h).

The binomial model is an approximation of the lognormal distribution. According to R. L. McDonald, "The lognormal distribution is the probability distribution that arises from the assumption that continuously compounded returns on the stock are normally distributed."

In a binomial tree, where the risk-neutral probability is p*, the probability of reaching the ith node can be expressed as

p_{ith node} = [n!/((n-i)!i!)](p*)^n-1(1-p*)ⁱ = C(n, i)(p*)^n-1(1-p*)ⁱ

Source: McDonald, R.L., Derivatives Markets (Second Edition), Addison Wesley, 2006, Ch. 11, pp. 354-358.

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

Problem SDRMPPBM1. The standard deviation of returns on Frivolous Co. stock over 1 year is 0.67. Find the standard deviation of returns on Frivolous Co. stock over 12 years.

Solution SDRMPPBM1. We use the formula σ_h = σ√(h), where h = 12, and σ = 0.67. Thus, σ₁₂ = 0.67√(12) = σ₁₂ = 2.320948082

Problem SDRMPPBM2. The standard deviation of returns on Meticulous Co. stock over 10 years is 0.02. The standard deviation of returns on Frivolous Co. stock over Z years is 0.15. Find Z.

Solution SDRMPPBM2.

We use the formula σ_h = σ√(h), but here h is expressed in 10-year periods. So Z = 10h.

√(h) = σ_h/σ = 0.15/0.02 = √(h) = 7.5, so h = 56.25 and Z = 562.5 years.

Problem SDRMPPBM3. The standard deviation of returns on Meticulous Co. stock over 10 years is 0.02; the annual continuously-compounded interest rate is 0.03, and the stock pays dividends at an annual continuously-compounded of 0.01. The stock price is currently $120/share. If the stock price increases in 10 years, what will it be?

Solution SDRMPPBM3. We use the formula

S_t+h = S_te^{(r- δ)h + σ√(h)}, where S_t = 120, r = 0.03, δ = 0.01, σ = 0.02, h = 10, so

S_t+10 = 120e^{(0.03- 0.02)10 + 0.02√(10)} = S_t+10 = $141.27909

Problem SDRMPPBM4. You can use a 15-period binomial tree to model the price movements of Stock Q. For each time period, the risk-neutral probability of an upward movement in the stock price is 0.54. Find the probability that stock price will be at the 8^th node of the binomial tree at the end of 15 periods.

Solution SDRMPPBM4. We use the formula p_{ith node} = C(n, i)(p*)^n-i(1-p*)ⁱ, where n = 15, i = 8, p = 0.54. Thus, p_{8th node} = C(15, 8) (0.54)⁷(0.46)⁸ = p_{8th node} = 0.172729907.

Problem SDRMPPBM5. You can use a 32-period binomial tree to model the price movements of Stock R. For each time period, the risk-neutral probability of an upward movement in the stock price is 0.78. Find the probability that stock price will be at the 11^th node of the binomial tree at the end of 32 periods.

Solution SDRMPPBM5. We use the formula p_{ith node} = C(n, i)(p*)^n-i(1-p*)ⁱ, where n = 32, i = 11, p = 0.78. Thus, p_{11th node} = C(32, 11)(0.78)²¹(0.22)¹¹ = p_{11th node} = 0.0408609315

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