Option Elasticity and Option Volatility: Practice Problems and Solutions

This is Section 42 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. See Section 29 here. See Section 30 here. See Section 31 here. See Section 32 here. See Section 33 here. See Section 34 here. See Section 35 here. See Section 36 here. See Section 37 here. See Section 38 here. See Section 39 here. See Section 40 here. See Section 41 here.

The formula for option elasticity Ω is

Ω = [% change in option price]/[% change in stock price] = S∆/C, where C is the option price, S is the stock price, and ∆ is the option delta.

For a call option, Ω ≥ 1. Ω decreases as the strike price K increases.

For a put option, Ω ≤ 0.

The volatility of an option can be expressed as

σ_option = σ_stock*│Ω│. That is, option price volatility is stock price volatility multiplied by the absolute value of option elasticity.

Source: McDonald, R.L., Derivatives Markets (Second Edition), Addison Wesley, 2006, Ch. 12, pp. 391-394.

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

Problem OEOV1. The stock of Assiduous Co. trades for $123 per share with a price volatility of 0.3. Certain call options on Assiduous Co. stock have a delta of 0.44 and a price of $20. Find the elasticity of such a call option.

Solution OEOV1. We use the formula Ω = S∆/C = 123*0.44/20 = Ω = 2.706

Problem OEOV2. The stock of Assiduous Co. trades for $123 per share with a price volatility of 0.3. Certain call options on Assiduous Co. stock have a delta of 0.44 and a price of $20. Find the volatility of such a call option.

Solution OEOV2. From Solution OEOV1, we know that Ω = 2.706 = │Ω│. Also, we are given that σ_stock = 0.3. Thus, σ_option = σ_stock*│Ω│ = 0.3*2.706 = σ_option = 0.8118

Problem OEOV3. The stock of Ingenious Co. trades for $550 per share with a prepaid forward price volatility (the volatility relevant for the Black-Scholes formula) of 0.2. Certain call options on Ingenious Co. stock have a strike price of $523 and a time to expiration of 3 years. The stock pays dividends at an annual continuously compounded yield of 0.03, and the annual continuously compounded interest rate is 0.07. Find the elasticity of such a call option.

Solution OEOV3. We can use the Black-Scholes formula to find the call option price.

First we find d₁ = [ln(S/K) + (r - ∂ + 0.5σ²)T]/[σ√(T)] =

[ln(550/523) + (0.07 - 0.03 + 0.5*0.2²)3]/[0.2√(3)] = d₁ = 0.6649251083

Now we find d₂ = d₁ - σ√(T) = 0.6649251083 - 0.2√(3) = d₂ = 0.3185149468

In MS Excel, using the input "=NormSDist(0.6649251083)", we find that N(d₁) = 0.746950872

In MS Excel, using the input "=NormSDist(0.3185149468)", we find that N(d₂) = 0.624952755

Now we use the Black-Scholes formula:

C(S, K, σ, r, T, ∂) = Se^-∂TN(d₁) - Ke^-rTN(d₂) = 550e^-0.03*30.746950872 - 523e^-0.07*30.624952755 = C = 110.5242361

We can also find ∆_call = e^-∂TN(d₁) = e^-0.03*30.746950872 = ∆_call = 0.6826616958

Now we use the formula Ω = S∆/C = 550*0.6826616958/110.5242361 = Ω = 3.39711855

Problem OEOV4. The stock of Ingenious Co. trades for $550 per share with a price volatility of 0.2. Certain call options on Ingenious Co. stock have a strike price of $523 and a time to expiration of 3 years. The stock pays dividends at an annual continuously compounded yield of 0.03, and the annual continuously compounded interest rate is 0.07. Find the volatility of such a call option.

Solution OEOV4. From Solution OEOV3, we know that Ω = 3.39711855 = │Ω│. Also, we are given that σ_{prepaid forward} = 0.2. But we need to find

σ_stock = (F^P/S)σ_{prepaid forward} = (Se^-∂T/S)σ_{prepaid forward} = e^-∂Tσ_{prepaid forward} = e^-0.03*30.2 =

σ_stock = 0.1827862371

Thus, σ_option = σ_stock*│Ω│ = 0.1827862371*3.39711855 = σ_option = 0.6209465166

Problem OEOV5. The stock of Ingenious Co. trades for $550 per share with a prepaid forward price volatility (the volatility relevant for the Black-Scholes formula) of 0.2. Certain put options on Ingenious Co. stock have a strike price of $523 and a time to expiration of 3 years. The stock pays dividends at an annual continuously compounded yield of 0.03, and the annual continuously compounded interest rate is 0.07. Find the elasticity of such a put option.

Solution OEOV5. Since the call price is known from Solution OEOV3, we can use put-call parity to calculate the put price: P(S, K, σ, r, T, ∂) = C(S, K, σ, r, T, ∂) + Ke^-rT - Se^-∂T =

110.5242361 + 523e^-0.07*3 - 550e^-0.03*3 = P = $31.79764484

Now we find ∆_put = ∆_call - e^-∂T = 0.6826616958 - e^-0.03*3 = ∆_put = -0.2312694895

Now we use the formula Ω = S∆/P = 550*-0.2312694895/31.79764484 = Ω = -4.000240265

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