Volatility and Early Exercise of American Options

This is Section 25 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.

On an American call option where the volatility is zero, it is optimal to defer exercise as long as the following condition holds:
rK > ∂S

It is optimal to exercise whenever

S > rK/∂

American "call options are early-exercised in order to capture dividends on the underlying stock."

With put options, the reverse holds. It is optimal to exercise early when

S < rK/∂

and it is optimal to defer exercise when rK < ∂S

Meaning of variables:

r = annual continuously-compounded risk-free interest rate.

∂ = annual continuously-compounded dividend yield.

K = strike price of the option.

S = underlying asset (stock) price.

When volatility is positive, the exercise bounds for lower volatility are lower on call options than the exercise bounds for higher volatility. The insurance effect of holding call options is greater the higher the volatility, and this effect is lost if the call is exercised. So if stock price volatility = σ = 0.1 and the lowest stock price at which it is optimal to exercise the call is $200, then if the stock price volatility were σ > 0.1, we would expect the lowest stock price at which it is optimal to exercise the call to be > $200.

With put options, lower volatility means that it is optimal to exercise at a higher stock price than would be optimal if volatility were higher. So with put options if stock price volatility = σ = 0.1 and the lowest stock price at which it is optimal to exercise the call is $200, then if the stock price volatility were σ > 0.1, we would expect highest stock price at which it is optimal to exercise the call to be < $200.

Source: McDonald, R.L., Derivatives Markets (Second Edition), Addison Wesley, 2006, Ch. 11, pp. 333-335, 365.

Problem VEEAO1. Impressive Co. stock currently trades for $456 per share. The price movements of this stock are not volatile at all. The stock pays dividends with an annual continuously compounded yield of 0.04. The annual continuously compounded interest rate is 0.05. You own American call options on Impressive Co. stock and you know that exercising early is not optimal right now. The strike price of the call options must at least be greater than some number X. Find X.

Solution VEEAO1. We are given that the following condition holds: rK > ∂S and S = 456, ∂ = 0.04, r = 0.05. Thus, 0.05K > 0.04*456, so K > 0.04*456/0.05 or K > 364.8. Thus, X = $364.8

Problem VEEAO2. You know that if the price of the non-volatile Precocious LLC stock increased at all, it would be a good idea to exercise an American call option on Precocious LLC stock right now. The stock currently trades for $430, and the annual continuously compounded interest rate is 0.09. The strike price of the option is $344. Find the annual continuously compounded yield on the dividends of Precocious LLC stock.

Solution VEEAO2. If the price of Precocious LLC stock increased at all, then the condition

S > rK/∂ would hold. Right now, S = rK/∂ is the case, so ∂ = rK/S, where r = 0.09, K = 344, and S = 430. Thus, ∂ = 0.09*344/430 = ∂ = 0.072

Problem VEEAO3. You own an American put option on the non-volatile Mysterious, Inc., stock. The strike price of the option is 23, and the stock's annual continuously compounded dividend yield is 0.4. The annual continuously compounded interest rate is 0.1. For which of these stock prices would it be optimal to exercise the option?

(a) S = 20
(b) S= 15.65
(c) S = 13.54
(d) S = 12.34
(e) S = 6.78
(f) S = 3.45

Solution VEEAO3. We know that for American put options, it is optimal to exercise early whenever S < rK/∂. We know that r = 0.1, K = 23, and ∂ = 0.4, so rK/∂ = 5.75. Of the prices listed, only 3.45 < 5.75. Thus, only (f) is the correct answer and only (f) describes a price for which early exercise would be optimal.

Problem VEEAO4. Volatile Co. stock prices currently have a volatility of σ = 0.3. For American call options with a strike price of $120 and time to expiration 1 year, you know that the lowest stock price where exercise is optimal is $160. If the stock price volatility changes, for which of these volatilities will exercise still be optimal at a stock price of $160? More than one correct answer is possible.

(a) σ = 0.1
(b) σ = 0.2
(c) σ = 0.4
(d) σ = 0.5
(e) σ = 0.6

Solution VEEAO4. Exercise will definitely not be optimal at $160 if the volatility increases, because the lowest bound for optimal exercise price will be pushed up by such an increase. This rules out choices (c), (d), and (e). On the other hand, exercise will definitely be optimal at $160 if the volatility decreases, because the lowest bound for optimal exercise price will be pushed down by such a decrease. Thus, both (a) and (b) are correct answers.

Problem VEEAO5. Volatile Co. stock prices currently have a volatility of σ = 0.3. For American put options with a strike price of $90 and time to expiration 1 year, you know that the highest stock price where exercise is optimal is $40. If the stock price volatility changes, for which of these volatilities will exercise still be optimal at a stock price of $40? More than one correct answer is possible.

(a) σ = 0.1
(b) σ = 0.2
(c) σ = 0.4
(d) σ = 0.5
(e) σ = 0.6

Solution VEEAO5. Exercise will definitely not be optimal at $40 if volatility increases, because the highest bound for optimal exercise price will be pushed up by such an increase. This rules out choices (c), (d), and (e). On the other hand, exercise will definitely be optimal at $40 if the volatility decreases, because the highest bound for optimal exercise price will be pushed up by such a decrease. Thus, both (a) and (b) are correct answers.

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