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Module 5: Option Pricing Practice Quiz
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Terms in this set (20)
What does an equity option's delta reflect?
The number of shares needed to replicate one call option
Suppose Ralph's stock price is currently $50. In the next six months it will either fall to $30 or rise to $80. What is the option delta of a call option with an exercise price of $50?
Option delta = (30 − 0)/(80 − 30) = 30/50 = 0.6.
Suppose ABCD's stock price is currently $50. In the next six months it will either fall to $40 or rise to $60. What is the current value of a six-month call option with an exercise price of $50? The six-month risk-free interest rate is 2 percent (periodic rate).
Replicating portfolio method: Call option payoff = 60 − 50 = 10 and zero;(60)(A) + (1.02)(B) = 10, (40)(A) + 1.02(B) = 0; Solving for A = 0.5 (option delta) &B = −19.6; call option price (current) = 0.5(50) − 19.61 = $5.39.
Risk-neutral valuation: Risk-neutral probability of a rise in value = (interest rate − % downside change)/(upside change − % downside change); [0.02 − (−10/50)]/[(10/50) − (−10/50)] = 0.55;Call option value = [(0.55)(10) + (0.45)(0)]/(1.02) = $5.39.
Suppose VS's stock price is currently $20. In the next six months it will either fall to $10 or rise to $30. What is the current value of a put option with an exercise price of $15? The six-month risk-free interest rate is 5 percent per six-month period.
Replicating portfolio method: Delta = (0 − 5)/(30 − 10) = −5/20 = −0.25. Value of put = (delta) × (share price) + PV(0.25 × 30) = −0.25 × 20 + 7.5/1.05 = $2.14.
Risk-neutral valuation: 20 = [x(30) + (1 − x)(10)]/1.05; x = 0.55 and (1 − x) = 0.45.Put option price = [(0.55)(0) + (0.45)(5)]/(1.05) = 2.14.
Suppose Carol's stock price is currently $20. In each six-month period it will either fall by 50 percent or rise by 100 percent. What is the current value of a one-year call option with an exercise price of $15? The six-month risk-free interest rate is 5 percent per six-month period. [Use the two-stage binomial method.]
20 = [40(x) + (10)(1 − x)]/1.05; x = 0.36667; (1 − x) = 0.6333;C1UP = [(65)(0.36667) + (5)(0.6333)]/1.05 = 25.714; C1DN = (5)(0.36667)/1.05 = 1.746;C = [25.714 × 0.36667 + 1.746 × 0.6333]/1.05 = 10.03.
The current price of a non-dividend-paying stock is $30. Over the next six months, it is expected to rise to $36 or fall to $26. Assume the risk-free rate is zero. What is the risk-neutral probability of that the stock price will be $36?
The formula for the risk-neutral probability of an up movement is
(e^rT - d)/(u-d)
In this case u = 36/30 or 1.2 and d = 26/30 = 0.8667. Also r = 0 and T = 0.5. The formula gives
p = (1 − 0.8667/(1.2 − 0.8667) = 0.4.
If the standard deviation of the continuously compounded returns (σ) on a stock is 40 percent, and the time interval is a year, then the upside change equals
1 + upside change = e^(0.4 × 1.0) = 1.4918, or upside change = 49.18%.
If the standard deviation of the continuously compounded returns on the asset is 40 percent and the interval is a year, then the downside change is equal to
1 + downside change = 1/(e^(0.4 × 1.0)) = 1/1.4918 = 0.67032; downside change = −32.96%.
The current price of a non-dividend-paying stock is $40. Over the next year, it is expected to rise to $42 or fall to $37. An investor buys one-year put options with a strike price of $41. Which of the following is necessary to hedge the position?
Buy 0.8 shares for each option purchased.
The key is to create a portfolio (i.e., holding some combination of the stock and the option) such that the value of the portfolio will be the same regardless of which way the stock moves (i.e., up or down).
Assume we hold 1 option and D shares of stock. If the stock price falls, then the value of the portfolio will be ($41 - $37) + $37D. If the stock price increases, then the value of the portfolio will be $0 + $42D (because the put option will expire worthless, but we'll still have D shares).
Now, set these scenarios equal to each other:
$4 + $37D = $0 + $42D
We now solve for D (the delta): this will tell us how many shares we need to hold in order to achieve the same payoffs, regardless of which way the stock moves:
$4 = ($42 - 37)D
$4/($42 - 37) = D = .8
The investor should therefore buy 0.8 shares for each option purchased.
Which of the following describes how American options can be valued using a binomial tree?
Check whether early exercise is optimal at all nodes where the option is in-the-money.
For an American option, we must check whether exercising is better than not exercising at each node where the option is in the money. (It is clearly not worth exercising when the option is out of the money)
The Black-Scholes formula represents the option delta as
N(d1).
A call option with an exercise price of $50 expires in six months, has a stock price of $54, and has a standard deviation of 80 percent. The risk-free rate is 9.2 percent per year annually compounded. Calculate the value of d1.
d1 = [ln(54/47.847)]/(0.8 × (0.5^0.5)) + (0.8 × (0.5^0.5))/2 = 0.4967.
A call option with an exercise price of $50 expires in six months, has a stock price of $54, and has a standard deviation of 80 percent. The risk-free rate is 9.2 percent per year annually compounded. Calculate the value of d2.
d2 = 0.4967 − 0.8(0.5^0.5) = −0.0690.
If the value of d1 is 1.25, then the value of N(d1) is equal to
Use the Cumulative Probabilities of the Standard Normal Distribution Table.Excel Spreadsheet: NORMSDIST(1.25) = 0.8944.
Assume the following data: Stock price = $50; Exercise price = $45; Risk-free rate = 6 percent per year (stated rate) compounded continuously; Continuously compounded variance = 0.2; Expiration = three months. Calculate the value of a European call option. (Use the Black-Scholes formula.)
d1 = 0.65; d2 = 0.4264; N(d1) = 0.742; N(d2 ) = 0.6651;Call option value = 50(0.742) − e^(−0.06)(0.25)[45][0.6651] = $7.62.
All else equal, if an option's strike price increases, then the
value of a put option increases and that of a call option decreases.
All else equal, if the volatility (variance) of the underlying stock increases, then the
value of both a put option and a call option increase.
The Black-Scholes option pricing model employs which five parameters?
Stock price, exercise price, risk-free rate, variance, and time to maturity
A stock is currently selling for $50. The stock price could go up by 10 percent or fall by 5 percent each month. The monthly risk-free interest rate is 1 percent. Calculate the price of an American put option on the stock with an exercise price of $55 and a maturity of two months. (Use the two-stage binomial method.)
p = (1 − (−5))/(−10 − (−5)) = 0.40 1 − p = 0.6.
End of one-month values: [(0)(0.4) + (2.75)(.6)]/1.01 = 1.6337, or zero and [(2.75)(.4) + (9.875)(0.6)]/1.01 = $6.9555, or $7.5 if exercised;
P = [0.4(1.6337) + (7.5)(0.6)]/1.01 = $5.10.
What was the original Black-Scholes-Merton model designed to value?
A European option on a stock providing no dividends.
The original Black-Scholes-Merton model was designed to value a European option on a stock paying no dividends.
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