Q1. Use DerivaGem to calculate the value of an American put option on a non-dividend-paying stock when the stock price is USD 30, the strike price is USD 32, the risk-free rate is 5%, the volatility is 30% and the time to maturity is 1.5 years. (Choose 'Binomial American' for the option type and 50 time steps.) (See Chapter 9 for supporting theory and materials)
a) What is the option's intrinsic value?
The option's intrinsic value is 32-30=$2.00
b) What is the option's time value?
Using DerivaGem, the value of the option is 4.5678, therefore the option's time value is 4.5678-2=2.5678.
c) What would a time value of zero indicate? What is the value of an option with zero time value?
A time value of 0 indicates that it is optimal to exercise the option immediately. Under this circumstances, we would have the value of option would be equal to the intrinsic value of option.
d) Using a trial and error approach, calculate how low the stock price would have to be for the time value of the option to be zero. (use 50 and 500 time steps)
50 steps:
When stock price = 25, we have option price = 7.53 still greater than intrinsic value of 7
When stock price = 23, we have option price = 9.12 still greater than intrinsic value of 9
When stock price = 22, we have option price = 10.02 still greater than intrinsic value of 10
When stock price = 21, we have option price = 11 thetime value = 0
When stock price = 21.5, we have option price = 10.5 the time value = 0
When stock price = 21.7 we have option price = 10.300535 the time value is fairly close to 0.
Therefore, under 50 steps, when the stock price could be as low as 21.7 to have a time value of zero.
500 steps
When stock price = 21.7, we have option price = 10.309 still greater than intrinsic value of 10.3
When stock price = 21.4, we have option price = 10.600973 the intrinsic value is fairly close to option value
When stock price = 21.3, we have option price = 10.7 the time value = 0
Therefore, under 500 steps, when the stock price could be as low as 21.3 to have a time value of zero.
Q2. Consider an option on a stock when the stock price is $41, the strike price is $40, the risk-free rate is 6%, the volatility is 35% and the time to maturity is one year. Assume that a dividend of $0.50 is expected after six months. (See Chapter 10 for supporting theory and materials)
a) Use DerivaGem to value the option assuming it is a European call.
Using DerivaGem we have price of European call is 6.97
b) Use DerivaGem to value the option assuming it is a European put.
Using DerivaGem we have price of European put of 4.124
c) Verify that put-call parity holds.
According to put-call Parity, we have
C + D + Ke-rT = p + S
d) Explore, using DerivaGem, what happens to the price of the options as the time to maturity becomes very large. For this purpose, assume there are no dividends. Explain the results you get.
Q3. Suppose that the price of a non-dividend-paying stock is $32, its volatility is 30% and the risk-free rate for all maturities is 5% per annum. Use DerivaGem to calculate the cost of setting up the following positions. In each case provide a table showing the relationship between profit and final stock price. Ignore the impact of discounting. Each table should have two columns, 'Stock Price Range' and 'Profit' (See Chapter 11 for supporting theory and materials)
a) a bull spread using European call options with strike prices of $25 and $30 and a maturity of six months
b) a bear spread using European put options with strike prices of $25 and $30 and a maturity of six months
c) a butterfly spread using European call options with strike prices of $25, $30 and $35 and a maturity of one year
d) a butterfly spread using European put options with strike prices of $25, $30 and $35 and a maturity of one year
e) a straddle using options with a strike price of $30 and a six-month maturity
f) a strangle using options with strike prices of $25 and $35 and a six-month maturity.
Q4. Consider a European call option on a non-dividend-paying stock where the stock price is AUD 40, the strike price is AUD 40, the risk-free rate is 4% per annum, the volatility is 30% per annum and the time to maturity is six months. (See Chapter 12 for supporting theory and materials)
a) Calculate u, d and p for a two-step tree
b) Value the option using a two-step tree.
c) Verify that DerivaGem gives the same answer.
d) Use DerivaGem to value the option with 5, 50, 100 and 500 time steps.
Q5. Consider an American call option when the stock price is $18, the exercise price is $20, the time to maturity is six months, the volatility is 30% per annum and the risk-free interest rate is 10% per annum. Two equal dividends of 40 cents are expected during the life of the option, with ex-dividend dates at the end of two months and five months. (See Chapter 13 for supporting theory and materials)
a) Use Black's approximation and the DerivaGem software to value the option.
b) Compare your approximate answer in a) against the American option price calculated using the Binomial model with 100 time steps.
Q6. It is 4 February. July call options on corn futures with strike prices of 260, 270, 280, 290 and 300 cost 26.75, 21.25, 17.25, 14.00 and 11.375, respectively. July put options with these strike prices cost 8.50, 13.50, 19.00, 25.625 and 32.625, respectively. The options mature on 19 June, the current July corn futures price is 278.25 and the risk-free interest rate is 1.1%. There are 135 days to maturity (assuming this is not a leap year, then 365 days in the year). Calculate implied volatilities for the options using DerivaGem using 500 time steps. Comment on the results you get. (hint: these options are American) (See Chapter16 for supporting theory and materials)
a) Complete the table for the calculated implied volatilities
|
Strike Price
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Call Price
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Put Price
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Call Implied Vol
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Put Implied Vol
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260
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26.75
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8.50
|
|
|
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270
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21.25
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13.50
|
|
|
|
280
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17.25
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19.00
|
|
|
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290
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14.00
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25.625
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|
|
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300
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11.375
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32.625
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|
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Q7. Consider a one-year European call option on a stock when the stock price is $30, the strike price is $30, the risk-free rate is 5% and the volatility is 25% per annum.
a) Use the DerivaGem software to calculate the price, delta, gamma, vega, theta and rho of the option.
b) Verify that delta is correct by changing the stock price to $30.1 and re-computing the option price.
c) Verify that gamma is correct by re-computing the delta for the situation where the stock price is $30.1.
d) Carry out similar calculations to verify that vega, theta and rho are correct.