Options, Futures and Other Derivatives - Assignment Example

Options, Futures and Other Derivatives Name Course Instructor Date Question 1 A) DerivaGem shows that an American- put option is 4.57. Put option’s intrinsic value is the difference between the strike price and the underlying price. The Strike price is $32, and the underlying price is $30. Therefore the Put option’s intrinsic value =$32 -$30 =$2. B) The option's time value is the difference between the option price and the intrinsic or exercise value. Therefore $4.57 -$2 = 2.57 C) A time value of zero indicates that it is best to exercise the option straightaway. The value of an option with zero-time value is equal to its instructive value in this case. D) DerivaGem shows that the option is $12(also its intrinsic value) when the stock value is 20. DerivaGem also shows that the option value is 7.54 when the stock price is 25. At this point, the time value is still positive that is 0.54. The closer the steps equal to 50, the trial and error approach shows that the time value fades as the stock price reduces to $21.69 and lower. When one takes 500 steps, the stock price reduces further to 21.35 (Hull, 2006). Question 2 (A) DerivaGem indicates that the value or the price of a the European call option is 6.9686. (B) The European put option’s value is 4.1244. (C) C + D + Ke -rT = 6.9686 + 0.5e -0.06*0.5 + 40e -0.06*1 =45.1244 Also P + s = 4.1244 + 41 = 45.1244 (D) Using DerivaGem, the price of the options approaches the stock price of 41 as a time to maturity becomes very large. The reason is for the time becoming large is that it does not have to take a long time to pay the price of a call option. Therefore, the present amount of what is supposed to be paid is close to zero. The European put option’s price becomes close to zero as the time to maturity becomes very large. The reason for the put option being close to zero is because of the present value of what is expected to be received is close to zero. The DerivaGem backs explanation as the test results that show when T = 100, the stock price is 40.94 which is close to 41. Therefore, the European put option becomes 0.04 that is close to zero (Hull, 2006). Question 3 When the non-dividend stock price is$30 and strike price is $25 using the DerivaGem, the (a) European call option with strike price of $25 has a cost of 7.90. The call option with a strike price of $30 will have a cost of 4.18. Therefore the cost of the bull spread is. The profits realised when one ignores the impact of discounting are: Range of the Stock Price The Profit 1.28 (b) A put option with $25 strike price will have a cost of 0.28 while a put option with $30 strike price will have a cost of 1.44. Therefore the cost of the bear spread is. The profits realised when one ignores the impact of discounting are: Range of the Stock Price The Profit (c)The Call option with a maturity of one year and a strike price of $25 will have a cost of 8.92. When the strike price is $30, the cost is 5.60. If the strike price is $35, the cost is 3.28. Therefore the cost of the butterfly spread is. The profits realised when one ignores the impact of discounting are: Range of the Stock Price The Profit (d) The European Put options with a maturity of one year and strike prices of $25, the cost is 0.70. When the strike price is $30, the cost is 2.14. If the strike price is $35, the cost is cost 4.57. Therefore the cost of the butterfly spread is. In allowing for the rounding off errors, the results are the same as in (c). The profits are also the same as that in (c). (e) A European call option with a strike price of $30 costs 4.18. A put option with that has a strike price of $30 costs 1.44. Therefore the cost of the straddle is. The profits realised when one ignores the impact of discounting are: Range of the Stock Price The Profit (f) A European call option with a time of six months and a strike price of $35 has a cost of 1.85. A European put option with a time of six months and a strike price of $25 has a cost of 0.28. Therefore the strangle cost is. The profits realised when one ignores the impact of discounting are: Range of the Stock Price The Profit −2.13 Question 4 In the case so that u = e 0.30 * √0.25 =1.1618, d = 1/u = 0.8607, and C) By the use of DerivaGem, the value of option using the two-step tree is 3.3739. When using the DerivaGem in the first worksheet, choose the Equity as the underlying type and Binomial European as the Option type. Then carrying out the calculation is the next step followed by the Displaying the tree that appears as the figure below. D) With a 5-time steps, the value of the option is 3.9229. With a 50-time steps, the value of the option is 3.7394. With a 100-time steps, the value of the option is 3.7478. Finally with 500 time-steps the value option is 3.7545 (Hull, 2006). Question 5 A) First of all, we assume that the value of the option is not exercised early. The time to maturity is 0.5, and there is a dividend of 0.4 in two and five months. Other parameters are , , , . Using the DerivaGem, the price is 0.7947. The DerivaGem also gives the price be 0.7668 assuming the value of the option is exercised at the point in five months before the final dividend. Therefore, the price given by using the Black’s approximation is 0.7947. B) The approximate American option price given by the DerivaGem is 0.7847. When calculating the American option using a binomial model with 100-time steps, the price is 0.8243. Therefore, the price increases when one uses a binomial model with 100-time steps as compared to Black approximation method (Hull, 2006). Question 6. Strike Price Call Price Put Price Call Implied Vol Put Implied Vol 260 26.75 8.50 24.69 24.59 270 21.25 13.50 25.40 26.14 280 17.25 19.00 26.85 26.86 290 14.00 25.625 28.11 27.98 300 11.375 32.625 29.24 28.57 Question 7 A) Using the DerivaGem software, the price = 3.7008, delta = 0.6274, gamma = 0.050, vega = 0.1135, theta = -0.00596 and rho = 0.1512. B) When the stock price is 30.1, the option price is 3.7638. The price option change is 3.7638 – 3.7008 = 0.0630. The delta in the situation predicts the option price change to be 0.6274 * 0.1 = 0.0627 which is close to 0.0630. C) When the stock price is 30.1, the delta is 0.6324. The change in delta is 0.6324 – 0.6274 = 0.005. The delta in the situation predicts the increase of price to 0.050 * 0.1 = 0.005 which is the same. D) The option price will increase by 0.1136 that is from 3.7008 to 3.6948 when the volatility change from 25% to 26%. The Vega value of 0.1135 shows the consistency and that the computing is correct. When the time to maturity changes from one year to 1 – (1/365), the option price reduces from 3.7008 to 3.6948 (3.7008 – 3.6948) with a margin of 0.006. The theta of -0.00596 shows the consistency of the computation. When the interest rate changes from 5% to 6%, the option value increases from 3.7008 to 3.8535 by 0.1527. The rho of 0.1512 shows the consistency of the computation (Hull, 2006). Reference Hull, J. C. (2006). Options, futures, and other derivatives. Pearson Education. Read More