Consider a 1-year (long) strangle on the Nasdaq-100 with strikes of 4,000 and 5,

Question

Consider a 1-year (long) strangle on the Nasdaq-100 with strikes of 4,000 and 5,000. The index spot level is 4,655 and its volatility is 20%. The risk-free rate is 4% and the index pays a dividend yield of 2%.

(a) Use a 6-step binomial tree to price this strategy (note: Up and down movements need to match the volatility. Show all the tree parameters).

b) What are the break-even point(s), the maximum profit and maximum loss of this strategy?

(c) Without using the binomial tree, what is the premium of the straddle with a 5000 strike price? Explain.

Explanation / Answer

5529.748 1529.748 5437.711 1246.538 For 4000 Strike call 5347.206 5347.206 Growth factor per step = a = e^(r% - D%)*dt At each node: 1014.723 1347.206 Here Upper value = Underlying Asset Price 5258.207 5258.207 r% = risk free rate = 4% Lower value = Option Price 825.1158 1093.747 As the time is 1.5 months Shading indicates where option is exercised 5170.69 5170.69 5170.69 So, dt = time interval = time step = 2/12 = 0.167 years Strike price = 4655 670.1547 886.8339 1170.69 5084.629 5084.629 5084.629 Therefore Discount factor each step = 0.845489 543.6141 718.0708 946 a = 1.006736 5000 5000 5000 5000 440.3732 580.5562 763.1669 1000 Up step size = u = e^(volatility * (dt)^1/2) 4916.78 4916.78 4916.78 468.6187 614.5595 803.13 Here volatility = s = 20% 4834.945 4834.945 4834.945 dt = 0.167 493.9155 643.5824 834.9451 4754.472 4754.472 Therefore 514.4652 664.9763 u = 1.016926 4675.339 4675.339 and d = 1/u = 0.983356 527.9454 675.3389 4597.523 hence the probability for the up move = p = (a-d)/(u-d) = 0.696468 531.3831 and probability of down move = 0.303532472 4521.001 521.0014 5148.196 0 5062.509 5.582086 For 5000 Strike put 4978.249 4978.249 Growth factor per step = a = e^(r% - D%)*dt At each node: 18.82996 21.75122 Here Upper value = Underlying Asset Price 4895.391 4895.391 r% = risk free rate = 4% Lower value = Option Price 33.28878 60.56471 As the time is 1.5 months Shading indicates where option is exercised 4813.912 4813.912 4813.912 So, dt = time interval = time step = 2/12 = 0.167 years Strike price = 4655 45.95547 86.50736 186.0881 4733.789 4733.789 4733.789 Therefore Discount factor each step = 0.845489 55.68937 102.6882 198.1174 a = 1.006736 4655 4655 4655 4655 62.31032 111.5532 201.6413 345 Up step size = u = e^(volatility * (dt)^1/2) 4577.522 4577.522 4577.522 115.0176 199.0572 331.1294 Here volatility = s = 20% 4501.334 4501.334 4501.334 dt = 0.167 192.2157 312.9745 498.6661 4426.414 4426.414 Therefore 292.245 459.7505 u = 1.016926 4352.74 4352.74 and d = 1/u = 0.983356 420.6326 647.2595 4280.293 hence the probability for the up move = p = (a-d)/(u-d) = 0.696468 584.1257 and probability of down move = 0.303532 4209.052 790.9477 b. Break even point = $4655 -$309.1654 = $4345.83 and $4655 - $62.31032 = $4592.689 The maximum loss is $309.1654 C. Premium would be = $440.3732

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