The risk neutral probability is 105 1 12 12 1 12 0 59 396 275 475 330 229 Thus

The risk neutral probability is 105 1 12 12 1 12 0 59

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The risk-neutral probability is: 105 1 12 12 1 12 0 59 . / . . / . . - - = 396 275 475 330 229
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Thus, the put price tree is: 059 0 0 41 27 6 105 10 75 . . . . . x x + = The price of the two-period European put on Bullmart is thus 10.75p. Put pay-off: max (300-475,0) = 0 max (300-330,0) = 0 max (300-229,0) = 71 059 0 0 41 71 105 27 6 . . . . x x + = 0
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Q3: You have estimated from financial market data that the annual volatility of XYZ is 15%. Today the price of XYZ is $100. Assume that XYZ does not pay any dividends. You are interested in valuing a call option on XYZ maturing in two year’s time. Assume that the annual interest rate is 10%. a) Construct a two period binomial tree (t= 0,1,2) for XYZ (that is, let D t =1) 100 b) Now consider a call option on XYZ with exercise price EX = $110 that expires in period 2. Find the replicating portfolio for this option first at time t=1 and then at t=0. First note that: C uu = max {135-110;0} = 25 C du = max {100-110;0} = 0 C dd = max {74-110;0} = 0 116 86 135 100 74 ࠵? = ࠵? %,’( = 1,16 d = 1/u = 0,86
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Now, at time 1 if the share price is 116 the replicating portfolio is: h C C S S H HH LH HH LH = - - = = 24 6 34 6 71 . . . B h S C r H H LH LH f = - + = = 1 71 100 11 64 5 . * . . So C u = .71 x 116 - 64.5 = $ 18 If share price is 86, there is no hope that the call option will end up in the money and so C d = 0. We can now find a replicating portfolio at time 0 such that its value at time 1 is sufficient to allow us to purchase the replicating portfolio at time 1. The time 0 replicating portfolio must have: h C C S S H L H L 0 17 9 116 86 2 6 = - - = - = . . . B h S C r L L f 0 0 1 6 86 2 11 = - + = = . * . . $47 So C 0 = .6 x 100 - 47 = $ 13
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c) What are the rebalancing trades? If the share price goes up to 116, we need to increase our hedge ratio. We purchase (.71-.6) =.11 shares and borrow .11 * 116 = (64.5 - 47*1.1) = $12.8 to do so. If the share price drops to 86 we sell all our shares and pay back our loan. d) Price the same call option using the “risk neutral pricing” method. In this case p r d u d = + - - = - - = 1 11 862 116 862 8 . . . . . We get: C x H = - = . ( . ) . . 8 134 6 110 11 17 9 ; C L = 0 ; C x 0 8 17 9 11 13 = = . . . ;
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Q4: You have estimated from financial market data that the annual volatility of QWE is also 15%. Similarly to the company in the previous example, the price of QWA is also $100. One difference exists, however, in that QWE is expected to pay a $6 dividend at time 1. Remember that the option holders are not entitled to receive the dividend. a) How does the binomial tree for QWE’s stock look like in the presence of this known $6 dollar dividend? 100 b) Use risk neutral pricing method to value a call option on QWE’s stock assuming that the exercise price of the option is still $110. Explain the difference between QWE’s option price and the price of the XYZ’s option. As before, p r d u d = + - - = - - = 1 11 862 116 862 8 . . . . . 127.6 94.6 68.8 116-6 = 110 86-6 = 80 92.8
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We get: C x H = - = . ( . ) . . 8 132 2 110 11 161 ; C L = 0 ; C x 0 8 161 11 117 = = . .
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