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Lecture 10

# Lecture 10.docx

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School
Department
Course
Professor
Nabil Tahani
Semester
Winter

Description
LECTURE 10 Example: slide 13 S0= 50, K = 52, r = 5% T = 2y , m = 2, ∆ t= 1 y, u = 1.2 ,, d = 0.8 Pricing European put option : 5% x 1 P = e − 0.8= 0.6282 1.2−0.8 The payoff for a put is K – S , Tr 0 Put = [ p x f + 2p(1-p)f + (1-p) f ] e 2 -r x T uu ud dd 2 2 -5% x 2 = [ p x 0 + 2p(1-p) 4 + (1-p) x 20 ] e = 4.1923 Now, to standardize everyone’s value of u and d , we can use sigma ( the closest factor that affects the stock price ) to get : - u = eσ √∆t t - d = e-σ √∆ er∆t−d - p = u−d Note that u and d are inversely related. For American put options: Note that you can exercise it at all possible times till the maturity date. You would have multiple step pay offs unlike the European options which miss out on the middle steps. What one would do is : - compare intrinsic value with the European option pay off - if intrinsic value ( ST– K) is more than the European pay off , you will choose intrinsic value as optimal and red it. - If it’s negative, choose the European value - Note: you would have to change the option value before that step by calculating intrinsic value and discounting it. - Note that the first time you hit intrinsic value, all the other numbers would be intrinsic and RED - - Stocks q≡ - Indices q ≡ div yield - Currencies q ≡ r Forfign risk free rate) - Futures q ≡r Four parameters to be calculated before any work: σ√∆t - u = e - d= 1/u (r-∆) t - a = e , note that q for stock is 0 - p = a−d u−d example: futures call option , S0= 1000, T = 1 y, K = 1000, m = 4, r = 4%, σ = 20% Answer: 0.20 √0.25 u= e = 1.1052 d = 1/u= 0.9048 a = e(r-q∆ = 1 1−0.9048 p = = 1
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