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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