European option pricing:
On a binomial tree:
• Call option price:
n
n i
n i
i n i


q
(
1

q
)
max
Su
d  X ,0

i
i 0  

• Put option price:
n
n i
n i
i n i


q
(
1

q
)
max
X

Su
d ,0

i
i 0  



American option pricing
• Backward procedure
• At each node we compute the maximum
between:
The payoff if exercised
The expected value under the risk neutral
probability of the payoffs in the two
subsequent nodes
Option price sensitivity
Call E
Put E
Call A
Put A
S
+
-
+
-
X
-
+
-
+
T
?
?
+
+
sigma
+
+
+
+
R
+
-
+
-
Div
-
+
-
+
Black & Scholes
• European Call: C  SN (d1 )  Xe rT N (d 2 )
ln( S / X )  (r   2 / 2)T
d1 
 T
d 2  d1   T
• European Put: P  SN (d1 )  Xe rT N (d 2 )
 rT
P

S

C

Xe
• Put Call Parity: