## Miami Dade College Futures and Options Other hi thank you for helping me everything is on the attached files, if you have more questions let me know. thank

Miami Dade College Futures and Options Other hi thank you for helping me everything is on the attached files, if you have more questions let me know. thank you! Day
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Close
20.69
21.24
22.08
21.42
21.08
21.75
21.87
21.93
21.85
21.97
21.18
20.79
21.09
21.86
22.27
22.13
21.75
21.09
21.75
20.85
20.66
21.28
21.12
21.2
20.12
20.07
19.87
20.2
20
20.01
19.97
20.49
20.77
20.69
19.94
19.82
20.3
20.38
20.16
20.11
20.46
Futures and Options
Homework 5
Due: Wednesday, April 22, 2020
Assume that on April 13 the ABC stock price was \$20.69. Assume that the option on
ABC stock expires in 1 month and the risk-free rate is 2.7% p.a.
a) Compute European call price with strike X=19 using Black Scholes closed form
solution. To compute the call price you will need the estimate of volatility.
Calculate historical volatility based on two months of data. Data is provided in
HW5_data.xls file.
b) Assume that the price of the call option on April 13 is \$2.12. Back out the
implied volatility from the Black and Scholes formula.
c) Compute European put price with strike X=19 using Black Scholes closed form
solution. For the estimate of volatility use implied volatility from part b.
d) Approximate prices of European put option with X=19 and European call option
with X=19 using 5-step binomial trees. For the estimate of volatility use implied
volatility from part b. Please, provide your u, d, p, stock tree, and option trees.
e) Approximate prices of American put option with X=19 and American call option
with X=19 using 5-step binomial trees. For the estimate of volatility use implied
volatility from part b. Please, provide your u, d, p, stock tree, and option trees.
Suffolk
Lecture 20
Introduction to Binomial
Trees
1
Homework 5
Suffolk
Due Wednesday, April 22
Should be done in Excel
Should be submitted on Blackboard
2
Option Pricing
Suffolk
Exact Solution
– Black Scholes
Approximation
– Trees (binomial, trinomial, etc.)
– Monte Carlo
– Finite Difference
3
Suffolk
A Simple Binomial Model
A stock price is currently \$20
In three months it will be either \$22 or \$18
Stock Price = \$22
Stock price = \$20
Stock Price = \$18
4
A Call Option
Suffolk
A 3-month call option on the stock has a
strike price of 21.
Stock Price = \$22
Option Price = \$1
Stock price = \$20
Option Price=?
Stock Price = \$18
Option Price = \$0
5
Movements in Time Δt
Suffolk
We choose the tree parameters p , u , and d
so that the tree gives correct values for the
mean and standard deviation of the stock
price changes in a risk-neutral world.
Su
S
Sd
6
Tree Parameters for a Stock w/o Dividends
Suffolk
When Δt is small a solution to the equations is
u = e
t
d = e − t
a−d
p=
u−d
r t
a=e
Note that Δt = T/N, where T – option
maturity, N – number of nodes in a tree
7
Suffolk
General Binomial Tree
3-step tree for the stock process:
S3,3=S0u3
S2,2=S0u2
S3,2=S0u2d
S1,1=S0u
S2,1=S0ud
S0
S3,1=S0ud2
S1,0=S0d
S2,0=S0d2
S3,0=S0d3
8
General Binomial Tree
Suffolk
Tree for computing call value at time t=0:
c3,3=max(0, S3,3 – X)
c2,2
c3,2=max(0, S3,2 – X)
c1,1
c2,1
c0
c3,1=max(0, S3,1 – X)
c1,0
c2,0
c3,0=max(0, S3,0 – X)
9
General Binomial Tree
Suffolk
Procedure for computing call value at time t=0:
– Start with the last level of the tree, N, (time T),
compute terminal values as
max(0, SN,k – X)
Note: for put option the terminal condition is:
max(0, X – SN,k)
– Values for all other nodes starting with level N1 and down to level 0 are computed as follows:
ci ,k = ( ci +1,k +1  p + ci +1,k  (1 − p ) )  e − rt
10
Example
Suffolk
Consider a 2-month European call option on
the Intel stock. Suppose that the current
stock price is 19.11, the strike \$20, the riskfree interest rate is 5% p.a., volatility is 20%.
Compute price of the European call option
using a 4-step binomial tree.
11
Example
Suffolk
dt = T/N = 0.16667/4 = 0.0417
u = eσ  sqrt(dt) = e0.20  sqrt(0.0417) = 1.0417
d = 1/u = 0.9600
a = er  dt = e0.05  0.0417 = 1.0021
p = (a – d)/(u – d) =
= (1.0021 – 0.96)/(1.0417 – 0.96) = 0.5153
12
Suffolk
Example
Stock tree:
22.50
21.60
20.74
20.74
19.91
19.91
19.11
19.11
18.35
19.11
18.35
17.61
17.61
16.91
16.23
First up node: S0u = 19.111.0417 = 19.91
13
Suffolk
Example
Terminal values for the call:
2.50
c(3,3)
c(2,2)
0.74
c(3,2)
c(1,1)
c(2,1)
c(0)
c(1,0)
0
c(3,1)
0
c(2,0)
c(3,0)
0
Up terminal node = max(0, 22.5 – 20) = 2.5
14
Example
Suffolk
For example, the value for node c(3,3) can be
computed as follows:
(2.500.5153 + 0.74(1- 0.5153))e-0.050.0417 =
1.6415
The value for node c(3,2):
(0.740.5153 + 0(1- 0.5153))e-0.050.0417 =
0.3784
15
Suffolk
Example
Roll back through the tree to compute c(0):
2.50
1.6415
1.0272
0.74
0.3784
0.6223
0.1946
0.3684
0.1001
0
0
0
0
0
0
European call price 36.84 cents
16
Black Scholes vs. Binomial Tree
N = 4, c =
0.3684
0.41
0.39
0.37
0.35
call price
Suffolk
N = 100, c =
0.3370
0.33
0.31
0.29
0.27
0.25
2
7
12
17
N
22
27
N = 1000, c =
0.3367
BS = 0.3367
17
Exercise
Suffolk
Consider a 2-month European put option on
the Intel stock. Suppose that the current
stock price is 19.11, the strike \$20, the riskfree interest rate is 5% p.a., volatility is 20%.
Compute price of the European put option
using a 3-step binomial tree.
18
Exercise
Suffolk
dt = T/N =
u = eσ  sqrt(dt) =
d = 1/u =
a = er  dt =
p = (a – d)/(u – d) =
19
Exercise
Suffolk
dt = T/N = 0.16667/3 = 0.0556
u = eσ  sqrt(dt) = e0.20  sqrt(0.0556) = 1.0483
d = 1/u = 0.9540
a = er  dt = e0.05  0.0556 = 1.0028
p = (a – d)/(u – d) =
= (1.0028 – 0.9540)/(1.0483 – 0.9540) = 0.5177
20
Exercise
Suffolk
Stock tree:
19.11
First up node: S0u =
21
Suffolk
Exercise
Stock tree:
22.01
21.00
20.03
20.03
19.11
19.11
18.23
18.23
17.39
16.59
First up node: S0u = 19.11  1.0483 = 20.03
22
Suffolk
Exercise
Terminal values for the put:
p(2,2)
p(1,1)
p(2,1)
p(0)
p(1,0)
p(2,0)
Up terminal node =
23
Suffolk
Exercise
Terminal values for the put:
0
p(2,2)
0
p(1,1)
p(2,1)
p(0)
p(1,0)
1.77
p(2,0)
3.41
Up terminal node = max(0, 20 – 22.01) = 0
24
Suffolk
Exercise
Roll back through the tree to compute p(0):
0
0
1.77
3.41
25
Exercise
Suffolk
The value for the node p(2,2):
The value for the node p(2,1):
The value for the node p(2,0):
26
Exercise
Suffolk
The value for the node p(2,2):
0
The value for the node p(2,1):
(00.5177 + 1.77(1- 0.5177))e-0.050.0556 =
0.8513
The value for the node p(2,0):
(1.770.5177 + 3.41(1- 0.5177))e-0.050.0556 =
2.5539
27
Suffolk
Exercise
Roll back through the tree to compute p(0):
0
0
0
0.8513
1.77
2.5539
3.41
28
Exercise
Suffolk
The value for the node p(1,1):
The value for the node p(1,0):
The value for the node p(0):
29
Exercise
Suffolk
The value for the node p(1,1):
(00.5177 + 0.8513(1- 0.5177))e-0.050.0556 =
0.4094
The value for the node p(1,0):
(0.85130.5177 + 2.5539(1- 0.5177))e-0.050.0556 =
1.6678
The value for the node p(0):
(0.40940.5177 + 1.6678(1- 0.5177))e-0.050.0556 =
1.0135
30
Suffolk
Exercise
Roll back through the tree to compute p(0):
0
0
0
0.4094
0.8513
1.0135
1.6678
1.77
2.5539
3.41
Put price \$1.0135
31
Suffolk
Lecture 21
Introduction to Binomial
Trees
1
Homework 5
Suffolk
Due Wednesday, April 22
Should be done in Excel
Should be submitted on Blackboard
2
Movements in Time Δt
Suffolk
We choose the tree parameters p , u , and d
so that the tree gives correct values for the
mean and standard deviation of the stock
price changes in a risk-neutral world.
Su
S
Sd
3
Tree Parameters for a Stock w/o Dividends
Suffolk
When Δt is small a solution to the equations is
u = e
t
d = e − t
a−d
p=
u−d
r t
a=e
Note that Δt = T/N, where T – option
maturity, N – number of nodes in a tree
4
Suffolk
General Binomial Tree
3-step tree for the stock process:
S3,3=S0u3
S2,2=S0u2
S3,2=S0u2d
S1,1=S0u
S2,1=S0ud
S0
S3,1=S0ud2
S1,0=S0d
S2,0=S0d2
S3,0=S0d3
5
General Binomial Tree
Suffolk
Tree for computing call value at time t=0:
c3,3=max(0, S3,3 – X)
c2,2
c3,2=max(0, S3,2 – X)
c1,1
c2,1
c0
c3,1=max(0, S3,1 – X)
c1,0
c2,0
c3,0=max(0, S3,0 – X)
6
General Binomial Tree
Suffolk
Procedure for computing call value at time t=0:
– Start with the last level of the tree, N, (time T),
compute terminal values as
max(0, SN,k – X)
Note: for put option the terminal condition is:
max(0, X – SN,k)
– Values for all other nodes starting with level N1 and down to level 0 are computed as follows:
ci ,k = ( ci +1,k +1  p + ci +1,k  (1 − p ) )  e − rt
7
Pricing American Options
Suffolk
American options can be exercised before
maturity.
American call option on a non-dividend stock
should never be exercised early.
Closed form solutions for pricing American
options do not exist.
American options can be priced using
binomial trees.