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
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
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
Please, use Excel for all your computations.
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
© Natalia Beliaeva, 2020
1
Homework 5
Suffolk
Due Wednesday, April 22
Should be done in Excel
Should be submitted on Blackboard
© Natalia Beliaeva, 2020
2
Option Pricing
Suffolk
Exact Solution
– Black Scholes
Approximation
– Trees (binomial, trinomial, etc.)
– Monte Carlo
– Finite Difference
© Natalia Beliaeva, 2020
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
© Natalia Beliaeva, 2020
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=?
© Natalia Beliaeva, 2020
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
© Natalia Beliaeva, 2020
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
© Natalia Beliaeva, 2020
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
© Natalia Beliaeva, 2020
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)
© Natalia Beliaeva, 2020
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
© Natalia Beliaeva, 2020
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.
© Natalia Beliaeva, 2020
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
© Natalia Beliaeva, 2020
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
© Natalia Beliaeva, 2020
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
© Natalia Beliaeva, 2020
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
© Natalia Beliaeva, 2020
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
© Natalia Beliaeva, 2020
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
© Natalia Beliaeva, 2020
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.
© Natalia Beliaeva, 2020
18
Exercise
Suffolk
dt = T/N =
u = eσ  sqrt(dt) =
d = 1/u =
a = er  dt =
p = (a – d)/(u – d) =
© Natalia Beliaeva, 2020
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
© Natalia Beliaeva, 2020
20
Exercise
Suffolk
Stock tree:
19.11
First up node: S0u =
© Natalia Beliaeva, 2020
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
© Natalia Beliaeva, 2020
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 =
© Natalia Beliaeva, 2020
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
© Natalia Beliaeva, 2020
24
Suffolk
Exercise
Roll back through the tree to compute p(0):
0
0
1.77
3.41
© Natalia Beliaeva, 2020
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):
© Natalia Beliaeva, 2020
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
© Natalia Beliaeva, 2020
27
Suffolk
Exercise
Roll back through the tree to compute p(0):
0
0
0
0.8513
1.77
2.5539
3.41
© Natalia Beliaeva, 2020
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):
© Natalia Beliaeva, 2020
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
© Natalia Beliaeva, 2020
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
© Natalia Beliaeva, 2020
31
Suffolk
Lecture 21
Introduction to Binomial
Trees
© Natalia Beliaeva, 2020
1
Homework 5
Suffolk
Due Wednesday, April 22
Should be done in Excel
Should be submitted on Blackboard
© Natalia Beliaeva, 2020
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
© Natalia Beliaeva, 2020
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
© Natalia Beliaeva, 2020
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
© Natalia Beliaeva, 2020
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)
© Natalia Beliaeva, 2020
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
© Natalia Beliaeva, 2020
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.
© Natalia Beliaeva, 2020
8
Binomial Tree for American Options
Suffolk
Build asset tree in the same way as before
Compute terminal values in the same way as
before
For every node from level N-1 to 0 compute the
option value as follows:
– If intrinsic value > discounted value, set option
value at the node equal to intrinsic value
– If intrinsic value < discounted value, set option value at the node equal to discounted value © Natalia Beliaeva, 2020 9 Binomial Tree for American Options Suffolk Formally, the node value for level N-1 through 0 can be computed as follows: American call: ( Ci ,k = max ( Ci +1,k +1  p + Ci +1,k  (1 − p ) )  e − rdt , Si ,k − X ) American call option on a non-dividend paying stock will never be exercised early American put: ( Pi ,k = max ( Pi +1,k +1  p + Pi +1,k  (1 − p ) )  e − rdt , X − Si ,k © Natalia Beliaeva, 2020 ) 10 Example Suffolk Consider a 2-month American 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 American put option using a 3-step binomial tree. © Natalia Beliaeva, 2020 11 Example 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 © Natalia Beliaeva, 2020 12 Suffolk Example 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 © Natalia Beliaeva, 2020 13 Suffolk Example 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 © Natalia Beliaeva, 2020 14 Example Suffolk The value for the node p(2,2): max(0, 20 – 20.9994) = 0 The value for the node p(2,1): max(0.8513, 20 – 19.11) = 0.89 The value for the node p(2,0): max(2.5539, 20 – 17.39) = 2.61 © Natalia Beliaeva, 2020 15 Suffolk Example Roll back through the tree to compute American put price P(0) (Note: top value – discounted value, bottom value – intrinsic value, value in red – option value at the node): 0 1.0723 0.4280 0.0000 1.7145 1.7700 0.0000 0.0000 0.8513 0.8900 2.5539 2.6094 0 1.77 3.41 American put price is $1.0723 © Natalia Beliaeva, 2020 16 Exercise Suffolk Consider a 3-month European and American put option on the Intel stock. Suppose that the current stock price is 19.11, the strike $22.5, the risk-free interest rate is 5% p.a., volatility is 20%. Compute price of the European and American put option using a 3-step binomial tree. © Natalia Beliaeva, 2020 17 Exercise Suffolk dt = T/N = u = eσ  sqrt(dt) = d = 1/u = a = er  dt = p = (a – d)/(u – d) = © Natalia Beliaeva, 2020 18 Exercise Suffolk dt = T/N = 0.25/3 = 0.0833 u = eσ  sqrt(dt) = e0.20  sqrt(0.0833) = 1.0594 d = 1/u = 0.9439 a = er  dt = e0.05  0.0833 = 1.0042 p = (a – d)/(u – d) = = (1.0042 – 0.9439)/(1.0594 – 0.9439) = 0.5217 © Natalia Beliaeva, 2020 19 Exercise Suffolk Stock tree: 19.11 © Natalia Beliaeva, 2020 20 Suffolk Exercise Stock tree: 22.72 21.45 20.25 20.25 19.11 19.11 18.04 18.04 17.03 16.07 © Natalia Beliaeva, 2020 21 Suffolk Exercise Terminal values for the put: p(2,2) p(1,1) p(2,1) p(0) p(1,0) p(2,0) © Natalia Beliaeva, 2020 22 Suffolk Exercise Terminal values for the put: 0 p(2,2) 2.2542 p(1,1) p(2,1) p(0) p(1,0) 4.4621 p(2,0) 6.4292 © Natalia Beliaeva, 2020 23 Exercise Suffolk Roll back through the tree to compute p(0): European put price © Natalia Beliaeva, 2020 24 Suffolk Exercise Roll back through the tree to compute p(0): 0 1.0737 2.2542 2.1279 3.2964 3.1419 4.2753 4.4621 5.3804 6.4292 European put price $3.1419 © Natalia Beliaeva, 2020 25 Exercise Suffolk Roll back through the tree to compute American put price P(0) (Note: top value – discounted value, bottom value – intrinsic value, value in red – option value at the node): American put price is © Natalia Beliaeva, 2020 26 Suffolk Exercise Roll back through the tree to compute American put price P(0) (Note: top value – discounted value, bottom value – intrinsic value, value in red – option value at the node): 0 3.2900 2.1795 2.2542 4.3685 4.4621 1.0737 1.0509 3.2964 3.3900 5.3804 5.4740 2.2542 4.4621 6.4292 American put price is $3.29 © Natalia Beliaeva, 2020 27 Purchase answer to see full attachment

Submit a Comment