Session 7a
Agenda
Binomial lattice models
Dynamic programming and valuation of investment opportunities
Applications:
–Value of American Options
–Value a Lease on Gold Mine
Binomial Lattice
Any model in which a stock price can only increase or decrease by
a certain amount during a period is called a binomial model.
Key Factors:
Two possible next states (up, down) at each stage
• Maybe in the form of a multiplier
Fixed independent probabilities from stage to stage (p, q)
• Where q = 1 - p
Time frame (Δt)
Discount rate
Period 0
Period 1
S
uS
dS
For spreadsheet layout
Up move: directly to the right
Down move: right and down one step
Single Period Example
Period 0
Period 1
S
uS
dS
A stock is currently selling for $40.
One period from now the stock will either
• increase to $50 with probability 0.5, or
• decrease in price to $32 with probability 0.5.
Single Period Call Option
What is a fair price for a European call option with an exercise
price of $40? Assume the risk free rate of interest is 1/9 ≈ 0.1111.
Payoff of the option:
Stock Up:
$10
with prob 0.5
Stock Down:
$0
with prob 0.5
Expected payoff of the option: $10.00 * 0.5 + $0.00 *0.5 = $5.00
Discounted expected payoff: $5.00 / (1 + 0.1111) = $4.50
A
1
2
3
4
5
6
7
8
9
B
C
D
E
Today Price
Discount Rate
$ 40.00
$ 0.11
Future Price
Probability Payout
$ 50.00
0.5 $ 10.00
$ 32.00
0.5 $
-
Expected Payout
Present Value
$
$
5.00
4.50
F
G
=MAX(C5-$C$1,0)
=MAX(C6-$C$1,0)
=SUMPRODUCT(D5:D6,E5:E6)
=C8/(1+C2)
H
Stock’s Growth Rate
In this example, the factors that influence the option price are
Current stock price ($40)
Two values of stock price in next period ($50, $32)
Risk free interest rate (1/9)
Exercise price of the call ($40)
How about the probability that the stock will go up or down?
The current price of the stock theoretically incorporates
information about the stock’s growth rate.
One benchmark: risk-neutral probability implied by the actual
option price.
Risk-Neutral Probability
Example: A stock is currently selling for $40. One period from now the
stock will either increase to $50 with probability 1 - q or decrease in
price to $32 with probability q. The risk free rate of interest is 1/9.
The arbitrage-free price for a European call option with an exercise
price of $40 is $56/9 ≈ $6.22. Assume this is the correct discounted
expected payoff.
Expected payoff of the option: $10 * (1 - q) + $0 * q = $10 * (1 - q)
Set the discounted expected payoff of the option equal to $56/9
$10 * (1 - q) / (1 + 1 / 9) = $6.22
Risk-Neutral Probability
$10 * (1 - q) / (1 + 1 / 9)
(1 - q) / (1 + 1 / 9)
1-q
q
= $6.22
= $6.22 / $10
= $0.622 * 1.1111
= 0.30865
Here q = 0.30865 = 25/81 is called the Risk-Neutral Probability.
Expected stock price next period: (($50*0.69135+$32*0.30865) = $44.44
Expected growth rate of the stock: ($44.44/$40)-1 = 1/9
American Option Pricing: Binomial Lattice
The Black-Scholes option pricing formula for pricing European puts
and calls was derived under the assumption that future stock prices
follow a lognormal distribution (continuous)
St  S 0 exp[ t   t Normal(0,1)]
where 𝑆0 is the initial stock price, v be the expected yearly growth rate,
and σ be the standard deviation of yearly growth rate.
To price American options, we need to approximate stock prices in
discrete time and work backwards from the option’s expiration date
(dynamic programming).
The trick is to approximate a continuous time stochastic process with a
discrete time stochastic process. One popular approximation is
binomial lattice.
Multi-period Binomial Lattice
Define a basic period length Δt (such as a day or a week).
If the price is known at the beginning of a period, the price at the
beginning of the next period is one of only two possible values.
Usually, these two possibilities are defined to be multiples of the price
at the previous period
• A multiple u > 1 (for up) with probability p
• A multiple d < 1 (for down) with probability q = 1 - p
• The parameters u, d, p are independent of the price at the
beginning of the period
If the price at the beginning of a period is S, then the new price is
• 𝑢 × 𝑆 with probability p
• 𝑑 × 𝑆 with probability q = 1 - p
Period 0
S
Binomial Lattice
Period 1
Period 2
Period 3
uS
u2S
u 3S
dS
duS
du 2 S
d 2S
d 2uS
For spreadsheet layout
Up move: directly to the right
Down move: right and down one step
d 3S
Risk Neutral Approach
When applying risk neutral approach, we set
u  e
ae
t
, d  e 
t
r f t
ad
p
,
ud
𝑟𝑓 is the risk free rate.
q  1 p
 1/ u
American Option Pricing: Example
Let’s price a 5-month American put option having
• Current stock price =$50
• Exercise price =$50
• Risk-free rate =10%
• Annual volatility =40%
• Δt=1 month=0.083 years
A
1
2
3
4
5
6
7
B
American Put Option
S
Current Price
K
Strike Price
r
sigma
t
delta t
C
$
$
50.00
50.00
0.1
0.4
0.4167
0.0833
D
=5/12
=1/12
A
1
2
3
4
5
6
7
8
9
10
11
12
American Put Option
S
K
r
sigma
t
delta t
u
d
a
p
q
B
Current Price
Strike Price
Up Multiplier
Down Multiplier
Prob. Up
Prob. Down
C
$
$
50.00
50.00
0.1
0.4
0.4167
0.0833
1.1224
0.8909
1.0084
0.5073
0.4927
D
E
F
 t
ue
d  e 
=EXP(C5*SQRT(C7))
=1/C8
=EXP(C4*C7)
=(C10-C9)/(C8-C9)
=1-C11
t
ae
ad
p
ud
q  1 p
 1/ u
r f t
Generating stock prices in next five periods:
A
8
9
10
11
12
13
14
15
16
17
18
19
20
21
u
d
a
p
q
Down Moves
B
C
Up Multiplier
Down Multiplier
Prob. Up
Prob. Down
D
E
F
G
H
I
1.1224
0.8909
1.0084
0.5073
0.4927
Stock Prices
0
1
0 $ 50.000 $ 56.120
1
$ 44.547
2
3
4
5
-
2
3
4
5
=$C$8*C16
$ 62.989 $ 70.699 $ 79.353 $ 89.066
=IF($B17<=D$15,($C$9/$C$8)*D16,"-")
$ 50.000
$ 39.689
-
$ 56.120
$ 44.547
$ 35.361
-
$ 62.989
$ 50.000
$ 39.689
$ 31.505
-
$ 70.699
$ 56.120
$ 44.547
$ 35.361
$ 28.069
Expected discounted value of future cash flows from the put
• At month 5, the option is just worth
Max(0, exercise price - stock price at month 5)
A
14
15
16 Down Moves
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
B
C
D
E
F
G
H
2
$ 62.989
$ 50.000
$ 39.689
-
3
$ 70.699
$ 56.120
$ 44.547
$ 35.361
-
4
$ 79.353
$ 62.989
$ 50.000
$ 39.689
$ 31.505
-
5
$ 89.066
$ 70.699
$ 56.120
$ 44.547
$ 35.361
$ 28.069
0
1
2
3
4.489 $ 2.163 $ 0.636 $
$ 6.960 $ 3.771 $ 1.302
$ 10.362 $ 6.378
$ 14.639
-
4
5
Stock Prices
0
1
0 $ 50.000 $ 56.120
1
$ 44.547
2
3
4
5
-
I
Put Value
0
1
2
3
4
5
$
-
$
$
$ 2.664
$ 10.311
$ 18.495
-
$
$
$
$ 5.453
$ 14.639
$ 21.931
=IF($B30<=H$24,MAX($C$3-H21,(1/(1+$C$4*$C$7))*($C$11*I30+(1-$C$11)*I31)),"-")
Optimal Policy via Conditional Formatting
How we do know whether to exercise or not if we are at certain state of a
stage (say in month 3, when the stock price is $44.55).
It is optimal to exercise the option at a stage (month) and state (price) if
and only if the value of the corresponding cell equals (exercise price stock price).
A
14
15
16 Down Moves
17
18
19
20
21
22
23
24
25
26
27
28
29
30
B
C
D
E
F
G
H
2
$ 62.989
$ 50.000
$ 39.689
-
3
$ 70.699
$ 56.120
$ 44.547
$ 35.361
-
4
$ 79.353
$ 62.989
$ 50.000
$ 39.689
$ 31.505
-
5
$ 89.066
$ 70.699
$ 56.120
$ 44.547
$ 35.361
$ 28.069
0
1
2
3
4.489 $ 2.163 $ 0.636 $
$ 6.960 $ 3.771 $ 1.302
$ 10.362 $ 6.378
$ 14.639
-
4
5
Stock Prices
0
1
0 $ 50.000 $ 56.120
1
$ 44.547
2
3
4
5
Put Value
0
1
2
3
4
5
$
-
$
$
$ 2.664
$ 10.311
$ 18.495
-
$
$
$
$ 5.453
$ 14.639
$ 21.931
Value a Lease of a Gold Mine
Goldco is trying to value a 10-year lease on a gold mine.
Each year up to 10,000 ounces of gold can be extracted at a cost of $200
per ounce.
The current price of gold is $400 and the future price of gold is uncertain.
Model the future price of gold with a binomial tree, assuming each year
that gold will increase by 20% or decrease by 10%.
The risk free rate is assumed to remain constant at 10%.
The price received for all gold extracted during a year is assumed to
equal the price of gold at the beginning of the year, but all cash flows
occur at the end of the year.
What is the value of the lease?
An Enhancement Option
Suppose that there is a possibility of enhancing the production rate by
purchasing a new mining machine and making some structural changes
in the mine.
(This option can be exercised any time before the lease is expired.)
This enhancement would cost $4 million but would raise the mine
capability by 25% to 12500 ounces per year, at a total operating cost of
$240 per ounce.
What is the value of this enhancement option?
Basic parameters, risk-neutral “up” probability
ad
p
ud
1
2
3
4
A
B
r
u
d
ounces per year
0.1
1.2
0.9
10000
C
D
Risk Neutral Prob
Cost
E
$
0.6667
200.00
F
G
=(1+B1-B3)/(B2-B3)
Gold prices after one period
1
2
3
4
5
6
7
8
9
10
A
B
r
u
d
ounces per year
0.1
1.2
0.9
10000
Gold Prices
0
1
2
3
$
0
400.00 $
$
-
C
D
Risk Neutral Prob
Cost
E
$
F
0.6667
200.00
1
2
3
=$B$2*B7
480.00 $
576.00 $
691.20 $
=IF($A8<=C$6,($B$3/$B$2)*C7,"-")
360.00 $
432.00 $
518.40 $
=IF($A9<=C$6,($B$3/$B$2)*C8,"-")
$
324.00 $
388.80 $
$
291.60 $
4
829.44
622.08
466.56
349.92
A
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
Gold Prices
0
1
2
3
4
5
6
7
8
9
10
Value Function
0
1
2
3
4
5
6
7
8
9
10
$
B
C
D
E
F
G
0
400.00 $
$
-
1
480.00 $
360.00 $
$
-
2
3
691.20
518.40
388.80
291.60
4
829.44
622.08
466.56
349.92
262.44
5
995.33
746.50
559.87
419.90
314.93
236.20
0
1
$ 24,074,547 $ 27,754,680 $
$ 17,936,647 $
$
-
576.00 $
432.00 $
324.00 $
$
-
$
$
$
$
$
-
2
3
31,221,057 $ 34,248,617 $
20,748,329 $ 23,252,253 $
12,894,277 $ 15,004,981 $
$ 8,821,155 $
$
-
4
36,531,660
25,221,115
16,738,206
10,376,024
5,609,762
$
$
$
$
$
$
-
$
$
$
$
$
$
-
5
37,660,608
26,350,063
17,867,154
11,504,972
6,733,336
3,172,343
H
$
$
$
$
$
$
$
-
$
$
$
$
$
$
$
-
6
1,194.39
895.80
671.85
503.88
377.91
283.44
212.58
6
37,092,764
26,234,640
18,091,047
11,983,353
7,402,582
3,967,004
1,448,845
I
$
$
$
$
$
$
$
$
-
$
$
$
$
$
$
$
$
-
7
1,433.27
1,074.95
806.22
604.66
453.50
340.12
255.09
191.32
7
34,115,541
24,343,230
17,013,996
11,517,071
7,394,377
4,302,357
1,983,342
437,213
J
$
$
$
$
$
$
$
$
$
-
$
$
$
$
$
$
$
$
$
-
8
1,719.93
1,289.95
967.46
725.59
544.20
408.15
306.11
229.58
172.19
8
27,800,322
19,982,473
14,119,086
9,721,546
6,423,391
3,949,774
2,094,562
703,153
36,497
'IF((number of down moves) <= (number of periods so far),
(1/(1 + r)*MAX((gold price - extraction cost)*ounces per year, 0)
+ (expected lease value next period) * (1/(1 + r)))
,"-")
Assuming this is a scenario that actually could happen,
Discounted lease profit this period or zero, whichever is greater
+ Discounted expected lease value next period
K
$
$
$
$
$
$
$
$
$
$
-
$
$
$
$
$
$
$
$
$
$
-
9
2,063.91
1,547.93
1,160.95
870.71
653.03
489.78
367.33
275.50
206.62
154.97
9
16,944,656
12,253,946
8,735,914
6,097,390
4,118,497
2,634,328
1,521,200
686,355
60,221
-
L
$
$
$
$
$
$
$
$
$
$
$
10
2,476.69
1,857.52
1,393.14
1,044.86
783.64
587.73
440.80
330.60
247.95
185.96
139.47
10
$
$
$
$
$
$
$
$
$
$
$
-
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
A
B
r
u
d
ounces per year
0.1
1.2
0.9
10000
Gold Prices
0
1
2
3
4
5
6
7
8
9
10
Value Function
0
1
$
0
400.00 $
$
-
C
D
E
Risk Neutral Prob
Cost
1
480.00 $
360.00 $
$
-
$
2
576.00 $
432.00 $
324.00 $
$
-
F
G
H
I
0.6667
200.00
3
691.20
518.40
388.80
291.60
$
$
$
$
$
-
4
829.44
622.08
466.56
349.92
262.44
$
$
$
$
$
$
-
5
995.33
746.50
559.87
419.90
314.93
236.20
$
$
$
$
$
$
$
-
6
1,194.39
895.80
671.85
503.88
377.91
283.44
212.58
$
$
$
$
$
$
$
$
-
7
1,433.27
1,074.95
806.22
604.66
453.50
340.12
255.09
191.32
0
1
2
3
4
5
6
7
=IF($A20<=B$19,(1/(1+$B$1)*MAX((B7-$E$2)*$B$4,0)+($E$1*C20+(1-$E$1)*C21)*(1/(1+$B$1))),"-")
$ 24,074,547 $ 27,754,680 $
31,221,057 $ 34,248,617 $ 36,531,660 $ 37,660,608 $ 37,092,764 $ 34,115,541
$ 17,936,647 $
20,748,329 $ 23,252,253 $ 25,221,115 $ 26,350,063 $ 26,234,640 $ 24,343,230