Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
28.1
Chapter 28
Real Options
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
28.2
An Alternative to the NPV Rule
for Capital Investments
? Define stochastic processes for the key
underlying variables and use risk-
neutral valuation
? This approach (known as the real
options approach) is likely to do a better
job at valuing growth options,
abandonment options,etc than NPV
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
28.3
The Problem with using NPV to
Value Options
? Consider the example from Chapter 10
? Suppose that the expected return required by
investors in the real world on the stock is 16%,What
discount rate should we use to value an option with
strike price $21?
Stock Price = $22
Stock price = $20
Stock Price=$18
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
28.4
Correct Discount Rates are
Counter-Intuitive
? Correct discount rate for a call option is
42.6%
? Correct discount rate for a put option is
–52.5%
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
28.5
General Approach to Valuation
? We can value any asset dependent on a
variable q by
– Reducing the expected growth rate of q by
ls where l is the market price of q-risk and
s is the volatility of q
– Assuming that all investors are risk-neutral
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
28.6
Extension to Many Underlying
Variables
? When there are several underlying
variable qi we reduce the growth rate of
each one by its market price of risk
times its volatility and then behave as
though the world is risk-neutral
? Note that the variables do not have to
be prices of traded securities
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
28.7
Estimating the Market Price of
Risk (equation 28.7,page 665)
r a t e f r e e-r i s k
t e r m-s h o r t t h e is and m a r k e t ; t h e on r e t u r n
e x p e c t e d t h e is r e t u r n ; sm a r k e t ' t h e of
v o l a t i l i t y t h e is m a r k e t ; t h e on r e t u r n s and
v a r i a b l e in c h a n g e s p e r c e n t a g e b e t w e e n
nc o r r e l a t i o ousi n s t a n t a n e t h e is w h e r e
by g i v e n is v a r i a b l e a of r i s k of p r i c e m a r k e t T h e
r
r
m
m
m
m
?
?
?
??
?
?
?l )(
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
28.8
Schwartz and Moon Have Applied the Real Options
Approach to Valuing Amazon.com
? They estimated stochastic processes for the
company’s sales revenue and its revenue growth
rate.
? They estimated the market prices of risk and other
key parameters (cost of goods sold as a percent of
sales,variable expenses as a percent of sales,fixed
expenses,etc.)
? They used Monte Carlo simulation to generate
different scenarios in a risk-neutral world.
? The stock price is the present value of the net cash
flows discounted at the risk-free rate.
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
28.9
Commodity Prices
? Futures prices can be used to define the
process followed by a commodity price
in a risk-neutral world.
? We can build in mean reversion and use
a process for constructing trinomial
trees that is analogous to that used for
interest rates in Chapter 23
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
28.10
Example (page 671)
A company has to decide whether to invest
$15 million to obtain 6 million barrels of oil at
the rate of 2 million barrels per year for three
years,The fixed operating costs are $6
million per year and the variable costs are
$17 per barrel,The spot price of oil $20 per
barrel and 1,2,and 3-year futures prices are
$22,$23,and $24,respectively,The risk-free
rate is 10% per annum for all maturities,
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
28.11
The Process for Oil
We assume that this is
d ln(S)=[q(t)-aln(S)] dt+?dz
Where a=0.1 and ?=0.2
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
28.12
Tree Assuming q(t)=0; Fig 28.1
E J
0,6 9 2 8 0,6 9 2 8
B F K
0,3 4 6 4 0,3 4 6 4 0,3 4 6 4
A C G L
0,0 0 0 0 0,0 0 0 0 0,0 0 0 0 0,0 0 0 0
D H M
- 0,3 4 6 4 - 0,3 4 6 4 - 0,3 4 6 4
I N
- 0,6 9 2 8 - 0,6 9 2 8
Node A B C D E F G H I
pu 0.1667 0.1217 0.1667 0.2217 0.8867 0.1217 0.1667 0.2217 0.0867
pm 0.6666 0.6566 0.6666 0.6566 0.0266 0.6566 0.6666 0.6566 0.0266
pd 0.1667 0.2217 0.1667 0.1217 0.0867 0.2217 0.1667 0.1217 0.8867
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
28.13
Final Tree for Oil Prices; Fig 28.2
E J
4 4,3 5 4 5,6 8
B F K
3 0,4 9 3 1,3 7 3 2,3 0
A C G L
2 0,0 0 2 1,5 6 2 2,1 8 2 2,8 5
D H M
1 5,2 5 1 5,6 9 1 6,1 6
I N
1 1,1 0 1 1,4 3
Node A B C D E F G H I
pu 0.1667 0.1217 0.1667 0.2217 0.8867 0.1217 0.1667 0.2217 0.0867
pm 0.6666 0.6566 0.6666 0.6566 0.0266 0.6566 0.6666 0.6566 0.0266
pd 0.1667 0.2217 0.1667 0.1217 0.0867 0.2217 0.1667 0.1217 0.8867
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
28.14
Valuation of Base Project; Fig 28.3
E J
4 2,2 4 0,0 0
B F K
3 8,3 2 2 1,4 2 0,0 0
A C G L
1 4,4 6 1 0,8 0 5,9 9 0,0 0
D H M
- 9,6 5 - 5,3 1 0,0 0
I N
- 1 3,4 9 0,0 0
Node A B C D E F G H I
pu 0.1667 0.1217 0.1667 0.2217 0.8867 0.1217 0.1667 0.2217 0.0867
pm 0.6666 0.6566 0.6666 0.6566 0.0266 0.6566 0.6666 0.6566 0.0266
pd 0.1667 0.2217 0.1667 0.1217 0.0867 0.2217 0.1667 0.1217 0.8867
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
28.15
Valuation of Option to Abandon; Fig 28.4
(No Salvage Value; No Further Payments)
E J
0,0 0 0,0 0
B F K
0,0 0 0,0 0 0,0 0
A C G L
1,9 4 0,8 0 0,0 0 0,0 0
D H M
9,6 5 5,3 1 0,0 0
I N
1 3,4 9 0,0 0
Node A B C D E F G H I
pu 0.1667 0.1217 0.1667 0.2217 0.8867 0.1217 0.1667 0.2217 0.0867
pm 0.6666 0.6566 0.6666 0.6566 0.0266 0.6566 0.6666 0.6566 0.0266
pd 0.1667 0.2217 0.1667 0.1217 0.0867 0.2217 0.1667 0.1217 0.8867
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
28.16
Value of Expansion Option; Fig 28.5 (Company Can
Increase Scale of Project by 20% for $2 million)
E J
6,4 5 0,0 0
B F K
5,6 6 2,2 8 0,0 0
A C G L
1,0 6 0,3 4 0,0 0 0,0 0
D H M
0,0 0 0,0 0 0,0 0
I N
0,0 0 0,0 0
Node A B C D E F G H I
pu 0.1667 0.1217 0.1667 0.2217 0.8867 0.1217 0.1667 0.2217 0.0867
pm 0.6666 0.6566 0.6666 0.6566 0.0266 0.6566 0.6666 0.6566 0.0266
pd 0.1667 0.2217 0.1667 0.1217 0.0867 0.2217 0.1667 0.1217 0.8867
28.1
Chapter 28
Real Options
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
28.2
An Alternative to the NPV Rule
for Capital Investments
? Define stochastic processes for the key
underlying variables and use risk-
neutral valuation
? This approach (known as the real
options approach) is likely to do a better
job at valuing growth options,
abandonment options,etc than NPV
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
28.3
The Problem with using NPV to
Value Options
? Consider the example from Chapter 10
? Suppose that the expected return required by
investors in the real world on the stock is 16%,What
discount rate should we use to value an option with
strike price $21?
Stock Price = $22
Stock price = $20
Stock Price=$18
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
28.4
Correct Discount Rates are
Counter-Intuitive
? Correct discount rate for a call option is
42.6%
? Correct discount rate for a put option is
–52.5%
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
28.5
General Approach to Valuation
? We can value any asset dependent on a
variable q by
– Reducing the expected growth rate of q by
ls where l is the market price of q-risk and
s is the volatility of q
– Assuming that all investors are risk-neutral
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
28.6
Extension to Many Underlying
Variables
? When there are several underlying
variable qi we reduce the growth rate of
each one by its market price of risk
times its volatility and then behave as
though the world is risk-neutral
? Note that the variables do not have to
be prices of traded securities
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
28.7
Estimating the Market Price of
Risk (equation 28.7,page 665)
r a t e f r e e-r i s k
t e r m-s h o r t t h e is and m a r k e t ; t h e on r e t u r n
e x p e c t e d t h e is r e t u r n ; sm a r k e t ' t h e of
v o l a t i l i t y t h e is m a r k e t ; t h e on r e t u r n s and
v a r i a b l e in c h a n g e s p e r c e n t a g e b e t w e e n
nc o r r e l a t i o ousi n s t a n t a n e t h e is w h e r e
by g i v e n is v a r i a b l e a of r i s k of p r i c e m a r k e t T h e
r
r
m
m
m
m
?
?
?
??
?
?
?l )(
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
28.8
Schwartz and Moon Have Applied the Real Options
Approach to Valuing Amazon.com
? They estimated stochastic processes for the
company’s sales revenue and its revenue growth
rate.
? They estimated the market prices of risk and other
key parameters (cost of goods sold as a percent of
sales,variable expenses as a percent of sales,fixed
expenses,etc.)
? They used Monte Carlo simulation to generate
different scenarios in a risk-neutral world.
? The stock price is the present value of the net cash
flows discounted at the risk-free rate.
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
28.9
Commodity Prices
? Futures prices can be used to define the
process followed by a commodity price
in a risk-neutral world.
? We can build in mean reversion and use
a process for constructing trinomial
trees that is analogous to that used for
interest rates in Chapter 23
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
28.10
Example (page 671)
A company has to decide whether to invest
$15 million to obtain 6 million barrels of oil at
the rate of 2 million barrels per year for three
years,The fixed operating costs are $6
million per year and the variable costs are
$17 per barrel,The spot price of oil $20 per
barrel and 1,2,and 3-year futures prices are
$22,$23,and $24,respectively,The risk-free
rate is 10% per annum for all maturities,
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
28.11
The Process for Oil
We assume that this is
d ln(S)=[q(t)-aln(S)] dt+?dz
Where a=0.1 and ?=0.2
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
28.12
Tree Assuming q(t)=0; Fig 28.1
E J
0,6 9 2 8 0,6 9 2 8
B F K
0,3 4 6 4 0,3 4 6 4 0,3 4 6 4
A C G L
0,0 0 0 0 0,0 0 0 0 0,0 0 0 0 0,0 0 0 0
D H M
- 0,3 4 6 4 - 0,3 4 6 4 - 0,3 4 6 4
I N
- 0,6 9 2 8 - 0,6 9 2 8
Node A B C D E F G H I
pu 0.1667 0.1217 0.1667 0.2217 0.8867 0.1217 0.1667 0.2217 0.0867
pm 0.6666 0.6566 0.6666 0.6566 0.0266 0.6566 0.6666 0.6566 0.0266
pd 0.1667 0.2217 0.1667 0.1217 0.0867 0.2217 0.1667 0.1217 0.8867
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
28.13
Final Tree for Oil Prices; Fig 28.2
E J
4 4,3 5 4 5,6 8
B F K
3 0,4 9 3 1,3 7 3 2,3 0
A C G L
2 0,0 0 2 1,5 6 2 2,1 8 2 2,8 5
D H M
1 5,2 5 1 5,6 9 1 6,1 6
I N
1 1,1 0 1 1,4 3
Node A B C D E F G H I
pu 0.1667 0.1217 0.1667 0.2217 0.8867 0.1217 0.1667 0.2217 0.0867
pm 0.6666 0.6566 0.6666 0.6566 0.0266 0.6566 0.6666 0.6566 0.0266
pd 0.1667 0.2217 0.1667 0.1217 0.0867 0.2217 0.1667 0.1217 0.8867
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
28.14
Valuation of Base Project; Fig 28.3
E J
4 2,2 4 0,0 0
B F K
3 8,3 2 2 1,4 2 0,0 0
A C G L
1 4,4 6 1 0,8 0 5,9 9 0,0 0
D H M
- 9,6 5 - 5,3 1 0,0 0
I N
- 1 3,4 9 0,0 0
Node A B C D E F G H I
pu 0.1667 0.1217 0.1667 0.2217 0.8867 0.1217 0.1667 0.2217 0.0867
pm 0.6666 0.6566 0.6666 0.6566 0.0266 0.6566 0.6666 0.6566 0.0266
pd 0.1667 0.2217 0.1667 0.1217 0.0867 0.2217 0.1667 0.1217 0.8867
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
28.15
Valuation of Option to Abandon; Fig 28.4
(No Salvage Value; No Further Payments)
E J
0,0 0 0,0 0
B F K
0,0 0 0,0 0 0,0 0
A C G L
1,9 4 0,8 0 0,0 0 0,0 0
D H M
9,6 5 5,3 1 0,0 0
I N
1 3,4 9 0,0 0
Node A B C D E F G H I
pu 0.1667 0.1217 0.1667 0.2217 0.8867 0.1217 0.1667 0.2217 0.0867
pm 0.6666 0.6566 0.6666 0.6566 0.0266 0.6566 0.6666 0.6566 0.0266
pd 0.1667 0.2217 0.1667 0.1217 0.0867 0.2217 0.1667 0.1217 0.8867
Options,Futures,and Other Derivatives,5th edition ? 2002 by John C,Hull
28.16
Value of Expansion Option; Fig 28.5 (Company Can
Increase Scale of Project by 20% for $2 million)
E J
6,4 5 0,0 0
B F K
5,6 6 2,2 8 0,0 0
A C G L
1,0 6 0,3 4 0,0 0 0,0 0
D H M
0,0 0 0,0 0 0,0 0
I N
0,0 0 0,0 0
Node A B C D E F G H I
pu 0.1667 0.1217 0.1667 0.2217 0.8867 0.1217 0.1667 0.2217 0.0867
pm 0.6666 0.6566 0.6666 0.6566 0.0266 0.6566 0.6666 0.6566 0.0266
pd 0.1667 0.2217 0.1667 0.1217 0.0867 0.2217 0.1667 0.1217 0.8867
