Option Pricing Models: Theoretical Justification

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WEMBA 2000 Real Options 37
Option Pricing Models Theoretical Justification
For all possible values of the underlying asset
at expiration (i) Calculate the option payoff
at that price (for a call max0,
S-K) (ii) Multiply the payoff by the probability
of that outcome (iii) Discount the
probability-weighted payoff at the riskless rate
of interest (iv) Add together all discounted
probability-weighted payoffs
Example
106 c 6 prob 1/8
104
102 c2 prob 3/8
102
100
100
98
98 c0 prob 3/8
96
94 c0 prob 1/8
Note error on handouts
call value (6 1/8 2 3/8) / (1.02)3
1.41
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WEMBA 2000 Real Options 38
The Lognormal Distribution of Asset Returns
Option pricing models assume that asset returns
are distributed lognormally If asset prices are
normally distributed, then returns are
lognormally distributed (mathematical
relationship) Empirically this has been shown
to be the case over the long-run Useful
characteristics of lognormal distribution (a)
returns cannot be negative (logarithms are never
negative) (b) volatility remains constant in
percentage terms
Frequency of returns
returns
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WEMBA 2000 Real Options 39
Valuation of Options Put-Call Parity
We construct two portfolios and show they always
have the same payoffs, hence they must cost the
same amount.
Portfolio 1 Buy 1 share of the stock today for
price S0 and borrow an amount PV(K) K e-rT How
much will this portfolio be worth at time T ?
Cashflow Cashflow Position Time 0 Time
T Buy Stock -S0 ST Borrow PV(K)
-K Net Portfolio 1 PV(K) - S0 ST - K
Portfolio payoff at time T
S
Payoff from stock
net payoff
K
ST
Payoff from borrowing
Payoff from borrowing
-K
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WEMBA 2000 Real Options 40
Valuation of Options Put-Call Parity
Portfolio 2 Buy 1 call option and sell 1 put
option with the same maturity date T and the same
strike price K. How much will this portfolio be
worth at time T ?
Cashflow Cashflow Time
T Position Time 0 ST lt K ST
gt K Buy Call - c 0 ST -
K Sell Put p - (K - ST )
0 Net Portfolio 2 p - c ST - K
ST - K
Portfolio payoff at time T
Payoff on long call
net payoff
K
ST
Payoff on short put
-K
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WEMBA 2000 Real Options 41
Valuation of Options Put-Call Parity
Payoff from Portfolio 1 and Portfolio 2 is the
same, regardless of level of ST , hence cost of
both portfolios (cashflows at time T 0 ) must
be the same. Hence S0 - PV(K) c -
p Put-Call Parity Rearranging c p S0 -
PV(K) (1)
Put-Call parity a worked example
Stock is selling for 100. A call option with
strike price 90 and maturity 3 months has a
price of 12. A put option with strike price 90
and maturity 3 months has a price of 2. The
risk-free rate is 5. Question Is there an
arbitrage? Test Put-Call parity Right-hand
side of (1) p S0 - PV(K) 2 100 - 90 e
-0.050.25
13.12 Left-hand side of (1) c 12
? 13.12 ! Market Price of c is too low
relative to the other three. Buy the call, and
Sell the "replicating portfolio".
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WEMBA 2000 Real Options 42
Valuation of Options Upper and Lower bounds
Upper Bounds c ? S p ? Ke -rT
Today Value Time
T Position ST lt K ST gt K Sell
Call c 0 -ST
K Buy Stock -S ST
ST Net c - S ? 0 ST ? 0 K
? 0
Today Value Time
T Position ST lt K ST gt
K Sell put p ST - K
0 lend money -Ke-T
K K Net p -
Ke-T ? 0 ST ? 0 K ? 0

Lower Bounds c gt S - K e-rT p gt K e-rT - S
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WEMBA 2000 Real Options 43
Early Exercise of American Options
Never optimal to exercise an American call (on a
non-dividend paying stock) early S 40 K
30 T 1 month (a) If you plan to hold the
stock beyond expiration then don't exercise
early (i) Earn 1 month interest on 30 (ii)
Purchase the stock at expiration if it is still
in-the-money (iii) If by chance it isn't in the
money, you have saved yourself K-ST (b) If you
plan to exercise and sell the stock immediately
You will earn S - K by exercising the option,
however. you should sell the option for c
instead Q How do you know c gt S - K ? A
See lower bound on previous slide c gt S - K e-rT
gt c gt S - K Hence camer ceuro on
non-dividend-paying stocks
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WEMBA 2000 Real Options 44
Options as Hedging Tools
Example 1 Portfolio Insurance
Value Time T Position ST lt K
ST gt K Own Portfolio ST
ST Buy Put (K - ST ) 0
Net K ST
Payoff on portfolio
Portfolio payoff at time T
Payoff on put
net payoff
K
ST
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WEMBA 2000 Real Options 45
Options as Hedging Tools
Example 2 Currency Hedging--A worked example
Polythene Providers Inc. has a global business
supplying polythene and other synthetic products
worldwide. The company's Treasurer, Pamela
Mann, has just been informed that Polythene
Providers Inc. may need to purchase supplies
from the UK in 2 months for 2 million, and is
concerned that the value of the pound may
appreciate against the dollar in the interim
period. So she purchases 2 million calls on
Sterling with a strike of 1.6 / (today's
exchange rate level), expiring in two months.
The call costs 10,800. If / appreciates above
1.6, Mann can purchase 2 million at the strike
of 1.6 for a cost of 3.2 mill. Suppose / is
1.75 in two months. Without the call, Mann
would have to pay 21.75 3.5 million With the
call, she pays 21.6 3.2 million, plus 0.01
for the call total 3.21 million The call has
saved 3.5 - 3.21 290,000 If / depreciates,
Mann will let the call expire worthless and
purchase 2 million at the market rate. Suppose
/ is 1.45 in two months. Mann pays 21.45
2.9 million plus 0.01 for the call total
2.91 million.
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WEMBA 2000 Real Options 46
Project Evaluation NPV vs. Real Option Valuation

• An electricity generator has the opportunity to
  build a new power project.
• Net cash flows are 100MM in the first year of
  operation.
• Net cash flows in the second year of operation
  depend upon whether an entrepreneurial link is
  built to bypass a transmission bottleneck.
• If the link goes ahead, demand for power from the
  new plant will be low and net cash flow will be
  80 mm.
• If the link does not go ahead, demand for power
  from the new plant will be high and net cash flow
  will be 125 mm.
• Similar uncertainty surrounds Year 3 net cash
  flows.
• Cash flows beyond Year 3 are perpetual.

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WEMBA 2000 Real Options 47
Electricity Generator Case
...
156
0.5
125
0.5
...
0.5
100
100
0.5
0.5
80
...
0.5
64
Expected Net Cash Flow
...
100
105
103
...
0
1
2
3
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WEMBA 2000 Real Options 48
Electricity Generator Case
Case 1
Case 2

• Now or never.
• Cost to build is 1,100.
• NPV1,044 - 1,100 -56.
• Negative NPV.
• Reject the project.

• Now or never.
• Cost to build is 1,000.
• NPV1,044 - 1,000 44.
• Positive NPV.
• Accept the project.

Case 3

• Option to delay for one year.
• During this one-year delay, the generator learns
  whether or not the new entrepreneurial link will
  proceed.
• Based on this additional information, a smarter
  decision can be made.
• Case 3a Cost to build is 1,100. Case 3b Cost
  to build is 1,000

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WEMBA 2000 Real Options 49
Electricity Generator Case
...
156
0.5
125
up state
...
0.5
100
0.5
80
down state
...
0.5
64
Expected Net Cash Flow in up state PV 1,277
...
128
125
Expected Net Cash Flow in down state PV 818
...
82
80
...
0
1
2
3
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WEMBA 2000 Real Options 50
Electricity Generator Case
Case 3

• Case 3a
• Cost to build is 1,100.
• proceed if up state
• NPV1277-1100177
• reject if down state
• NPV0.
• Expected NPV today is
• Compare with NPV without 1yr delay
• NPV without delay - 56
• Difference 136

• Case 3b
• Cost to build is 1,000.
• proceed if up state
• NPV1277-1000277
• reject if down state
• NPV0.
• Expected NPV today is
• Compare with NPV without 1 yr delay
• NPV without delay 44
• Difference 82

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WEMBA 2000 Real Options 51
Electricity Generator Case
Case 4

• Plant can be abandoned at any time for 800. Cost
  of building plant is 1000.
• This option will be exercised whenever the PV of
  future cash flows falls below 800.
• This only happens at the lowest node, where
  perpetual cash flows are 64.

• When the abandonment option is incorporated, the
  NPV of building the project now is 77.
• The NPV of waiting for one year is 126.
• It is still optimal to delay for one year in this
  case, although the incremental value of delaying
  has decreased.
• The value of the option to delay is lower if it
  is easy to exit a bad investment.

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WEMBA 2000 Real Options 52
Electricity Generator Case Conclusions

• The option to delay can be valuable, even if the
  project has positive NPV if started immediately.
• The value of these options is ignored by standard
  DCF techniques.
• Proper analysis of these options is needed not
  just for project valuation, but also for project
  timing.

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WEMBA 2000 Real Options 53
Case Study Rigby Oil
Rigby Oil owns the drilling rights for a small
oil field in the North Sea. A drilling platform
has been constructed, but extraction has not yet
commenced. Rigby owns the drilling rights for
the next five years. We have the following
information The current spot price of oil is
28 per barrel The annualized standard
deviation of percentage changes in the price of
oil is 40 p.a. The 3 month government bond
rate is 6.00 p.a. and the 10yr government bond
rate is 6.5 The estimated oil reserve in
Rigby's oil field is 1.2 million
barrels Extraction can proceed at the rate of
100,000 barrels per month The forward market
for oil is highly liquid hence oil can be sold
forward at fair value (which implies that, for
the purposes of the option model, you can sell
all the oil that you extract at the spot
price as of the day you begin extraction). The
existing drilling platform uses out-of-date
technology resulting in extraction costs of
25/barrel Before extraction can commence,
startup costs of 6 million will be required to
prepare the drilling equipment for operation
A competitor, McKensey Oil, has offered Rigby
8 million for the drilling rights in their
entirety.
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WEMBA 2000 Real Options 54
Case Study Rigby Oil (2)
Traditional NPV analysis Cashflows from
extraction - 6 (28 - 25) 1.2 -2.4
million reject Cashflow from selling the
lease 8 million accept?
Option-Adjusted Present Value analysis S
K T r ? BS call value
Option cost Option-adjusted
PV
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WEMBA 2000 Real Options 55
Case Study Rigby Oil (2)
Traditional NPV analysis Cashflows from
extraction - 6 (28 - 25) 1.2 -2.4
million reject Cashflow from selling the
lease 8 million accept?
Option-Adjusted Present Value analysis S
K T r ? BS call value
Option cost Option-adjusted
PV
33.6 (28/barrel for 1.2 MM barrels)
30 (25 extraction costs per barrel over
1.2 MM barrels)
4 (you need to start drilling in 4 years
if you are to complete extraction within 5 years)
6.25 (we need a 4-year rate. Try interpolating
between the 3 month and 10 year rates,
and test sensitivity of results to this
assumption)
40
15
6
9
Keep the Option!
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WEMBA 2000 Real Options 56
Case Study Rigby Oil (3)
Option to wait
Suppose the decision facing Rigby Oil were
changed as follows startup costs were only 3
million no option to sell the lease to the
competitor NPV analysis - 3 (28 - 25) 1.2
0.6 million Accept? No! Should not
exercise early! If you want to exercise the
call and immediately sell the underlying asset
(the oil), what should you do instead? Sell
the option! There may not be a buyer for the
lease at a fair market price (this is not a
liquid financial option) How do we earn the
fair market value of the option if there isn't a
buyer? HINT remember the "replicating
portfolio" method of valuing options
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WEMBA 2000 Real Options 57
Caveats for using Financial Options Models on
Real Options
Binomial pricing methods require the potential to
buy or sell the underlying asset to create
replicating or riskless portfolio. Note it is
not necessary to actually buy/sell the underlying
asset. Options are priced relative to the price
of the underlying security--relying on accurate
valuation of that security by the financial
markets.
What if the underlying asset when the company is
not publicly traded? Can we use alternative,
traded assets to proxy for the real underlying
asset? If so Tracking risk How
closely do they mimic the performance of the real
asset? Transactions costs It may be costly to
create and dynamically update a
replicating portfolio of assets
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WEMBA 2000 Real Options 58
Tracking Portfolio Risks
Basis Risk Northern Farms expands into organic
bread baking, and enters into a supply contract
on organic flour from midwestern flour mill.
Northern Farms hedges the flour price risk with
wheat options since flour is not a traded
commodity. Massive floods in the midwest take
out rail transportation lines. Cost of organic
flour increases substantially, while wheat prices
are relatively unaffected. Leakage Owning a
physical commodity may have benefits or costs
that do not accrue to owners of derivatives on
the commodity. If you own aluminum and there is
an unexpected price increase because of
shortages, you can earn a convenience yield
by feeding some of your supply into the market.
Alternatively, increased storage costs may work
against you if there is a short-term glut on a
commodity. (The resultant price drop is reflected
in the derivative value, but the increased
storage costs are not). Private Risk Sun King
Technology is considering a radical new design
for Sun workstation chips. They have two
concerns (1) whether the chip will be developed
in time and on budget, and (2) whether the market
demand for Sun workstations will be buoyant when
they bring the new chip to market. (2) is market
risk, and can be hedged (e.g. by purchasing
options on other Sun workstation stocks, or other
assets closely allied with the market for
computer hardware). However, (1) is private
risk, and cannot be hedged.
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WEMBA 2000 Real Options 59
Using Financial Options Models on Real Options
When can we use options models in the "real"
world? When the project outcomes closely mimic
the price performance of a liquid, traded
security whose returns are distributed
lognormally, so that the "replicating portfolio"
option pricing theory is justified Why do we
use financial options models in the "real"
world? Options theory provides insight into the
uncertain and changing nature of capital pricing
decisions, and offers a better method for
evaluating projects in the face of uncertainty
than traditional "static" models (such as
NPV) What are the principal differences between
the options approach and the NPV
approach? Options are more valuable when
projects are risky (i.e. cash flows are
volatile) Option theory enables us to use a
single, riskless discount rate throughout