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Version 1/9/2001
FINANCIAL ENGINEERING:
DERIVATIVES AND RISK MANAGEMENT
(J. Wiley, 2001)
K. Cuthbertson and D. Nitzsche
Lecture
Real Options
Copyright K. Cuthbertson, D.Nitzsche
Topics
Basic Concepts
Valuation of real options using BOPM
Extension of Tree to Many Periods
Valuation of Internet Company using continuous time
method
Copyright K. Cuthbertson, D.Nitzsche
Basic Concepts
Copyright K. Cuthbertson, D.Nitzsche
OPTIONALITY
Conventional NPV is ‘passive’
Black Gold and Crude Hole - oilfields
- both have negative NPV taken separately
- call option to expand, with strike= additional investme
Option to Abandon
- BMW purchase of Rover
- put option to sell off assets
Option to defer
- don’t build plant today, wait
- American call with strike = investment cost
- Is the call worth more ‘alive’ (ie. Postpone start) or ‘d
that is exercise now (pay K and collect revenues)
Copyright K. Cuthbertson, D.Nitzsche
OPTIONALITY
Table 19.1 : Similarities Between Financial and Real Options
Param Financial Option
Real Option
S
Stock price
Present value of expected cashflows
K
Strike price
Investment cost
R
Riskless interest rate
Riskless interest rate

Share-price volatility
Volatility of project cashflows
 = T-t Time-to-maturity
Time until oppnity to invest disappears
Copyright K. Cuthbertson, D.Nitzsche
‘Drivers’ of NPV and real options
Conventional NPV
PV {Fixed costs}
NPV
PV {Expected cash flows}
Payoffs = Max { PV - I0 , 0}
Real Options
Cash flow volatility
PV {Fixed costs}=‘strike’
Interest rate
Time to maturity
RO
PV {Expected cash flows}
Value lost over option’s life
Copyright K. Cuthbertson, D.Nitzsche
Valuation of real options
using BOPM
Copyright K. Cuthbertson, D.Nitzsche
Steps in Valuation (BOPM)
Measure the volatility of the value of the firm - from observed
stock prices
Assume this represents ‘outcomes’ SU or SD for existing proje
without ‘optionality’, in the investment decision (ie.‘passive’)
= conventional NPV
Apply the (abandonment) option and get new payoffs for the ‘t
including optionality’ . Discount, using risk neutral valuation to
obtain ‘adjusted NPV including optionality’, then
adjusted NPV* = conventional NPV + value of option
Copyright K. Cuthbertson, D.Nitzsche
‘Share Price’ Tree
Calculate discount rate k, from observed share price
U= 2, D = 1/U = 0.5
pu =0.5
S0=18
£ 36 =SU
pu = ‘real world probability
£ 9=SD
What is the cost of capital, k ?
18 = [0.5 x 36 + 0.5 x 9] / (1+k)
(1+k) = [0.5 x (36) + 0.5 x (9)] /18
Hence k = 25%
Discounting all future outcomes (SU, SU2 etc) using k,
will always ‘reproduce’ the current value of the stock S0 (firm)
Copyright K. Cuthbertson, D.Nitzsche
‘Baseline/existing’ projects
Assume projects, without any optionality have the same risk as
the firm’s existing projects, so U = 2, p = 0.5 and k is the
appropriate discount rate (with no optionality/passive).
Risk Neutrality
Surprisingly:
Discounting the above outcomes SU and SD using risk free rate
r = 5.25% (ignore contin. comp) and q = (R - D)/(U-D) = 0.3693
will also give the same value for S0 (=18):
S0 = [ q SU + ( 1- q) SD ]/R = 18
where R = (1+r).
Thus:
Instead of discounting the risky outcomes SU,SD using real
probabilities p, we can use q ,and then discount using the risk
free rate
Copyright K. Cuthbertson, D.Nitzsche
NPV of (Simple-One Period) Abandonment Option
Calculate NPV of the ‘abandonment option’, by
1) using ‘real world’ probabilities p and discounting the payoffs
using the cost of capital k (‘decision tree analysis’ DTA)
2) using ‘risk neutral’ probabilities q and discounting the payoffs
using the risk free rate, r
We find the two NPV are different
The DTA approach is incorrect because it uses the discount rate
k, which is calculated from the payoffs without any optionality.
This discount rate is too large, given the lower riskiness of
outcomes with the abandonment option
The risk neutral approach gives the correct answer for the NPV
(with optionality)
Copyright K. Cuthbertson, D.Nitzsche
1) One period Abandonment Option- using DTA (ie. k and p)
Investment cost (at t=0 ) = 105,
U=2, k = 25%, p = 0.5
Note: ‘Baseline’ tree is perfectly correlated with ‘share price’
tree, since U=2.
Baseline value of firm
180= Su
S0= 90
Abandon for K = 100 at t=1
180 = Vu
V0 = ??
45=Sd
S0 = [ pu Su + (1-pu) Sd ]/(1+k)
Conventional NPV
= 90 - 105 = -15
100 = Vd
Payoff = max {ST , K }
Discount ‘optionality’ using k and p
V0 = [ pu Vu + (1-pu) Vd ]/(1+k) = 112
NPV* = 112 - 105 = 7
Note: Spread of outcomes with ‘optionality’ is less than
‘baseline’
Copyright K. Cuthbertson, D.Nitzsche
2) NPV Abandonment Option - using RNV
Investment cost(at t=0 ) =110 and r = 5.25%,U= 2, q = 0.3693
Baseline value of firm
Abandon for K = 100 at t=1
180
90
180
??
45
Conventional NPV = -15
100
Payoff = max {ST , K }
V0 = [ q Vu + (1-q) Vd ]/R = 123
NPV* = 123 - 105 =18
Value of abandonment option = 123 - 90 = 33
Copyright K. Cuthbertson, D.Nitzsche
NPV Abandonment Option - using RNV
If, using RNV gives the correct value for the project (with
optionality) , (ie. V = 122.89)
then
what is the implicit discount rate k* applicable in the ‘real
world’ (ie using the real world probabitily, p) ?
123 = [ pu Vu + (1-pu) Vd ]/(1+k* )
= [0.5 x 180 + 0.5 x 100] / (1+k* )
Hence
k* = 13.8%
Why is k* = 13.8% less than k = 25% ? And which is correct ?
Copyright K. Cuthbertson, D.Nitzsche
NPV Abandonment Option - Comparison
Baseline/existing projects (no optionality), has V = 90
Abandonment using k=25%,
V = 112
Abandonment using options theory
V= 123 (and
k*=13.8%)
Valuation using k=25% understates the value of the
project with optionality, since the latter has less risky
outcomes
and should be discounted at a lower rate than ‘existing’
projects
RNV implicitly uses the ‘correct’ (lower) discount rate of
k*=13.68%
Copyright K. Cuthbertson, D.Nitzsche
Where does the ‘baseline’ tree come from?
1) Assume the ‘baseline’ project is ‘scale enhancing’
so its risk is the same as the riskiness of all the firms
existing projects
2) Then we can use the (annual) volatility of the firms
stock returns  as a measure of ‘risk’ of the baseline
project.
3) Then take U = exp[ x sqrt(dt) ] and D = 1/U
4) The risk neutral probabilities q can be shown to
be:
q = (R - D) / ( U-D)
- these are use to value the project with optionality
Copyright K. Cuthbertson, D.Nitzsche
Summary: Optionality and NPV
1) Work out the ‘tree’ for project (ie. SU, SD etc)
without optionality
Value of project without optionality (S0) equals the PV
of value in ‘up’and ‘down’ states(SU, SD etc), using
either p and k or RNV (ie. q and r ) ~ both give S0
2) Reconstruct the tree including optionality
Obtain the ‘adjusted’ PV of project by working back
through this tree using ‘risk neutral valuation’
Copyright K. Cuthbertson, D.Nitzsche
When is it worth using real options?
When conventional NPV is close to zero
When there is great uncertainty in future outcomes
-options have more value the higher is 
When project has a long life
- options have higher value as T increases
Copyright K. Cuthbertson, D.Nitzsche
Extension of Tree to Many Periods
3
Copyright K. Cuthbertson, D.Nitzsche
Figure 19.5 : Evolution of ‘baseline’ valuation
U= 1.9937, q = 0.3693
I0 = 105.4 (or 105 rounded)
713.2
357.7
179.4
179.4
90
90
45.1
45.1
22.6
11.4
0
1
2
Copyright K. Cuthbertson, D.Nitzsche
3
Time
Fig 19.6 : Option to expand (at t=3, only)
Payoff= Max{1+e)S3 - I3, S3 }
e=50%, I3 = 45
European Call
1024.8
493.9
224.1
236.6
105.7
V0* = 113.4
45.1
50.7
22.6
11.4
0
1
2
Copyright K. Cuthbertson, D.Nitzsche
3
Time
19.7 : Contraction option (at t=1 only)
Original I0 =105, is invested 50 at t=0 and 58 at t=1- same PV
Option is to invest smaller amount I*=10 at t=1, with 55%
lower revenues
Payoff= Max{S1 - I1, (1-c%) S1- I* }
c%=55%,
I1 = 58,
I*=10
121.4
(Baseline Su= 179.4)
10.3
(Baseline Sd =45.1)
48.8
0
1
2
Copyright K. Cuthbertson, D.Nitzsche
3
Time
19.8 : American abandonment option
Payoff= Max{Si , A}
A=60
At T=3, only abandon on 2 lower nodes
Tree: Vt-1 = max { Vt rec, A }
where Vrec = [qVu + (1-q) Vd] /R
713.2
357.7
179.4
184.7
98.9
107.1
60.0 (was 45.1)
70.6
60
60.0 (was 11.4)
0
1
2
Copyright K. Cuthbertson, D.Nitzsche
3
Time
19.9 : Investment default option
At t=1 (only) can choose whether to invest or not
Payoff= Max{S1 - I1 , 0 }
I1 =58 and I0 = 50
121.4(was 179.4, now 179.4-58)
42.6
0
0
(was 45.1, now max{45.1-58,0 )
1
2
3
NPV* = 42.6 - 50 = -7.4
Value of inv. Default option = -7.4 - (-15.4) = 8
Copyright K. Cuthbertson, D.Nitzsche
Time
19.10 : Default on debt repayment
At t=3 (only) can default on debt of D3 = 91
Payoff= Max{S3 - D3 , 0 } D3 = 91 and I0 = 55
622.2
271.3
88.3
113.8
31.0
46.5
0
10.9
0
0
0
1
2
3
*
NPV = 46.5 - 55 = -8.5 (-8.9 in text)
Time
Value of inv. Default option = -8.5 - (-15.4) = 6.9 (6.5 in te
Copyright K. Cuthbertson, D.Nitzsche
19.11 :Investment default (t=1) and debt default(t=3)
At =1, Payoff= Max{S1 - I1 , 0 }
At t=3 Payoff= Max{S3 - D3 , 0 }
I1 =58 (ie.t=1 value of 55)
D3 =91 (ie. t=3 value of 50
622.1
271.3
88.3
55.8
31.0
19.6
0
0
0
0
NPV*
= 19.6
Value of COMBINED option = 19.6- (-15.4) = 35
Copyright K. Cuthbertson, D.Nitzsche
Time
COMBINED OPTIONS
Value of investment default option = 8
Value of debt default option = 6.9
Value of combined option = 35
Option values are not additive
Copyright K. Cuthbertson, D.Nitzsche
VALUATION OF INTERNET COMPANIES
Copyright K. Cuthbertson, D.Nitzsche
VALUATION OF INTERNET COMPANIES
Have value because of expansion option and option to set up
allied sites (option on an option)
Given the terminal value of the firm, the value today is
V = (1/R) E*( VT)
- this is risk neutral valuation
- use MCS to obtain alternative values for VT based on
assessment of revenues minus costs, in a stochastic enviro
- key parameters: mean growth of revenues and their volatili
rate of change of average growth rate, mark-up over costs
Copyright K. Cuthbertson, D.Nitzsche
VALUATION OF INTERNET COMPANIES
Stochastic representation of revenues and costs
Actual changes in revenues
Change in expected growth in revenues
(Risk-adjusted) process for revenue R is mean reverting:
dRt
 (  t  1 t )dt   t dz1*
Rt
dt  [k (    )  2t ]dt  t dz2*
Costs: fixed F and costs proportional to revenues, (+)R
Ct = F + ( + )Rt
( + ) = 94% ‘baseline case’, hence profit margin= 6%
The cashflow of the firm (ignoring taxes) is:
Copyright K. Cuthbertson, D.Nitzsche
VALUATION OF INTERNET COMPANIES
The cashflow of the firm (ignoring taxes) is:
Yt = Rt - Ct
And the change in cash balances dX are:
dX = -Yt dt
All cash is ‘retained’ and earns the risk-free rate, r
If cash balances fall to zero, firm is assumed to go bankrupt
Given the terminal value of the firm, the value today is
V = (1/R) E*( XT)
MCS generates values for error terms and gives different val
for XT
Copyright K. Cuthbertson, D.Nitzsche
VALUATION OF INTERNET COMPANIES
December 1999: Amazon.com = $5.5bn
Sensitivity Analysis
If the profit margin is reduced from 6% to 5%
then Amazon’s value falls from $5.5bn to $4.3bn.
Copyright K. Cuthbertson, D.Nitzsche
END OF SLIDES
Copyright K. Cuthbertson, D.Nitzsche