copyright k. cuthbertson, d.nitzsche financial engineering: derivatives and risk management (j....

Copyright K. Cuthbertson, D.Nitzsche

FINANCIAL ENGINEERING:DERIVATIVES AND RISK MANAGEMENT(J. Wiley, 2001)

K. Cuthbertson and D. Nitzsche

Lecture

Real Options

Version 1/9/2001


Topics

Basic Concepts

Valuation of real options using BOPM

Extension of Tree to Many Periods

Valuation of Internet Company using continuous time method


Basic Concepts


OPTIONALITY

Conventional NPV is ‘passive’

Black Gold and Crude Hole - oilfields - both have negative NPV taken separately- call option to expand, with strike= additional investment

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 ‘dead’ that is exercise now (pay K and collect revenues)


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


‘Drivers’ of NPV and real options

NPV PV {Expected cash flows}PV {Fixed costs}

RO PV {Expected cash flows}PV {Fixed costs}=‘strike’

Interest rate Value lost over option’s life

Cash flow volatility Time to maturity

Conventional NPV

Real Options

Payoffs = Max { PV - I0 , 0}


Valuation of real options using BOPM


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 projects, without ‘optionality’, in the investment decision (ie.‘passive’) = conventional NPV

Apply the (abandonment) option and get new payoffs for the ‘treeincluding optionality’ . Discount, using risk neutral valuation toobtain ‘adjusted NPV including optionality’, then

adjusted NPV* = conventional NPV + value of option


‘Share Price’ Tree

S0=18

£ 36 =SU

£ 9=SD

U= 2, D = 1/U = 0.5pu =0.5

What is the cost of capital, k ?

18 = [0.5 x 36 + 0.5 x 9] / (1+k)

Hence k = 25%(1+k) = [0.5 x (36) + 0.5 x (9)] /18

Discounting all future outcomes (SU, SU2 etc) using k, will always ‘reproduce’ the current value of the stock S0 (firm)

pu = ‘real world probability

Calculate discount rate k, from observed share price


‘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 NeutralitySurprisingly: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


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)


1) One period Abandonment Option- using DTA (ie. k and p)

S0= 90

45=Sd

180= Su

Baseline value of firm

V0 = ??

100 = Vd

180 = Vu

Abandon for K = 100 at t=1

Investment cost (at t=0 ) = 105, U=2, k = 25%, p = 0.5

Conventional NPV= 90 - 105 = -15

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’

Payoff = max {ST , K }S0 = [ pu Su + (1-pu) Sd ]/(1+k)

Note: ‘Baseline’ tree is perfectly correlated with ‘share price’ tree, since U=2.


2) NPV Abandonment Option - using RNV

90

45

180

Baseline value of firm

??

100

180

Abandon for K = 100 at t=1

Investment cost(at t=0 ) =110 and r = 5.25%,U= 2, q = 0.3693

Conventional NPV = -15 Payoff = max {ST , K }

V0 = [ q Vu + (1-q) Vd ]/R = 123

NPV* = 123 - 105 =18

Value of abandonment option = 123 - 90 = 33


NPV Abandonment Option - using RNV

If, using RNV gives the correct value for the project (with optionality) , (ie. V = 122.89)thenwhat 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 ?


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%


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


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’


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


Extension of Tree to Many Periods

3


Figure 19.5 : Evolution of ‘baseline’ valuation

90

45.1

179.4

357.7

90

22.6

713.2

179.4

45.1

11.4

Time0 1 2 3

U= 1.9937, q = 0.3693I0 = 105.4 (or 105 rounded)


Fig 19.6 : Option to expand (at t=3, only)

V0* = 113.4

50.7

236.6

493.9

105.7

22.6

1024.8

224.1

45.1

11.4

Time0 1 2 3

Payoff= Max{1+e)S3 - I3, S3 } e=50%, I3 = 45

European Call


19.7 : Contraction option (at t=1 only)

48.8

10.3

121.4

Time0 1 2 3

Payoff= Max{S1 - I1, (1-c%) S1- I* }

c%=55%, I1 = 58, I*=10

Original I0 =105, is invested 50 at t=0 and 58 at t=1- same PVOption is to invest smaller amount I*=10 at t=1, with 55% lower revenues

(Baseline Sd =45.1)

(Baseline Su= 179.4)


19.8 : American abandonment option

107.1

70.6

184.7

357.7

98.9

60

713.2

179.4

60.0 (was 45.1)

60.0 (was 11.4)

Time0 1 2 3

Payoff= Max{Si , A} A=60

At T=3, only abandon on 2 lower nodesTree: Vt-1 = max { Vt rec, A } where Vrec = [qVu + (1-q) Vd] /R


19.9 : Investment default option

42.6

0 (was 45.1, now max{45.1-58,0 )

121.4(was 179.4, now 179.4-58)

Time0 1 2 3

Payoff= Max{S1 - I1 , 0 } I1 =58 and I0 = 50

At t=1 (only) can choose whether to invest or not

NPV* = 42.6 - 50 = -7.4

Value of inv. Default option = -7.4 - (-15.4) = 8


19.10 : Default on debt repayment

46.5

10.9

113.8

271.3

31.0

0

622.2

88.3

0

0

Time0 1 2 3

At t=3 (only) can default on debt of D3 = 91

Payoff= Max{S3 - D3 , 0 } D3 = 91 and I0 = 55

NPV* = 46.5 - 55 = -8.5 (-8.9 in text)Value of inv. Default option = -8.5 - (-15.4) = 6.9 (6.5 in text)


19.11 :Investment default (t=1) and debt default(t=3)

19.6

0

55.8

271.3

31.0

0

622.1

88.3

0

0

Time

At =1, Payoff= Max{S1 - I1 , 0 } I1 =58 (ie.t=1 value of 55)At t=3 Payoff= Max{S3 - D3 , 0 } D3 =91 (ie. t=3 value of 50)

NPV* = 19.6Value of COMBINED option = 19.6- (-15.4) = 35


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


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 environment

- key parameters: mean growth of revenues and their volatility, rate of change of average growth rate, mark-up over costs



Stochastic representation of revenues and costsActual changes in revenuesChange in expected growth in revenues(Risk-adjusted) process for revenue R is mean reverting:

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:

Yt = Rt - Ct

And the change in cash balances dX are:

dX = -Yt dt

All cash is ‘retained’ and earns the risk-free rate, rIf 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)

- this is risk neutral valuation

December 1999, the model gave a value for Amazon.com of $5.5bn,If the profit margin is reduced from 6% to 5% then Amazon’s value falls from $5.5bn to $4.3bn. - use MCS to obtain alternative values for VT based on assessment of revenues minus costs, in a stochastic environment - key parameters - mean growth and vol of revenues, rate of change of average growth rate, mark-up over costs

*11 )( dzdt

R

dRttt

t

t

*22 ])([ dzdtkd ttt



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, rIf cash balances fall to zero, firm is assumed to go bankruptGiven the terminal value of the firm, the value today is

V = (1/R) E*( XT)

MCS generates values for error terms and gives different valuesfor XT



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.


END OF SLIDES

copyright k. cuthbertson, d.nitzsche financial engineering: derivatives and risk management (j....

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