Chap 6

Compound and Switching Options

Compound and Switching Options

Compound options are options whose value is contingent on other options.

Switching options allow the owner to start up and shut down operations, to switch from one mode of operation to another, or to enter and exit an industry.

Valuing compound options

Compound options are options whose value is contingent on the value of other options.

This type of compounding is called a simultaneous compound option because the equity and the call option on equity are alive simultaneously.

Compound options can also be sequential. Any type of phased investment fits this

category.

Methodology for simultaneous compound options

The key feature of “simultaneous” compound options is the underlying option and the option on it are simultaneous available.

They are not sequential in time. The solution proceeds in two steps.

First, we value the equity as an American call on the value of the firm with its exercise price equal to the face value of debt.

30

20

20

30

: [ ,0] [1,433.3415 800,0] [633.3415,0] 633.3415

: [ ,0] [1,127.4772 800,0] [327.4772,0] 327.4772

: [ ,0] [886.8820 800,0] [86.8820,0] 86.8820

: [ ,0]

A MAX u V X MAX MAX

B MAX u dV X MAX MAX

C MAX ud V X MAX MAX

D MAX d V X

[697.6281 800,0] 0MAX 2

0

30

20

(1 ) 86.8820 ,

[ (1 ) 0]

( ) 86.8820

f

f

mud V r B in the up state and

md V r B in the down state

md V u d

86.920.4591

786.63(0.2406)m

30 (1 ) 0

0.4591(697.6281) 1.08 0

fmd V r B

B

320.44

296.55651.08

B

20 0.4591(786.5916) 296.5565 64.5677md V B

0

1127.5 (1 ) 441.629 ,

[886.9 (1 ) 204.4490] ,

( ) 237.18

f

f

m r B in the up state node H and

m r B in the down state node I

mV u d

237.15

0.9858240.60

m

0 (1 ) 441.629

0.9857(1,127.50) 1.08 441.629

fmuV r B

B

(441.629 1111.3767)

620.24121.08

B

0 0.9858(1,000) 620.2412 365.5588mV B

Note that since the entity value of the firm was assumed to be $1,000, the market value of its risky debt must be $1,000 - $365.5588 = $634.4412.

2

2

2 2

(1 ) 233.3414 ,

[ (1 ) 0] ,

( ) 233.3414

f

f

mS A r B in the up state node A

mS B r B in the down state node B

m S A S B

233.3414 233.34140.7629

633.3414 327.4772 305.8642231.3572

0.7629*530.5156 231.3572 173.3991

m

B

Enode

mv B

2

2

2

(1 ) 173.3991 ,

[ (1 ) 0] ,

( 2 ) 173.3991

(530.5156 259.2391) 173.3991

f

f

mS E r B in the up state node E

mS F r B in the down state node F

m S E S F

m

173.39910.6392

271.2765m

2 (1 ) 0

0.6392(259.2391) 1.08 0

fmS F r B

B

165.7056

153.43111.08

B

1 0.6392(441.629) 153.4311 282.37 153.4311

128.8581

mS H B

1

1

1 1

(1 ) 128.8581 ,

[ (1 ) 0] ,

( ) 128.8581

f

f

mS H r B in the up state node H

mS I r B in the down state node I

m S H S I

128.8581 128.85810.5433

441.629 204.449 237.18m

1 (1 ) 0

0.5434(204.47) 1.08 0

fmS H r B

B

111.109102.8787

1.08B

0 0.5433(365.5588) 102.8787 198.6081 102.8787

95.7294

mS B Since both options are alive during the same

interval of time, the call contingent on the equity is a simultaneous compound option.

20

0

0

(1 ) 530.5156 ,

[ (1 ) 259.2391] ,

( ) 271.2765

f

f

mu V r B in the up state node E

mudV r B in the down state node F

muV u d

125.271

25.271m

20 (1 ) 530.5156

1(1,271.2563) 1.08 530.5156

fmu V r B

B

0

530.5156 1,271.2563 740.7407658.8710

1.08 1.081(1,127.50) 685.8710 441.629

B

muV B

0

20

0

(1 ) 259.2391 ,

[ (1 ) 64.5635] ,

( ) 194.6756

886.9(0.2406) 194.6756

f

f

mudV r B in the up state node F

md V r B in the down state node G

mdV u d

m

9123.039.213

67.194m

20 (1 ) 0

0.9123(786.5916) 1.08 64.5635

fmd V r B

B

0

653.044604.6690

1.080.9123(886.9) 604.6690 204.4499

B

mdV B

0

0

(1 ) 441.629 ,

[ (1 ) 204.4499] ,

f

f

muV r B in the up state node H

mdV r B in the down state node I

0.9858m 620.2412B

0 0.9858*1000 620.2412 365.5588mV B

0

441.629 1.08 41.629

[204.4499 1.08 0]

m=0.1755,B=-33.2193

mC 0.1755*365.5588 33.2193 30.9363

m B

m B

B

Review of compound options

With sequential compound options the order of economic priority is the opposite of the time sequence.

In our example, the compound option was worth $30.9363.

Valuing switching options

Switching options give their owner the right to switch between two modes of operation at a fixed cost.

Assume that a company already has a manufacturing facility operating that uses technology X.

Due to increased demand, we are considering a new factory with the following options : to use technology X again, to use alternative technology Y, or to invest in a flexible technology Z that allows us to switch from X to Y for $15 and from Y to X for $10.

These switching costs are designed Cxy and Cyx respectively.

The flexible technology Z requires a higher investment of $110.

XBX

XEXDXB FCF

WACC

PVpPVpPV

)1(

)1(

0.5 196 (1 0.5)100.0025140 135.7807 140 275.7807

(1 0.09)

291.0495 100 191.0495X X XNPV PV I

YDfXD PVBrmPV )1(

YBfXB PVBrmPV )1(

2187.010001.196

100121

XEXD

YEYD

PVPV

PVPVm

121 0.2187 196.0174.4059

1 1.05YD XD

f

PV nPVB

r

0.2187 135.7807 74.4059 110

104.107 110 214.1070YB XB YBPV mPV B FCF

284.2154 100 184.2154Y Y YNPV PV I

291.0495 100 191.0495X X XNPV PV I 191.0495 184.2154X YNPV NPV

We ask the same question twice : Assuming we have been in mode X at the

previous state, would we stay in X or would we switch to Y and pay the switching cost?

Assuming we have been in mode Y at the previous state, would we stay in Y or would we switch to X and pay the switching cost?

( , ) (196,121 15) 196 XD XD YD XYS MAX PV PV C MAX Stay in X

( , ) (121,196 10) 186 YD YD XD YXS MAX PV PV C MAX Switch to X

8958.0100196

100186

XEXD

YEYD

PVPV

SSm

186 0.8958 1969.9191

1 1.05YD XD

f

S mPVB

r

0.8958 135.7807 9.9191 110 241.5571fYB XB YBPV mPV B FCF

( , ) (275.7807,214.1070 15) 275.7807f fXB XB YB XYS MAX PV PV C MAX

Stay in X( , ) (241.5571,275.7807 10) 265.7807f f

YB YB XB YXS MAX PV PV C MAX

Switch in X

AY AX XAROA ROA PV

302.3096 110 192.3096ZX XA ZROA ROA I

306.0376 110 196.0376ZY YA ZROA ROA I

291.0495 100 191.0495ZY ZX XROA ROA NPV