Option Pricing Models: Theoretical Justification For all possible values of the underlying asset at expiration: (i)Calculate the option payoff at that

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: max[0, 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

100

102

98

104

100

96

106 c = 6 prob = 1/8

102 c=2 prob = 3/8

98 c=0 prob = 3/8

94 c=0 prob = 1/8

call value = (6 * 1/8 + 2 * 3/8) / (1.02)3

= 1.41

WEMBA 2000 Real Options 37

Note: error on handouts

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 lognormallydistributed (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

returns

Frequencyof returns


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 CashflowPosition Time = 0 Time = T

Buy Stock -S0 ST

Borrow PV(K) -K

Net: Portfolio 1 PV(K) - S0 ST - K

Portfoliopayoff

at time T

ST

K

Payoff from borrowing

Payoff from borrowing

Payoff from stock

net payoff

-K

S

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 = TPosition Time = 0 ST < K ST > K

Buy Call - c 0 ST - K

Sell Put p - (K - ST ) 0

Net: Portfolio 2 p - c ST - K ST - K


Portfoliopayoff

at time T

STK

Payoff on short put

Payoff onlong call

net payoff

-K



Payoff from Portfolio 1 and Portfolio 2 is the same, regardless of level of ST , hence costof 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 hasa 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 -0.05*0.25

= 13.12Left-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".


Valuation of Options: Upper and Lower bounds

Upper Bounds:

c S p Ke -rT

Today Value Time = TPosition ST < K ST > K

Sell Call c 0 -ST + K

Buy Stock -S ST ST

Net: c - S 0 ST 0 K 0

Today Value Time = TPosition ST < K ST > 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 > S - K e-rT

p > K e-rT - S


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 > S - K ?A: See lower bound on previous slide: c > S - K e-rT => c > S - K

Hence camer = ceuro on non-dividend-paying stocks


Options as Hedging Tools

Example 1: Portfolio Insurance

Value Time = TPosition ST < K ST > K

Own Portfolio ST ST

Buy Put (K - ST ) 0

Net: K ST

Portfoliopayoff

at time T

STK

Payoff on put

Payoff onportfolio

net payoff


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 thatthe 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 2*1.75 = $3.5 millionWith the call, she pays 2*1.6 = $3.2 million, plus $0.01 for the call: total = $3.21 millionThe 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 2*1.45 = $2.9 million plus $0.01 for the call: total = $2.91 million.


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.


0 1 2 3

100

64

100

156

80

125

100 105103Expected NetCash Flow

...

...

...

...

...

0.5

0.5

0.5

0.5

0.5

0.5

.044,1

10.1

1

10.0

105

10.1

103

10.1

100220 PV

Electricity Generator CaseWEMBA 2000 Real Options 47

• Now or never.

• Cost to build is 1,100.

• NPV=1,044 - 1,100 = -56.

• Negative NPV.

• Reject the project.

Electricity Generator Case

Case 1

• Now or never.


• NPV=1,044 - 1,000 = 44.

• Positive NPV.

• Accept the project.

Case 2

• 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

Case 3


0 1 2 3

64

100

156

80

125

128125Expected Net Cash Flowin “up” state PV = 1,277

...

...

...

...

...

8280Expected Net Cash Flowin “down” state PV = 818 ...

“up” state

“down” state

0.5

0.5

0.5

0.5


Case 3a


– proceed if “up state”

NPV=1277-1100=177

– reject if “down state”

NPV=0.

• Expected NPV today is:

• Compare with NPV without 1yr delay:

NPV without delay = - 56

Difference: 136

.80

10.1

05.01775.00

NPV

Case 3

Case 3b


– proceed if “up state”

NPV=1277-1000=277

– reject if “down state”

NPV=0.

• Expected NPV today is:

• Compare with NPV without 1 yr delay:

NPV without delay = 44

Difference: 82

.126

10.1

05.02775.00

NPV


Electricity Generator Case

• 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.

Case 4

100

800

100

156

80

125

...

...

Yearly CashFlows

One-time Liquidation

Value

• 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.


• 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.

Electricity Generator Case: Conclusions



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 thenext 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 impliesthat, for the purposes of the option model, you can sell all the oil that you extract at the spot priceas 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 preparethe drilling equipment for operation

A competitor, McKensey Oil, has offered Rigby $8 million for the drilling rights in their entirety.

Case Study: Rigby Oil

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 =



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!


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 financialoption)

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]



Caveats for using Financial Options Models on Real Options

Binomial pricing methods require the potential to buy or sell the underlying asset to create replicatingor riskless portfolio. [Note: it is not necessary to actually buy/sell the underlying asset. Options are pricedrelative 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

(a) Near perfect tracking (b) Imperfect tracking


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 themidwest 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 byfeeding some of your supply into the market. Alternatively, increased storage costsmay work against you if there is a short-term glut on a commodity. (The resultant pricedrop 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 thenew chip to market. (2) is market risk, and can be hedged (e.g. by purchasing options on otherSun workstation stocks, or other assets closely allied with the market for computer hardware). However, (1) is private risk, and cannot be hedged.


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 thatthe "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 natureof capital pricing decisions, and offers a better method for evaluatingprojects 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

Documents

Option Pricing Models: Theoretical Justification For all possible values of the underlying asset at expiration: (i)Calculate the option payoff at that