Black and Scholes and Beyond

Professor XXXXXCourse Name / Number

2

Black-Scholes Call Options

Reasoning behind Black and Scholes is similar to that of the binomial model:

Does a combination of option and shares exist that provides a risk-free payoff?

Assumptions of the model

• Stock prices can move at every moment in time.

• Movements of stock prices are random, and therefore unpredictable.

• Volatility (standard deviation) of stock’s movements is known.

3

Calculating B-S Call Options

– S = current market price of underlying stock– X = strike price of option– t = amount of time before option expires (in years)– r = annual risk-free interest rate– = annual standard deviation of underlying stock’s

returns– e = 2.718– N(X) = probability of drawing a value less than or equal

to X from standard normal distribution

tdd

t

trX

S

d

12

2

1

2ln

C = SN(d1) – Xe-rt N(d2)

4

An Example

• Price of Stock Holm Inc. is currently $28– A European call option on Stock Holm Inc. has expiration

date three months in the future and strike price of $25.– Estimate of standard deviation on Stock Holm Inc. is 40%

and the risk-free rate is 3%.

5041.0

7041.04/14.0

41

24.0

03.02528

ln

12

2

1

tdd

d

What should the call price be?

N(0.7041) = 0.7593 N(0.5041) = 0.6929

C = 28(0.7593) – 25( 2.718-(.03)(0.25))(0.6929) = $4.07

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B-S Call Values for Stock Holm Inc.

0 15 28 45 600

5

10

15

20

25

30

35

40

45

50

Stock Price ($)

Val

ue o

f C

all O

ptio

n ($

)

Call Price Now Call Payoff at expiration

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Option’s Delta

Measures how much the call price changes as the underlying stock price changes:

– Delta is equal to the slope of the solid line in previous graph.– Delta equals the value N(d1) from B-S formula.– When call is far out of the money, delta is close to zero.– When call is deep in the money, delta is close to one.

1/N(d1) is equivalent to hedge ratio (h) from binomial model.

– Investors who own Stock Holm Inc. stock and want to create a hedged portfolio need to sell roughly 1/N(d1) = 1.316 calls for each share in their portfolio.

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• Replace PV(X) with Xe-rt and C with Black-Scholes equation.

Using B-S to Value Puts

Use put-call parity to determine the price of the put:

S + P = PV(X) + C

P = Xe-rt [1 – N(d2)] – S [1 - N(d1)]

Put option does not always sell for more than its intrinsic value, as can be observed on next slide.

– If stock price is very low, almost certain that option will be exercised at expiration.

– Amount by which put price falls below intrinsic value = time value of money investor loses by having to wait until expiration.

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B-S Put Values for Stock Holm Inc.

0 15 28 45 60 750

5

10

15

20

25

30

35

40

45

50

Stock Price ($)

Val

ue o

f P

ut O

ptio

n ($

)

Put Price Put Payoff

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Volatility

Can use historical data to estimate volatility:

• Use daily data if stock traded frequently.

• Otherwise, use weekly or monthly data.

Implied volatility

• From the observed price of an option in the market,we can compute implied volatility by “inverting” the B-S formula.

• Studies showed that out-of-the-money options have higher implied volatility than at-the-money options.

Speculative trading strategies seek to exploit differences in implied volatilities for a single

underlying asset.

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Corporate bondholders have a position equivalent to a portfolio of risk-free debt and a short put

option.

Options Embedded in Ordinary Corporate Bonds

Bond p

ayoff

at

matu

rity $25

0

Value of assets at maturity

0

Long Corporate Bond

$250

• If firm’s assets are not sufficient to repay its debt at maturity, the firm puts its assets to the bondholders.

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Options Embedded in the Stock of Levered Firm

Shareholders of a levered firm hold a call option.

Share

hold

ers

’ payoff

Value of assets at maturity

0$250

• If firm’s assets are worth more than the value of its debt, the shareholders keep the surplus.

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WarrantsSecurities issued by firms that grant the right to buy shares of stock at a fixed price for a given

period of time

Similar to call options, with some differences:

1. Warrants are issued by firms. Call options might be issued by other investors.

2. Number of outstanding shares increases when warrants are exercised. No new shares are issued when calls are exercised.

3. Warrants usually have longer maturities than options.4. Firms frequently attach warrants to other securities they

issue.The firm receives capital from warrants once when they are issued and again if they are exercised.

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Pricing Warrants

Use Black and Scholes model to price warrants.

Must take into account dilution that occurs when firms issue new shares to warrant holders:

where ,NN

N$C valueWarrant

21

1

• N1 : number of old shares outstanding• N2 : number new shares issued when warrants are exercised

• An example…– On April 12, 2002, UQM Technology issued 1,160,095

additional shares at $4.31/share.– For every five shares, investors received a warrant with

strike price of $5.73 and expiration date on April 12, 2004.

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Pricing Warrants

$1.52 019,322848,829,81

18,829,848$1.54 lueWarrant va

• On April 12, 2002, UQM Technologies had 18,829,848 common shares outstanding after issuing new shares.– Stock price = $4.40; st. deviation = 77%; risk-free rate = 2%

750.0772.0339.0

339.0277.0

2277.0

02.073.540.4

ln

12

2

1

tdd

d

N(0.339) = 0.633 N(-0.750) = 0.226

C = 4.40(0.633) – 5.73( 2.718-(.02)(2))(0.2263) = $1.54

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Convertibles

Convertible bonds grant the owner the option to convert the bonds for a certain number of shares.

Conversion ratio

• Number of shares bondholders receive if they convert

Conversion value

• Current stock price x conversion ratio• Provides lower bound on the market

value of convertible bond

• For example…– Amgen Inc. issued in Feb 2002, a 30-year, zero-coupon

convertible bonds with ytm of only 1.125%.– Price of convertible bond = $1,000/(1.01125)30 =

$714.9Investors accepted the low yield because they also received the option to convert each bond with face value of $1,000 into

8.8601 shares of Amgen.

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The Value of an Amgen Convertible Bond

Bon

d pr

ice/

Val

ue

Stock Price0

$714.9

$80.69

69.80$8.8601

$714.90 Price Conversion

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Real Options: Drilling Rights Example

Company analyzes whether it should bid for oil drilling rights for one year on a tract of land.

– Cost of extracting the oil is $18 per barrel.– Current and expected future price of oil is $16 per barrel.NPV approach gives a negative value for this project

What if we take into consideration the volatility of oil prices?

• Could use option models to determine the value of drilling rights: The right to drill is like a call option with exercise price of $18.– Assume volatility of the oil price is 30% and risk free rate is 4%.– Given this volatility, the firm can expect at some point the price

of oil could rise to make drilling profitable.

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Real Options: Drilling Rights Example

Initial approach: use B-S model to price the option

Bid value: call option value x number of barrels expected in this field

– B-S model is an oversimplification in this case: It assumes that the options can be exercised only at the end of the year.

– More accurate approach: Use binomial model to price the option.

409.030.0109.0

109.0130.0

1230.0

04.01816

ln

12

2

1

tdd

d

C = $1.40

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Real Options: McTreat Spot Example

Uncertainties related to this project:

– Level of sales the kiosk will achieve: Would costumers like to purchase McDonald's dessert products from the new locations?– The sensitivity of the McTreat Spot enterprise value to the overall economy

McDonalds evaluates the possibility of investing in McTreat Spot, small kiosks offering ice cream and

frozen yogurt.

What if McDonald’s launches program without testing?

McDonald’s invests in a few product test stores in prime locations.

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NPV of McTreat Spot

– Potential market is evaluated at 1,000 storefronts.– Cost for setting up each storefront is $750,000: total

investment of $750 million.– Rollout period is 2 years.

• Management estimates the following outcomes:– With 0.4 probability, the project is a hit and generates

free cash flow of $1.5 billion in two years.– With 0.6 probability, the project is a bust. PV of free

cash flows in this case is $500 million in two years.• Management assigns a 13% discount rate to this project.

Compute NPV of the project

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NPV of McTreat Spot

million $45.2- million $750 -million $704.8

million $7501.13

million $5000.6billion $1.50.4NPV 2

NPV of project

Immediate rollout would destroy shareholder value.

McDonald’s can learn the value of the 1,000 store chain before investing $750 million by using the test

program.

How much should the company spend for the test program?

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Test Program as a Call Option

Investing in test stores is similar to a call option:

– Underlying asset: the cash inflows from a 1,000-kiosks– Strike price: the $750 million needed to build the kiosk– McDonalds will exercise the option if the testing program

shows that value of the business is greater than $750 million..

Use binomial model to price the option:

– Underlying asset is always what you get if you exercised the option: In June, 2001, estimated value of McTreat Spot is $704.8 million.

– If value of the option is higher than the cost of the test stores (their negative NPV), invest in the test program.

– Assume risk free rate is 5%.

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Test Program as a Call Option

• Use risk-neutral method• Expected return on the underlying asset is the risk-free

rate.• Expected value of the asset in 2 years = $704.8(1.05)2 =

$777.0.• p = risk-neutral probability of an up movement• 1 - p = the probability of a down move

$704.8 million

$1.5 billion

$500 million

Probability = 0.4

Probability = 0.6

Real option payoff

$750 million

$0

June 2001 June 2003

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Test Program as a Call Option

• Expected payoff on underlying asset equals risk-free rate:• $1,500p + $500(1 – p) = $777• p = 0.277• 1 – p = 0.723.

• Use risk-neutral probabilities to calculate expected cash flows in 2003 from the call option.

• Expected cash flow from option = $750p + $0(1 – p)= $750(0.277) = $207.8

• PV of the call option = $207.8/(1.05)2 = $188.4

As long as the cost of the testing program is less than $188.4 million, McDonald’s should invest in it.

B-S formula can be used to value a European call on a stock that pays no dividends.

Of the five inputs of the model, only volatility is an unknown that analysts need to estimate.

Both debt and equity of a firm have option-like characteristics.

Capital investments with option-like characteristics may be valued using binomial

model or B-S model.

Black and Scholes and Beyond