Simon Benninga. Financial Modelling 3-rd edition

Подождите немного. Документ загружается.

518 Chapter 19

is a monotonic function of the sigma. Here’s a Data|Table from our basic

Black-Scholes spreadsheet:

AB C D

S 45 Current stock price

X 50 Exercise price

T 1 Time to maturity of option (in years)

r 8.00% Risk-free rate of interest

Sigma 30.00% Stock volatility

Call price 4.88 <-- =BSCall(B2,B3,B4,B5,B6)

Data table: Call price as function of volatility

4.8759 <-- =B8, table header

15% 2.1858

16% 2.3646

17% 2.5437

18% 2.7229

19% 2.9023

20% 3.0817

21% 3.2612

22% 3.4407

23% 3.6202

24% 3.7997

25% 3.9792

26% 4.1587

27% 4.3381

Black-Scholes Option Price Is Monotonic in Sigma

Call Price and Volatility

1.80

2.10

2.40

2.70

3.00

3.30

3.60

3.90

4.20

4.50

15 18 21 24 27 30

Volatility, (%)

BS call price

We use VBA to deﬁ ne a function CallVolatility that ﬁ nds the σ for a

call option. The function is deﬁ ned as CallVolatility(Stock, Exercise,

Time, Interest, Target), where the deﬁ nitions are as follows:

Stock is the stock price S.

Exercise is the option’s exercise price X.

Time is the time to the option’s maturity T.

Interest is the interest rate r.

Target is the call price C.

The function ﬁ nds σ for which the Black-Scholes formula is equal to C.

519 The Black-Scholes Model

The technique used by the function is very similar to the technique

used in trial and error: We start with two estimates for the possible σ: a

High estimate of 100 percent and a Low estimate of 0 percent. We now

do the following:

•

Plug the average of the High and the Low into the Black-Scholes

formula. This gives us CallOption(Stock, Exercise, Time, Interest, (High

+ Low) / 2). (Note that the function CallVolatility assumes that the func-

tion CallOption is available to the spreadsheet.)

•

If CallOption(Stock, Exercise, Time, Interest, (High + Low) / 2) >

Target, then the current σ estimate of (High + Low) / 2 is too high, and

we replace High by (High + Low) / 2.

•

If CallOption(Stock, Exercise, Time, Interest, (High + Low) / 2) <

Target, then the current σ estimate of (High + Low) / 2 is too low and

we replace Low by (High + Low) / 2.

We repeat this procedure until the difference High-Low is less than

0.0001 (or some other arbitrary constant).

Here’s an example of this function, including a data table and a graph

that show the implied volatility as a function of the call price:

Function CallVolatility(Stock, Exercise, Time,

Interest, Target)

High = 1

Low = 0

Do While (High - Low) > 0.0001

If CallOption(Stock, Exercise, Time,

Interest, (High + Low) / 2) > Target

Then

High = (High + Low) / 2

Else Low = (High + Low) / 2

End If

Loop

CallVolatility = (High + Low) / 2

End Function

520 Chapter 19

19.6 Dividend Adjustments to the Black-Scholes

The Black-Scholes formula assumes that the option’s underlying security

pays no dividends prior to the exercise date T. In certain cases it is easy

to make an adjustment to the model for dividends. This section surveys

two such adjustments: We ﬁ rst look at option pricing when future divi-

dends are known with certainty, and we then examine option pricing

when the underlying security pays out a continuous dividend. The prin-

ciple underlying both cases is the same: The options are priced on an

adjusted underlying value that nets out the present value of dividends

paid between the option purchase date and the option exercise date.

FBA G

5S1.00

5X0.00

Interest 8.00%

Target call price 6.00

Implied call volatility 15.35% <-- =CallVolatility(B3,B4,B5,B6,B7)

Data Table: Implied volatility as a function of the call price

Call price

Implied

volatility

15.35% <-- =B9, table header

5.00

7.51%

5.50

11.96%

6.00

15.35%

6.50

18.45%

7.00

21.39%

7.50

24.25%

8.00

27.07%

8.50

29.84%

9.00

32.59%

9.50

35.33%

10.00

38.05%

10.50

40.77%

BLACK-SCHOLES IMPLIED VOLATILITY

The VBA module attached to this spreadsheet defines a function called

CallVolatility(S,X,T,interest,target_call_price)

. To use this function fill in the relevant rows (in boldface).

The cell labeled "Implied call volatility" contains the function.

Implied Call Volatility as Function of

Call Price

567891011

Call price

Volatility, s (%)

521 The Black-Scholes Model

19.6.1 A Known Dividend to Be Paid before the Option Expiration

It often happens that a stock’s future dividend is known at the time the

option is traded. This is most commonly the case when a dividend has

already been announced, but it can also happen because many stocks

pay quite regular and relatively inﬂ exible dividends. In this case the

option should be priced not on the current stock price S but on the stock

price minus the present value of the dividend or dividends anticipated

before the option expiration date T:

Here’s an example: Coca-Cola (stock symbol KO) pays quarterly divi-

dends in March, June, September, and November of each year. Dividends

(Figure 19.2) seem quite stable; on 28 July 2006, the last two dividends

were $0.31 per share.

Computing the implied volatility for the January 2007 calls and puts

on Coca-Cola shows that taking account of anticipated dividends makes

a signiﬁ cant difference in the pricing. We can also deduce from the prox-

imity of the prices that take dividends into account (cells B19:B20)

versus the distance between the prices that do not take the dividends

into account (cells B22:B23) that the former are correct.

Time 0

Time t

dividend

payment

Time T

option

expiration

Stock price = S Div

max[S

– X, 0]

Stock price minus

PV(dividend) =

S – Div*exp[–rt]

Figure 19.2

Yahoo data for Coca-Cola. Note the stability of the dividends. On 28 July 2006, the closing stock price was $44.52.

523 The Black-Scholes Model

19.6.2 Dividend Adjustments for Continuous Dividend Payouts: The Merton

Model

In the preceding subsection we looked at the case of known future divi-

dends. This subsection discusses a model due to Merton (1973) for pricing

options on a stock that pays continuous dividends. Continuous dividends

may seem an odd assumption. But a basket of stocks such as the S&P

500 index or the Dow-Jones 30 can best be approximated by the assump-

tion of a continuous dividend payout, since there are many stocks and

since the index components more or less pay out their dividends through-

out the year.

Assuming the continuous dividend yield of k, Merton proved the fol-

lowing call-option-pricing formula:

CSe Nd XeNd

kT rT

=−

−−

() ()

where

SX r k T

+−+ln / /2)()(

dd T

=−

EDCBA

Current date 28-Jul-06

Option expiration date 19-Jan-07

Current stock price 44.52

Interest rate 5.00%

Date

Anticipated

dividend

Present

value

Mid September 13-Sep-06 0.31 0.31 <-- =C8*EXP(-$B$5*((B8-$B$2)/365))

End November 29-Nov-06 0.31 0.30 <-- =C9*EXP(-$B$5*((B9-$B$2)/365))

Stock price net of PV(dividends) 43.91

-- =B4-SUM(D8:D9)

Exercise price, X 45.00 <-- Approximately at the money

Time to maturity, T 0.4795 <-- =(B3-B2)/365

Interest rate, r 5.00% Risk-free rate of interest

Call price 1.80 <-- Call price on 28jul06

Put price 1.85 <-- Put price on 28jul06

Implied volatility

Call, S net of dividends 14.95% <-- =CallVolatility(B11,B12,B13,B14,B15)

Put, S net of dividends 15.15% <-- =PutVolatility(B11,B12,B13,B14,B16)

Call, S with dividends 12.19% <-- =CallVolatility(B4,B12,B13,B14,B15)

Put, S with dividends 17.45% <-- =PutVolatility(B4,B12,B13,B14,B16)

PRICING THE COCA-COLA JAN07 CALLS AND PUTS

524 Chapter 19

This model is used in the next spreadsheet to price the exchange-

traded fund that tracks the S&P 500 index:

AB C

S 127.98 Current stock price

X 127.00 Exercise price

T 0.6329 <-- Option expires 16-Mar-07, today's date 28-Jul-06

r 5.00% Risk-free rate of interest

k 1.70% Dividend yield

Sigma 14% Stock volatility

0.3122 <-- =(LN(B2/B3)+(B5-B6+0.5*B7^2)*B4)/(B7*SQRT(B4))

0.2008 <-- =B8-B7*SQRT(B4)

N(d

)

0.6226

<--- Uses formula NormSDist(d

)

N(d

)

0.5796

<--- Uses formula NormSDist(d

)

Call price 7.51

<-- S*Exp(-k*T)*N(d

)-X*exp(-r*T)*N(d

)

Put price 3.94 <-- call price - S*Exp(-k*T) + X*Exp(-r*T): by Put-Call parity

3.94

<-- X*exp(-r*T)*N(-d

)-S*Exp(-k*T)*N(-d

): direct formula

Merton's Dividend-Adjusted Option-Pricing Model

Used Here to Price S&P 500 Spiders (symbol: SPY)

The Merton model is often used to price currency options. Suppose

we take an option on the euro. The option speciﬁ es a dollar exchange

rate for euros (in the following example, the call option lets us buy 10,000

euros in 0.0575 years for $1.285 per euro). The asset underlying the

option is a euro interest-bearing security with interest rate r

BACD

S 1.276 Current exchange rate: U.S. dollar price of one Euro

X 1.285 Exercise price

5.00% U.S. interest rate

€

5.50% Euro interest rate

T 0.0575 Time to maturity of option (in years)

Sigma 4.70% Euro volatility in dollars

-0.6095

<--(LN(S/X)+(r

-r

€

+0.5*sigma^2)*T)/(sigma*SQRT(T))

-0.6208

<-- d

- sigma*SQRT(T)

Number of Euros per call contract 10,000

N(d

)

0.2711

<--- Uses formula NormSDist(d

)

N(d

)

0.2674

<--- Uses formula NormSDist(d

)

Call price 23.69

<-- (S*Exp(-r

€

*T)*N(d

)-X*exp(-r

*T)*N(d

))*B11

Put price 112.23

<-- (X*exp(-r

*T)*N(-d

)-S*Exp(-r

€

*T)*N(-d

))*B11: direct formula

Pricing Currency Options

Intuition: The underlying asset of the

currency option is a euro. The euro

pays a dividend, which is the euro

interest rate. Therefore the Merton

model applies, with the underlying

asset price being S*exp(-r

€

*T), where

€

is the interest rate on euros. Note

also the change in d

, where r

-r

€

appears instead of r

as in the regular

Black-Scholes formula.

525 The Black-Scholes Model

19.7 Using the Black-Scholes Formula to Price Structured Securities

A “structured security” is Wall Street parlance for securities that incor-

porate combinations of stocks, options, and bonds. In this section we give

three examples of such securities and show how to price them using the

Black-Scholes model.

In the process we also return to the discussion of

Chapter 16, showing how the proﬁ t diagrams of option strategies can

help us understand such securities.

19.7.1 A Simple Structured Security: Principal Protection plus Participation in

Market Upside Moves

A simple and popular structured offers guaranteed return of the inves-

tor’s principal plus some participation in the upside moves of the market.

Here’s an example: Homeside Bank offers its customers the following

“principal-protected, upside potential” security (PPUP).

•

Initial investment in the security: $1,000.

•

No interest paid on the security.

•

In ﬁ ve years the PPUP pays back the $1,000 plus 50 percent of the

increase in the S&P 500 index. Writing the index price today as S

and the

index price in ﬁ ve years as S

, the payoff on the PPUP can be written as

$1,000 1 max ,+∗ −

⎛

⎝

⎜

⎞

⎠

⎟

⎡

⎣

⎢

⎤

⎦

⎥

50 1 0

To analyze the PPUP, we ﬁ rst rewrite the maturity payment as

$1,000 max , $1,000

Payoff on

zero cou

150 10

+∗ −

⎛

⎝

⎜

⎞

⎠

⎟

⎡

⎣

⎢

⎤

⎦

⎥

↑

ppon

bond

Payoff on at-the-money

max ,



+∗ −

↑

*( )1 000

ccall option

 

This shows that the PPUP payoff is composed of two parts:

•

A $1,000 return of principal. Since no interest is paid on this principal,

its value today is the present value of the payment at the risk-free

4. Not all structureds can be priced using Black-Scholes; more complicated, path-

dependent securities often need to be priced using the Monte Carlo methods discussed

in Chapters 22 and 23.

526 Chapter 19

interest rate, $1,000*e

−rT

, where r is the interest rate and T = 5 is the

maturity of the PPUP.

•

$1,000*

50%

times the value today of an at-the-money call on the S&P

500.

We can use the following spreadsheet to price this security:

CBA

Initial S&P 500 price, S

950 <-- The price of the S&P 500 at PPUP issuance

Structured exercise price, X 950

Risk-free interest rate for 5 years, r 5.00%

Time to maturity, T 5

Volatility of S&P 500,

25%

Participation rate 50% <-- Percentage of increase in the S&P going to PPUP owner

Strutured components, value today

Bond paying $1,000 at maturity 778.80 <-- =EXP(-B4*B5)*1000

Participation rate /S

*at-the-money call on S&P 500

162.52 <-- =1000*B7/B2*BSCall(B2,B3,B5,B4,B6)

Value of structured security today 941.32 <-- =SUM(B10:B11)

ANALYZING A SIMPLE STRUCTURED PRODUCT

$1,000 Deposit with 50% Participation in S&P Increase over 5 Years

The value of the structured security (cell B12) is $941.42. This valua-

tion has two parts:

•

The present value of the bond part of the PPUP is $778.80 (cell B10).

•

The value of $1,000*

950

at-the-money calls on the S&P is $162.52.

Given the parameters in cells B2:B7, the PPUP is overpriced—it sells

for $1,000, whereas its market value ought to be $941.32. Another way

to think about the structured is to compute its implied volatility: What

(cell B6) will give the market valuation (cell B12) of the PPUP

to equal the $1,000 price being asked by the Homeside Bank? Either

GoalSeek or Solver will solve this problem:

CBA

Initial S&P 500 price, S

950 <-- The price of the S&P 500 at PPUP issuance

Structured exercise price, X 950

Risk-free interest rate for 5 years, r 5.00%

Time to maturity, T 5

Volatility of S&P 500,

42.00%

Participation rate 50% <-- Percentage of increase in the S&P going to PPUP owner

Strutured components, value today

Bond paying $1000 at maturity 778.80 <-- =EXP(-B4*B5)*1000

Participation rate /S

*At-the-money call on S&P 500

221.20 <-- =1000*B7/B2*BSCall(B2,B3,B5,B4,B6)

Value of structured security today 1000.00 <-- =SUM(B10:B11)

ANALYZING A SIMPLE STRUCTURED PRODUCT

$1,000 Deposit with 50% Participation in S&P Increase over 5 Years

527 The Black-Scholes Model

19.7.2 More Complicated Structured Products

Suppose you want to create a security with the following payoff

pattern:

Strategy

payoff

The payoff pattern increases (dollar for dollar) as the terminal price

of the underlying asset increases from 0 ≤ S

≤ X

. Between X

and X

the payoff pattern is ﬂ at, and for X

≤ S

, the payoff again increases dollar

for dollar with the price of the underlying. The algebraic formula for this

payoff pattern is

XXS SX

TT11 2

00−−+−max , max ,()()

To prove that this formula creates the graph,

XXS SX

TT11 2

00−−+−

↑

max , max ,

Payoff of written

put

()()



↑↑

−+= <

≤<

Payoff of bought

call

 

XXSS S X

XXSX

TT T

11 1

112

1 TTT

XXS−≤

⎧

⎨

⎪

⎩

⎪

A slightly more complicated payoff pattern is the following: