the black-scholes model chapter 13

1

The Black-ScholesModel

Chapter 13

2

Pricing an European Call

The Black&Scholes model

Assumptions:

1. European options.

2. The underlying stock does not pay dividends during the option’s life.

Time linet T

ct Max{0, ST – K}

3

Pricing an European Call

The Black&Scholes model

ct = NPV[Expected cash flows]

ct = NPV[E (max{0, ST – K})].

ct = e-r(T-t)max{E (0, ST – K)}.

4

Pricing an European Call The Black&Scholes model

Ct = e-r(T-t){[0STK]Pr(STK)

+ E [(ST–

K)ST>K]Pr(ST>K)}

ct = e-r(T-t)[E (ST–K)ST>K]Prob(ST>K)

5

Pricing an European Call The Black&Scholes model

ct = e-r(T-t)E [(ST)ST>K]Prob(ST > K)

– e-r(T-t)KProb(ST > K)

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The Stock Price Assumption(p282)

• In a short period of time of length t the return on the stock is normally distributed:

• Consider a stock whose price is S

tσt,μφS

S

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The Lognormal Property

• It follows from this assumption that

• Since the logarithm of ST is normal, ST is lognormally distributed

Tσ T,2

σμlnSφlnS

or

Tσ T,2

σμφlnSlnS

2

0T

2

0T

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The Lognormal Distribution

1)(eeS)var(S

e S)E(STσμT2

0T

μT0T

2

9

Black&Scholes formula:

ct = StN(d1) – Ke-r(T-t) N(d2)

d1 = [ln(St/K) + (r +.52)(T – t)]/√(T – t)

d2 = d1 - √(T – t),

N(d) is the cumulative standard normal distribution.

All the parameters are annualized and continuous in time

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pt = Ke-r(T-t) N(- d2) – StN(- d1).

d1 = [ln(St/K) +(r +.52)(T – t)]/√(T – t)

d2 = d1 - √(T – t),

N(d) is the cumulative standard normal distribution.

All the parameters are annualized and continuous in time

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INTEL Thursday, September 21, 2000. S = $61.48

CALLS - LAST PUTS - LAST

K OCT NOV JAN APR OCT NOV JAN APR

40 22 --- 23 --- --- --- 0.56 ---

50 12 --- --- --- 0.63 --- --- ---

55 8.13 --- 11.5 --- 1.25 --- 3.63 ---

60 4.75 --- 8.75 --- 2.88 4 5.75 ---

65 2.50 3.88 5.75 8.63 6.00 6.63 8.38 10

70 0.94 --- 3.88 --- 9.25 --- 11.25 ---

75 0.31 --- --- 5.13 13.38 --- --- 16.79

80 --- --- 1.63 --- --- --- --- ---

90 --- --- 0.81 --- --- --- --- ---

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S = 61.48: K = 65: JAN: T-t = 121/365 = .3315yrs;

R = 4.82%r = ln[1.0482] = .047. = 48.28%

d1={ln(61.48/65) + [.047

+.5(.48282)].3315}/(.4828)(.3315)

= -.0052

d2 = d1 – (.4828)(.3315) = -.2832

N(d1) = .4979; N(d2) = .3885

c= 61.48(.4979) – 65e-(.047)(.3315)(.3885)= 5.75

p = 5.75 -61.48 + 65e-(.047)(.3315) = 8.26

d

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S = 61.48: K = 70: JAN: T-t = 121/365 = .3315yrs;

R = 4.82%r = ln[1.0482] = .047. = 48.28%.

d1={ln(61.48/70) + [.047 +.5(.48282)].3315}/(.4828)(.3315)

= -.2718.

d2 = d1 – (.4828)(.3315) = -.5498

Next, we follow the extrapolations suggested in the text:

N(-.2718) = ?

N(-.5498) = ?

d

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N(d1) = N(-.2718) = N(-.27) - .18[N(-.27) – N(-.28)]

= .3936 -.18[.3936 -.3897] = .392274

N(d2) = N(-.5498) = N(-.54) - .98[N(-.54) – N(-.55)]

= .2946 - .98[.2946 - .2912] = .291268

ct = StN(d1) – Ke-r(T-t)N(d2)

c = 61.48(.392274) – 65e-(.047)(.3315)(.292268) = 4.04

pt = Ke-r(T-t)N(- d2) – StN(- d1)BUT

Employing the put-call parity for European options on a non dividend paying stock, we have:

p = 4.04 - 61.48 + 70e-(.047)(.3315) = 11.81

d

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Black and Scholes prices satisfy the put-call parity for European options on a non dividend paying stock:

ct – pt = St - Ke-r(T-t) .

Substituting the Black&Scholes values:


pt = Ke-r(T-t) N(- d2) – StN(- d1).

into the put-call parity yields:

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ct – pt = StN(d1) – Ke-r(T-t) N(d2)

- [Ke-r(T-t) N(- d2) – StN(- d1)]

ct - pt = StN(d1) – Ke-r(T-t) N(d2)

- Ke-r(T-t) [1 - N(d2)] + St[1 - N(d1)

ct - pt = St – Ke-r(T-t)

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The Inputs

St = The current stock priceK = The strike priceT – t = The years remaining to

expirationr = The annual, continuously

compounded risk-free rate

= The annual SD of the returns on the underlying asset

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The Inputs

St = The current stock price

Bid price?Ask price?Usually: mid spread

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The InputsT – t = The years remaining to

expirationBlack and Scholes: continuous markets

1 year = 365 days.

Real world : the markets are open for trading only 252 days.

1 year = 252 days.

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The inputs

1. r continuously compounded

2. R simple. r = ln[1+R].

3. Ra; Rb; n = the number of days to the T-bill maturity.

360n

2RR

1

1ln

n

365r

ba

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The Volatility (VOL)

• The volatility of an asset is the standard deviation of the continuously compounded rate of return in 1 year

• As an approximation it is the standard deviation of the percentage change in the asset price in 1 year

22

Historical Volatility (page 286-9)

1. Take n+1observed prices S0, S1, . . . , Sn at the end of the i-th time interval, i=0,1,…n.

2. is the length of time interval in years.

For example, if the time interval between i and i+1 is one week,

then = 1/52, If the time interval is

one day, then = 1/365.

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Historical Volatility

1. Calculate the continuously compounded return in each time interval as:

1.n1,2,...,i ,S

Slnu

1i

ii

24


Calculate the standard deviation of the ui´s:

2

1

)]([1-n

1S

n

ii uavru

25


The estimate of the Historical Volatility is:

τ

sσ̂

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Implied Volatility (VOL) (p.300)

• The implied volatility of an option is the volatility for which the Black-Scholes price equals the market price

• There is a one-to-one correspondence between prices and implied volatilities

• Traders and brokers often quote implied volatilities rather than dollar prices

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d1 = [ln(St/K) + (r +.52)(T – t)]/√(T – t),

d2 = d1 - √(T – t).

Inputs: St ; K; r; T-t; and ct

The solution yields: Implied Vol.

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Dividend adjustments (p.301)

1. European options

When the underlying asset pays dividends the adjustment of the Black and Scholes formula depends on the information at hand.

Case 1. The annual dividend payout ratio, q, is known. Use:

Where St is the current asset market price

t)q(TtFormula eSS

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Dividend adjustment

2. European options

Case 2. It is known that the underlying asset will pay a series of cash dividends, Di, on dates ti during the option’s life.

Use:

Where St is the current asset market price

i

t)r(titFormula

ieD - SS

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Dividend adjustment2. American options: T = the option expiration date.tn = date of the last dividend payment

before T.Black’s Approximation c=Max{1,2}1. Compute the dividend adjusted

European price.2. Compute the dividend adjusted

European price with expiration at tn;

i.e., without the last dividend.

the black-scholes model chapter 13

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