Notes on derivatives

7/30/2019 Notes on derivatives

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Options and Futures

Note # 7

November 2011

The Black Scholes Option Pricing Formula

1. The Behavior of Stock Prices

What are desirable properties of a process for stock

price?

(i) random (stochastic)

(ii) reflects the level of market efficiency

(iii) has an expected upward trend

(iv) has volatility proportional to time horizon

(v) cannot go below zero

The assumed process will be referred to as the

stochastic process for stock price.

1.1. The Markov Property

Stock price process depends on the stocks currentvalue, not past value.

Price history is irrelevant. =Weak-form efficiency.

Price distribution at any time is independent of pastpath.

By extension, changes in stock price depend onlyon the current price. independent S's.

1.2. Brownian Motion

A variable z follows a Brownian motion if:

(i) z = where (0,1); and

(ii) the values ofz for any two t's are independent (i.e.

Markov)

t

It follows that for a short time period, t:

(i) E[z] =0.

(ii) Var[z] =t.

What about the change during a longer time period,

T?

[z(T) z(0)] (0, )T


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1.3. The Price Process for Non-Dividend-Paying

Stocks

A variable S follows a geometric Brownian motionprocess if:

S =S t +S z (discrete)

=t +z

dS =S dt +S dz (continuous)

=dt +dz

S

S

S

dS

Change =expected drift term +random term where:

dt =a very small interval of time (an instant)

dS =change in stock price by the end ofdt.

=expected return p.a. on the stock

=volatility of the stock price

dz =the random term = where (0,1).dt

Note the following:

(i) dS is random through the term dz 1st property.

(ii) dS depends on current price, S, not past prices as dzs

are not correlated with one another. 2nd property.

(iii) E (dS) =S dt 3rd property

(iv) Var (dS) =2S2 dt. 4th property.

(v) IfS =0, dS =0.5th property

In other words,

E ( ) =dt

Var( ) =2 dt Std ( ) = .

Ex. Consider a non-dividend-paying stock whose = 12% p.a.

and =30% p.a. Its price process is:

S

dS

S

dS

S

dSdt

It can be shown that under GBM, stock price at any

future time has a lognormal distribution.

lnST (ln S +( )T, ).

Ex. The probdist of the above stock in a weeks time is:

2

2 T

Aside Lognormal Distribution

A variable has a lognormal distribution if its natural

log is normally distributed.

Ex.y has a lognormal dist. if ln y (,).

A lognormallydistributed variable can only take on

positive values.


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It can also be shown that:

(i) E (ST

) =SeT

(ii) Var (ST) = 1222 TT eeS

Ex. For the above stock,

END ASIDE

2. The Price Process for Options on Non-

Dividend-Paying Stocks

With the assumed price process for a stock, we can

derive the price process for any derivatives on thatstock.

Itos Lemma (modified version):

IfS follows a Geometric Brownian Motion, then f(S) also

follows a GBM, with different parameters for and .

Since c =f(S), we can use Itos Lemma to derive

the price process for option:

df=

=[] dt +[]dz.

SdzS

fdtS

S

f

t

fS

S

f

222

2

2

1

Note that:

- The option price process consists of a non-random term

and a random term.

- The random term consists of the same dz term as in the

stock price process. The randomness of option price

comes solely from the randomness of stock price.

3. The Black-Scholes Option Pricing formula

With this insight, Black and Scholes developed an

option pricing formula.

Assumptions underlying Black-Scholes:

- Options are European.

- Stock price follows a GBM with and constant. (i.e. Price

is lognormally distributed).

- No transaction costs or taxes

- No arbitrage opportunities


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- Borrow or lend at the same risk-free rate

-The risk-free rate, r, is constant and the same for all

maturities.

- No dividends during the life of the options

- Stock trading is continuous.

Based on the price processes of stock and option,

what about a portfolio of:

-Short 1 unit of option; and

- Long unit of stock.

Cost of this port based on its composition = S f.

S

f

S

f

Payoff of this port after a small interval, dt,

=Stock payoff Option payoff

= [S dt +S dz]

=- No dz term risk-free.

S

f

Sdz

S

fdtS

S

f

t

fS

S

f

22

2

2

2

1

dtSS

f

t

f

222

2

2

1

By holding option and stock in the right proportion, a

risk-free port is created.

Cost of port based on composition = S f

Risk-free payoff =[ S f] r dt.

S

f

S

f

Therefore,

- = [ S f] r dt.

Or,

B-S PDE.

dtSS

f

t

f

222

2

2

1

S

f

.2

12

222

rfS

fS

S

frS

t

f

Note that the B-S PDE has to be satisfied by all

derivatives, f, whose value depends on S.

We can solve the PDE to come up with a pricing

formula by specifying boundary conditions; e.g.,

- c S

- c =max{0, ST

- K) at maturity


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B-S obtained:

c =SN(d1) Ke-rTN(d2)

p =Ke-rTN(-d2) SN(-d1)

where

d1 =

d2 =d1 .

.)2/()/ln( 2

T

TrKS

T

N (y) =cumulative probability function of a standard

normal random variable. y (0,1).

N (-y) =1 - N (y).

Ex. Consider a stock whose S is $30 and is 20% p.a. The

rf is 10% p.a. What are the prices of a 6-month European

call and a 3-month European put on this stock where K of

both options is $31?

S =30; K =31; r=0.1;=0.2; T =0.5 (call), 0.25 (put).

3.1. Properties of the B-S prices

Do the B-S prices conform to our understanding of

how the factors (S, K, T, r, ) affect option prices?

3.2. Hedge Ratio in the B-S Framework

Note that to construct a risk-free port, a short option

is paired with unit of stock.

So, is the hedge ratio in the B-S framework.

S

f

S

f

It can be shown that:

- for call, =N (d1 ) >0.

- for put, = - N (-d1) =N (d1 ) 1


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3.3. Risk-Neutral Valuation

Recall from the Binomial model that we can value

options as if we were risk-neutral.

Risk-neutral investors expect a risk-free return on all

investments.

We can price options as if we were risk-neutral

because we can mathematically construct the risk-

neutral probabilities.

In the risk-neutral framework, the expected return of

any asset under these prob is the risk-free rate.

We can do the same in the B-S framework byassuming that the stocks expected return is the risk-

free rate:

dS =rS dt +S dz

Then, we:

(i) calculate the E(payoff) of the option given the

lognormal dist of the stock price; and

(ii) discount the E(payoff) by the risk-free rate. will

get the same B-S pricing formula.

3.4. Binomial VS. B-S

If we keep increasing the #of periods under the

Binomial model, the price process will converge to

the GBM. Binomial price B-S price.

Generally, convergence occurs around 100 periods.

3.5. Options on Dividend-Paying Stocks

Assume that future dividends can be correctly

predicted.

- This is reasonable for short-life options.

- For long-life options, it is more realistic to assume that the

dividend yield is known.

3.6.1. European Options

The B-S is still correct if the PV of dividends is

subtracted from the current stock prices.

Ex. Consider the above 6-m European call. Suppose:

Div 1 =$0.25 in 2 months

Div 2 =$0.25 in 5 months


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3.6.2. American Options

We will separate our discussion between calls and

puts.

For American call options on dividend-paying stocks,

early exercise is possible.

- We cannot apply the original B-S formula directly.

- We have to use an approximation technique.

The approximation technique involves:

1. Determine if early exercise can be completely ruledout.

2. If so, the call price will equal the European call price

with the original maturity but with the stock price

reduced by the PVs of all dividends during the

option's life.

3. If not, repeat the calculation in step 2. Also, calculate

the price of a European call with maturity date on

each of the ex-dividend dates (on which early

exercise cannot be ruled out) but with the stock price

reduced by the PVs of all dividends up to that date.

The highest of these prices is the approximated price

for the American call.

Ex. Consider a 6-month American call on a stock whose S is $20

and is 20% p.a. The is 10% p.a. The call's K is $18. The rfrate is 6% p.a. Suppose there will be two dividends during the

option's life. The first dividend is $0.25 with an ex-dividend date

in 2 months, while the second is $0.25 with an ex-dividend date

in 5 months.

For American puts, they will be exercised whenever

they are sufficiently deep in the money.

- The price drop just after the stock goes ex-dividend

increases the likelihood.

There are some approximation techniques to use,

but it is better to use the binomial framework to

value American puts.

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