02 BlackScholes(f)

7/31/2019 02 BlackScholes(f)

1/18

Primbs, MS&E 345 1

A First Look at the Black-Scholes

Equation


2/18


3/18

Primbs, MS&E 345 3

Assumptions (to be used throughout most of the course)

There are no transaction costs (i.e. markets are frictionless)

Trading may take place continuously

There is no prohibition on short selling

The risk free rate is the same for borrowing and lending

Assets are perfectly divisible.

These are the standard assumptions.

When I deviate from them, I will mention it specifically,

otherwise assume that they are always in force.


4/18

Primbs, MS&E 345 4

The Set-up:

Securities:

Bond: rBdtdB

Stock: SdzSdtdS

0 5 10 15 20 25 30 35 40 45 500

2

4

6

8

10

12

14

Bond:

-Deterministic

-Exponential Growth

-Continuous compounding

rt

teBB

0


5/18

Primbs, MS&E 345 5

The Set-up:

Securities:

Bond: rBdtdB

Stock: SdzSdtdS

0 1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

8

Stock:

-Geometric Brownian Motion

-Log-Normal

-Always positive

tzt

teSS

)(

0

2

21


6/18

Primbs, MS&E 345 6

The Set-up:

Consider a derivative security whose price depends on St and t.

We will call it: ),( tSc t

Securities:

Bond: rBdtdB

Stock: SdzSdtdS

By Itos lemma:

dzScdtcSSccdcSSSSt

)( 2221


7/18Primbs, MS&E 345 7

Now we have 3 price processes:

Bond:rBdtdB

Stock:SdzSdtdS

dzScdtcSSccdcSSSSt

)( 2221 Derivative:

Here comes the Black-Scholes argument:

Lets form a portfolio using two of the assets, so that it

looks exactly like the third.

Then this portfolio must have the same price as the third.

We can choose any two assets for our portfolio. Lets choose

the stock and derivative, and create a bond.



Bond:rBdtdB

Stock:SdzSdtdS

dzScdtcSSccdcSSSSt

)( 2221 Derivative:

Our portfolio will consist ofD shares of the stock and b of the derivative.tttttcSP bD


To create a bond, we can dynamically choose D and b so that the

portfolio is riskless (i.e. dP has no dz term).

nothing

Since this portfolio will be riskless, it must earn the same rate of returnas the bond. Hence, we must have dP=rPdt. (Otherwise, we can arbitrage

by shorting the one with the smaller return, and using that money to buy

the one with the larger return).

Lets perform these calculations...



Bond:rBdtdB

Stock:SdzSdtdS

dzScdtcSSccdcSSSSt

)( 2221 Derivative:

Our portfolio will consist ofD shares of the stock and b of the derivative.

To compute dP, we could use Itos lemma:

......)()( DDD bbb cddcSddScdSddP

But Wait!!!, We want out portfolio to be self financing.

Lets think about this...


The first thing we need to do is compute dP and choose D and b to

eliminate the dz term.

nothing

tttttcSP bD


10/18Primbs, MS&E 345 10

You purchase Dt shares of stock

and bt of the derivative.

dt period

Lets think about how a portfolio works:

Your portfolio is

worthtttttcSP bD

Now your portfolio is worth

dtttdtttdttcSP

D b

If you want, you can

rebalance your portfolio

now. But if you dont addor take out any money, then

dttdttdttdtt

dtttdtttdtt

cS

cSP

D

D

b

b

So: )( ttttdtttdttttdttt cScSPPdP bb DD )()(

tdttttdtttccSS D

b

ttttdcdS bD


11/18Primbs, MS&E 345 11


Bond:rBdtdB

Stock:SdzSdtdS

dzScdtcSSccdcSSSSt

)( 2221 Derivative:

The equation for dP is known as the self-financing constraint.

As long as no money is added or taken from the portfolio,

it will have the above dynamics.

Our portfolio will consist ofD shares of the stock and b of the derivative.cSP bD

dcdSdP bDSo, we have:

nothing


12/18Primbs, MS&E 345 12

Bond:rBdtdB

Stock:SdzSdtdS

Derivative:

D

b+

dzScSdtcSSccSdPSSSSt)())((

22

21 bb DD

nothing

dzScdtcSSccdcSSSSt

)( 2221

0D SScS b


cSP bD

dcdSdP bDSo, we have:

Lets do some portfolio algebra to compute dP.

To make the portfolio riskless, eliminate the dz term

ScbD

Substitute in


13/18Primbs, MS&E 345 13

Bond:rBdtdB

Stock:SdzSdtdS

Derivative:

D

b+

nothing

dzScdtcSSccdcSSSSt

)( 2221

dtcScdPSSt

)(22

2

1b nothing

rPdt

must look like the bond

dtcSr )( bD substitute cSP bD

ScbDdtSccr

S)( b substitute

)(

22

2

1

SStcSc

b

)( Sccr S b

rccSrSccSSSt

22

21 The Black-Scholes Equation



14/18Primbs, MS&E 345 14

Which derivative was this?

If it was a European call option with strike K and maturity T:

If it was a European put option with strike K and maturity T:

In general, the boundary condition determines

which derivative it is.

)(),( KSTSc

is the boundary condition.0),0( tc

)(),( SKTSc

is the boundary condition.0),( tc


15/18Primbs, MS&E 345 15

The Black-Scholes equation

(European Call Option)

Solution:

)()(),(2

)(

1 dNKedSNtSctTr

tT

tTrKSd

))(()/ln(2

21

1

tTdd 12

where:

)(N distribution function for a standard Normal (i.e.N(0,1))

We will derive this solution later in the course...(If you like, you can verify it now.)

rccSrScc SSSt 22

21

)(),( KSTSc 0),0( tc


16/18Primbs, MS&E 345 16

Other properties of the Black-Scholes solution:

-It doesnt depend on the mean return of the stock, .

These are properties that we will understand later...

]|),([),()(

tT

tTr

tSTScEetSc

SdzrSdtdS

The solution can also be written as:

where

Risk Neutral Pricing.

Not the true dynamics of the stock!


17/18Primbs, MS&E 345 17

This was our first look at the basic Black-Scholes argument.

Next, we will take a bit more abstract look at some

of the basic arguments hidden in this derivation and

see how far this approach can be generalized...

This will lead to a nice methodology for computing

partial differential/difference equations for a

surprisingly large number of derivative securities.


18/18

References

Black, F. and M. Scholes, The pricing of options and

corporate liabilities, Journal of Political Economy, 81, 637-659, 1973.

Hull, J. Options, Futures, and Other Derivatives, 4th Ed. Prentice Hall,

2000.

Luenberger, D. G. Investment Science, Oxford Press, 1998.

Merton, R. C., Theory of rational option pricing, Bell Journal of

Economics and Management Science, 4, 141-183, 1973.

Wilmott, P. Paul Wilmott on quantitative finance, Vol. 1 & 2, Wiley,

2000.

02 BlackScholes(f)

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