02 BlackScholes(f)
Transcript of 02 BlackScholes(f)
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Primbs, MS&E 345 1
A First Look at the Black-Scholes
Equation
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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.
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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
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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
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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
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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.
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Bond:rBdtdB
Stock:SdzSdtdS
dzScdtcSSccdcSSSSt
)( 2221 Derivative:
Our portfolio will consist ofD shares of the stock and b of the derivative.tttttcSP bD
Now we have 3 price processes:
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...
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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...
Now we have 3 price processes:
The first thing we need to do is compute dP and choose D and b to
eliminate the dz term.
nothing
tttttcSP bD
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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
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Now we have 3 price processes:
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
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Bond:rBdtdB
Stock:SdzSdtdS
Derivative:
D
b+
dzScSdtcSSccSdPSSSSt)())((
22
21 bb DD
nothing
dzScdtcSSccdcSSSSt
)( 2221
0D SScS b
Now we have 3 price processes:
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
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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
Now we have 3 price processes:
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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
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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
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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!
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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.
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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.