A First Look at the Black-Scholes Equation

= ∆ t dS t + β t dct
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Now we have 3 price processes:
nothing dB = rBdt dS = µSdt + σSdz dc = (ct + µScS + 1 σ 2 S 2 cSS )dt + σScS dz 2
Bond: Stock: Derivative:
Here comes the Black-Scholes argument:
Let’s 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. Let’s choose the stock and derivative, and create a bond.
Our portfolio will consist of ∆ shares of the stock and β of the derivative.
Pt = ∆ t St + β t ct
The first thing we need to do is compute dP and choose ∆ and β to eliminate the dz term. To compute dP, we could use Ito’s lemma:
Primbs, MS&E 345
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Now we have 3 price processes:
nothing dB = rBdt dS = µSdt + σSdz dc = (ct + µScS + 1 σ 2 S 2 cSS )dt + σScS dz 2
Bond: Stock: Derivative:
ct + rScS + 1 σ 2 S 2 cSS = rc 2
= rPdt
∆ = − βcS
The Black-Scholes Equation
Primbs, MS&E 345
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Which derivative was this? If it was a European call option with strike K and maturity T:
dc = (ct + µScS + 1 σ 2 S 2 cSS )dt + σScS dz 2
c( St , t )
Primbs, MS&E 345
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Now we have 3 price processes:
dB = rBdt dS = µSdt + σSdz dc = (ct + µScS + 1 σ 2 S 2 cSS )dt + σScS dz 2
dP = β (ct + 1 σ 2 S 2 cSS )dt 2
nothing must look like the bond substitute P = ∆S + βc substitute
= r (∆S + βc)dt = rβ (c − cS S )dt
β (ct + 1 σ 2 S 2 cSS ) = rβ (c − cS S ) 2
Primbs, MS&E 345
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Now we have 3 price processes:
nothing dB = rBdt dS = µSdt + σSdz dc = (ct + µScS + 1 σ 2 S 2 cSS )dt + σScS dz 2
Bond: Stock: Derivative:
6 5 4 3
St = S0e
( µ − 1 σ 2 ) t +σzt 2
2 1 0
0
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Primbs, MS&E 345
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The Set-up: Securities: Bond: Stock:
dB = rBdt
dS = µSdt + σSdz
Consider a derivative security whose price depends on St and t. We will call it: By Ito’s lemma:
Primbs, MS&E 345
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The Set-up: Securities: Bond: Stock:
dB = rBdt
dS = µSdt + σSdz
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Bond: -Deterministic -Exponential Growth -Continuous compounding
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Bond: Stock: Derivative:
Our portfolio will consist of ∆ shares of the stock and β of the derivative. So, we have:
dP = ∆dS + βdc
P = ∆S + βc
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.
Primbs, MS&E 345
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Now we have 3 price processes: ∆ + β
nothing dB = rBdt dS = µSdt + σSdz dc = (ct + µScS + 1 σ S 2 cSS )dt + σScS dz 2
Bond: Stock: Derivative:
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Let’s think about how a portfolio works:
Now your portfolio is worth
You purchase ∆t shares of stock and βt of the derivative.
Pt + dt = ∆ t St + dt + β t ct + dt
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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.
If you want, you can rebalance your portfolio now. But if you don’t add or take out any money, then
Your portfolio is worth Pt = ∆ t St + β t ct
dt period
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Bt = B0 e rt
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Primbs, MS&E 345
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The Set-up: Securities: Bond: Stock:
dB = rBdt
dS = µSdt + σSdz
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Stock: -Geometric Brownian Motion -Log-Normal -Always positive
Our portfolio will consist of ∆ shares of the stock and β of the derivative.
Pt = ∆ t St + β t ct
To create a bond, we can dynamically choose ∆ and β so that the portfolio is riskless (i.e. dP has no dz term). Since this portfolio will be riskless, it must earn the same rate of return as 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). Let’s perform these calculations...