1. The Black- Scholes Equation
Chapter 21
The Black- Scholes Equation

2. Differential Equations and Valuation Under Uncertainty
An example of transformation of a valuation equation into a differential equation: stock valuation
Consider the familiar stock valuation setup
which can be rewritten as
Dividing by h and letting h  0
We obtain the differential equation that describes the evolution of the stock price over time to generate an appropriate rate of return
Change in stock price
Cash payout
Return on Stock
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3. Differential Equations and Valuation Under Uncertainty (cont’d)
Dividend-paying stocks
If the value of the stock at time T is , that is the terminal boundary condition is , the solution is
Bonds
Let S(t) represent the price of a zero-coupon bond that pays $1 at T
The general solution to this differential equation is S(t)=Ae-r(T-t)
With the terminal boundary condition S(T)=$1, the particular solution for the bond value is S(t)= $1 x e-r(T-t)
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4. The Black-Scholes Equation
Consider the problem of creating a riskless hedge for an option position through trading in shares and bonds
Assume that the stock price follows geometric Brownian motion
where a is the expected return in the stock, s is the stock’s volatility, and d is the continuous dividend yield
If we invest W in these bonds, the change in value of the bond is
dW = rWdt
Let I denote the total investment in the option, N stocks, and W in the risk-free bonds so that the total investment is zero
I = V (S, t) + NS + W =0
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5. The Black-Scholes Equation (cont’d)
Applying Itô’s Lemma
Since the option D is VS , set N = –VS ; this will make W = VS S – V
Also, dI = 0, because with zero investment, the return should also be zero. Substituting all these and dividing by dt
This is the Black-Scholes PDE for any contingent claim, assuming
Underlying asset follows const. volatility geometric Brownian motion
Underlying asset pays a continuous proportional dividend at d rate
The contingent claim itself pays no dividend
The interest rate is fixed with equal borrowing and lending rates
There are no transaction costs
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6. The Black-Scholes Equation (cont’d)
The call price formula
satisfies the Black-Scholes PDE using the appropriate boundary condition.
Similarly, it can be shown that the put pricing formula, and the formulas for asset-or-nothing options, cash-or-nothing options, and gap options satisfy the Black-Scholes PDE using the appropriate boundary condition in each case
Black-Scholes PDE can be generalized to the case when one uses equilibrium expected return on the underlying asset instead of the risk-free return
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7. The Black-Scholes Equation (cont’d)
When the underlying asset is not an investment asset
where the dividend yield d in the original PDE is replaced by , the “lease rate” of the asset
is the difference between the equilibrium expected return and the actual expected return m on the noninvestment asset
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8. Risk-Neutral Pricing
The original Black-Scholes PDE does not contain the expected stock return a, but only the risk-free rate. The following version of the Black-Scholes equation
indicates that the option appreciates on average at the risk-free rate
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9. Risk-Neutral Pricing (cont’d)
Clearly related to the above equation is the following equation
For the risk-neutral process the above equation becomes the Kolmogorov backward equation
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10. Changing the Numeraire
What happens when the number of options received at expiration is random?
Currency translation: a cash flow originating in yen can be valued in yen, or in some other currency (more on this in Chapter 22)
Quantity uncertainty: an agricultural producer who wants to insure production of an entire field must hedge total revenue rather than quantity alone
All-or-nothing options: they can be structured either to pay cash if a certain event occurs or in shares.
The value of an all-or-nothing option is the first term in the BS formula, which can be viewed as a risk-neutral probability with a change in numeraire
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