Steven R. Dunbar
Department of Mathematics
203 Avery Hall
University of Nebraska-Lincoln
Lincoln, NE 68588-0130
Stochastic Processes and
Advanced Mathematical Finance
Derivation of the Black-Scholes Equation
Mathematically Mature: may contain mathematics beyond calculus with proofs.
What is the most important idea in the derivation of the binomial option pricing model?
For mathematical modeling of a market for a risky security we will ideally assume:
The ﬁrst assumption is the essence of what economists call the eﬃcient market hypothesis. The eﬃcient market hypothesis leads to the second assumption as a conclusion, called the random walk hypothesis. Another version of the random walk hypothesis says that traders cannot predict the direction of the market or the magnitude of the change in a stock so the best predictor of the market value of a stock is the current price. We will make the second assumption stronger and more precise by specifying the probability distribution of the changes with a stochastic diﬀerential equation. The remaining hypotheses are simplifying assumptions that can be relaxed at the expense of more diﬃcult mathematical modeling.
We wish to ﬁnd the value of a derivative instrument based on an underlying security that has value . Mathematically, we assume:
or equivalently that is a Geometric Brownian Motion with parameters and .
The ﬁrst assumption is a mathematical translation of a strong form of the eﬃcient market hypothesis from economics. It is a reasonable modeling assumption but ﬁner analysis strongly suggests that security prices have a higher probability of large price swings than Geometric Brownian Motion predicts. Therefore the ﬁrst assumption is not supported by data. However, it is useful since we have good analytic understanding of Geometric Brownian Motion.
The second assumption is a reasonable assumption for a modeling attempt although good evidence indicates neither interest rates nor the volatility are constant. On reasonably short times scales, say a period of three months for the lifetime of most options, the interest rate and the volatility are approximately constant. The third and fourth assumptions are mathematical translations of the assumptions that securities can be bought and sold in any amount and that trading times are negligible, so that standard tools of mathematical analysis apply. Both assumptions are reasonable for modern security trading.
We consider a simple derivative instrument, an option written on an underlying asset, say a stock that trades in the market at price . A payoﬀ function determines the value of the option at expiration time . For , the option value should depend on the underlying price and the time . We denote the price as . So far all we know is the value at the ﬁnal time . We would like to know the value so that we know an appropriate buying or selling price of the option.
As time passes, the value of the option changes, both because the expiration date approaches and because the stock price changes. We assume the option price changes smoothly in both the security price and the time. Across a short time interval we can write the Taylor series expansion of :
The neglected terms are of order , , and and higher. We rely on our intuition from random walks and Brownian motion to explain why we keep the terms of order but neglect the other terms. More about this later.
To determine the price, we construct a replicating portfolio. This will be a speciﬁc investment strategy involving only the stock and a cash account that will yield exactly the same eventual payoﬀ as the option in all possible future scenarios. Its present value must therefore be the same as the present value of the option and if we can determine one we can determine the other. We thus deﬁne a portfolio consisting of shares of stock and units of the cash account . The portfolio constantly changes in value as the security price changes randomly and the cash account accumulates interest.
In a short time interval, we can take the changes in the portfolio to be
since , where is the interest rate. This says that short-time changes in the portfolio value are due only to changes in the security price, and the interest growth of the cash account, and not to additions or subtraction of the portfolio amounts. Any additions or subtractions are due to subsequent reallocations ﬁnanced through the changes in the components themselves.
The diﬀerence in value between the two portfolios’ changes by
This represents changes in a three-part portfolio consisting of an option and short-selling units of the security and units of bonds.
Next come a couple of linked insights: As an initial insight we will eliminate the ﬁrst order dependence on by taking . Note that this means the rate of change of the derivative value with respect to the security value determines a policy for . Looking carefully, we see that this policy removes the “randomness” from the equation for the diﬀerence in values! (What looks like a little “trick” right here hides a world of probability theory. This is really a Radon-Nikodym derivative that deﬁnes a change of measure that transforms a diﬀusion, i.e. a transformed Brownian motion with drift, to a standard Wiener measure.)
Second, since the diﬀerence portfolio is now non-risky, it must grow in value at exactly the same rate as any risk-free bank account:
This insight is actually our now familiar no-arbitrage-possibility argument: If , then anyone could borrow money at rate to acquire the portfolio , holding the portfolio for a time , and then selling the portfolio, with the growth in the diﬀerence portfolio more than enough to cover the interest costs on the loan. On the other hand if , then short-sell the option in the marketplace for , purchase shares of stock and lend the rest of the money out at rate . The interest growth of the money will more than cover the repayment of the diﬀerence portfolio. Either way, the existence of risk-free proﬁts in the market will drive the inequality to an equality.
Recall the quadratic variation of Geometric Brownian Motion is deterministic, namely ,
Divide out the factor, and recall that , and , so that on the left . The terms on left and right cancel, and we are left with the Black-Scholes equation:
Note that under the assumptions made for the purposes of the modeling the partial diﬀerential equation depends only on the constant volatility and the constant risk-free interest rate . This partial diﬀerential equation (PDE) must be satisﬁed by the value of any derivative security depending on the asset .
There are two things to immediately notice about the PDE.
The derivation above follows reasonably closely the original derivation of Black, Scholes and Merton. Option prices can also be calculated and the Black-Scholes equation derived by more advanced probabilistic methods. In this equivalent formulation, the discounted price process is shifted into a “risk-free” measure using the Girsanov Theorem, so that it becomes a martingale. The option price is then the discounted expected value of the payoﬀ in this measure, and the PDE is obtained as the backward evolution equation for the expectation. The probabilistic view is more modern and can be more easily extended to general market models.
The derivation of the Black-Scholes equation above uses the fairly intuitive partial derivative equation approach because of the simplicity of the derivation. This derivation:
The disadvantages are that:
This derivation of the Black-Scholes equation is drawn from “Financial Derivatives and Partial Diﬀerential Equations” by Robert Almgren, in American Mathematical Monthly, Volume 109, January, 2002, pages 1–11.
Explain in ﬁnancial terms what each of these solutions represents. That is, describe a simple “claim”, “derivative” or “option” for which this solution to the Black Scholes equation gives the value of the claim at any time.
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Last modiﬁed: Processed from LATEX source on August 10, 2016