Steven R. Dunbar
Department of Mathematics
203 Avery Hall
University of Nebraska-Lincoln
Lincoln, NE 68588-0130
http://www.math.unl.edu
Voice: 402-472-3731
Fax: 402-472-8466
Stochastic Processes and
Advanced Mathematical Finance
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Single Period Binomial Models
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Student: contains scenes of mild algebra or calculus that may require guidance.
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If you have two unknown values, how many equations will you need to derive to ﬁnd the two unknown values? What must you know about the equations?
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The single period binomial model is the simplest possible ﬁnancial model, yet it has the elements of all future models. It is strong enough to be a conceptual model of ﬁnancial markets. It is simple enough to permit pencil-and-paper calculation. It can be comprehended as a whole. It is also structured enough to point to natural generalization. The single period binomial model is an excellent place to start studying mathematical ﬁnance.
Begin by reviewing some deﬁntions:
The quantiﬁable elements of the single period binomial ﬁnancial model are:
A realistic ﬁnancial assumption would be $D<exp\left(rT\right)<U$. Then investment in the risky security may pay better than investment in a risk free bond, but it may also pay less! The mathematics only requires that $U\ne D$, see below.
We can try to ﬁnd the value of the derivative by creating a portfolio of the stock and the bond that will have the same value as the derivative itself in any circumstance, called a replicating portfolio. Consider a portfolio consisting of $\varphi $ units of the stock worth $\varphi S$ and $\psi $ units of the bond worth $\psi B$. Note the assumption that the stock and bond are divisible, so we can buy them in any amounts including negative amounts that are short positions. If we were to buy the this portfolio at time zero, it would cost
One time period of length $T$ on the trading clock later, the portfolio would be worth
after a down move and
after an up move. You should ﬁnd this mathematically meaningful: there are two unknown quantities $\varphi $ and $\psi $ to buy for the portfolio, and we have two expressions to match with the two values of the derivative! That is, the portfolio will have the same value as the derivative if
$$\begin{array}{rcll}\varphi SD+\psi Bexp\left(rT\right)& =& f\left(SD\right)& \text{}\\ \varphi SU+\psi Bexp\left(rT\right)& =& f\left(SU\right).& \text{}\end{array}$$
The solution is
and
Note that the solution requires $SU\ne SD$, but we have already assumed this natural requirement. Without this requirement there would be no risk in the stock, and we would not be asking the question in the ﬁrst place. The value (or price) of the portfolio, and therefore the derivative should then be
$$\begin{array}{rcll}V& =& \varphi S+\psi B& \text{}\\ & =& \frac{f\left(SU\right)-f\left(SD\right)}{SU-SD}\cdot S+\left[\frac{f\left(SD\right)}{Bexp\left(rT\right)}-\frac{\left(f\left(SU\right)-f\left(SD\right)\right)SD}{\left(SU-SD\right)Bexp\left(rT\right)}\right]\cdot B& \text{}\\ & =& \frac{f\left(SU\right)-f\left(SD\right)}{U-D}+\frac{1}{exp\left(rT\right)}\frac{f\left(SD\right)U-f\left(SU\right)D}{\left(U-D\right)}.& \text{}\end{array}$$We can make one ﬁnal simpliﬁcation that will be useful in the next section. Deﬁne
so then
so that we write the value of the derivative as
(Here $\pi $ is not used as the mathematical constant giving the ratio of the circumference of a circle to its diameter. Instead the Greek letter for p suggests a similarity to the probability $p$.)
Now consider some other trader oﬀering to sell this derivative with payoﬀ function $f$ for a price $P$ less than $V$. Anyone could buy the derivative in arbitrary quantity, and short the $\left(\varphi ,\psi \right)$ stock-bond portfolio in exactly the same quantity. At the end of the period, the value of the derivative would be exactly the same as the portfolio. So selling each derivative would repay the short with a proﬁt of $V-P$ and the trade carries no risk! So $P$ would not have been a rational price for the trader to quote and the market would have quickly mobilized to take advantage of the “free” money on oﬀer in arbitrary quantity. This ignores transaction costs. For an individual, transaction costs might eliminate the proﬁt. However for large ﬁrms trading in large quantities, transaction costs can be minimal.
Similarly if a seller quoted the derivative at a price $P$ greater than $V$, anyone could short sell the derivative and buy the $\left(\varphi ,\psi \right)$ portfolio to lock in a risk-free proﬁt of $P-V$ per unit trade. Again the market would take advantage of the opportunity. Hence, $V$ is the only rational price for the derivative. We have determined the price of the derivative through arbitrage.
Note that we did not determine the price of the derivative in terms of the expected value of the stock or the derivative. A seemingly logical thing to do would be to say that the derivative will have value $f\left(SU\right)$ with probability $p$ and will have value $f\left(SD\right)$ with probability $1-p$. Therefore the expected value of the derivative at time $T$ is
The present value of the expectation of the derivative value is
Except in the peculiar case that the expected value just happened to match the value $V$ of the replicating portfolio, arbitrage drives pricing by expectation out of the market! The problem is that the probability distribution $\left(p,1-p\right)$ only takes into account the movements of the security price. The expected value is the value of the derivative over many identical iterations or replications of that distribution, but there will be only one trial of this particular experiment, so expected value is not a reasonable way to weight the outcomes. Also, the expected value does not take into account the rest of the market. In particular, the expected value does not take into account that an investor has the opportunity to simultaneously invest in alternate combinations of the risky stock and the risk-free bond. A special combination of the risky stock and risk-free bond replicates the derivative. As such the movement probabilities alone do not completely assess the risk associated with the transaction.
Nevertheless, we still might have with a nagging feeling that pricing by arbitrage as done above ignores the probability associated with security price changes. One may legitimately ask if there is a way to value derivatives by taking some kind of expected value. The answer is yes, a diﬀerent probability distribution associated with the binomial model correctly takes into account the rest of the market. In fact, the quantities $\pi $ and $1-\pi $ deﬁne this probability distribution. This is called the risk-neutral measure or more completely the risk-neutral martingale measure. Economically speaking, the market assigns a “fairer” set of probabilities $\pi $ and $1-\pi $ that give a value for the option compatible with the no-arbitrage principle. Another way to say this is that the market changes the odds to make option pricing fairer. The risk-neutral measure approach is the modern, sophisticated, and general way to approach derivative pricing.
Robert Merton summarized this derivative pricing strategy succinctly in [5]:
“The basic insight underlying the Black-Scholes model is that a dynamic portfolio trading strategy in the stock can be found which will replicate the returns from an option on that stock. Hence, to avoid arbitrage opportunities, the option price must always equal the value of this replicating portfolio.”
This section is adapted from: “Chapter 2, Discrete Processes” in Financial Calculus by M. Baxter, A. Rennie, Cambridge University Press, Cambridge, 1996, [2].
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The aim is to set up and solve for the replicating portfolio and the value of the corresponding derivative security in a single period binomial model. The scripts output the portfolio amounts and the derivative security value. First set values of $S$, $U$, $D$, $r$, $T$, and $K$. Deﬁne the derivative security payoﬀ function. Set the matrix of coeﬃcients in the linear system. Set the right-hand-side payoﬀ vector. Solve for the portfolio using a linear solver. Finally, take the dot product of the portfolio with the security and bond values to get the derivative security value.
Octave script for singleperiod..
Perl PDL script for singleperiod..
Scientiﬁc Python script for singleperiod..
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[1] Marco Allavenada and Peter Laurence. Quantitative Modeling of Derivative Securities. Chapman and Hall, 2000. HG 6024 A3A93 2000.
[2] M. Baxter and A. Rennie. Financial Calculus: An introduction to derivative pricing. Cambridge University Press, 1996. HG 6024 A2W554.
[3] S. Benninga and Z. Wiener. The binomial option pricing model. Mathematical in Education and Research, 6(3):27–33, 1997.
[4] Freddy Delbaen and Walter Schachermayer. What is a …free lunch. Notices of the American Mathematical Society, 51(5), 2004.
[5] Robert C. Merton. Inﬂuence of mathematical models in ﬁnance on practice: past, present and future. In S. D. Howison, F. P. Kelly, and P. Wilmott, editors, Mathematical Models in Finance. Chapman and Hall, 1995. popular history.
[6] Paul Wilmott, S. Howison, and J. Dewynne. The Mathematics of Financial Derivatives. Cambridge University Press, 1995.
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Steve Dunbar’s Home Page, http://www.math.unl.edu/~sdunbar1
Email to Steve Dunbar, sdunbar1 at unl dot edu
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