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

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Rating

Student: contains scenes of mild algebra or calculus that may require guidance.

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Question of the Day

Two items can be purchased in any amounts each, call the amounts $x$ and $y$. (Think about stocks and bonds.) The two items each contribute revenue at rates specific to the current financial environment. (Think about stock profits in good times and bad, call them $r_{\text{ g}x}$ and $r_{\text{ b}x}$ and bond interest at rates $r_{\text{ g}y}$ and $r_{\text{ b}y}$.) Two specific outcomes must be achieved, one in good times, one in bad times (call them $f_g$ and $f_b$). What mathematical set-up is required to find the specific amounts of each item?

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Key Concepts

1. The simplest model for pricing an option is based on a market having a single period, a single security having two uncertain outcomes, and a single bond.
2. Replication of the option payouts with the single security and the single bond leads to pricing the derivative by arbitrage.

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Vocabulary

1. Security: A promise to pay, or an evidence of a debt or property, typically a stock or a bond. Also referred to as an asset.
2. Bond: Interest bearing securities, which can either make regular interest payments, or a lump sum payment at maturity, or both.
3. Stock: A security representing partial ownership of a company, varying in value with the value of the company. Also known as shares or equities.
4. Derivative: A security whose value depends on or is derived from the future price or performance of another security. Also known as financial derivatives, derivative securities, derivative products, and contingent claims.
5. A portfolio of the stock and the bond which will have the same value as the derivative itself in any circumstance is a replicating portfolio.

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Mathematical Ideas

Single Period Binomial Model

The single period binomial model is the simplest possible financial model, yet it contains the elements of all future models. The single period binomial model is an excellent place to start studying mathematical finance. It is strong enough to be a somewhat realistic 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 quantifiable elements of the single period binomial financial model are:

1. A single interval of time, from $t=0$ to $t=T$.
2. A single stock of initial value $S$, in the time interval $\left[0,T\right]$ it can either increase by a factor $U$ to value $SU$ with probability $p$, or it can decrease in value by factor $D$ to value $SD$ with probability $q=1-p$.
3. A single bond with a continuously compounded interest rate $r$ over the interval $\left[0,T\right]$. If the initial value of the bond is $B$, then the final value of the bond will be $Bexp\left(rT\right)$.
4. A market for derivatives (such as options) dependent on the value of the stock at the end of the period. The payoff of the derivative to some investor would be the rewards (or penalties) $f\left(SU\right)$ and $f\left(SD\right)$. For example, a futures contract with strike price $K$ would have value $f\left(S_T\right)=S_T-K$. A call option with strike price $K$, would have value $f\left(S_T\right)=max\left(S_T-K,0\right)$.

A realistic financial assumption would be that $D<exp\left(rT\right)<U$. 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 attempt to find the value of the derivative by creating a portfolio of the stock and the bond which 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 we are making the assumption that the stock and bond are divisible, we can buy them in any amounts including negative amounts which 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

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

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.

How NOT to price the derivative and a hint of a better way.

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, pricing by expectation would be driven out of the market by arbitrage! 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 are left with a nagging feeling that pricing by arbitrage as done above ignores the probability associated with security price changes. One could legitimately ask if there is a way to value derivatives by taking some kind of expected value. The answer is yes, there is another probability distribution associated with the binomial model that 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 and we will talk more about it later. 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 very modern, sophisticated, and general way to approach derivative pricing. However it is too advanced for us to approach just yet.

Summary

From R. C. Merton in “Influence of mathematical models in finance on practice: past, present and future”, in Mathematical Models in Finance, edited by S.D. Howison, F. P. Kelly, and P. Wilmott, Chapman and Hall, London, 1995, pages 1-15.

Sources

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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Problems to Work for Understanding

1. Consider a stock whose price today is $50. Suppose that over the next year, the stock price can either go up by 10%, or down by 3%, so the stock price at the end of the year is either $55 or $48.50. The interest rate on a $1 bond is 6%. If there also exists a call on the stock with an exercise price of $50, then what is the price of the call option? Also, what is the replicating portfolio?
2. A stock price is currently $50. It is known that at the end of 6 months, it will either be $60 or $42. The risk-free rate of interest with continuous compounding on a $1 bond is 12% per annum. Calculate the value of a 6-month European call option on the stock with strike price $48 and find the replicating portfolio.
3. A stock price is currently $40. It is known that at the end of 3 months, it will either $45 or $34. The risk-free rate of interest with quarterly compounding on a $1 bond is 8% per annum. Calculate the value of a 3-month European put option on the stock with a strike price of $40, and find the replicating portfolio.
4. Your friend, the financial analyst comes to you, the mathematical economist, with a proposal: “The single period binomial pricing is all right as far as it goes, but it is certainly is simplistic. Why not modify it slightly to make it a little more realistic? Specifically, assume the stock can assume three values at time $T$, say it goes up by a factor $U$ with probability $p_U$, it goes down by a factor $D$ with probability $p_D$, where $D<1<U$ and the stock stays somewhere in between, changing by a factor $M$ with probability $p_M$ where $D<M<U$ and $p_D+p_M+p_U=1$.” The market contains only this stock, a bond with a continuously compounded risk-free rate $r$ and an option on the stock with payoff function $f\left(S_T\right)$. Make a mathematical model based on your friend’s suggestion and provide a critique of the model based on the classical applied mathematics criteria of existence of solutions to the model and uniqueness of solutions to the model.

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References

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Outside Readings and Links:

1. A video lesson on the binomial option model from Hull.

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