Steven R. Dunbar
Department of Mathematics
203 Avery Hall
Lincoln, NE 68588-0130
http://www.math.unl.edu
Voice: 402-472-3731
Fax: 402-472-8466

Stochastic Processes and

__________________________________________________________________________

Single Period Binomial Models

_______________________________________________________________________

Note: These pages are prepared with MathJax. MathJax is an open source JavaScript display engine for mathematics that works in all browsers. See http://mathjax.org for details on supported browsers, accessibility, copy-and-paste, and other features.

_______________________________________________________________________________________________

### Rating

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

_______________________________________________________________________________________________

### Section Starter Question

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?

_______________________________________________________________________________________________

### 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.

__________________________________________________________________________

### Vocabulary

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

__________________________________________________________________________

### Mathematical Ideas

#### Single Period Binomial Model

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:

1. A security is a promise to pay, or an evidence of a debt or property, typically a stock or a bond. A security is also referred to as an asset.
2. A bond is an interest bearing security that can either make regular interest payments or a lump sum payment at maturity or both.
3. A stock is a security representing partial ownership of a company, varying in value with the value of the company. Stocks are also known as shares or equities.
4. A derivative is a security whose value depends on or is derived from the future price or performance of another security. Derivatives are also known as ﬁnancial derivatives, derivative securities, derivative products, and contingent claims.

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 $S$ and ﬁnal value ${S}_{T}$. In the time interval $\left[0,T\right]$ it may either increase by a factor $U$ to value ${S}_{T}=SU$ with probability $p$, or it may decrease in value by factor $D$ to value ${S}_{T}=SD$ 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 the investor would be $f\left(SU\right)$ 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 $D. 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

$\varphi S+\psi B.$

One time period of length $T$ on the trading clock later, the portfolio would be worth

$\varphi SD+\psi Bexp\left(rT\right)$

after a down move and

$\varphi SU+\psi Bexp\left(rT\right)$

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

$\varphi =\frac{f\left(SU\right)-f\left(SD\right)}{SU-SD}$

and

$\psi =\frac{f\left(SD\right)}{Bexp\left(rT\right)}-\frac{\left(f\left(SU\right)-f\left(SD\right)\right)SD}{\left(SU-SD\right)Bexp\left(rT\right)}.$

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

$\pi =\frac{exp\left(rT\right)-D}{U-D}$

so then

$1-\pi =\frac{U-exp\left(rT\right)}{U-D}$

so that we write the value of the derivative as

$V=exp\left(-rT\right)\left[\pi f\left(SU\right)+\left(1-\pi \right)f\left(SD\right)\right].$

(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

$𝔼\left[f\right]=pf\left(SU\right)+\left(1-p\right)f\left(SD\right).$

The present value of the expectation of the derivative value is

$exp\left(-rT\right)𝔼\left[f\right]=exp\left(-rT\right)\left[pf\left(SU\right)+\left(1-p\right)f\left(SD\right)\right].$

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.

#### Summary

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.”

#### Sources

This section is adapted from: “Chapter 2, Discrete Processes” in Financial Calculus by M. Baxter, A. Rennie, Cambridge University Press, Cambridge, 1996, [2].

_______________________________________________________________________________________________

### Algorithms, Scripts, Simulations

#### Algorithm

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.

#### Scripts

R
1S <- 50
2up <- 0.1
3down <- 0.03
4B <- 1
5r <- 0.06
6T <- 1
7K <- 50
8
9f <- function(x, strike) {
10    max(x - strike, 0)
11}
12
13m <- rbind(c(S * (1 - down), B * exp(r * T)), c(S * (1 + up), B * exp(r * T)))
14payoff <- c(f(S * (1 - down), K), f(S * (1 + up), K))
15
16portfolio <- solve(m, payoff)
17value <- portfolio %*% c(S, B)
18
19cat("portfolio: phi=", portfolio[1], "psi=", portfolio[2], "\n")
20cat("value = ", value, "\n")
Octave
1S = 50;
2up = .10;
3down = 0.03;
4B = 1;
5r = 0.06;
6T = 1;
7K = 50;
8
9function retval = f(x, strike)
10retval = max(x - strike, 0);
11endfunction
12
13m = [S * (1 - down), B * exp(r * T); S * (1 + up), B * exp(r * T)];
14payoff = [f(S * (1 - down), K); f(S * (1 + up), K)];
15
16portfolio = m \ payoff;
17value = transpose(portfolio) * [S; B];
18
19disp("portfolio: phi ="), portfolio(1), disp("psi ="), portfolio(2)
20disp("derivative value:"), value
Perl
1use PDL::NiceSlice;
2
3$S = 50; 4$up   = .10;
5$down = 0.03; 6$B    = 1;
7$r = 0.06; 8$T    = 1;
9$K = 50; 10 11sub f { 12 my ($x, $strike ) = @_; 13 return max( pdl [$x - $strike, 0 ] ); 14} 15 16$m = pdl [
17    [ $S * ( 1 -$down ), $B * exp($r * $T ) ], 18 [$S * ( 1 + $up ),$B * exp( $r *$T ) ]
19];
20$payoff = transpose( 21 pdl [ f($S * ( 1 - $down ),$K ), f( $S * ( 1 +$up ), $K ) ] ); 22 23$portfolio = inv($m) x$payoff;
24$value = ( pdl [$S, $B ] ) x$portfolio;
25
26print "portfolio: phi=", $portfolio->range( [0] )->range( [0] ), 27 "psi=",$portfolio->range( [0] )->range( [1] ), "\n";
28print "value = ", $value->range( [0] )->range( [0] ), "\n"; SciPy 1import scipy 2from scipy import linalg 3 4S = 50 5up = .10 6down = 0.03 7B = 1 8r = 0.06 9T = 1 10K = 50 11 12 13def f(x, strike): 14 return max(x - strike, 0) 15 16 17m = scipy.array([[S * (1 - down), B * scipy.exp(r * T)], [S * (1 + up), 18 B * scipy.exp(r * T)]]) 19payoff = scipy.array([f(S * (1 - down), K), f(S * (1 + up), K)]) 20 21portfolio = scipy.linalg.solve(m, payoff) 22value = scipy.vdot(portfolio, scipy.array([S, B])) 23 24print portfolio: phi=, portfolio[0], psi=, portfolio[1], \n 25print derivative value: , value, \n __________________________________________________________________________ ### 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 continuously compounded 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 and the stock stays somewhere in between, changing by a factor $M$ with probability ${p}_{M}$ where $D 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.
5. Modify the scripts to accept interactive inputs of important parameters and to output results nicely formatted and rounded to cents.

__________________________________________________________________________

### References

[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.

__________________________________________________________________________

__________________________________________________________________________

I check all the information on each page for correctness and typographical errors. Nevertheless, some errors may occur and I would be grateful if you would alert me to such errors. I make every reasonable eﬀort to present current and accurate information for public use, however I do not guarantee the accuracy or timeliness of information on this website. Your use of the information from this website is strictly voluntary and at your risk.

I have checked the links to external sites for usefulness. Links to external websites are provided as a convenience. I do not endorse, control, monitor, or guarantee the information contained in any external website. I don’t guarantee that the links are active at all times. Use the links here with the same caution as you would all information on the Internet. This website reﬂects the thoughts, interests and opinions of its author. They do not explicitly represent oﬃcial positions or policies of my employer.

Information on this website is subject to change without notice.