The Discrete Time Model



Two Alternative Binomial Option Pricing Model Approaches to Derive Black-Scholes Option Pricing Model[1]

CHENG-FEW LEE

Department of Finance and Economics, Rutgers Business School

Rutgers University, New Brunswick

New Jersey, U.S.

CARL S. LIN

Department of Economics

Rutger University, New Brunswick

New Jersey, U.S.

Abstract

In this chapter, we review two famous models on binomial option pricing, Rendleman and Barter (RB, 1979) and Cox, Ross, and Rubinstein (CRR, 1979). We show that the limiting results of the two models both lead to the celebrated Black-Scholes formula. From our detailed derivations, CRR is easy to follow if one has the advanced level knowledge in probability theory but the assumptions on the model parameters make its applications limited. On the other hand, RB model is intuitive and does not require higher level knowledge in probability theory. Nevertheless, the derivations of RB model are more complicated and tedious. For readers who are interested in the binomial option pricing model, they can compare the two different approaches and find the best one which fits their interests and is easier to follow.

1. Introduction

The main purpose of this chapter is to review two famous binomial option pricing model: Rendleman and Barter (RB, 1979) and Cox, Ross, and Rubinstein (CRR, 1979). First, we will give an alternative detailed derivation of the two models and show that the limiting results of the two models both lead to the celebrated Black-Scholes formula. Then we will make comparisons of the two different approaches and analyze the advantages of each approach.

Hence, this chapter can help to understand the statistical aspects of option pricing models for Economics and Finance professions. Also, it gives important financial and economic intuitions for readers in statistics professions. Therefore, by showing two alternative binomial option pricing models approaches to derive the Black-Scholes model, this chapter is useful for understanding the relationship between the two important optional pricing models and the Black-Scholes formula.

2. The Two-State Option Pricing Model of Rendleman and Bartter

In Rendleman and Bartter (1979), a stock price can either advance or decline during the next period. Let [pic]and [pic] represent the returns per dollar invested in the stock if the price rises (the + state) or falls (the - state), respectively, from time T-1 to time T. And [pic]and [pic] the corresponding end-of-period values of the option.

Let R be the riskless interest rate, Rendleman and Bartter (1979) show that the price of the option can be represented as a recursive form

[pic]

that can be applied at any time T-1 to determine the price of the option as a function of its value at time T.

2.1 The Discrete Time Model

From the above equation, the value of a call option at maturing date T-1 is given by

[pic] (2.1)

Similarly,

[pic] (2.2)

Substituting (2.1) into (2.2) can get,

[pic]

[pic] (2.3)

Noting that [pic], so (2.3) can be simplified as:

[pic] (2.4)

We can use this recursive form to get [pic]:

Since after T periods, there are [pic] ways that a sequence of (T) pluses can occur, [pic] ways that (T-1) pluses can occur, [pic] ways that (T-2) pluses can occur, and so on….

Hence, by Binomial Theorem [pic]can be represented as:

[pic]

[pic](2.5)

Next to determine the value of the option at maturity. Suppose that stock increases [pic] times and declines [pic] times, then the price of the stock will be [pic] on the expiration date. So the option will be exercised if

[pic][pic]

The maturity value of the option will be

[pic] (2.6)

Let [pic] denote the minimum integer value of [pic] in (2.6) for which the inequality is satisfied.

[pic] (2.7)

where[pic] is the integer operator.

i.e., taking natural logarithm of RHS of (2.6),

[pic]

Hence, the maturing value of the option is given by

[pic] (2.8)

Substituting (2.8) into (2.5), then the generalized option pricing equation for the discrete time is

[pic] (2.9)

2.2 The Continuous Time Model

For (2.9), we can write is as:

[pic](2.10)

Since [pic]+[pic], [pic], therefore, can interpret it as “pseudo probability”.

Let [pic] and [pic], we can restate (2.10) as:

[pic] (2.11)

where [pic] is the cumulative binomial probability function, the number of successes will fall between [pic] and [pic] after [pic] trials.

As [pic], [pic] (2.12)

where[pic]is the probability of a normally distributed random variable with zero mean and variance 1 taking values between a lower limit [pic] and a upper limit [pic]. And by the property of binomial pdf,

[pic]

Thus, [pic] (2.13)

Let [pic], then [pic]. And since [pic], the remaining things to be determined are [pic] and [pic].

From equation (10) and (11) of the text in the Rendleman and Bartter (1979),

[pic]

substituting [pic] and [pic] into (2.7), so

[pic]

In the limit, the term [pic] will be simplify to [pic]. So,

[pic] (2.14)

Substituting [pic], [pic] and [pic] into [pic],

[pic]

Now expanding in Taylor’s series in T,

[pic]

where,

[pic]

and

[pic]

where [pic] denotes a function tending to zero more rapidly than [pic].(when we expanding in Taylor’s series in T, the rest of the terms tending to zero more rapidly than [pic] so regard them as a function[pic].) Hence,

[pic]

and,

[pic]

After canceling terms,

[pic]

Now substituting [pic] for [pic] and [pic] for [pic] into (2.14),

[pic]

Similarly,

[pic]

Since [pic], let [pic], [pic], the continuous time version of the two-state model is obtained:

[pic]

The above equation is identical to the Black-Scholes model.

3. The Binomial Option Pricing Model of Cox, Ross and Rubinstein

In this section we will concentrate on the limiting behavior of the binomial option pricing model proposed by Cox, Ross and Rubinstein (CRR, 1979).

3.1 The Binomial Option Pricing Formula of CRR

Let [pic] be the current stock price, [pic] the option exercise price, [pic] the riskless rate. It is assumed that the stock follows a binomial process, from one period to the next it can only go up by a factor of [pic] with probability [pic] or go down by a factor of [pic]with probability [pic]. After n periods to maturity, CRR showed that the option price C is:

[pic] (3.1)

An alternative expression for C, which is easier to evaluate, is

[pic] (3.2)

where [pic]and m is the minimum number of upward stock movements necessary for the option to terminate in the money, i.e., m is the minimum value of k in (3.1) such that [pic]

3.2 Limiting Case

We now show that the binomial option pricing formula as given in Equation (3.2) will converge to the celebrated Black-Scholes option pricing model. The Black-Scholes formula is

[pic] (3.3)

where

[pic] (3.4)

[pic] = the variance of stock rate of return

t = the fixed length of calendar time to expiration date, such that [pic].

We wish to show that Equation (3.2) will coincide with Equation (3.3) when [pic].

In order to show the limiting result that the binomial option pricing formula converges to the continuous version of Black-Scholes option pricing formula, we suppose that h represents the lapsed time between successive stock price changes. Thus, if t is the fixed length of calendar time to expiration, and n is the total number of periods each with length h, then [pic]. As the trading frequency increases, h will get closer to zero. When [pic], this is equivalent to [pic].

Let [pic] be one plus the interest rate over a trading period of length h. Then, we will have

[pic] (3.5)

for any choice of n. Thus, [pic], which shows that [pic] must depend on n for the total return over elapsed time t to be independent of n. Also, in the limit, [pic] tends to [pic] as [pic].

Let S* be the stock price at the end of the nth period with the initial price S. If there are j up periods, then

[pic] (3.6)

where j is the number of upward moves during the n periods.

Since j is the realization of a binomial random variable with probability of a success being q, we have expectation of log (S*/S)

[pic] (3.7)

and its variance

[pic] (3.8)

Since we divide up our original longer time period t into many shorter subperiods of length h so that [pic], our procedure calls for making n longer, while keeping the length t fixed. In the limiting process we would want the mean and the variance of the continuously compounded log rate of return of the assumed stock price movement to coincide with that of actual stock price as [pic]. Let the actual values of [pic]and [pic] respectively. Then we want to choose u, d, and q in such a manner that[pic]and [pic] as [pic]. It can be shown that if we set

[pic]

[pic] (3.9)

[pic]

then [pic] and [pic]as [pic]. In order to proceed further, we need the following version of the central limit theorem.

Lyapounov’s Condition. Suppose [pic] are independent and uniformly bounded with [pic], [pic] and [pic]

If [pic] for some [pic] then the distribution of [pic] converges to the standard normal as [pic].

Theorem 1. If

[pic] (3.10)

then

[pic] (3.11)

where N(z) is the cumulative standard normal distribution function.

Proof. See Appendix.

It is noted that the condition (3.10) is a special case of the Lyapounov’s condition which is stated as follows. When [pic] we have the condition (3.10).

This theorem says that when the fixed length t is divided into many subperiods, the log rate of return will approach to the normal distribution when the number of subperiods approached infinity. For this theorem to hold, the condition stated in Equation (3.10) has to be satisfied. We next show that this condition is indeed satisfied.

We will next show that the binomial option pricing model as given in Equation (3.2) will indeed coincide with the Black-Scholes option pricing formula as given in Equation (3.3). Observe that[pic] is always equal to[pic], as evidenced from Equation (3.5). Thus, comparing the two option pricing formulae given in Equations (3.2) and (3.3), we see that there are apparent similarities. In order to show the limiting result, we need to show that as [pic],

[pic] and [pic]

In this section we will only show the second convergence result, as the same argument will hold true for the first convergence. From the definition of [pic], it is clear that

[pic] (3.12)

Recall that we consider a stock to move from S to uS with probability p and dS with probability (1-p). During the fixed calendar period of t=nh with n subperiods of length h, if there are j up moves, then

[pic]. (3.13)

The mean and variance of the continuously compounded rate of return for this stock are [pic]and [pic] where

[pic] and [pic].

From Equation (3.13) and the definitions for [pic] and [pic], we have

[pic]. (3.14)

Also, from the binomial option pricing formula we have

[pic]

where is a real number between 0 and 1.

From the definitions of [pic] and [pic], it is easy to show that

[pic]

Thus from Equation (3.12) we have

[pic] (3.15)

We will now check the condition given by Equation (3.10) in order to apply the central limit theorem. Now recall that

[pic],

with [pic], and d and u are given in Equation (3.9).

We have

[pic]

Hence, the condition given by Equation (10) is satisfied because

[pic]

Finally, in order to apply the central limit theorem, we have to evaluate [pic], [pic] and [pic] as [pic] It is clear that

[pic] and [pic].

Hence, in order to evaluate the asymptotic probability in Equation (3.12), we have

[pic]

Using the fact that [pic], we have, as [pic]

[pic]

Similar argument holds for[pic], and hence we completed the proof that the binomial option pricing formula as given in equation (3.2) includes the Block-Scholes option pricing formula as a limiting case.

4. Comparison of the Two Approaches

From the results of last two sections, we show that both RB and CRR models lead to the celebrated Black-Scholes formula. The following table shows the comparisons of the necessary mathematical and statistical knowledge and assumptions for the two models.

|Model |Rendleman and Bartter (1979) |Cox, Ross and Rubinstein (1979) |

|Mathematical and |Basic Algebra |Basic Algebra |

|Probability Theory |Taylor Expansion |Taylor Expansion |

|Knowledge |Binomial Theorem |Binomial Theorem |

| |Central Limit Theorem |Central Limit Theorem |

| |Properties of Binomial Distribution |Properties of Binomial Distribution |

| | |Lyapounov’s Condition |

|Assumption |1. The distribution of returns of the stock is |The stock follows a binomial process from one period to|

| |stationary over time and the stock pays no |the next it can only go up by a factor of “u” with |

| |dividends.(Discrete Time Model) |probability “p” or go down by a factor of “d” with |

| | |probability “1-p”. |

| |2. The mean and variance of logarithmic returns of the |In order to apply the Central Limit Theorem, “u”, “d”, |

| |stock are held constant over the life of the |and “p” are needed to be chosen. |

| |option.(Continuous Time Model) | |

|Advantage |1. Readers who have undergraduate level training in |1. Readers who have advanced level knowledge in |

|and |mathematics and probability theory can follow this |probability theory can follow this approach; but for |

|Disadvantage |approach. |those who don’t, CRR approach may be difficult to |

| |2. The approach of RB is intuitive. But the derivation |follow. |

| |is more complicated and tedious than the approach of |2. The assumption on the parameters “u”, “d”, “p” makes|

| |CRR. |CRR approach more restricted than RB approach. |

Hence, like we indicate in the table, CRR is easy to follow if one has the advanced level knowledge in probability theory but the assumptions on the model parameters make its applications limited. On the other hand, RB model is intuitive and does not require higher level knowledge in probability theory. However, the derivation is more complicated and tedious.

For readers who are interested in the binomial option pricing model, they can compare the two different approaches and find the best one which fits their interests and is easier to follow.

Appendix

The Binomial Theorem

[pic]

Lindberg-Levy Central Limit Theorem

If [pic] are a random sample from a probability distribution with finite mean[pic] and finite variance [pic] and [pic], then

[pic]

Proof of Theorem 1.

Since

[pic]

And

[pic]

We have [pic] .

Thus

[pic]

Hence the condition for the theorem to hold as stated in Equation (3.10) is satisfied.

References

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Cox, J., S. A. Ross and M. Rubinstein. “Option Pricing: A Simplified Approach”. Journal of Financial Economics, 7 (1979, pp. 229-263.)

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MacBeth, J. and L. Merville. “An Empirical Examination of the Black-Scholes Call Option Pricing Model,” The Jollrnal of Finance, 34 (December 1979, pp. 1173-86).

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[1] Section 3 of this chapter is essentially drawing from the paper by Lee et al.(2004).

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