From binomial expectations to the Black-Scholes formula : the main ideas

We show how a discrete random variable on a finite probability space endowed with a binomial distribution may be close to a random variable on the continuum, in a way which respects the expectation. As an application, we approximate the random variables of a discrete multiplicative binomial process by continuous exponentials, and thus derive an option pricing formula, which contains the formula of Black and Scholes as a special case.


Introduction
We study continuous approximations of discrete expressions in the context of elementary probability theory and option pricing. The main result is a sort of extension of the De Moivre-Laplace central limit theorem, and concerns the approximation of the expectation of a random variable with respect to a binomial distribution by an expectation with respect to the standard normal distribution.
Our study is motivated by the derivation of the Black-Scholes formula (see [?]) for the pricing of European call options. In [?], J. C. Cox, S. A. Ross and NI.
presented an option pricing formula in the form of a discrete binomial expectation, and then they showed that in the limit it converged to the Black-Scholes formula.
As a consequence of our main theorem we obtain a pricing formula for continuous options, of which the Black-Scholes formula is a special case. Our derivation is both more direct and more general than the derivation of Cox, Ross and Rubinstein: we reduce their sum formula to a Riemann-sum of the Black-Scholes integral formula. However, our setting is still their simple discrete pricing model, and thus avoids entirely the complications of limits of stochastic processes, continuous stochastic processes and measure theory. Instead, we apply nonstandard analysis, and following N. G. Cutland, E. Kopp and W. Willinger [6], we assume that the time steps of the discrete model are infinitesimal. With respect to their approach to option pricing, we obtained a further simplification, by avoiding the transitions between a standard and a nonstandard model, and Loeb-measure theory.
where h is a discrete random variable defined on the xi's.
We show that under a suitable condition the above reasoning can be extended to this sum, leading to the approximations where h is a continuous real function, closely related to h. Indeed we have the following main result Theorem 2.1 (Main Theorem). Let +oo, 0 p 1 and Qp be the probability space given by ( 2.I ) and endowed with the binomial distribution BN,p. Let h : : Qp --~ R be a random variable of class S°, and of S-exponential order in +oo.
The theorem transforms an expectation with respect to the binomial distribution into an expectation with respect to the standard normal distribution. We remark that the formal nonstandard proof is very similar to the observations above. See [3].
3 Discrete arithmetic and geometric Brownian motions Our application concerns the approximation of the expectation of a random variable with respect to a discrete geometric binomial process S(t, x). This process will be . defined on an arithmetic binomial network. Let T > 0, N E N, and dt > 0 be such that Ndt = T. . Then WT,dt is the network given by Then indeed the process is defined on WT,dt. ~Ve assume that the upper increment of (3.7) has conditional probability p, and the lower increment has conditional probability 1p, and that the increments are independent in time. Then S(t, x) is properly defined as a stochastic process, and up to elementary transformations its random variables S(t) = S(t, . ) have binomial distributions. In particular Pr = So (I + pdt + 03C3dt)j 1 + pdt -03C3dt)N-j} = BN,p(j). . Note that if p = 1/2, then is the relative conditional expectation, or drif t rate of the process and o~2 its relative conditional variance, or volatility.

Expectations and option pricing
In the economic context of option pricing, the process S(t, x) endowed with the conditional probability p = ~ is considered as a model describing the possible movements in time of the price of a share of some stock; trading is allowed at the times 0, dt, 2dt, ... , T }, the drift rate of the stock price being equal to ~u, and its volatility cr. Given a real-valued function f, the random variable f(S(T)) models a claim on that share at the future time T. For instance, let K > 0. Then the claim f(S(T)) = (S(T) -K)+ is called the European call option with exercise date T and with striking price K. It models the payoff of a contract giving its owner the right to buy the share S at time T for the price K. In fact, we described a stochastic process which is suitable for the discrete option pricing model of Cox, Ross and Rubinstein. They argue (see also [4]) that if r is the risk-free rate of interest, the correct price Cdt of the claim f must be the Present Value (henceforth P1g) of the expectation of the random variable j(S(T)) in a risk-neutral world (that is, the drift rate of the process S must be r). Let then Er f(S(T)) (4.8) denote the expectation of the random variable f (S(T)) in a risk-neutral world. Then Cdt = (4.9) Recall that the present value in a risk-neutral world of an asset A equals its future value A(T) at time T discounted at the risk-free rate of interest. That is to say

=.=1(T)/(1 + rdt)T/dt
If the process S(t, x) is in a risky world, (that is, its drift rate /z is different from r) then it is always possible to adjust its conditional probability p to some value p(r) which will change its drift rate to the prescribed risk-free rate of interest r E R. Note that p(r) must satisfy so p(r) 2 + r 2Q~ dt   The integral of the right-hand side of ( 4.13 ) is the Feynman-Kac formula (see [8]). From this we obtain the Black-Scholes formula by a straightforward standard transformation. Theorem 4.1 (Black-Scholes formula). . Let T > 0 be appreciable and WT be an infinitesimal arithmetic binomial network. Let S(t, x) be the discrete geometric Brownian motion on WT with appreciable initial value So > 0, limited drift rate ,u and appreciable volatility 03C32. Let r be a limited risk-free rate of interest. Let Co = ° (PYr(Er f (S(T)))) be the shadow of the price of a the European call option (S(T) -K)+ with striking price K and exercise date T. . Put _ log(So/K) -(r -a2/2)T x0 = 03C3T Then C0 ~ So . N xo + 03C3T) -.N (xo). . (4.14) Notice that ( 4.14 ) becomes an identity if So, K, and T are standard.
There are three main differences between the work of Cutland, Kopp and Willinger [6] and our approach in [3]. First to estimate S(T) they use a nonstandard continuous they use the Loeb-measure and Loeb-spaces [12], while we use Riemannsums, such as sketched above, and the external numbers of [10] and [11]. Third, their setting is Robinsonian nonstandard analysis [16], while our setting is axiomatic nonstandard analysis IST [14]. The main difference is that in the latter approach the infinitesimals are included within the set of real numbers R, while in the former approach they are included in a nonstandard extension of R.