Part 3: Bridging Binomial Logic with Black-Scholes Precision
How Option Pricing Evolves from Discrete Steps to Continuous-Time Dynamics
In the previous post, we described a particular type of process in which variables change in a way that cannot be predicted with certainty. In statistics, we call such uncertainty a stochastic process because we try to rationalize it by assigning different probabilities to the possible outcomes1.
Specifically, we considered a price at time t0 (the initial step) equal to 100, and a scenario where, at time t1 (the final step), the price could either fall to 90 or rise to 110. This is an example of a discrete-time stochastic process, where changes occur only at predefined time intervals. Moreover, it is also discrete in space, meaning the underlying asset can only take on certain specific values within a defined range.
We determined the fair value of a call option under this model by applying the no-arbitrage principle.
The Binomial Model is stochastic under the Risk-Neutral Probabilities (see part 2).
A stochastic process, by definition, models uncertainty over time by assigning a probability distribution to future outcomes.
In the standard ("real-world") version of a stochastic process, these probabilities might reflect empirical observations or subjective beliefs.
When we introduce risk-neutral probabilities, we are not removing the randomness of the process. Instead, we are changing the probability measure under which we evaluate expected values. Specifically, we are moving from the real-world measure to the risk-neutral measure, as we explained in the previous post.
Thus, even though we are using different probabilities for valuation purposes, the process is still stochastic because the outcome remains governed by chance. The switch to risk-neutral probabilities simply allows us to value the call option (and, nevertheless, the put option) in a way that reflects market consistency and arbitrage-free pricing, not to eliminate randomness itself.
Although the underlying assets of options are not traded 24/7 and are quoted at discrete intervals (e.g., in increments of 5 cents), it is generally considered acceptable to model indices, futures, stocks, and ETFs as continuous variables that follow a continuous-time stochastic process.
The opposite of a stochastic process is a deterministic system, which always yields the same result given a specific set of inputs.
A classic example is planetary motion: if we know the current positions and velocities of the planets, we can calculate their positions and velocities 1,000 years into the future with precision.
This is not the case with prices. Knowing today’s prices does not allow us to determine tomorrow’s prices with certainty. Financial markets are inherently uncertain, and prices are modeled as random variables within a probabilistic framework rather than deterministic ones.
A continuous stochastic process is formally defined using a mathematical tool that is widely employed to describe physical, social, and economic phenomena: the differential equation.
A differential equation resembles a regular equation, but instead of unknown variables (as in a typical equation such as x+4=0 ), the unknowns are functions2.
This formalism allows us to model how variables change over time in a structured yet probabilistic way, which is essential when working with continuous-time.
There are several ways to derive the Black-Scholes equation. The approach we present here mirrors the logic used in the binomial model: it starts from the no-arbitrage principle.
As we did with the binomial model, we assume we have two different portfolios that are guaranteed to have the same value at expiration. According to the One Price Law, these portfolios must have the same value today. If they didn't, an arbitrage opportunity would exist: one could buy the cheaper portfolio and sell the more expensive one for a risk-free profit.
In the binomial model, this principle leads to an algebraic equation system, solved by backward induction: we know the two possible values at expiration, and we derive the call value at inception.
The B&S model introduces 2 complications:
There are infinitely many possible outcomes, rather than just two.
The hedge needs to be continuously adjusted as prices evolve over time (not just period zero and period 1).
So the equation to derive the call value goes through a little modification:
In the binomial model, the call option's value is a single number.
In contrast, in the Black-Scholes model, the option's value is a function: C = C(S,t); here’s our differential equation.
This means that the value of the call is not a fixed number but instead depends continuously on two variables:
The current price of the underlying asset (S)
The time to expiration (t)
So, it is no longer a point value as in the binomial model, but a set of values defined over the entire space of possible combinations of S and t. It fully satisfies the mathematical definition of a function: it assigns a unique output (option value) to every possible input pair S and t.
The value of the call must now take into account that the outcomes are infinite, while the outcome probabilities are still defined in the risk-neutral probability framework.
It is important to note that these infinite outcomes are, however, contained within a predefined constant volatility, as in the binomial model.
This is the limitation of Black & Scholes (see the introduction to this series).
We also note in the equation the presence of P instead of the L.
In contrast to the binomial model, where we express the option value as a linear combination of (Delta x S) and a fixed cash position L, the Black-Scholes model describes a continuously evolving system. The hedge must be adjusted at every instant because the underlying asset follows a stochastic (random) process in continuous time.
In this setting, a fixed amount of cash L isn't sufficient to replicate the changing value of the option over time. Instead, we use P, which is not a constant but a continuously changing liquid risk-free position. This position earns the risk-free3 rate and adjusts to match the evolving option price as the market moves.
Think of P as the dynamic counterpart to L.
P must be interpreted as the dynamically adjusted component that changes as S and t change, which, in turn, impact the option value.
In the binomial model, a fixed cash balance L offsets the risk from holding Delta shares.
In the Black-Scholes model, P evolves with time and does the same job, but in a continuously updating way.
This leads us to a special type of differential equation known as a partial differential equation (PDE), because it involves multiple independent variables (in our case, S and t) and captures how small changes in those variables affect the function C(S,t). We explore these infinitesimal variations using partial derivatives, which tell us how the option value responds to small changes in the underlying price and in time.
Below is a visualization of these concepts.
In the next article, we will delve into these two key variables, S and t, in detail and examine how the Black-Scholes framework models their impact on option prices, all while avoiding mathematical formalities.
A stochastic process is a mathematical model that describes how a system evolves with inherent randomness. In such processes, the outcome at any given time is not fully determined by the previous state, but instead follows a probability distribution.
Brief definition of a function: A function establishes a relationship between a set of inputs (the independent variable) and a set of possible outputs (the dependent variable). Each input corresponds to a single output, although multiple inputs can correspond to the same output.
In other words, for a given value of x, there is only one corresponding value of y. However, the same y value may be associated with different values of x.
Remember, we consider a zero risk-free rate for simplicity.