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Binomial CRR model

The Cox–Ross–Rubinstein (CRR) binomial tree is the classic discrete-time option-pricing model. It is valuable because it makes replication and risk-neutral probabilities completely explicit, and because it converges to Black–Scholes as the time step shrinks.

Objectives

  • State the assumptions that make the one-period binomial market arbitrage-free and complete.
  • Derive the CRR risk-neutral probability and the pricing recursion.
  • See how the tree converges to Black–Scholes for European options.
  • Understand why binomial trees are convenient for early-exercise features (American options).

1. One-period market recap

Over one time step \(\Delta t\), assume:

  • A risk-free asset grows by \(R = e^{r\Delta t}\).
  • The stock moves from \(S\) to \(Su\) (up) or \(Sd\) (down), with \(u>d>0\).

No-arbitrage condition

To avoid arbitrage you need

\[ d < R < u. \]

If \(R\ge u\), borrow at \(r\) and buy the stock (dominates the bond). If \(R\le d\), short the stock and invest in the bond.

2. Replication and risk-neutral probability

Let a derivative pay \(V_u\) in the up state and \(V_d\) in the down state. Hold \(\Delta\) shares and \(B\) in the bond so that

\[ \Delta Su + BR = V_u,\qquad \Delta Sd + BR = V_d. \]

Solving gives

\[ \Delta = \frac{V_u - V_d}{S(u-d)}, \qquad B = \frac{uV_d - dV_u}{R(u-d)}. \]

The time-0 price is \(V_0=\Delta S + B\), which can be rearranged into a risk-neutral expectation:

\[ \boxed{ V_0 = \frac{1}{R}\left(p^* V_u + (1-p^*)V_d\right), } \qquad p^* := \frac{R-d}{u-d}. \]

Here \(p^*\in(0,1)\) precisely when \(d<R<u\). This is the CRR risk-neutral probability.

3. Multi-period tree and backward induction

With \(n\) steps, you build the tree of possible stock prices and compute option values backwards:

  1. Set terminal values \(V_{n,j} = g(S_{n,j})\) at maturity.
  2. For \(k=n-1,\dots,0\),
\[ V_{k,j} = \frac{1}{R}\left(p^* V_{k+1,j+1} + (1-p^*)V_{k+1,j}\right). \]

For American options, replace step 2 with

\[ V_{k,j} = \max\left\{ g(S_{k,j}),\; \frac{1}{R}\left(p^* V_{k+1,j+1} + (1-p^*)V_{k+1,j}\right)\right\}, \]

which is where trees shine: early exercise is handled naturally by the max operator.

CRR chooses

\[ u = e^{\sigma\sqrt{\Delta t}},\qquad d = e^{-\sigma\sqrt{\Delta t}}, \]

so the tree variance matches \(\sigma^2\Delta t\) to first order. Then

\[ p^* = \frac{e^{r\Delta t}-e^{-\sigma\sqrt{\Delta t}}}{e^{\sigma\sqrt{\Delta t}}-e^{-\sigma\sqrt{\Delta t}}}. \]

As \(\Delta t\to 0\) (i.e., \(n\to\infty\) for fixed \(T\)), the binomial price of a European call converges to the Black–Scholes price. Intuitively:

  • the log-stock performs a random walk with step size \(\sigma\sqrt{\Delta t}\),
  • the central limit theorem turns that walk into Brownian motion,
  • and the risk-neutral drift matches the continuous-time \(r\) drift.

References

  • Cox, Ross & Rubinstein (1979), “Option Pricing: A Simplified Approach”.
  • Shreve, Stochastic Calculus for Finance I, binomial-tree chapters.
  • Hull, Options, Futures, and Other Derivatives, chapters on binomial trees.