# A binomial option-pricing scheme for knock-out barrier options

Dr. Burak Yildiz,

Industrial Engineering Department,

Baskent University, Ankara, Türkiye

[buraky@baskent.edu.tr](mailto:buraky@baskent.edu.tr)

[Alternative mail: tr.burak.yildiz@gmail.com](mailto:tr.burak.yildiz@gmail.com)

Preprint - November 25, 2025

## Abstract.

We propose a combinatorial framework that extends the classical binomial option pricing model to incorporate knock-out barrier features. By employing combinatorial analysis, our approach constructs a discrete-time pricing scheme capable of valuing barrier options within the Cox-Ross-Rubinstein (CRR) lattice. This framework aims to provide an alternative formulation that complements existing approaches in the literature.

**Keywords:** Barrier Options, Binomial Model, Option Pricing

**JEL Codes:** G12, C63

## Introduction.

An option is a financial derivative that grants the holder the right, but not the obligation, to buy (call) or sell (put) an underlying asset at a specified strike price on or before a predetermined expiration date. Among the diverse family of derivatives, two fundamental categories are *vanilla options* and *barrier options*. Vanilla options, such as European-style calls and puts, are the most basic and widely traded instruments. Their valuation is well-established through discrete-time frameworks such as the Cox-Ross-Rubinstein (CRR) binomial model and its continuous-time counterpart, the Black-Scholes model.

Barrier options, in contrast, are *path-dependent* derivatives whose payoff depends not only on the terminal value of the underlying asset but also on whether the asset price has breached a pre-specified barrier during the option's life. Knock-out barrier options, for instance, become void once the underlying price crosses the barrier level. Despite their simplicity in definition, the valuation of barrier options poses additional challenges because their payoffs depend on the entire price trajectory, not solely on the final state. Consequently, a straightforward application of the CRR model is insufficient, as it does not natively accommodate the barrier constraint.

In this paper, we present a combinatorial extension of the CRR model designed to price knock-out barrier options in discrete time. The proposed approach introduces a counting-based mechanism that enforces the barrier condition directly within the binomial lattice, preserving its recombining nature and computational tractability. By leveraging combinatorial reasoning, the framework provides an alternative and systematic perspective for representing path restrictions. Rather than positioning this work as a definitive innovation, we aim to offer a *complementary formulation* that unifies binomial pricing intuition with combinatorial structure. To the best of our knowledge, such an explicit combinatorial construction has not been extensively discussed in the literature, and we hope it may serve as a foundation for further analytical developments or extensions to other path-dependent derivatives.

## Basic Literature.

Building upon the Black-Scholes framework (Merton, 1973), barrier options were first systematically studied by Reiner and Rubinstein (1991), who derived closed-form solutions for various knock-in and knock-out structures under continuous monitoring.
## Binomial option pricing formulas.

### Cox-Ross-Rubinstein (CRR), binomial option pricing formula

$T$ = Time to maturity (fraction of a year), $v$ = Volatility, $r$ = Risk-free interest rate

$K$ = Strike price, $S$ = Underlying security price

$H_u$ = Upside barrier, $H_d$ = Downside barrier

$$u = e^{v \sqrt{\frac{T}{n}}} > 1$$

$$d = \frac{1}{u} = e^{-v \sqrt{\frac{T}{n}}} < 1$$

$$p = \frac{e^{r \frac{T}{n-d}}}{u-d}$$

$$q = 1 - p$$

$$C_{\text{vanilla}} = e^{-rT} \sum_{j=0}^{n} \binom{n}{j} p^j q^{n-j} \max\{Su^j d^{n-j} - K, 0\}$$

$$P_{\text{vanilla}} = e^{-rT} \sum_{j=0}^{n} \binom{n}{j} p^j q^{n-j} \max\{K - Su^j d^{n-j}, 0\}$$

### Knock-out barrier options

We modified the above binomial coefficients slightly, and obtained formulas for knock-out barrier options using a counting argument as follows:

Down and out options pricing:

$$\text{Let, } H(n) = -\left(\left\lfloor \frac{\log(H_d/S)}{\log(d)} \right\rfloor + 1\right)$$

And introduce a new notation: $\ll A \gg = \max\{A, 0\}$

$$C_{\text{downout}} = e^{-rT} \sum_{j=0}^{n} \ll \binom{n}{j} - \ll \binom{n}{j-H(n)} \gg \gg p^j q^{n-j} \ll Su^j d^{n-j} - K \gg$$

$$P_{\text{downout}} = e^{-rT} \sum_{j=0}^{n} \ll \binom{n}{j} - \ll \binom{n}{j-H(n)} \gg \gg p^j q^{n-j} \ll K - Su^j d^{n-j} \gg$$

Up and out options pricing:

$$\text{Let, } H(n) = \left(\left\lfloor \frac{\log(H_u/S)}{\log(u)} \right\rfloor + 1\right)$$

$$C_{\text{upout}} = e^{-rT} \sum_{j=0}^{n} \ll \binom{n}{j} - \ll \binom{n}{j-H(n)} \gg \gg p^j q^{n-j} \ll Su^j d^{n-j} - K \gg$$

$$P_{\text{upout}} = e^{-rT} \sum_{j=0}^{n} \ll \binom{n}{j} - \ll \binom{n}{j-H(n)} \gg \gg p^j q^{n-j} \ll K - Su^j d^{n-j} \gg$$
A sketch of proof.

Observe that, the number of length-*n* paths with *j* up-moves equals the sum of the number of length-*n* − 1 paths with *j* up-moves ending in a down-move and the number of length-*n* − 1 paths with *j* − 1 up-moves ending in an up-move. This yields the Pascal’s Identity. In a similar way, we define the following:

$$
V(n,j) := \ll \binom{n}{j} - \ll \binom{n}{j-H(n)} \gg
$$

V(n, j) is the total valid up j moves. If we can also show that V(n, j) = V(n - 1, j) + V(n - 1, j - 1), then
proof is done.

$$
V(n-1,j) + V(n-1,j-1) = \ll \binom{n-1}{j} - \ll \binom{n-1}{j-H(n-1)} \gg + \ll \binom{n-1}{j-1} - \ll \binom{n-1}{j-1-H(n-1)} \gg
$$

Also using the Pascal’s Identity,

$$
\binom{n}{j} = \binom{n-1}{j} + \binom{n-1}{j-1}
$$

We will have the following identity,

$$
\binom{n}{j-H(n-1)} = \binom{n-1}{j-H(n-1)} + \binom{n-1}{j-H(n-1)-1} = \binom{n}{j-H(n)}
$$

Since it is not difficult to show that for $n \gg 0$, $H(n-1) = H(n)$.

$$
\text{If } 0 < \binom{n-1}{j} < \binom{n-1}{j-H(n-1)} \text{ and } H(n-1) > 0, \text{ then } \frac{n-1+H(n-1)}{2} < j.
$$

$$
\text{If } 0 < \binom{n-1}{j} < \binom{n-1}{j-H(n-1)} \text{ and } H(n-1) < 0, \text{ then } \frac{n-1+H(n-1)}{2} > j.
$$

$$
\text{If } 0 < \binom{n-1}{j-1} < \binom{n-1}{j-1-H(n-1)} \text{ and } H(n-1) > 0, \text{ then } \frac{n-1+H(n-1)}{2} < j-1.
$$

$$
\text{If } 0 < \binom{n-1}{jj-1} < \binom{n-1}{j-1-H(n-1)} \text{ and } H(n-1) < 0, \text{ then } \frac{n-1+H(n-1)}{2} > j-1.
$$

In these above 4 cases, $V(n-1,j) + V(n-1,j-1) = 0 + 0 = 0 = V(n,j)$

$$
\text{If } 0 \le j < H(n-1), \text{ then } \ll \binom{n-1}{j} - \ll \binom{n-1}{j-H(n-1)} \gg \gg = \binom{n-1}{j} \text{ and } \ll \binom{n-1}{j-1} - \ll \binom{n-1}{j-1-H(n-1)} \gg \gg = \binom{n-1}{j-1}
$$

$$
V(n-1,j) + V(n-1,j-1) = \binom{n-1}{j} + \binom{n-1}{j-1} = \binom{n}{j} = V(n,j)
$$

If $0 \le H(n-1) \le j-1 < n-1$ or $H(n-1) < 0 \le j-1 < n-1$ then

$$
\ll \binom{n-1}{j} - \ll \binom{n-1}{j-H(n-1)} \gg = \binom{n-1}{j} - \binom{n-1}{j-H(n-1)} \quad \text{and} \quad \ll \binom{n-1}{j-1} - \ll \binom{n-1}{j-1-H(n-1)} \gg = \binom{n-1}{j-1} - \binom{n-1}{j-1-H(n-1)}
$$

$$
V(n-1,j) + V(n-1,j-1) = \binom{n-1}{j} - \binom{n-1}{j-H(n-1)} + \binom{n-1}{j-1} - \binom{n-1}{j-1-H(n-1)} = V(n,j)
$$

An interesting case would be when $0 \le j = H(n-1) = H(n)$ then,

$$
\ll \binom{n-1}{j} - \ll \binom{n-1}{j-H(n-1)} \gg = \binom{n-1}{j} - 1 \quad \text{and} \quad \ll \binom{n-1}{j-1} - \ll \binom{n-1}{j-1-H(n-1)} \gg = \binom{n-1}{j-1}
$$

$$
V(n-1,j) + V(n-1,j-1) = \binom{n-1}{j} - 1 + \binom{n-1}{j-1} = V(n,j)
$$
# Numerical Comparisons with MATLAB and C++ library QuantLib.

## Scenario 1

$T = 0.1$ time to maturity

$\nu = 0.45$ volatility

$r = 0.04$ interest rate

$K = 140$ strike price

$S = 130$ spot price

$H_u = 170$ upside barrier

$H_d = 100$ downside barrier

<table><thead><tr><th></th><th>n=1000</th><th>n=10000</th><th>QuantLib</th><th>MATLAB</th></tr></thead><tbody><tr><th>C<sub>upout</sub></th><td>2.208811</td><td>2.191758</td><td>2.17958</td><td>2.1796</td></tr><tr><th>C<sub>downout</sub></th><td>3.828745</td><td>3.828053</td><td>3.77964</td><td>3.7701</td></tr><tr><th>P<sub>downout</sub></th><td>10.580378</td><td>10.498082</td><td>10.5036</td><td>10.5109</td></tr><tr><th>P<sub>upout</sub></th><td>13.267248</td><td>13.266388</td><td>13.2258</td><td>13.2177</td></tr></tbody></table>

**Notes:**

1. The C++ implementation of the proposed binomial summation formulas presented in this paper is available at:

<figure><img src="image_1.png" /></figure>

https://github.com/trproftrader/singlebarrier_options

2. Numerical results for $n = 1000$ and $n = 10000$ were generated using the file **barrier_quant_revised.cpp** in the above repository.

3. For comparison purposes, the **barrierbybls** function from **MATLAB** were used.

4. The **QuantLib** C++ library was installed on macOS via Homebrew using the command: "brew install quantlib"

## Scenario 2

If we narrow down the barriers, then

$H_u = 155$ upside barrier

$H_d = 120$ downside barrier

<table><thead><tr><th></th><th>n=1000</th><th>n=10000</th><th>QuantLib</th><th>MATLAB</th></tr></thead><tbody><tr><th>C<sub>upout</sub></th><td>0.584756</td><td>0.535966</td><td>0.533869</td><td>0.5349</td></tr><tr><th>C<sub>downout</sub></th><td>3.429163</td><td>3.429816</td><td>3.3806</td><td>3.3737</td></tr><tr><th>P<sub>downout</sub></th><td>1.362969</td><td>1.370508</td><td>1.35519</td><td>1.3597</td></tr><tr><th>P<sub>upout</sub></th><td>13.112414</td><td>13.089205</td><td>13.0522</td><td>13.0456</td></tr></tbody></table>

## Conclusion.

In practice, barrier options in the CRR model are most commonly priced using backward induction. This paper's formula bypasses this recursive process by providing a direct, closed-form summation at maturity $T$. Also, the paper presents the final result as a fully combinatorial framework built directly into the CRR formula. This provides an elegant and explicit structure that unifies the valuation into a single, path-counting expression that is not typically found.

## Funding Declaration.

The author declares that no specific funding or assistance was received for this study.
**Acknowledgement.**

I would like to express my deepest gratitude to Joe Pimbley for carefully reading and commenting constructively on all versions of this manuscript, for drawing my attention to the relevance of the reflection principle, and for providing valuable insights on the paper’s position within the literature. All remaining errors are my own.

**References.**

Black, F. & Scholes, M. (1973). *The Pricing of Options and Corporate Liabilities*. Journal of Political Economy, 81(3), 637–654.

Reiner, E. S. & Rubinstein, M. (1991). *Breaking Down the Barriers*. Risk, 4(8), 28–35.