Binomial options pricing model - AbsoluteAstronomy.com

In finance

Finance

"Finance" is often defined simply as the management of money or “funds” management Modern finance, however, is a family of business activity that includes the origination, marketing, and management of cash and money surrogates through a variety of capital accounts, instruments, and markets created...

, the binomial options pricing model (BOPM) provides a generalizable numerical method

Numerical analysis

Numerical analysis is the study of algorithms that use numerical approximation for the problems of mathematical analysis ....

for the valuation of options

Option (finance)

In finance, an option is a derivative financial instrument that specifies a contract between two parties for a future transaction on an asset at a reference price. The buyer of the option gains the right, but not the obligation, to engage in that transaction, while the seller incurs the...

. The binomial model was first proposed by Cox

John C. Cox

John Carrington Cox is the Nomura Professor of Finance at the MIT Sloan School of Management. He is one of the world's leading experts on options theory and one of the inventors of the Cox–Ross–Rubinstein model for option pricing, as well as of the Cox–Ingersoll–Ross model for interest rate dynamics...

, Ross

Stephen Ross (economist)

Stephen Alan "Steve" Ross is the inaugural Franco Modigliani Professor of Financial Economics at the MIT Sloan School of Management. He is known for initiating several important theories and models in financial economics...

and Rubinstein

Mark Rubinstein

Mark Edward Rubinstein is a leading financial economist and financial engineer. He is currently Professor of Finance at the Haas School of Business of the University of California, Berkeley, where he is involved in teaching courses in the , an academic program that is focused on equipping...

in 1979. Essentially, the model uses a “discrete-time” (lattice based

Lattice model (finance)

In finance, a lattice model can be used to find the fair value of a stock option; variants also exist for interest rate derivatives.The model divides time between now and the option's expiration into N discrete periods...

) model of the varying price over time of the underlying

Underlying

In finance, the underlying of a derivative is an asset, basket of assets, index, or even another derivative, such that the cash flows of the derivative depend on the value of this underlying...

financial instrument. In general, binomial options pricing models do not have closed-form solutions.

Use of the model

The Binomial options pricing model approach is widely used as it is able to handle a variety of conditions for which other models cannot easily be applied. This is largely because the BOPM is based on the description of an underlying instrument over a period of time rather than a single point. As a consequence, it is used to value American options that are exercisable at any time in a given interval as well as Bermudan options that are exercisable at specific instances of time. Being relatively simple, the model is readily implementable in computer software (including a spreadsheet

Spreadsheet

A spreadsheet is a computer application that simulates a paper accounting worksheet. It displays multiple cells usually in a two-dimensional matrix or grid consisting of rows and columns. Each cell contains alphanumeric text, numeric values or formulas...

).

Although computationally slower than the Black–Scholes formula, it is more accurate, particularly for longer-dated options on securities with dividend

Dividend

Dividends are payments made by a corporation to its shareholder members. It is the portion of corporate profits paid out to stockholders. When a corporation earns a profit or surplus, that money can be put to two uses: it can either be re-invested in the business , or it can be distributed to...

payments. For these reasons, various versions of the binomial model are widely used by practitioners in the options markets.

For options with several sources of uncertainty (e.g., real option

Real option

Real options valuation, also often termed Real options analysis, applies option valuation techniques to capital budgeting decisions. A real option itself, is the right — but not the obligation — to undertake some business decision; typically the option to make, abandon, expand, or contract a...

s) and for options with complicated features (e.g., Asian option

Asian option

An Asian option is a special type of option contract. For Asian options the payoff is determined by the average underlying price over some pre-set period of time...

s), binomial methods are less practical due to several difficulties, and Monte Carlo option model

Monte Carlo option model

In mathematical finance, a Monte Carlo option model uses Monte Carlo methods to calculate the value of an option with multiple sources of uncertainty or with complicated features....

s are commonly used instead. When simulating a small number of time steps Monte Carlo simulation will be more computationally time-consuming than BOPM (cf. Monte Carlo methods in finance

Monte Carlo methods in finance

Monte Carlo methods are used in finance and mathematical finance to value and analyze instruments, portfolios and investments by simulating the various sources of uncertainty affecting their value, and then determining their average value over the range of resultant outcomes. This is usually done...

). However, the worst-case runtime of BOPM will be O(2ⁿ), where n is the number of time steps in the simulation. Monte Carlo simulations will generally have a polynomial time complexity, and will be faster for large numbers of simulation steps.

Methodology

The binomial pricing model traces the evolution of the option's key underlying variables in discrete-time. This is done by means of a binomial lattice (tree), for a number of time steps between the valuation and expiration dates. Each node in the lattice represents a possible price of the underlying at a given point in time.

Valuation is performed iteratively, starting at each of the final nodes (those that may be reached at the time of expiration), and then working backwards through the tree towards the first node (valuation date). The value computed at each stage is the value of the option at that point in time.

Option valuation using this method is, as described, a three-step process:

price tree generation,
calculation of option value at each final node,
sequential calculation of the option value at each preceding node.

STEP 1: Create the binomial price tree

The tree of prices is produced by working forward from valuation date to expiration.

At each step, it is assumed that the underlying instrument will move up or down by a specific factor (

) per step of the tree (where, by definition,

and

). So, if

is the current price, then in the next period the price will either be

.

The up and down factors are calculated using the underlying volatility

Volatility (finance)

In finance, volatility is a measure for variation of price of a financial instrument over time. Historic volatility is derived from time series of past market prices...

, and the time duration of a step,

, measured in years (using the day count convention

Day count convention

In finance, a day count convention determines how interest accrues over time for a variety of investments, including bonds, notes, loans, mortgages, medium-term notes, swaps, and forward rate agreements . This determines the amount transferred on interest payment dates, and also the calculation of...

of the underlying instrument). From the condition that the variance

Variance

In probability theory and statistics, the variance is a measure of how far a set of numbers is spread out. It is one of several descriptors of a probability distribution, describing how far the numbers lie from the mean . In particular, the variance is one of the moments of a distribution...

of the log of the price is

, we have:

The above is the original Cox, Ross, & Rubinstein (CRR) method; there are other techniques for generating the lattice, such as "the equal probabilities" tree. The Trinomial tree

Trinomial Tree

The Trinomial tree is a lattice based computational model used in financial mathematics to price options. It was developed by Phelim Boyle in 1986. It is an extension of the Binomial options pricing model, and is conceptually similar...

is a similar model, allowing for an up, down or stable path.

The CRR method ensures that the tree is recombinant, i.e. if the underlying asset moves up and then down (u,d), the price will be the same as if it had moved down and then up (d,u) — here the two paths merge or recombine. This property reduces the number of tree nodes, and thus accelerates the computation of the option price.

This property also allows that the value of the underlying asset at each node can be calculated directly via formula, and does not require that the tree be built first. The node-value will be:

Where

is the number of up ticks and

is the number of down ticks.

STEP 2: Find Option value at each final node

At each final node of the tree — i.e. at expiration of the option — the option value is simply its intrinsic

Option time value

In finance, the time value of an option is the premium a rational investor would pay over its current exercise value , based on its potential to increase in value before expiring. This probability is always greater than zero, thus an option is always worth more than its current exercise value...

, or exercise, value.

Max [ (

), 0 ], for a call option

Call option

A call option, often simply labeled a "call", is a financial contract between two parties, the buyer and the seller of this type of option. The buyer of the call option has the right, but not the obligation to buy an agreed quantity of a particular commodity or financial instrument from the seller...

Max [ (

–

), 0 ], for a put option

Put option

A put or put option is a contract between two parties to exchange an asset, the underlying, at a specified price, the strike, by a predetermined date, the expiry or maturity...

Where

is the strike price

Strike price

In options, the strike price is a key variable in a derivatives contract between two parties. Where the contract requires delivery of the underlying instrument, the trade will be at the strike price, regardless of the spot price of the underlying instrument at that time.Formally, the strike...

and

is the spot price of the underlying asset at the

period.

STEP 3: Find Option value at earlier nodes

Once the above step is complete, the option value is then found for each node, starting at the penultimate time step, and working back to the first node of the tree (the valuation date) where the calculated result is the value of the option.

In overview: the “binomial value” is found at each node, using the risk neutrality

Risk-neutral measure

In mathematical finance, a risk-neutral measure, is a prototypical case of an equivalent martingale measure. It is heavily used in the pricing of financial derivatives due to the fundamental theorem of asset pricing, which implies that in a complete market a derivative's price is the discounted...

assumption; see Risk neutral valuation. If exercise is permitted at the node, then the model takes the greater of binomial and exercise value at the node.

The steps are as follows:

(1) Under the risk neutrality assumption, today's fair price

Fair value

Fair value, also called fair price , is a concept used in accounting and economics, defined as a rational and unbiased estimate of the potential market price of a good, service, or asset, taking into account such objective factors as:* acquisition/production/distribution costs, replacement costs,...

of a derivative

Derivative (finance)

A derivative instrument is a contract between two parties that specifies conditions—in particular, dates and the resulting values of the underlying variables—under which payments, or payoffs, are to be made between the parties.Under U.S...

is equal to the expected value

Expected value

In probability theory, the expected value of a random variable is the weighted average of all possible values that this random variable can take on...

of its future payoff discounted by the risk free rate

Risk-free interest rate

Risk-free interest rate is the theoretical rate of return of an investment with no risk of financial loss. The risk-free rate represents the interest that an investor would expect from an absolutely risk-free investment over a given period of time....

. Therefore, expected value is calculated using the option values from the later two nodes (Option up and Option down) weighted by their respective probabilities—“probability” p of an up move in the underlying, and “probability” (1-p) of a down move. The expected value is then discounted at r, the risk free rate

Risk-free interest rate

corresponding to the life of the option.

The following formula to compute the expectation value is applied at each node:

Binomial Value = [ p × Option up + (1-p) × Option down] × exp (- r × Δt), or

where

is the option's value for the

node at time

is chosen such that the related binomial distribution simulates the geometric Brownian motion

Geometric Brownian motion

A geometric Brownian motion is a continuous-time stochastic process in which the logarithm of the randomly varying quantity follows a Brownian motion, also called a Wiener process...

of the underlying stock with parameters r and σ,

is the dividend yield

Dividend yield

The dividend yield or the dividend-price ratio on a company stock is the company's total annual dividend payments divided by its market capitalization, or the dividend per share, divided by the price per share. It is often expressed as a percentage...

of the underlying corresponding to the life of the option. It follows that in a risk-neutral world futures price should have an expected growth rate of zero and therefore we can consider

for futures.

Note that for

to be in the interval

the following condition on

has to be satisfied

(2) This result is the “Binomial Value”. It represents the fair price of the derivative at a particular point in time (i.e. at each node), given the evolution in the price of the underlying to that point. It is the value of the option if it were to be held—as opposed to exercised at that point.

(3) Depending on the style of the option, evaluate the possibility of early exercise at each node: if (1) the option can be exercised, and (2) the exercise value exceeds the Binomial Value, then (3) the value at the node is the exercise value.

For a European option, there is no option of early exercise, and the binomial value applies at all nodes.
For an American option, since the option may either be held or exercised prior to expiry, the value at each node is: Max (Binomial Value, Exercise Value).
For a Bermudan option, the value at nodes where early exercise is allowed is: Max (Binomial Value, Exercise Value); at nodes where early exercise is not allowed, only the binomial value applies.

In calculating the value at the next time step calculated—i.e. one step closer to valuation—the model must use the value selected here, for “Option up”/“Option down” as appropriate, in the formula at the node.

The following algorithm

Algorithm

In mathematics and computer science, an algorithm is an effective method expressed as a finite list of well-defined instructions for calculating a function. Algorithms are used for calculation, data processing, and automated reasoning...

demonstrates the approach computing the price of an American put option, although is easily generalized for calls and for European and Bermudan options:



 function americanPut(T, S, K, r, sigma, q, n) {

     '        T... expiration time

     '        S... stock price

     '        K... strike price

     '        n... height of the binomial tree

  

     deltaT := T / n;

     up := exp(sigma * sqrt(deltaT));

  

     p0 := (up * exp(-r * deltaT) - exp(-q * deltaT)) * up / (up^2 - 1);

     p1 := exp(-r * deltaT) - p0;

  

     ' initial values at time T

     for i := 0 to n {

         p[i] := K - S * up^(2*i - n);

         if p[i] < 0 then p[i] := 0;

     }

  

     ' move to earlier times

     for j := n-1 down to 0 {

         for i := 0 to j {

             p[i] := p0 * p[i] + p1 * p[i+1];    ' binomial value

             exercise := K - S * up^(2*i - j);  ' exercise value

             if p[i] < exercise then p[i] := exercise;

         }

     }

  

     return americanPut := p[0];

 }

Discrete dividends

In practice, the use of continuous dividend yield,

, in the formula above can lead to significant mis-pricing of the option near an ex-dividend date. Instead, it is common to model dividends as discrete payments on the anticipated future ex-dividend dates.

To model discrete dividend payments in the binomial model, apply the following rule:

At each time step, , calculate , for all where is the present value of the -th dividend. Subtract this value from the value of the security price at each node (, ).

Relationship with Black–Scholes

Similar assumptions underpin both the binomial model and the Black–Scholes model, and the binomial model thus provides a discrete time approximation to the continuous process underlying the Black–Scholes model. In fact, for European options without dividends, the binomial model value converges on the Black–Scholes formula value as the number of time steps increases. The binomial model assumes that movements in the price follow a binomial distribution; for many trials, this binomial distribution approaches the normal distribution assumed by Black–Scholes. In addition, when analyzed as numerical procedure, the CRR binomial method can be viewed as a special case of explicit finite difference method for Black–Scholes PDE.

In 2011, Georgiadis shows that the binomial options pricing model has a lower bound on complexity that rules out a closed-form solution.

Use of the model

Methodology

STEP 1: Create the binomial price tree

STEP 2: Find Option value at each final node

STEP 3: Find Option value at earlier nodes

Discrete dividends

Relationship with Black–Scholes

See also

Discussion

Variations

Computer implementations