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In finance , the binomial options pricing model BOPM provides a generalizable numerical method for the valuation of options. The binomial model was first proposed by Cox , Ross and Rubinstein in In general, Georgiadis showed that binomial options pricing models do not have closed-form solutions. The Binomial options pricing model approach is widely used since 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. Although computationally slower than the Black—Scholes formula, it is more accurate, particularly for longer-dated options on securities with dividend 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. 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. However, the worst-case runtime of BOPM will be O 2 n , 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. Monte Carlo simulations are also less susceptible to sampling errors, since binomial techniques use discrete time units. This becomes more true the smaller the discrete units become. 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. At each step, it is assumed that the underlying instrument will move up or down by a specific factor or 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 or.
The up and down factors are calculated using the underlying volatility , , and the time duration of a step, , measured in years using the day count convention of the underlying instrument. From the condition that the variance of the log of the price is , we have:. The Trinomial tree is a similar model, allowing for an up, down or stable path.
The CRR method ensures that the tree is recombinant, i. 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. At each final node of the tree — i. Where is the strike price and is the spot price of the underlying asset at the period.
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.
If exercise is permitted at the node, then the model takes the greater of binomial and exercise value at the node. The expected value is then discounted at r , the risk free rate corresponding to the life of the option. It represents the fair price of the derivative at a particular point in time i. It is the value of the option if it were to be held—as opposed to exercised at that point. In calculating the value at the next time step calculated—i. The following algorithm demonstrates the approach computing the price of an American put option, although is easily generalized for calls and for European and Bermudan options:.
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. 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 lognormal distribution assumed by Black—Scholes. In addition, when analyzed as a numerical procedure, the CRR binomial method can be viewed as a special case of the explicit finite difference method for the Black—Scholes PDE; see Finite difference methods for option pricing.
In , Georgiadis shows that the binomial options pricing model has a lower bound on complexity that rules out a closed-form solution. Binomial options pricing model In finance , the binomial options pricing model BOPM provides a generalizable numerical method for the valuation of options. Journal of Financial Economics. Energy derivative Freight derivative Inflation derivative Property derivative Weather derivative.