Option binomial model
Rearranging the above three equations to solve for parameters p , u and d leads to, Equation 4: The unique solution for parameters p , u and d given in Equation 4 ensures that over a short period of time the binomial model matches the mean and variance of an asset in a risk free world, and as will be seen shortly, ensures that for a multi-step model the price of the underlying asset is symmetric around the starting price S 0.
Before considering the more general case of a many-step model, consider the two-step model shown in Figure 2 Figure 2: A Two-Step Binomial Model. As with the one-step model of Figure 1 , over the first period of time in the two-step model the asset price may move either up to S u or down to S d. Over the second period, if the price moved up to S u in the first period then the price may move to either S uu or S ud.
However if the price moved down in the first period to S d then in the second period it may move to either S du or S dd. However if they are not equal then the price tree is said to be non-recombining or bushy. Since there are typically tens if not hundred or thousands of time steps taken when pricing an option the amount of data and hence computer memory, and computation time required to calculate a non-bushy tree is typically prohibitively and hence they are rarely used.
The third equation of the CRR model ensures that it generates a recombining tree that is centred around the original stock price S 0. Taking multiple time steps leads to the tree shown in Figure 3. A Multi-Step Binomial Model. In general the time period between today and expiry of the option is sliced into many small time periods. A tree of potential future asset prices is then calculated. Each point in the tree is refered to as a node.
The tree contains potential future asset prices for each time period from today through to expiry. The second step in pricing options using a binomial model is to calculate the payoffs at each node corresponding to the time of expiry. This corresponds to all of the nodes at the right hand edge of the price tree.
In general the payoff may depend on many different factors. As an example, the payoffs of simple put and call options will use the standard formulae. The third step in pricing options using a binomial model is to discount the payoffs of the option at expiry nodes back to today. This is achieved by a process called backwards induction , and involves stepping backwards through time calculating the option value at each node of the lattice in a sequential manner.
This is achieved using the appropriate following formulae. It is critical to notice that with backwards inducton the counter n starts at N i. Following the three step procedure described above the value of the option V 0 may be calculated. A companion option pricing tutorial discusses the mathematics behind several alternative binomial models. Back To Top Option Pricing. Option Pricing Using The Binomial Model Binomial models and there are several are arguably the simplest techniques used for option pricing.
Calculate potential future prices of the underlying asset s at expiry and possibly at intermediate points in time too. Calculate the payoff of the option at expiry for each of the potential underlying prices. Discount the payoffs back to today to determine the option price today. Each of those steps is discussed in the following sections.
Calculating a Tree for the Underlying Asset Price The first step in pricing options using a binomial model is to create a lattice, or tree, of potential future prices of the underlying asset s. The notation used is, S 0: One Step Binomial Model The essence of the model is this: These include methods developed by, Cox-Ross-Rubinstein: This is the method most people think of when discussing the binomial model, and the one discussed in this tutorial.
This is commonly called the equal-probability model. This is commonly called the moment matching model. This is a modification of the original Judd-Yarrow model that incorporates a risk-neutral probablity rather than an equal probability. This is a modification of the original Cox-Ross-Runinstein model that incorporates a drift term that effects the symmetry of the resultant price lattice.
This uses a completely different approach to all the other methods, relying on approximating the normal distrbution used in the Black-Scholes model. A Risk-Neutral World Three equations are required to be able to uniquely specify values for the three parameters of the binomial model. Third Equation for the Cox-Ross-Rubinstein Binomial Model Rearranging the above three equations to solve for parameters p , u and d leads to, Equation 4: The Binomial options pricing model approach has been 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. 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:. At each final node of the tree—i.