Digital option pricing model
To get a slope of , you buy calls at and you sell calls at. How much will the above portfolio cost? You earn from selling the calls, and pay for the calls. The net cost is: Many complicated payoffs can be re-created as combinations of vanilla puts and calls. Digital Call Options A digital call option with is similar - it pays off one dollar if at expiration, and pays off zero otherwise: As a starting point, consider buying a call with and selling a call with: Consider buying two calls with and selling two calls at: Given that the slope is , to get an infinite slope, we take the limit as goes to zero.
It might look more familiar if I re-wrote it as: Conclusion Many complicated payoffs can be re-created as combinations of vanilla puts and calls.
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 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.