A derivative is a contract whose payoff depends on the value of some underlying asset. In cases where closed-form derivative pricing may be complex or even impossible, numerical procedures excel. A numerical procedure is the use of iterative computational methods in attempting to converge to a solution. One such basic implementation is a binomial tree. In a binomial tree, a node represents the state of an asset at a certain point of time associated with a price. Each node leads to two other nodes in the next time step. Similarly, in a trinomial tree, each node leads to three other nodes in the next time step. However, as the number of nodes or the time steps of trees increase, so do the computational resources consumed. Lattice pricing attempts to solve this problem by storing only the new information at each time step, while reusing values where possible.
In finite difference pricing, the nodes of the tree can also be represented as a grid. The terminal values on the grid consist of terminal conditions, while the edges of the grid represent boundary conditions in asset pricing. We will discuss the explicit method, implicit method, and the Crank-Nicolson method of the finite differences schemes to determine the price of an asset.
Although vanilla options and certain exotics such as European barrier options and lookback options can be found to have a closed-form solution, other exotic products such as Asian options do not contain a closed-form solution. In these cases, the pricing of options can be used with numerical procedures.
In this chapter, we will cover the following topics:
- Pricing European and American options using a binomial tree
- Using a Cox-Ross-Rubinstein (CRR) binomial tree
- Pricing options using a Leisen-Reimer (LR) tree
- Pricing options using a trinomial tree
- Pricing options using a binomial and trinomial lattice
- Deriving Greeks from a tree for free
- Finite differences with the explicit, implicit, and Crank-Nicolson method
- Implied volatility modelling using a LR tree and the bisection method
An option is a derivative of an asset that gives an owner the right but not the obligation to transact the underlying asset at a certain date for a certain price, known as the maturity date and strike price respectively.
A call option gives the buyer the right to buy an asset by a certain date for a certain price. A seller or writer of a call option is obligated to sell the underlying security to the buyer at the agreed price, should the buyer exercise his/her rights on the agreed date. A put option gives the buyer the right to sell the underlying asset by a certain date for a certain price. A seller or writer of a put option is obligated to buy the underlying security from the buyer at the agreed price, should the buyer exercise his/her rights on the agreed date.
The most common options available are the European options and American options. Other exotic options include Bermudan options and Asian options. This chapter will deal mainly with European and American options. An European option can only be exercised on the maturity date. An American option may be exercised at any time throughout the lifetime of the option.