# CHAPTER 13

# MONTE CARLO SIMULATIONS ON TREES

In this chapter we introduce an important methodology for pricing interest rate securities and their derivatives: Monte Carlo Simulations. In Monte Carlo simulations, we use computers to simulate several interest rate scenarios in the future, and then obtain the value of the security by averaging an appropriate discounted value of the payoff. In this chapter, we develop the Monte Carlo simulation methodology on binomial trees, and show that we can value quite complex path-dependent interest rate derivatives. In Chapter 17 the Monte Carlo simulation methodology is extended to a more general framework.

## 13.1 MONTE CARLO SIMULATIONS ON A ONE-STEP BINOMIAL TREE

Consider a one-step risk neutral binomial tree in which the interest rate has an equal chance to move up or down the tree. We want to price an interest rate option that pays at time *T* = 0.5 (i.e. *i* = 1) if the interest rate increases. That is, the payoff of the option is *c*_{1} = 100 × max*(r*_{1} − *r*_{K}, 0), where *r*_{K} is the strike rate and *r*_{1} is the rate at time *i* = 1. Using risk neutral pricing, the value of the option is:

Table 13.1 computes the value of the option for a given binomial tree.

An alternative way of computing the expected future payoff is to *simulate* upward and downward movements in the tree using a computer. For instance, in Excel the function *RAND*() simulates a realization from ...