# CHAPTER 17

# RISK NEUTRAL PRICING AND MONTE CARLO SIMULATIONS

Risk neutral pricing is by far the most utilized methodology to price derivative securities. In this chapter, we review its foundation – no arbitrage and the Feynman-Kac theorem – and apply it to obtain prices and hedge ratios of relatively complex interest rate securities. One important implication of risk neutral pricing is that it provides the theoretical foundation for using Monte Carlo simulations as a tool for pricing.

## 17.1 RISK NEUTRAL PRICING

Consider the general process

where *m(r, t)* and *s(r, t)* are functions of *r* and time *t.* For instance, in the case of the Vasicek model, we had *m(r, t)= λ( – r)* and *s(r, t)= σ.* In Chapter 15 we obtained the Fundamental Pricing Equation by following these steps:

- Consider a portfolio of two interest rate securities;
- Choose the number of units in the portfolio in order to make it
*riskless;*
- Use Ito’s lemma to find the dynamics of the portfolio capital gains;
- Impose no arbitrage, and thus, the portfolio return must equal the risk-free rate;
- Obtain a Partial Differential Equation that needs to be satisfied by every security.

The result was the following: Let *V (r, t)* be the value of any security that depends on the interest rater. Let *T* be the maturity of the interest rate security ...