Refresher Reading
The ArbitrageFree Valuation Framework
2021 Curriculum CFA Program Level II Fixed Income
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The idea that market prices will adjust until there are no opportunities for arbitrage underpins the valuation of fixedincome securities, derivatives, and other financial assets. It is as intuitive as it is wellknown. For a given investment, if the net proceeds are zero (e.g., buying and selling the same dollar amount of stocks) and the risk is zero, the return should be zero. Valuation tools must produce a value that is arbitrage free. The purpose of this reading is to develop a set of valuation tools for bonds that are consistent with this notion.
The reading is organized around the learning objectives. After this brief introduction, Section 2 defines an arbitrage opportunity and discusses the implications of no arbitrage for the valuation of fixedincome securities. Section 3 presents some essential ideas and tools from yield curve analysis needed to introduce the binomial interest rate tree. In this section, the binomial interest rate tree framework is developed and used to value an optionfree bond. The process used to calibrate the interest rate tree to match the current yield curve is introduced. This step ensures that the interest rate tree is consistent with pricing using the zerocoupon (i.e., spot) curve. The final topic presented in the section is an introduction of pathwise valuation. Section 4 describes a Monte Carlo forwardrate simulation and its application. A summary of the major results is given in Section 5.
Learning Outcomes
The member should be able to:

explain what is meant by arbitragefree valuation of a fixedincome instrument;

calculate the arbitragefree value of an optionfree, fixedrate coupon bond;

describe a binomial interest rate tree framework;

describe the backward induction valuation methodology and calculate the value of a fixedincome instrument given its cash flow at each node;

describe the process of calibrating a binomial interest rate tree to match a specific term structure;

compare pricing using the zerocoupon yield curve with pricing using an arbitragefree binomial lattice;

describe pathwise valuation in a binomial interest rate framework and calculate the value of a fixedincome instrument given its cash flows along each path;

describe a Monte Carlo forwardrate simulation and its application.
Summary
This reading presents the principles and tools for arbitrage valuation of fixedincome securities. Much of the discussion centers on the binomial interest rate tree, which can be used extensively to value both optionfree bonds and bonds with embedded options. The following are the main points made in the reading:

A fundamental principle of valuation is that the value of any financial asset is equal to the present value of its expected future cash flows.

A fixedincome security is a portfolio of zerocoupon bonds.

Each zerocoupon bond has its own discount rate that depends on the shape of the yield curve and when the cash flow is delivered in time.

In wellfunctioning markets, prices adjust until there are no opportunities for arbitrage.

The law of one price states that two goods that are perfect substitutes must sell for the same current price in the absence of transaction costs.

An arbitrage opportunity is a transaction that involves no cash outlay yet results in a riskless profit.

Using the arbitragefree approach, viewing a security as a package of zerocoupon bonds means that two bonds with the same maturity and different coupon rates are viewed as different packages of zerocoupon bonds and valued accordingly.

For bonds that are option free, an arbitragefree value is simply the present value of expected future values using the benchmark spot rates.

A binomial interest rate tree permits the short interest rate to take on one of two possible values consistent with the volatility assumption and an interest rate model.

An interest rate tree is a visual representation of the possible values of interest rates (forward rates) based on an interest rate model and an assumption about interest rate volatility.

The possible interest rates for any following period are consistent with the following three assumptions: (1) an interest rate model that governs the random process of interest rates, (2) the assumed level of interest rate volatility, and (3) the current benchmark yield curve.

From the lognormal distribution, adjacent interest rates on the tree are multiples of e raised to the 2σ power.

One of the benefits of a lognormal distribution is that if interest rates get too close to zero, then the absolute change in interest rates becomes smaller and smaller.

We use the backward induction valuation methodology that involves starting at maturity, filling in those values, and working back from right to left to find the bond’s value at the desired node.

The interest rate tree is fit to the current yield curve by choosing interest rates that result in the benchmark bond value. By doing this, the bond value is arbitrage free.

An optionfree bond that is valued by using the binomial interest rate tree should have the same value as discounting by the spot rates.

Pathwise valuation calculates the present value of a bond for each possible interest rate path and takes the average of these values across paths.

The Monte Carlo method is an alternative method for simulating a sufficiently large number of potential interest rate paths in an effort to discover how the value of a security is affected and involves randomly selecting paths in an effort to approximate the results of a complete pathwise valuation.