This is the second tutorial of two covering derivatives for the Level II CFA Exam. The topics of options, swaps, and credit default swaps can be intimidating for some candidates, particularly given the volume of associated formulas. One recommended prep approach is to start with the conceptual aspects of derivatives and then apply the formulas. […]

# Derivatives Part 2

## Introduction to Options

An option is a derivative instrument that gives its holder a right to buy or sell an asset, at a set price (the strike price) on a set date (contract expiration). A call option gives its owner the right to buy; it is not a promise to buy. For example, a store holding an item […]

## Synthetic Options and Rationale

The prices of put and call options have an identity relationship through the concept of put-call parity. c0 + X/(1+rF)T = p0 + S0 c0 = Current price of the European call p0 = Current price of the European put X = Strike price of the put and the call T = Time to expiration […]

## One Period Binomial Option Pricing Model

The following model can be used for options on stocks, currencies, and commodities; points on interest rate option pricing will be made at the end of this section. A critical component for option pricing with the one period binomial model is the notion of constructing a hedged portfolio. H = nS – c H = […]

## Call Option Price Formula

Call option price formula for the single period binomial option pricing model: c = (πc+ + (1-π) c–) / (1 + r) π = (1+r-d) / (u-d) “π” and “1-π” can be called the risk neutral probabilities because these values represent the price of the underlying going up or down when investors are indifferent to […]

## Binomial Interest Rate Options Pricing

Similar to the applications of the interest rate tree discussed in fixed income, a variation of the binomial option pricing model as presented can be used for options on bonds and interest rates. The analyst will need to: Create an interest rate tree of future spot rates. Calibrate the interest rate tree so current prices […]

## Black-Scholes-Merton (BSM) Option Pricing Model

Commonly called “Black-Scholes” outside the CFA exam world. BSM is a model for deriving the price of an option. Assumptions Stock returns are lognormally distributed. The risk free rate is known and stays constant during the option term. The stock’s volatility is known and stays constant during the option term. Transaction costs are omitted from […]

## Black-Scholes-Merton Model and the Greeks

The Black-Scholes-Merton model has six inputs (or five, if gamma is considered a sub-part of delta); five are known as the Greeks. Delta: The change in the option price per one dollar change in the underlying stock’s price; alternatively, the change in the option price equals the change in the underlying multiplied by the option’s […]

## Dynamic Delta Hedging & Gamma Related Issues

Traders and securities dealers can use an option’s delta to create hedges for the price risk exposure that they have in other option or underlying asset positions. Example, assume that a securities dealer has sold (short position) 100 call options on Ford Motors (NYSE: F) and that each option represents 100 shares. Thus, the dealer’s […]

## Estimating Volatility for Option Pricing

A key challenge in using either Black-Scholes-Merton or a binomial option pricing model is accurately estimating the stock’s return volatility. Two common approaches for estimating volatility: Historical Approach: This assumes that past volatility is representative of future volatility. For BSM, the annualized standard deviation of price returns is applied. σannual = σperiodic * √periods per […]