Valuation of Contingent Claims
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Introduction
A contingent claim is a derivative instrument that provides its owner a right but not an obligation to a payoff determined by an underlying asset, rate, or other derivative. Contingent claims include options, the valuation of which is the objective of this reading. Because many investments contain embedded options, understanding this material is vital for investment management.
Our primary purpose is to understand how the values of options are determined. Option values, as with the values of all financial instruments, are typically obtained using valuation models. Any financial valuation model takes certain inputs and turns them into an output that tells us the fair value or price. Option valuation models, like their counterparts in the forward, futures, and swaps markets, are based on the principle of no arbitrage, meaning that the appropriate price of an option is the one that makes it impossible for any party to earn an arbitrage profit at the expense of any other party. The price that precludes arbitrage profits is the value of the option. Using that concept, we then proceed to introduce option valuation models using two approaches. The first approach is the binomial model, which is based on discrete time, and the second is the Black–Scholes–Merton (BSM) model, which is based on continuous time.
The reading is organized as follows. Section 2 introduces the principles of the noarbitrage approach to pricing and valuation of options. In Section 3, the binomial option valuation model is explored, and in Section 4, the BSM model is covered. In Section 5, the Black model, being a variation of the BSM model, is applied to futures options, interest rate options, and swaptions. Finally, in Section 6, the Greeks are reviewed along with implied volatility. Section 7 provides a summary.
Learning Outcomes

describe and interpret the binomial option valuation model and its component terms;

calculate the noarbitrage values of European and American options using a twoperiod binomial model;

identify an arbitrage opportunity involving options and describe the related arbitrage;

calculate and interpret the value of an interest rate option using a twoperiod binomial model;

describe how the value of a European option can be analyzed as the present value of the option’s expected payoff at expiration;

identify assumptions of the Black–Scholes–Merton option valuation model;

interpret the components of the Black–Scholes–Merton model as applied to call options in terms of a leveraged position in the underlying;

describe how the Black–Scholes–Merton model is used to value European options on equities and currencies;

describe how the Black model is used to value European options on futures;

describe how the Black model is used to value European interest rate options and European swaptions;

interpret each of the option Greeks;

describe how a delta hedge is executed;

describe the role of gamma risk in options trading;

define implied volatility and explain how it is used in options trading.
Summary
This reading on the valuation of contingent claims provides a foundation for understanding how a variety of different options are valued. Key points include the following:

The arbitrageur would rather have more money than less and abides by two fundamental rules: Do not use your own money and do not take any price risk.

The noarbitrage approach is used for option valuation and is built on the key concept of the law of one price, which says that if two investments have the same future cash flows regardless of what happens in the future, then these two investments should have the same current price.

Throughout this reading, the following key assumptions are made:

Replicating instruments are identifiable and investable.

Market frictions are nil.

Short selling is allowed with full use of proceeds.

The underlying instrument price follows a known distribution.

Borrowing and lending is available at a known riskfree rate.


The twoperiod binomial model can be viewed as three oneperiod binomial models, one positioned at Time 0 and two positioned at Time 1.

In general, Europeanstyle options can be valued based on the expectations approach in which the option value is determined as the present value of the expected future option payouts, where the discount rate is the riskfree rate and the expectation is taken based on the riskneutral probability measure.

Both Americanstyle options and Europeanstyle options can be valued based on the noarbitrage approach, which provides clear interpretations of the component terms; the option value is determined by working backward through the binomial tree to arrive at the correct current value.

For Americanstyle options, early exercise influences the option values and hedge ratios as one works backward through the binomial tree.

Interest rate option valuation requires the specification of an entire term structure of interest rates, so valuation is often estimated via a binomial tree.

A key assumption of the Black–Scholes–Merton option valuation model is that the return of the underlying instrument follows geometric Brownian motion, implying a lognormal distribution of the return.

The BSM model can be interpreted as a dynamically managed portfolio of the underlying instrument and zerocoupon bonds.

BSM model interpretations related to N(d1) are that it is the basis for the number of units of underlying instrument to replicate an option, that it is the primary determinant of delta, and that it answers the question of how much the option value will change for a small change in the underlying.

BSM model interpretations related to N(d2) are that it is the basis for the number of zerocoupon bonds to acquire to replicate an option and that it is the basis for estimating the riskneutral probability of an option expiring in the money.

The Black futures option model assumes the underlying is a futures or a forward contract.

Interest rate options can be valued based on a modified Black futures option model in which the underlying is a forward rate agreement (FRA), there is an accrual period adjustment as well as an underlying notional amount, and that care must be given to daycount conventions.

An interest rate cap is a portfolio of interest rate call options termed caplets, each with the same exercise rate and with sequential maturities.

An interest rate floor is a portfolio of interest rate put options termed floorlets, each with the same exercise rate and with sequential maturities.

A swaption is an option on a swap.

A payer swaption is an option on a swap to pay fixed and receive floating.

A receiver swaption is an option on a swap to receive fixed and pay floating.

Long a callable fixedrate bond can be viewed as long a straight fixedrate bond and short a receiver swaption.

Delta is a static risk measure defined as the change in a given portfolio for a given small change in the value of the underlying instrument, holding everything else constant.

Delta hedging refers to managing the portfolio delta by entering additional positions into the portfolio.

A delta neutral portfolio is one in which the portfolio delta is set and maintained at zero.

A change in the option price can be estimated with a delta approximation.

Because delta is used to make a linear approximation of the nonlinear relationship that exists between the option price and the underlying price, there is an error that can be estimated by gamma.

Gamma is a static risk measure defined as the change in a given portfolio delta for a given small change in the value of the underlying instrument, holding everything else constant.

Gamma captures the nonlinearity risk or the risk—via exposure to the underlying—that remains once the portfolio is delta neutral.

A gamma neutral portfolio is one in which the portfolio gamma is maintained at zero.

The change in the option price can be better estimated by a deltaplusgamma approximation compared with just a delta approximation.

Theta is a static risk measure defined as the change in the value of an option given a small change in calendar time, holding everything else constant.

Vega is a static risk measure defined as the change in a given portfolio for a given small change in volatility, holding everything else constant.

Rho is a static risk measure defined as the change in a given portfolio for a given small change in the riskfree interest rate, holding everything else constant.

Although historical volatility can be estimated, there is no objective measure of future volatility.

Implied volatility is the BSM model volatility that yields the market option price.

Implied volatility is a measure of future volatility, whereas historical volatility is a measure of past volatility.

Option prices reflect the beliefs of option market participant about the future volatility of the underlying.

The volatility smile is a two dimensional plot of the implied volatility with respect to the exercise price.

The volatility surface is a three dimensional plot of the implied volatility with respect to both expiration time and exercise prices.

If the BSM model assumptions were true, then one would expect to find the volatility surface flat, but in practice, the volatility surface is not flat.
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