8.3 Valuation of Contingent Claims - 8.3 Valuation of Contingent Claims

Key Concepts

• Contingent Claims
• Options
• Futures and Forwards
• Black-Scholes Model
• Binomial Option Pricing Model
• Risk-Neutral Valuation

Contingent Claims

Contingent Claims are financial instruments whose value depends on the value of an underlying asset. These claims provide the holder with the right, but not the obligation, to buy or sell the underlying asset under certain conditions. Common examples include options, futures, and swaps.

Example: A call option on a stock gives the holder the right to buy the stock at a specified price (strike price) before a certain date (expiration date). The value of this option depends on the stock's price movements.

Options

Options are financial derivatives that give the holder the right, but not the obligation, to buy (call option) or sell (put option) an underlying asset at a specified price (strike price) on or before a specified date (expiration date). Options are used for hedging, speculation, and arbitrage.

Example: A call option with a strike price of $50 on a stock currently trading at $45 has no intrinsic value but has time value. If the stock price rises above $50, the option's intrinsic value increases, and the holder can exercise the option to buy the stock at $50.

Futures and Forwards

Futures and Forwards are contracts that obligate the buyer to purchase and the seller to sell an asset at a specified future date and price. Futures are traded on exchanges, while forwards are traded over-the-counter (OTC). These contracts are used to hedge against price fluctuations and for speculation.

Example: A farmer enters into a futures contract to sell corn at $4 per bushel in six months. If the market price of corn rises to $5 per bushel by the contract's expiration, the farmer will sell at the agreed-upon price of $4, effectively hedging against price risk.

Black-Scholes Model

The Black-Scholes Model is a mathematical model used to price European options. It considers the underlying asset's price, the option's strike price, the time to expiration, the risk-free interest rate, and the volatility of the underlying asset. The model assumes that the price of the underlying asset follows a lognormal distribution.

Example: Using the Black-Scholes Model, an analyst calculates the price of a European call option on a stock with a current price of $50, a strike price of $55, a risk-free rate of 2%, a time to expiration of 6 months, and an annualized volatility of 20%. The model provides a theoretical option price.

Binomial Option Pricing Model

The Binomial Option Pricing Model is a numerical method used to price options by constructing a binomial tree that represents possible future prices of the underlying asset. The model assumes that the asset's price can move up or down at each time step, and it calculates the option's value by backward induction.

Example: A two-period binomial tree is constructed for a stock with an initial price of $50. In each period, the stock price can either increase by 20% or decrease by 10%. The model calculates the option's value at each node of the tree, starting from the expiration date and working backward to the present.

Risk-Neutral Valuation

Risk-Neutral Valuation is a method used to price derivatives by assuming that investors are indifferent to risk. It involves discounting the expected payoff of the derivative at the risk-free rate, rather than the asset's expected return. This approach simplifies the valuation process by eliminating the need to estimate the asset's expected return.

Example: To value a call option using risk-neutral valuation, the expected payoff at expiration is calculated under the assumption that investors are risk-neutral. This expected payoff is then discounted at the risk-free rate to find the option's present value.