Notes on Options Valuation and Pricing Models

Option Valuation Models

Overview

The standard discounted cash flow (DCF) method fails to effectively value options, necessitating more complex valuation techniques. This lecture introduces two principal approaches:

1. Option Equivalent Approach: also known as the replicating portfolio method which replicates the payoffs from the option by creating a strategic investment involving stocks and borrowed funds.

2. Risk-Neutral Method: a more theoretical approach that assumes investors are indifferent to risk, simplifying the valuation process by calculating expected future value and discounting back at the risk-free rate.

Replicating Portfolio

Concept

A replicating portfolio consists of a combination of investments in the underlying asset (stock) and risk-free borrowing. Its net cost yields the option's value.

Example

Consider purchasing a six-month call option on Amazon shares with an exercise price of $1,830, where the stock also trades at this price. Assuming a short-term risk-free interest rate of 2%, the price could either rise to $2,196 (20% increase) or drop to $1,525 (20% decrease). The possible payoffs from the call option would be:

• If stock price = $1,525, the payoff = $0.

• If stock price = $2,196, the payoff = $366 (Max[0, (2,196 - 1,830)]).

Comparing these with a direct investment strategy, purchasing 0.54545 shares and borrowing the PV of the exercise price shows identical payoffs to those from the call, perpetuating the law of one price - both investments must have the same value today.

Value Calculations

Using the formulas for option payoffs and replication, you can ascertain:

• Value of the Call Option = Value of 0.54545 shares - Value of the loan (including interest).

• Consequently, both payoffs are equal, signifying the effectiveness of the replicating portfolio in option valuation.

Option Delta and Hedging

Definition

The hedge ratio, or option delta (δ), is the number of shares required to replicate the option's payoffs. Mathematically, it can be expressed as:
[ δ = \frac{Cu - Cd}{Su - Sd} ]
Where Cu and Cd represent the call option values in the up and down states, and Su and Sd indicate the stock prices in those states.

Implementation

You create your payoff structure by calculating delta shares and borrowing needs. This leverages the reduced capital requirement compared to direct stock investment, providing higher risk but potential reward.

Risk-Neutral Valuation

Framework

1. All investors are assumed to be indifferent to risk, simplifying expected stock return calculations.

2. The expected return from stock growth must equal the risk-free rate during evaluation.

3. Probabilities (p) allow for the straightforward calculation of expected option values: [ E(option) = puPayoff + (1-p)*dPayoff ]

4. Risk-neutral valuation leads to the conclusion that the Amazon Call Option's expected value can be calculated without direct reference to investor risk preferences.

Calculation Example

For the previous Amazon Call Option, expected future value is computed as:
[ E(C) = (p)(366) + ((1-p)(0)) = (0.5091)(366) ]
This results in a present value assessment, aligning consistent calculations across valuation methods.

Valuation of Put Options

Framework

The process of calculating put options parallels that of call options. It's essential to compute expected payoffs similar to the replicating portfolio approach. With an Amazon put option:

• If stock falls to $1,525, the put payoff = $305.

• Conversely, if stock rises to $2,196, the payoff = $0.

Delta Calculation

The delta of a put option is inherently negative, requiring the configuration of negatively correlated shares. The relationship between call and put options, derived from put-call parity, further aids in validating option prices across valuation approaches:
[ P = C + PV(X) - S0 ]

The Binomial Method

Overview

This method simplifies valuation further by allowing for variations in stock price at discrete intervals instead of continuous models, reflecting real market situations. This often provides a more granular way to assess option value by systematically analyzing two potential outcomes at each interval.

Application

Using scenarios where stock changes are represented as discrete rises or falls (up/down), the method can manage probabilities, allowing for extensive historical data utilization to evaluate stock behavior over specified periods.
 

1. Create a binomial price tree demonstrating possible future prices.

2. Assess risk-neutral probabilities at each node to calculate option payoffs under each outcome.

Black-Scholes Pricing Model (BSOPM)

Fundamentals

Developed by Fischer Black and Myron Scholes, this model provides a closed-form solution for pricing European options. Key assumptions include no dividends, a constant risk-free rate, and efficient market conditions promoting randomness in asset pricing.

Assumptions

• Perfect capital markets: all investors can borrow and lend at the same risk-less interest.

• Risk-less rate of interest is known and constant over the life of option (same for variance of underlying asset)

• No transaction costs or taxes when buying and selling option/assets

• No restrictions or penalties for short selling

• Underlying assets is efficiently priced and pay no dividends

• Options are European

Applications

BSOPM is prevalent for valuing European-style options due to its flexibility and efficiency in adapting to continuous asset price models. Most importantly, it makes assumptions conducive for real-world applications, facilitating market analysis and profitability assessment for investors.

BSOPM Formula

The value of a call option can be computed using:
[ C = N(d1)S - N(d2)PV(X) ]

Interpretation

Understanding N(d1) and N(d2) helps gauge how likely an option is to expire in-the-money, with high values indicating strong positioning for an option's profitability.