Security Valuation via Option Pricing

Security valuation using option pricing is like figuring out how much a financial product is worth by looking at options (buying or selling agreements). Options give the owner the choice to buy (call option) or sell (put option) an asset at a set price within a specific time. Options are a right and not an obligation.

Using option pricing to value security is like peeking into its story beyond the usual numbers. It’s like seeing what it is now and what people think it might be in the future, giving a more complete picture for investors.

Imagine you’re selling your old video game. Option pricing is like considering not just the game itself but also how popular it is, how quickly people buy or sell it, and how long it will be before a new version comes out.

Let’s discuss the basics of option pricing and different methods to determine the value of various Financial instruments.

Options Pricing 101

Market participants widely use option pricing to assess the value of a company’s share, particularly in the case of private companies with intricate ownership structures. This is common for private companies with complicated ownership structures.

Option Pricing is a way to distribute the overall value of a company among its different types of investments. It helps allocate the value among different types of investments. Option Pricing doesn’t determine the total value of the company. So, a company’s total value has to be found first before using the option pricing model. This model works well for companies with longer liquidity and many cash-out ways.

How Does the Option Pricing Model Work?

Option pricing models, such as the Black-Scholes and binomial models, are mathematical tools used to estimate the fair value of financial options. These models help in valuing financial securities, particularly options.

Option pricing is the process of assigning a value to financial choices. It’s a tool that helps us understand the market’s expectations for the future and make informed decisions, allowing us to navigate the complexities of risk in finance.

Black-Scholes Model:

It was introduced in 1973 and has been widely adopted in finance for valuing options and other derivatives. The model provides a formula to estimate an option’s fair market value at a given time and the risk-free interest rate.

Below are the other assumptions made in this model:

Exercise of options is only permissible at expiration for European call options.

The underlying asset does not pay a dividend during the option’s life.

The volatility and risk-free rate of the underlying asset’s returns remain constant over the option’s lifespan.

The underlying asset exhibits a normal distribution of returns.

No transaction costs or taxes are associated with trading options or the underlying asset.

This takes into consideration 5 Key Variables:

• Call Option Price © — Price paid to buy the right, not the obligation.
• Underlying Price (S) — Current market price of the asset.
• Strike Price (K) — Price at which the option holder can buy or sell the asset.
• Time to expiration (t) — Time remaining until the option’s end.
• Risk-Free Interest Rate (r) — The rate of return on a risk-free investment.
• Volatility (σ) — Measure of the asset’s price fluctuations.

How to calculate the value using Black Scholes Formula:

C = SN(d1) — Ke-rt N (d2)

Where:

d1 = ln(S/K)+(r+(σ2)/2)⋅T

​ σ⋅√T

d2=d1−σ⋅√T

Call Option Price ©

C=S0⋅N(d1)−K⋅e−rT⋅N(d2)

Put Option Price (P)

P = K ⋅ e−rT ⋅ N(−d2) — S0 ⋅ N(−d1)

Example:

Suppose you want to calculate the theoretical price of a call option for a stock with the following parameters:

- Current stock price (S): $100

- Strike price (K): $105

- Time to expiration (T): 3 months (0.25 years)

- Risk-free interest rate (r): 5% per annum

- Volatility (σ): 20% per annum

Understand the Components

S = 100, K = 105, T = 0.25, r = 0.05, sigma = 0.20

Step 2: Calculate d1 and d2

d1 = ln(100/105) + (0.05 + (0.202)/2). 0.25 = -0.0458

0.20.√0.25

d2 = -0.0458–0.20 . √0.25 = -0.2958

Calculate Call Option Price ©

C = 100 . N(-0.0458) — 105 . e -0.05.0.25 . N (-0.2958)

Using standard normal distribution tables or a calculator, N(-0.0458) = 0.4872 and N(-0.2958) = 0.3829.

C =100 . 0.4872–105 . e-0.05.0.25 0.3829

C = 4.59

So, the theoretical price of the call option is approximately $4.59.

According to the Black-Scholes Model, the fair value of the European call option under these parameters is $4.59.

Binomial Models

The Binomial Pricing Model is a mathematical model for pricing financial derivatives, particularly options. Cox, Ross, and Rubinstein introduced the model in 1979. The model assumes the underlying asset’s price follows a binomial distribution over time.

Basic Assumptions:

1. The model divides time into discrete periods, typically equal, during which the underlying asset’s price can move.
2. The underlying asset’s price can only increase or decrease during each period.
3. The model assumes a risk-neutral world, meaning investors are indifferent to risk when valuing securities.

Parameters:

• Initial Stock Price (S0) — The underlying asset’s current price.
• Upward Movement Factor (u) — The factor by which the stock price increases upward.
• Downward Movement Factor (d) — The factor that decreases the stock price in a downward movement.
• Risk-Free rate (r) — The rate at which an investor can lend or borrow money without risk.
• Time to Maturity (T) — The total period until the option expires.

Binomial Tree:

The model creates a binomial tree to represent the possible price movements of the underlying asset over time. Each node in the tree represents a possible price at a specific point in time.

At each node, two branches represent the upward and downward movements. The values of the nodes are calculated based on the up and down factors.

Option Pricing:

• The model is commonly used to price European and American options. At each node, the option value is calculated based on the probability-weighted average of the values in the next period.
• The model determines the option value at each node by discounting the expected future payoffs at the risk-free rate.

Limitations of Option Pricing Model:

The Black-Scholes and binomial option pricing models are widely used in finance to value options, but they have limitations and challenges. Here are some of the key issues with these models and insights into when alternative valuation approaches may be necessary:

Assumptions:

• Both models assume constant variability over the option’s life, which may not hold in real-world scenarios. It can change due to market events, and taking consistent variability can lead to inaccurate pricing.
• The Black-Scholes model assumes continuous trading, while the binomial model assumes discrete time intervals. In reality, markets may not trade continuously, and the discrete nature of trading could affect the accuracy of the models.
• The models assume a constant risk-free interest rate, which may not reflect market conditions accurately. Changes in interest rates can impact option prices, and using a regular rate may lead to mispricing.

Market Frictions:

• Both models do not account for transaction costs, which can be important in real-world trading. Ignoring transaction costs can lead to overestimation of potential profits.
• Large trades can impact market prices, especially in markets with low trading volume. The models do not consider the market impact of large transactions, which can result in unrealistic valuations.

Behavioral Factors:

• Both models assume a continuous price movement, neglecting sudden jumps or discontinuities in the underlying asset’s price. In reality, markets can experience unexpected events that lead to significant price jumps, and the models do not capture these.
• The models assume that asset returns follow a normal distribution, which may be false. Extreme events, such as financial crises, can lead to non-normal distributions, impacting the accuracy of the models.

Numerical Challenges:

The binomial model can be computationally intensive, especially for many time steps. More advanced numerical methods or closed-form solutions may be preferable in such cases.

Manage your Stock Options with Eqvista!

Often, pricing options provide a way for companies to show their commitment to Investors. Pricing the options allows the value of the options to be accurate for different stock categories. Eqvista assists companies in managing the stock options for their investors. The team at Eqvista is well-versed in the issues surrounding stock option pricing. Contact Eqvista today to get started.