CXD Analytics - Options on Financial Securities

What are Options?

Financial options, commonly referred to as "options," are financial instruments that give the buyer the right, but not the obligation, to buy or sell an underlying asset at a predetermined price (called the "strike price") on or before a specific date (called the "expiration date"). Options are a type of derivative security, as their value is derived from the price movement of an underlying asset, such as stocks, commodities, currencies, or market indexes.

Option Types

Options are contracts that give the holder the right, but not the obligation, to buy or sell an underlying asset at a predetermined price, known as the strike price, within a specific time frame. There are two main types of options: call options and put options. A call option gives the holder the right to buy the underlying asset, while a put option gives the holder the right to sell the underlying asset.

American and European options are types of financial options contracts that give the option holder the right, but not the obligation, to buy or sell an underlying asset at a predetermined price (called the "strike price") before or at a specific date (called the "expiration date"). The primary difference between the two lies in the timing of when the option can be exercised.

American options: These options can be exercised at any time before or on the expiration date. This flexibility allows the option holder to react to change in the market, potentially capturing profits earlier or limiting losses. American options are commonly used for trading stocks and stock indices.
European options: These options can only be exercised on the expiration date itself, not before. This limitation means that the option's value can only be realized on a specific date, which can be less favorable for the option holder if market conditions change significantly prior to expiration. European options are typically used for trading foreign currencies and interest rate products.

Both American and European options can be either call options (the right to buy) or put options (the right to sell). The choice of option type depends on the investor's outlook on the underlying asset's future price movement and their specific investment strategy.

Payoffs of European Options

European options are options that can only be exercised at the expiration date. The payoff of a call option is max(0, S-K), where S is the price of the underlying asset and K is the strike price. The payoff of a put option is max(0, K-S).

Pricing Bounds for European Options

The pricing of European options is bounded between the value of the underlying asset and the present value of the strike price. The lower bound is zero, while the upper bound is the value of the underlying asset. This is because the holder of a call option can always choose not to exercise the option if the price of the underlying asset is lower than the strike price, while the holder of a put option can always choose to exercise the option if the price of the underlying asset is higher than the strike price.

Put-Call Parity for European Options

Put-call parity is a fundamental relationship between the prices of European call and put options. The put-call parity equation is C + PV(K) = P + S, where C is the price of a call option, PV(K) is the present value of the strike price, P is the price of a put option, and S is the price of the underlying asset. This relationship holds true because if the price of the call option is higher than the price of the put option, then an arbitrage opportunity exists.

Early Exercise of American Options

Early exercise of an American option is optimal when exercising the option would yield a higher value than waiting for the option to expire. Let's consider the optimal time for early exercise of American put and call options separately.

American Put Option: Early exercise of an American put option is optimal when the present value of the dividends (if any) that the option holder would receive by exercising the option and holding the underlying asset is less than the value of exercising the put option immediately. This situation can occur when the underlying asset is trading significantly below the strike price, the time value of the option is low, and the present value of dividends is not significant. In this case, the investor would be better off exercising the option early, selling the underlying asset at the strike price, and potentially reinvesting the proceeds at a higher return.

American Call Option: Early exercise of an American call option is generally not optimal for non-dividend-paying stocks. This is because the option's time value usually provides a better return than exercising the option early. However, for dividend-paying stocks, early exercise can be optimal just before the ex-dividend date if the present value of the dividend exceeds the time value of the option. By exercising the call option early, the investor can capture the dividend, which might compensate for the loss of the option's time value.

Corporate Securities as Options

Corporate securities can be thought of as options on the underlying value of the company. For example, common stock can be viewed as a call option on the value of the company's assets. If the company's assets are worth more than its liabilities, then the common stock has value.

Exotic Options

Exotic options are types of financial options contracts that have unique or complex features that make them distinct from standard options. These features may include non-standard expiration dates, non-standard strike prices, and non-standard underlying assets.

Exotic options may also have payout structures that are different from standard options, such as contingent premiums, barrier options, and binary options. These types of options are often customized to meet the specific needs of the buyer or seller, and they are typically traded over-the-counter rather than on public exchanges. Exotic options are generally more complex and risky than standard options and require a higher level of expertise to understand and trade effectively.

Exotic options are options that have more complex payoffs than standard European and American options. Two popular types of exotic options are Asian options and lookback options.

Asian Options

Asian options have a payoff that depends on the average price of the underlying asset over a specified time period. The price of Asian options is usually calculated using Monte Carlo simulation techniques.

Lookback Options

Lookback options have a payoff that depends on the highest or lowest price of the underlying asset over a specified time period. These options are often used to hedge against extreme price movements in the underlying asset.

Option Strategies

Option strategies are combinations of buying and selling options that can be used to manage risk and potentially generate profits. Some popular option strategies include buying call options, buying put options, selling covered call options, and buying protective put options.

The following are option strategies: protective put, covered call, collar, bull spread, and straddle. For each strategy, we will explain why it would be used, how one would implement it, and provide the math for how the payoff is calculated.

Protective Put

A protective put is a strategy that combines a long position in an underlying asset with a long put option. The purpose of this strategy is to protect the investor against a decline in the asset's value.

To implement a protective put, an investor will purchase the underlying asset and simultaneously purchase a put option with a strike price near the current price of the asset.

The payoff of a protective put is calculated as follows:

$\begin{aligned} P a y o f f & = M a x (0, S t r i k e - A s s e t P r i c e) \\ + (A s s e t P r i c e - I n i t i a l A s s e t P r i c e) \\ - P u t P r e m i u m \end{aligned}$

The following is an example of the payoff of a protective put with a current stock price of $100, strike price of $100, and an option price of $7.50.

Covered Call

A covered call is a strategy that involves holding a long position in an underlying asset while simultaneously selling a call option on the same asset. This strategy is used to generate income from the option premium and is typically implemented when the investor expects the asset's value to remain relatively stable.

To implement a covered call, an investor will purchase the underlying asset and simultaneously sell a call option with a strike price near the current price of the asset.

The payoff of a covered call is calculated as follows:

$\begin{aligned} P a y o f f & = M i n (0, A s s e t P r i c e - S t r i k e) \\ + (A s s e t P r i c e - I n i t i a l A s s e t P r i c e) \\ + C a l l P r e m i u m \end{aligned}$

The following is an example of the payoff of a covered call with a current stock price of $100, strike price of $100, and an option price of $7.50.

Collar

A collar is a strategy that combines a protective put and a covered call to limit the investor's exposure to large price movements. The goal is to hedge against significant losses while still participating in potential gains.

To implement a collar, an investor will purchase the underlying asset, purchase a put option with a strike price below the current price of the asset, and sell a call option with a strike price above the current price of the asset.

The payoff of a collar is calculated as follows:

$\begin{aligned} P a y o f f & = M a x (0, S t r i k e_{P u t} - A s s e t P r i c e) \\ - M a x (0, A s s e t P r i c e - S t r i k e_{C a l l}) \\ + (A s s e t P r i c e - I n i t i a l A s s e t P r i c e) \\ - P u t P r e m i u m \\ + C a l l P r e m i u m \end{aligned}$

The following is an example of the payoff of a collar call with a current stock price of $100, call strike price of $110, put strike price of $90, and a call option price of $7.50, and put option price of $4.25.

Bull Spread

A bull spread is a strategy that benefits from a moderate increase in the price of the underlying asset. It can be implemented using either call or put options.

To implement a bull spread using call options, an investor will purchase a call option with a lower strike price and sell a call option with a higher strike price. Both options should have the same expiration date. To implement a bull spread using put options, an investor will sell a put option with a higher strike price and buy a put option with a lower strike price, again with the same expiration date.

The payoff of a bull spread using call options is calculated as follows:

$\begin{aligned} P a y o f f_{C a l l} & = M a x (0, A s s e t P r i c e - S t r i k e_{C a l l 1}) \\ - M a x (0, A s s e t P r i c e - S t r i k e_{C a l l 2}) \\ - C a l l P r e m i u m_{C a l l 1} \\ + C a l l P r e m i u m_{C a l l 2} \end{aligned}$

The payoff of a bull spread using put options is calculated as follows:

$\begin{aligned} P a y o f f_{P u t} & = M a x (0, S t r i k e_{P u t 1} - A s s e t P r i c e) \\ - M a x (0, S t r i k e_{P u t 2} - A s s e t P r i c e) \\ + P u t P r e m i u m_{P u t 1} \\ - P u t P r e m i u m_{P u t 2} \end{aligned}$

The following is an example of the payoff of a bullish spread implemented with the underlying stock and two call options. The current stock price of $100, sell a call with strike price of $110, buy a call with strike price of $90, and the short call price is $8.25 and the long call option price is $6.50.

Straddle

A straddle is a strategy that benefits from significant price movements in the underlying asset, regardless of the direction. It involves simultaneously buying a call option and a put option with the same strike price and expiration date.

To implement a straddle, an investor will purchase a call option and a put option with the same strike price (usually close to the current price of the underlying asset) and the same expiration date.

The payoff of a straddle is calculated as follows:

$\begin{aligned} P a y o f f & = M a x (0, A s s e t P r i c e - S t r i k e) \\ - M a x (0, S t r i k e - A s s e t P r i c e) \\ - C a l l P r e m i u m \\ - P u t P r e m i u m \end{aligned}$

The following is an example of the payoff of a straddle implemented with the underlying stock, a call option and a put option. The current stock price of $100, the call and the put strike prices are both $100, the put price is $1.85 and the call price is $2.25.

Pricing Options via Replicating Portfolios

The pricing of options can be achieved through the creation of a replicating portfolio that mimics the payoff of the option. This replicating portfolio can be constructed using the underlying asset and a risk-free bond.

American Options: Dynamic Replication

Dynamic replication is a technique used to price American options by continuously adjusting the portfolio's holdings of the underlying asset and the risk-free bond to replicate the option's payoff.

Binomial Tree Model and its Relation to Black-Scholes-Merton Model

The Binomial Tree Model, also known as the Cox-Ross-Rubinstein Model, is a discrete-time method for valuing options and other financial instruments. It uses a tree-like structure to represent the potential future price movements of an underlying asset over a specified time period. At each node of the tree, the asset price can either move up or down, according to pre-determined probabilities.

The Binomial Tree Model can be used to derive the Black-Scholes-Merton (BSM) Model, which is a continuous-time model for option pricing. As the number of time steps in the Binomial Model approaches infinity, the discrete-time model converges to the continuous-time BSM Model. The BSM Model provides a closed-form solution for European-style options, making it computationally more efficient compared to the Binomial Tree Model for large numbers of time steps.

In the finance industry, the Binomial Model is commonly used for:

Option pricing: Valuing options, including American-style options, which can be exercised at any time before expiration, unlike European-style options.
Risk management: Assessing the potential impact of various risk factors on a portfolio, and using this information to hedge positions and minimize risk exposure.
Capital budgeting: Evaluating investment projects with uncertain cash flows and determining the optimal investment strategies.

Single Period Binomial Tree Model

The Binomial Tree Model is a mathematical model used to price options. In the single-period Binomial Tree Model, the underlying asset can only take two possible values: uS and dS, where u and d are the up and down factors and S is the current price of the underlying asset.

Multi-Period Binomial Tree Model

In the multiple-period Binomial Tree Model, the underlying asset can take multiple possible values at each point in time. In the Cox-Ross-Rubinstein model this is achieved by chaining together numerous single period models that visually fan out as the number of periods increase. Over time, the greater number of periods that have passed the wider the variance of possible outcomes. This model assumes that the price of the underlying asset can move up or down with fixed probabilities and that the movements are independent of each other. Keep in mind that in a multi-period binomial tree, given a node in the binary tree, the only randomness is either an up move or down move.

Risk-Neutral Pricing

The Binomial Tree Model assumes that the market is complete, which means that every possible future event is known and that there are no arbitrage opportunities. Risk-neutral pricing is a technique used in the Binomial Tree Model to price options, assuming that the probability of an up movement in the underlying asset is equal to the risk-neutral probability of an up movement.

State Prices

In a Binomial Tree Model, the price of a security can move up or down in discrete time periods. We can derive risk-neutral probabilities and state prices from the given parameters. The risk-neutral probabilities, $q_{u}$ and $q_{d}$ , represent the probabilities of an up and down movement, respectively, in a risk-neutral world where the expected return on the security equals the risk-free rate. The state prices, $ϕ_{u}$ and $ϕ_{d}$ , reflect the present values of the probabilities of the up and down states, respectively.

To derive the risk-neutral probabilities, we use the following formula:

$q_{u} = \frac{(1 + r) - d}{u - d}$

$q_{d} = 1 - q_{u} = \frac{u - (1 + r)}{u - d}$

where $u$ and $d$ are the factors by which the security price increases or decreases, and $r$ is the risk-free rate per time period.

Once we have the risk-neutral probabilities, we can derive the state prices using the following equations:

$ϕ_{u} = \frac{1}{1 + r} q_{u}$

$ϕ_{d} = \frac{1}{1 + r} q_{d}$

The state prices, $ϕ_{u}$ and $ϕ_{d}$ , represent the present value of the risk-neutral probabilities of the up and down states, respectively, discounted by the risk-free rate. These equations help us price various financial instruments, such as options and futures, in a Binomial Tree Model framework.

American Options: Pricing Using Binomial Tree Model

American options can be priced using the Binomial Tree Model by considering the option holder's decision to either exercise the option or hold it until the next period. The price of the option at each node of the binomial tree is calculated by comparing the expected value of exercising the option to the expected value of holding the option.

Empirical Implementation of the Binomial Tree Model

The Binomial Tree Model can be empirically implemented by parameterizing the model using $u = e^{(σ \sqrt{(\frac{T}{n}}))}$ and $d = \frac{1}{u}$ , where sigma is the volatility of the underlying asset, T is the time to expiration, and n is the number of periods. The first two moments of stock returns, such as the mean => $E_{0} [\frac{S_{T}}{S_{0}}] = e^{μ T}$ and variance => $V a r (\ln \frac{S_{T}}{S_{0}}) = σ^{2} T$ , can also be used to parameterize the model.

The Black-Scholes-Merton Model

The Black-Scholes-Merton model is a mathematical model used to price options by assuming that the price of the underlying asset follows a geometric Brownian motion and that the market is complete. The model provides closed-form solutions for the prices of European call and put options.

The Black-Scholes-Merton model is used to determine the theoretical price of European call and put options. In this explanation, we will focus on the formula for the value of a call option using the variables:

$S_{0}$ is the current price of the underlying asset
$K$ is the strike price of the option
$T$ is the time to maturity (in years)
$σ$ is the volatility of the underlying asset (annualized standard deviation of returns)
$r$ is the risk-free interest rate (annualized)

We will also use the notation $N (\cdot)$ to represent the standard normal cumulative distribution function. The formula for the value of a call option is:

$C (S_{0}, K, T, σ, r) = S_{0} N (X) - K e^{- r T} N (X - σ \sqrt{T})$

where:

$X = \frac{\ln (\frac{S_{0}}{K e^{- r T}})}{σ \sqrt{T}} + \frac{1}{2} σ \sqrt{T}$
$N (X)$ is the standard normal cumulative distribution function evaluated at $X$ .
The value of $N (X)$ is the option's $δ$ , described in the next section.
$N (X - σ \sqrt{T})$ is the standard normal cumulative distribution function evaluated at $X - σ \sqrt{T}$

Implementing the BSM Model - Implied Volatility

The Black-Scholes-Merton model requires the input of the volatility of the underlying asset. However, this volatility is not directly observable and must be estimated from market prices. Implied volatility is the volatility that equates the market price of an option to the price predicted by the Black-Scholes-Merton model. Implied volatility can be used to gauge the market's expectation of future volatility and to compare the pricing of options with different strike prices and expirations.

In the world of options trading, the Black-Scholes-Merton (BSM) model has become an essential tool for traders and investors alike. The model, first proposed by Fischer Black, Myron Scholes, and Robert Merton in the early 1970s, has revolutionized how option prices are calculated, making it easier for market participants to gauge the value of various financial instruments. One of the key components of the BSM model is implied volatility, which is a critical aspect to understand and implement for successful options trading. Next we will explore implied volatility, how it's calculated using the BSM model, and its importance in options pricing.

Understanding Implied Volatility

Implied volatility is a measure of the expected future volatility of an underlying asset's price. It represents the market's consensus on the magnitude of price fluctuations over a specific period. Implied volatility is an essential input in the BSM model, which helps traders estimate the fair price of an option.

Calculating Implied Volatility Using the BSM Model

The BSM model is a mathematical formula that calculates the theoretical price of an option based on several inputs, including the current price of the underlying asset, the option's strike price, time to expiration, risk-free interest rate, and implied volatility. However, the implied volatility is not directly observable, so it needs to be derived from the model itself.

To calculate implied volatility, traders use a numerical method called the Newton-Raphson method or a simple iterative search, such as a binary search. The process involves solving the BSM equation for implied volatility, given the current market price of the option:

Start with an initial guess for implied volatility.
Plug the guess into the BSM model to calculate the theoretical option price.
Compare the theoretical option price with the actual market price.
If the difference between the two prices is within an acceptable tolerance, the implied volatility is found. If not, adjust the implied volatility guess and repeat steps 2-4 until convergence is achieved.

Importance of Implied Volatility in Options Pricing

Implied volatility plays a vital role in options trading for several reasons:

Fair Pricing: It helps traders estimate the fair price of an option, allowing them to make informed decisions about whether an option is overpriced or underpriced.
Risk Assessment: Implied volatility serves as an indicator of the market's perception of risk, enabling traders to assess the potential risk associated with holding or trading a specific option.
Strategy Selection: Understanding implied volatility can help traders choose the most appropriate options strategies based on their market outlook and risk tolerance.
Portfolio Management: Implied volatility can be used as an input in portfolio risk management models, helping investors monitor and manage the risk associated with their options positions.

Implementing the BSM model and understanding implied volatility is essential for options traders and investors. By calculating implied volatility, traders can better estimate the fair value of an option, assess market risk, and make more informed trading decisions. As you dive deeper into the world of options trading, mastering the concept of implied volatility and its application in the BSM model will prove to be an invaluable asset.

Option Greeks

Option Greeks are a set of financial measures used to evaluate the sensitivity of an option's price to various factors, such as changes in the price of the underlying asset, time decay, and changes in market conditions. The most common Option Greeks include:

Delta (Δ): Delta measures the sensitivity of an option's price to a change in the price of the underlying asset. It represents the expected change in the option's price for a $1 change in the underlying asset's price. Call options have a positive delta (0 to 1), while put options have a negative delta (-1 to 0).

Example: If a call option has a delta of 0.6, the option's price would be expected to increase by $0.60 for a $1 increase in the underlying asset's price.

Omega (Ω): Also known as Lambda (λ) or elasticity, is an option Greek that measures the percentage change in an option's price relative to a percentage change in the underlying asset's price. It is a measure of leverage, indicating how much the option's value will change for a given percentage change in the underlying asset's price. Omega is a higher-order Greek, meaning it is derived from other Greeks, such as Delta and Gamma. Omega is calculated as the product of an option's Delta and the underlying asset's price, divided by the option's price. It represents the percentage change in the option's price for a 1% change in the underlying asset's price.

Example: If an option has an Omega of 5, the option's price would be expected to increase by 5% for a 1% increase in the underlying asset's price.

Gamma (Γ): Gamma measures the rate of change of an option's delta with respect to changes in the price of the underlying asset. It helps investors understand how the option's sensitivity to price changes (delta) will be affected as the underlying asset's price moves.

Example: If an option has a gamma of 0.05, its delta will increase by 0.05 for each $1 increase in the underlying asset's price.

Theta (Θ): Theta measures the sensitivity of an option's price to the passage of time, or time decay. It represents the expected change in the option's price for a one-day decrease in time to expiration. Theta is typically negative, as options lose value over time.

Example: If an option has a theta of -0.10, the option's price is expected to decrease by $0.10 per day due to time decay.

Vega (ν): Vega measures the sensitivity of an option's price to changes in implied volatility, which represents the market's expectation of future price movements in the underlying asset. Vega indicates how much the option's price will change for a 1% change in implied volatility.

Example: If an option has a vega of 0.15, the option's price would be expected to increase by $0.15 for a 1% increase in implied volatility.

Rho (ρ): Rho measures the sensitivity of an option's price to changes in interest rates. It represents the expected change in the option's price for a 1% change in interest rates. Rho is generally more important for long-term options, as interest rate changes have a greater impact on their value.

Example: If a call option has a rho of 0.08, the option's price would be expected to increase by $0.08 for a 1% increase in interest rates.

Option Greeks are used by traders and investors to better understand and manage the risks associated with options trading. They help in adjusting positions to account for changing market conditions, making informed decisions, and creating more effective hedging strategies.

Conclusion

Options on financial securities are a versatile investment tool that allows investors to manage risk and potentially generate profits. The pricing of options can be achieved using various mathematical models, including the Binomial Tree Model and the Black-Scholes-Merton model. Option strategies, such as buying call options and selling covered call options, can be used to tailor an investor's risk exposure to their specific investment objectives. Additionally, options can be viewed as corporate securities and can be used to hedge against changes in a company's value.

Exotic options, such as Asian and lookback options, provide investors with additional flexibility in managing their portfolio risk. The pricing of American options can be more complex due to the option holder's ability to exercise the option at any time before expiration. However, dynamic replication techniques can be used to price American options.

Finally, option Greeks provide investors with a way to measure the sensitivity of an option's price to various factors. Implied volatility is a particularly useful tool for estimating future volatility and comparing the pricing of options with different strike prices and expirations. Overall, options on financial securities provide investors with a powerful tool for managing their investment risk and achieving their financial objectives.

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