Options are powerful financial tools that offer flexibility and the potential for significant gains, but they can also be quite complex. There are a lot of different parts to options contracts and specific strategies when it comes to options trading or investing, and options pricing is just another part of the options contract cycle. Options pricing isn’t just a number; it reflects multiple factors like the underlying asset, time, volatility, and market conditions. Whether you’re new to options or looking to refine your strategy, understanding how options are priced and how the Greeks come into play is an essential step in building confidence as an investor. There are a number of options pricing models used by traders and investors, in this article, we take a look at three of the most common pricing models used. By acquiring this knowledge, you’ll gain insight into the logic behind the numbers, and gain an understanding of how strategic decisions are made.
How options are priced
Options pricing might seem complex, but with some explaining, it becomes easier to grasp. At its core, the current options price is the live bid offer and last trade on your trading screen. Here, we’ll explore some of the most widely used methods for pricing options, including the Black-Scholes model, the Binomial model, and Monte Carlo simulations.
Options pricing is the process of determining how much an options contract should cost. To work this out several factors come into play:
Price of the underlying asset (like stocks or commodities).
Strike price: the agreed price for exercising the option.
Time to expiration: how long until the option expires.
Volatility: the expected fluctuation in the asset’s price.
Risk-free interest rate: the return on risk-free investments like government bonds. However, with government bonds, it's likely to be short-term cash rate and not long-term bond rate. In essence, the term of interest rate should match the time to expiry.
To calculate an option’s price, experts use mathematical models. You can find the most popular ones below:
1. The Black-Scholes Model
The Black-Scholes model also known as the Black-Scholes-Merton (BSM model) was the first widely used model for options pricing. It is often considered the gold standard for pricing European-style options. It calculates the option’s value by combining the asset price, strike price, time to expiration, volatility, and risk-free rate. While simple and efficient, this model assumes constant volatility, no dividends and works best for options that can only be exercised at expiration.
Formula:
Source: Investopedia
2. The Binomial Model
The Binomial model approaches pricing step-by-step, dividing an option’s life into intervals to calculate possible price movements. This creates a tree of potential outcomes, making it especially useful for American-style options, which can be exercised at any point. However, it can get computationally intensive for longer timeframes. The key inputs for this model are the stock price, strike price, time to expiration, risk-free interest rate, and the asset's volatility. The benefit of using this model is that, unlike the BSM model, the binomial model can handle complex situations such as varying volatility and dividend payments.
The basic calculation method of the binomial model is to use the same probability for each period for success and failure until the option contract expires. However, different probabilities can be incorporated for each period based on any newly obtained information as time passes. To do this, traders or investors use what is called a binomial tree when pricing American options.
Here’s a basic example:
Suppose we want to price a call option on a stock using a one-period binomial model.-
• Stock Price: $100 • Strike Price: $105
• Time to Expiration (T): 1 year
• Risk-Free Interest Rate (r): 5%
• Up Factor (u): 1.2 (Stock price increases by 20%)
• Down Factor (d): 0.8 (Stock price decreases by 20%)
• Probability of Up Move (p): Derived using risk-neutral valuation.
First, determine stock prices at expiration. For instance, the stock price is $100
After one period, the stock price can either:
• Go up (20%): $120
• Go down(-20%): $80
You now have to determine the payoffs against the strike price. The option payoff at expiration depends on whether the stock price is above or below the strike price ($105). For a call option:
• If the stock price goes up: $120 -105 = $15
• If the stock price goes down: $80-105 = 0
Taking into account the above, you would also need to calculate the risk-neutral probability. The risk-neutral probability is calculated as:
p = (e^(rT) - d) / (u - d)
Now substitute the values:
p = (e^(0.05 × 1) - 0.8) / (1.2 - 0.8) = (1.0513 - 0.8) / 0.4 = 0.6283
Once you have worked this out, you can work out the expected payoff. The discounted expected payoff is the value of the option price at time 0 (today):
C₀ = e^(-rT) × (p × Payoff_u + (1 - p) × Payoff_d)
Substituting the values:
C₀ = e^(-0.05) × (0.6283 × 15 + (1 - 0.6283) × 0)
C₀ = 0.9512 × (0.6283 × 15)
C₀ = 0.9512 × 9.4245 = 8.96
Therefore, the theoretical price of the call option is $8.96. This is a simple one-period example. For real-world applications, multi-period binomial trees are used to capture more granular price movements.
Monte Carlo Simulation
Monte Carlo simulation is a method used to predict possible outcomes in situations with uncertainty. In options trading, it helps estimate future prices by running thousands of simulations based on different possible market conditions. Instead of relying on a single guess, this model provides a range of outcomes, helping traders understand risk and probability better.
How it Works
Instead of assuming a fixed future price for an asset, Monte Carlo simulation randomly generates thousands of potential price movements. These simulations consider factors like volatility and time to expiration. The results create a probability distribution, giving traders a clearer picture of possible profits and losses.
The 4 Key Steps
1. Define the variables – Identify the key factors that influence the outcome, such as stock price (S), volatility (σ), risk-free rate (r), and time to expiration (T).
2. Assign probabilities – Model the stock price using a random process, typically the Geometric Brownian Motion:
S_(t+Δt) = S_t × e^((r - ½σ²)Δt + σϵ√Δt)
where ϵ is a randomly generated number from a normal distribution.
3. Run simulations – Generate thousands of possible price paths by iterating the equation above over time steps Δt, creating a distribution of potential future prices.
4. Analyze the results – Compute the average of all simulated option payoffs, discounting them back to present value:
Option Price = e^(-rT) × (1/N) Σ Payoff_i
where N is the number of simulations.
Monte Carlo simulation is a powerful tool that helps investors prepare for uncertainty. By understanding different possible scenarios, investors can make more informed choices and manage risk more effectively. A tip here is to do the calculations in an EXCEL spreadsheet and repeat the calculation for the desired number of times with each calculation representing one day. The result is the simulation of the assets' future price movement
Understanding the Greeks
Now we will take a closer look at the Greeks, which are essential tools for understanding how different factors influence an option’s value. Here’s a breakdown of what they mean and why they matter:
1. Delta: Price sensitivity
The delta (Δ) of an option is the rate of change in the price of the option compared to changes in the price of the underlying asset. Delta tells you how much an option’s price changes when the underlying asset moves upward or downward by $1.
The formula is:
Delta=(∆ Option Price)/(∆ Share Price)
For example, if a call option’s Delta is 0.7, a $1 increase in the stock price would raise the option’s price by $0.70. Put options have negative Delta values since they move in the opposite direction of the stock price.
It's important to note that Delta changes over time and with changes in volatility. Using Delta is useful and has several different uses/interpretations/approximations:
The rate of change in the price of the call or put with respect to the underlying asset.
Hedge ratio required to remain delta-neutral.
The probability that the call will end up in-the-money at expiry.
The negative of the rate of change in the price of the call as the strike price increases.
To learn more about using Delta and how to choose options strategies effectively, check out the blog on our Options Screener feature.
2. Gamma: Tracking delta’s moves
Options Guide - Basics1.docx
Gamma (Γ) measures how much Delta changes as the stock price moves. Think of it as Delta’s sensitivity. Higher Gamma means Delta adjusts more quickly, which is particularly relevant for short-term options that can experience rapid price changes as they reach expiry.
An investor who holds many option positions will use Gamma to see how quickly their options will gain or lose value. Alternatively, Gamma can help create a Delta-neutral position, which means the total option value won’t change when the stock price moves.
Gamma changes over time and with volatility, affecting how quickly Delta responds to price movements. As an option gets closer to expiration, its Gamma tends to increase, because Delta shifts more abruptly with small stock price changes. This happens particularly for at-the-money options as they will swing from delta 1 to 0 very quickly for a call, i.e. they will expire worthless of ITM. This means that short-term options can experience rapid swings in value, making them riskier but also more responsive to price movements.
However, if volatility is high, Gamma tends to be lower because Delta changes more smoothly over a wider range of prices. In contrast, when volatility is low, Gamma is higher since Delta shifts more sharply. This is why investors should pay close attention to Gamma—it helps to anticipate how much their option’s sensitivity to price changes will fluctuate, allowing them to manage risk more effectively.
3. Theta: Time decay
Theta (Θ) shows how much an option’s price decreases as time passes. Options lose value as expiration nears, even if everything else stays the same. For instance, an option with a Theta of -0.05 loses $0.05 in value each day.
Theta is useful for the trader or investor when planning how long to hold an option. By convention, theta is negative, which means that if you are long an option, it loses value over time.
4. Vega: Impact of volatility
Vega (V) measures how much an option’s price changes with a 1% change in implied volatility. Higher Vega means the option is more sensitive to market uncertainty, which can be a critical factor during major events. The vega of an option is always positive because when volatility goes up, option prices increase and buyers will pay more for the protection/insurance that the options offer.
5. Rho: Interest rate sensitivity
Rho (Ρ) indicates how much an option’s price changes with a 1% change in the risk-free interest rate. While less impactful for short-term options, Rho can matter for long-term ones where interest rate shifts have a bigger effect.
Implied volatility and its impact on options prices
Implied volatility (IV) plays a major role in determining an option's premium—the price you pay to buy the option. Here's why:
Higher implied volatility = higher premiums If the market expects large price movements, the likelihood increases that the option will be profitable for the buyer. This potential for higher returns makes the option more valuable, driving up its price.
Lower implied volatility = lower premiums When the market expects smaller price movements, the chances of an option becoming profitable decrease. As a result, its price (or premium) tends to be lower.
Implied volatility is rooted in probability, meaning it offers an estimate rather than a definitive prediction of future price movements. This probabilistic nature can indirectly influence option prices, as investor behaviour is often shaped by these expectations. It’s important to note that an option’s price doesn’t always align with anticipated trends. However, observing how other investors interact with the option can provide valuable insights. Since implied volatility is closely tied to market sentiment, it plays a significant role in shaping option prices.
You can learn more about IV in our previous blog.
Understanding how options are priced is crucial for making informed decisions in the options market. Visit our Options Trading page on the website to explore more opportunities. Join Tiger Trade and practice options trading with no real capital risk by using the Tiger Trade demo account. Plus, if you open an account and make a deposit, you'll get four $0 brokerage monthly trades on ASX, US stocks or ETFs or US options.*
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