Exploring Option Pricing Methods: The Black-Scholes, Binomial, and Trinomial Models
You learn the most important thing to be known in the financial markets and trading: pricing options. Anyone planning to start trading or students of economics: nothing is as fundamental as knowing the price of an option so they can make more informed decisions. Today, in this tutorial, let's review three of the most popular pricing models for an option: the Black-Scholes Model, Binomial Option Pricing, and Trinomial Option Pricing. Need a hand with your accounting assignment? At Assignment In Need, we offer the best Accounting Assignment Help to make your work easier and stress-free.
Each of these models has its strengths and weak points and can be used in certain situations. But do not worry, as we are here to break them down simply and interestingly without going deep into heavy mathematical jargon. So, grab your coffee, sit back, and let's get started step by step.
What Are Option Pricing Models?
Let's get ahead and know exactly what these are, option pricing models. In just a few words, they are the mathematical model used in computing the theoretical price of the financial options. An option in the domain of finance confers upon the buyer the right but not an obligation to either sell or buy the asset for the price defined prior within the definite time. It should hence have a probable correct price as by it, one is at an instance whereby it will specify the right moments for one to purchase and sell while at it determines the most opportune moments for hedging.
Methods to express the choices that have been devised are numerous. Included in these, are: Black-Scholes, Binomial models, and Trinomial models amongst others. And it's not a list of the few industry-applied models but here's the list of each one of them with further elucidation.
Black-Scholes Model: The Old Trusted Solution
The most popularly used models in option pricing include the Black-Scholes Model. Economists Fischer Black, Myron Scholes, and Robert Merton developed the very first model early in the 1970s. That created a lot of buzz and drastically changed the way options were priced.
What is the Black-Scholes Formula?
Another form, included in the theoretical estimate would be used for calculating the cost of a European-style option. This is an option that exercises only at expiration and includes other things such as:
- the present stock price
- the strike price of the option
- time till expiration
- risk-free rate of interest
- and the volatility in the stock
This is a pretty simple formula, but magic comes with variables. If you did some financial modeling, this for example could represent the Black-Scholes formula.
C=S0N(d1)−Xe−rtN(d2)C = S_0 N(d_1) - X e^{-rt} N(d_2)C=S0N(d1)−Xe−rtN(d2)
Where:
- C represents the call option price
- S_0 represents the current stock price
- X represents the strike price
- t represents time to expiration
- r represents the risk-free rate
- N() represents the cumulative distribution function of the standard normal distribution
- D_1 and D_2 in terms of other variables
Do not bother about the mathematics, but essentially what the Black-Scholes Model offers to us is that, theoretically how an option would be worth having all this in place. Then it makes the options trading make sense on the side of an investor.
Advantages of the Black-Scholes Model
- Elegant and simple
- Widely used in pricing European-style options
Disadvantages of the Black-Scholes Model
While there is so much elegance and simplicity in the Black-Scholes Model, highly praised as it may be, so also is its limitation equally weighted. For example, it assumes markets are efficient, assets volatility constant, and that options can only be exercised at maturity. Now, in real life, these assumptions cannot always be guaranteed to come to pass, and because of this reason, the Black-Scholes model may not yield the best price in dynamic markets.
This is where the Binomial Option Pricing and Trinomial Option Pricing models come in.
Binomial Option Pricing: Much More Flexible Model
The Black-Scholes formula is fairly slick for trivial problems, while this model allows greater flexibility of this Binomial Option Pricing. More precisely, most applications were the pricing for the American option-which one could exercise early but not exactly at expiration.
How Does The Binomial Option Pricing Work?
This is a Binomial Option Pricing model that breaks up the life of the option into many short intervals of time. Now at each one of those periods, the underlying asset price increases or decreases—and therefore the word "binomial".
The model works by building up a binomial tree. Each node of that tree is assumed to be the possible price of the underlying asset at some point in time. You then compute the option's price at the final nodes—the expiration points—and work back from there to the present, taking into account the probability of an upward or downward movement.
Binomial Model Assumptions
To apply the Binomial model you have to assume:
- At each step the asset price moves either to the right or to the left.
- The size and direction of the up and down moves are given by the volatilities.
- There needs to be a nonzero constant risk-free rate.
While much more flexible than the Black-Scholes formula, full of some simplifying assumptions it is useful for a variety of types of options, including American options, where the Black-Scholes formula fails to give the right answer.
Advantages and Disadvantages of the Binomial Option Pricing Model
The Binomial Option Pricing model can be viewed as highly flexible and therefore applied to a larger option. Specifically, this is handy for the pricing of early exercise options that feature American options. However, breaking into numerous periods can make this model too computationally intense.
Trinomial Option Pricing Model: Get Flexible
If you thought the Binomial model was flexible enough, then you would only see how flexible it is in the Trinomial Option Pricing model. It just takes the same premise of the binomial approach but extends that to three movements: up, down, or no change instead of just up or down alone.
How does the Trinomial Option Pricing work?
There is also a Trinomial Option Pricing model where each node carries in its vision the presence of three branches; upward, coming downwards and a "do-nothing" holding an equivalent to staying the same, about the asset.
The trinomial model is much more complex than the binomial. Indeed, it is much more accurate when applied in the pricing of options displaying complexity characteristics or over a long term. The dynamics of the price for the underlying asset under the trinomial model are modeled in a way to accommodates real-life considerations.
Advantages and Disadvantages of the Trinomial Tree Model
With difficult options features the Trinomial Tree Model is more accurate than a binomial one. Not less computer costly than the Binomial, but now from the number of time steps used.
Which Option Pricing Model to choose?
Black-Scholes, or Binomial, each according to purpose.
This way, if one needs relatively simple European options, Black-Scholes might be the best choice for the job because it is both efficient and pretty simple to handle. But if you have American-type options or some that are more heavily featured, you would want the Binomial Option Pricing or even the Trinomial Option Pricing model. But to get a little more accurate without having to do so much more calculations, try the Trinomial.
Conclusion
Of most importance in the understanding of applying any option pricing model is important when a person goes deeper into finance. Out of those that we discussed; namely, the Black-Scholes Model, Binomial Option Pricing, and Trinomial Option Pricing, are bound to come up with varied methods that would be followed when pricing options thus different strengths and weaknesses.
Whether you are a finance student or just looking to get a better grasp of financial derivatives pricing, knowing how to use these models will give you a significant edge in understanding how the market works.
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