Option pricing models are mathematical models used for the purpose of valuing the options. Through this article, we attempt to explain the most prevalent and widely acknowledged option’s pricing models.

Table of Contents

## What are Options?

Before getting into the depths of an option pricing model, it is important to first understand what an option is. Imagine your favorite mango season is around the corner and you can’t wait to eat them! However, due to the uncertainty of rain this season it is difficult to estimate the price at which mangoes shall be available this season. In case of a good rainfall, they may be appropriately priced. A bad monsoon may, however, jack up the prices and you may have to wait for a whole another year before you can get the taste of it.

You go to the market wondering what to do. One of the fruit sellers senses your dilemma and calls out to you. You explain to him your worry regarding the monsoons and mango prices. He comes up with an innovative solution to your situation. He offers to sell the mangoes to you (when they arrive) at a per-fixed price of $5 per dozen. This offer will prevail no matter what the actual prices in the markets are. You contemplate that $5 for a dozen is the fair price of mangoes and it is a good deal. But there lies a twist to the situation. To book the price of $5 you will have to pay $1 upfront. This amount of $1 is called the option price.

**An option creates a right (not an obligation) to buy or sell a certain asset at a predetermined price, on or before a predetermined date.**

In the above situation, $5 is called the exercise price.

Consider the following situations

### Out of the Money

The actual price of the mangoes is $4. Thus, the option held is rendered worthless. (Why would you pay $5 for an article currently worth $4?) However, the maximum loss is capped at $1 (option price).

### At the Money

The actual price of mangoes is $6. This will be the price you pay when you don’t exercise the option. When exercised, total cost borne equals ($1+$5, i.e., the option price plus the exercise price). In this situation, you end up in a break even or indifferent position.

### In the Money

The actual price turns out to be $8. In this situation exercising the option makes complete sense. You would be able to purchase the mangoes at a price point of $5($6 in total considering the option price) in a market where the prevailing price is $8. Therefore, you will be in a position of obvious advantage as compared to the rest of buyers.

## Call and Put Options

Another concept which needs to be crystal clear before going understanding an option pricing model is that of call and put options.

### Call Option

An option contract that casts a right (not an obligation) to buy the underlying asset at a predetermined price in or before expiry.

### Put Option

An option contract that casts a right (not an obligation) to sell an underlying asset at a predetermined price on or before expiry.

## Option Pricing Models

There exist several option pricing models. It is nearly impossible to traverse the length and breadth of the entire volume of option pricing theories. Through this article, an attempt is made to condense and explain the most prevalent and widely acknowledged option pricing models.

## Binomial Model

A binomial model is an option pricing model that is easily understandable and less complex when compared to black and Scholes model or a Monte Carlo simulation.

**As per the binomial option pricing model, the price of an option is equal to the difference between the present value of the stock (as computed through a binomial tree) and the spot price.**

### Assumptions in Binomial Model

The following assumptions in a binomial option pricing model

- Based on the efficient markets hypothesis.
- There exist only two possible prices for the forthcoming period, hence the name binomial.
- The two prices are the ones realized on an uptick or downtick.
- No arbitrage is possible.
- The rate of interest remains unchanged throughout the period under consideration.
- The investors are risk neutral.
- There does not exist any transaction cost.

### Two-Period Binomial Model

There exists an asset with a spot price of S_{0}. Now, in one years time, the price of this asset will either increase by u%(uptick) or fall by d% (downtick). The probability of uptick is indicated by “p” and that of downtick by “1-p”.

#### Formula

The formula is expressed as follows:

4 variables have already been bought up! Do not get overwhelmed. The derivation of each of them is here below.

Formula keys:

e^^{(rt/n)}= Risk Free Rate, e= exponential, σ = Standard deviation, √t/n= time period

Let us construct a binomial option pricing model

The current spot price of the asset (S_{0}) = $100, RFR= 10%, and Standard Deviation σ = 20%

Therefore,

Uptick = e0.0.20√1 = **1.2214**

Downtick = 1/u = 1/1.2214 = **0.8187**

Therefore, probability of uptick (p) = (1+10%)-0.8187/1.2214-0.8187 =0.698 or **0.7**

Therefore probability of downtick (1-p) = 1-0.7=**0.3**

The first branch of the binomial tree will look like this. The prices at either end of the node indicate the two possible and only outcomes given the set of assumptions.

Since we are studying a two-period binomial model, we shall further build branches for one more period.

After the values for every node at the end of year 2 are derived, they are multiplied by their respective probabilities. The relevant probabilities are the product of probability value for every branch. For example, Branch 1 with a node value of $149.18 has arrived through two branches (uptick * uptick). Therefore the relevant probability is (0.7*0.7). So on and so forth for the remaining branches.

The price of stock derived is two years hence. Therefore, the present value of the stock as per binomial model is derived. The same will be computed as follows.

= 121.1282/ (1.10)2

=$100.10

Therefore, the maximum price of the option equals $(100.10 -100)

= 10 cents

## Black and Scholes Option Pricing Model

This model is particularly used to value European options that are held to maturity. This formula was derived by Fischer Black and Myron Scholes who went on to win the Nobel Prize for this discovery. Before the discovery of this formula, options trading was considered as a gamble having no mathematical or scientific basis. It was this formula which explained the rationale behind option trading. Immediately following the release of this formula, a dramatic surge in the volume of options trading was noticed. Though dated, present-day analysts and brokers borrow heavily from the B&S option pricing model. This is a testimony to its accuracy and precision behind the formula.

### Assumptions in B&S Model

#### Constant Volatility

This option pricing model assumes the volatility (amplitude of movement in stock prices) to be constant through the life of the option. While in the short term the volatility may oscillate around a small range, in the long run, it is highly unlikely for the volatility to remain constant. This is also a limitation of the B&S model. Since it does not account for the movement in one of the most significant variables of the B&S model.

#### Constant Risk-Free Interest Rate

Like the volatility, the B&S option pricing model also assumes a constant risk-free interest rate. Constant implies the same rate for borrowing and lending which is highly improbable in practice. However, the magnitude of the impact of this assumption is not as large as that of assuming constant volatility. This is because the two rates differ only by a few basis points. Moreover, the interest is not subject to widespread changes in the long run.

#### Random Walk

This price of the underlying asset is assumed to be moving in accordance with the random walk theory. The random walk theory states that at any given moment, the price of an asset may move up or down with equal probability. Implicitly, the price of underlying at time (t+1) is completely independent of its price at time (t).

#### Normally Distributed Returns

The returns on the underlying risky asset are said to follow a normal distribution. A normal distribution is nothing but a bell curve when translated graphically. A bell curve in this context represents that the probability of smaller changes in price in the near future is greater than extreme changes in price. Thus, the bell shape of the curve.

Further, owing to the normal distribution of returns and the undeniable relationship between the returns and prices, the price of underlying tends to be lognormally distributed.

### Formula of Black & Scholes Model

Example:

Let us go through a simple B&S problem to crystallize the concept in our mind.

Following information is available for Co X’s shares and Call Options:

Current Price (S)= $185, Option Exercise Price (E)= $170, Risk Free Rate = 7%, Time to expiry = 3 years and σ= 0.18

Computing the Value of Option

d1= {ln(S/E) + (r+ σ 2 /2 )t} / σ √t

= {ln(185/170) + (0.07+ 0.182/2) 3}/0.18√3

= (0.08452 + 0.2586)/0.18√3 = 0.34312/0.31177 = 1.1006

Therefore, d1 = 1.1006

d2 = d1 – σ √t

=1.1006 – 0.31177 = 0.7888

N(d1) = 0.8644 (from table)

N(d2 ) = 0.7848

Value of option = SN(d1) – N (d2).Ee-rt

=185(0.8644) – 170/e 0.21 (0.7848)

= 159.914 – 170/1.2336 * 0.7848

=159.91-108.15 = $51.76

^{1,2}