The derivatives market is an exciting field to study in Finance. In my academic path, I first encountered derivatives during the last year of my Bachelor’s program. In the Master’s program, I gained a deeper understanding of the different financial products, how they can be combined for diverse investment strategies and also how to build a prototype of such a contract in a programming environment.
I have to admit that it was a tough learning journey; however, now, looking back at this theory, I find it fascinating and essential for any Finance professional.
In this article, I want to get back to the roots of the derivatives market and talk through the call option contract.
What are derivatives?
As the name suggests, derivatives derive value from the underlying asset. Essentially, derivatives are contracts between a buyer and a seller, where the conditions of the contact refer to the underlying asset. Derivatives market is approximately 5 times bigger than the traditional financial market.
It is important to understand, that while the underlying assets have a specific value such as a share in a company, derivatives do not have value without the underlying asset and therefore, cannot exist without it.
How to use derivatives?
Alright, the whole idea of the derivatives market might be clear but how would one use it and why it is necessary? The short answer would be investment strategy optimization and speculation.
To illustrate this idea, let us take a call option as an example. An option is a contract where the two parties, namely, the buyer and the seller agree on the following conditions:
- The buyer pays some money (premium) for the option. To put it differently, she pays for the possibility to enter into the buying agreement.
- The buyer and the seller agree that on a certain day T, the buyer can choose whether she wants to buy the underlying asset from the seller at the price K (also called the strike price).
- The seller owns the underlying asset. She receives the premium paid by the buyer and is obligated to sell the asset to the buyer if she chooses to buy it.
Note here, that the buyer has a choice whether to buy (an option to buy); however, the seller has an obligation to sell.
Now, what is the motivation and what are the beliefs of both parties?
- The buyer must believe that the asset will be worth more at time T than it is worth at the current moment. She expects to make a profit because she will pay K for something that will be worth more than that, let’s say, K+x. To bet on this value increase, she is ready to pay a premium to enter into the call option contract. Hence, her total expected profit is K+x-K-premium = x-premium.
If her forecast is wrong, i.e., if the price is lower than K, she should choose not to buy the asset. In that case, her expected profit = -premium. Namely, she loses the premium.
- The seller receives a guaranteed premium from the buyer. The belief of the seller should be that the asset will not increase in value at the time of the call option maturity, i.e., at time T. If her belief is correct, she gains the premium and keeps the asset.
If her forecast is wrong, she would have to sell the asset worth K+x at a price of K. Thus, she would make a loss. In that case, her profit would be premium+K-(K+x) = premium-x, where x > premium.
The graph below captures the call option contract value from both perspectives.
Derivatives valuation (option)
Black-Scholes model
One of the most commonly used method in option valuation is the Black-Scholes-Merton model. This model values the option price based on several risk factors and a time component.
The model is based on the following assumptions:
- No dividends on the underlying asset are paid out during the life of the option
- Market movement cannot be predicted
- There are no transaction costs in buying the option
- The implied volatility and the risk-free rate parameters are known and are constant
- The returns on the underlying asset are log-normally distributed
- The option is European and can only be exercised at expiration
| Parameter | Relation to CALL option price | Explanation |
|---|---|---|
|
Time (t) |
Option price is higher when the time to expiration is higher |
The longer the time to expiration, the more probability there is that the underlying asset value will change, and, as the result, the option will end up in-the-money |
|
Risk-free rate (r) |
Option price decreases with an increase in risk-free rate |
The higher the risk-free rate, the lower is the net present value of the underlying asset. Option price decreases with the decrease in the value of the underlying asset |
|
Implied volatility (sigma) |
Option price is higher for an asset with a higher implied volatility |
The higher the volatility of the underlying asset, the higher probability that the underlying asset price will change, and, as the result, the option will end up in-the-money |
|
Option’s delta (d1) |
Option price rises with the rise in delta |
Option’s delta is the amount by which an option price is expected to change with 1 monetary unit change in the underlying asset. The more the option price is exposed to the volatility in the underlying asset, the higher probability that the underlying asset price will change, and, as the result, the option will end up in-the-money |
|
Normal distribution corresponding to the probability that the option will be exercised at expiration (N(d2)) |
Option price dcreases with the rise in N(d2) |
The more the option price is exposed to the volatility in the underlying asset, the higher probability that the underlying asset price will change, and, as the result, the option will end up in-the-money |
|
Asset price (S) |
Option price rises with the rise in the value of the underlying asset |
The (call) option value increases when the value of the underlying asset increases. This shift suggests that the belief of the option contract buyer is correct |
|
Strike rice (K) |
Option price decreases as strike price increases |
The option holder intends to buy the underlying asset at price K and make a profit of S1 - K. The increase in K leads to lower profit for the option holder which is not advantegeous |
The complete model is as follows:
Binomial tree
Another method of option valuation is the binomial tree which can be used to value both option types – American and European. This valuation allows for the price to increase (by u) or decrease (with d) at each node thus forming a tree with multiple outcomes. The number of possible outcomes is proportional to the number of periods (t) in which we observe the price movements.
When working with binomial trees, the risk neutrality assumption is essential, i.e., we assume that the present value of the derivative equals its future value discounted at the risk-free rate.
Looking at the tree scheme, we can see that at each node we allow the price to increase or decrease with probability u or d, respectively. The payoff is therefore dependent on the risk-neutral probability p, the initial price S0, the number of periods and the risk-free rate.
Given the two-period binomial tree, the concept can be wrapped in the model as follows:
Finally, the option price f is can be found as follows:
f=e^{-2r\Delta t}\times(p^{2}\times f_{uu} + 2p(1-p)\times f_{ud} + (1-p)^{2}\times f_{dd})I hope yo enjoyed reading through the introduction to derivatives! In the next post, I will expand on the derivatives’ valuation and its implementation in Python.