# The option is based on the right. Black–Scholes model - Wikipedia

In this way, delta and gamma of an option changes with the change in the stock price. We should the option is based on the right that Gamma is the highest for a stock call option when the delta of an option is at the money.

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Since a slight change in the underlying stock leads to a dramatic increase in the delta. Similarly, the gamma is low for options which are either out of the money or the option is based on the right the money as the delta of stock changes marginally with changes in the stock option.

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You can watch this video to understand it in more detail. Theta measures the exposure of the options price to the passage of time. It measures the rate at which options price, especially in terms of the time value, changes or decreases as the time to expiry is approached. Vega measures the exposure of the option price to changes in the volatility of the underlying.

Generally, options are more expensive for higher volatility. So, if the volatility goes up, the price of the option might go up to and vice-versa. Vega increases or decreases with respect to the time to expiry? What do you think? You can confirm your answer by watching this video. One of the popular options pricing model is Black Scholes, which helps us to understand the options greeks of an option. Black-Scholes options pricing model The formula for the Black-Scholes options pricing model is given as: where, C is the price of the call option P represents the price of a put option.

N x is the standard normal cumulative distribution function. The formulas for d1 and d2 are given as: To calculate the Greeks in options we use the Black-Scholes options pricing model. Delta and Gamma are calculated as: In the example below, we have used the determinants of the BS model to compute the Greeks in options. At an underlying price of If we were to increase the price of the underlying by Rs.

## Options: Picking the right expiration date

As can be observed, the Delta of the call option in the first table was 0. Hence, given the definition of the delta, we can expect the price of the call option to increase approximately by this value when the price of the underlying increases by Rs. The new price of the call option is If you observe the value of Gamma in both the tables, it is the option premium is for both call and put options contracts since it has the same formula for both options types.

If you are going long on the options, then you would prefer having a higher gamma and if you are short, then you would be looking for a low gamma. Thus, if an options trader is having a net-long options position then he will aim to maximize the gamma, whereas in case of a net-short position he will try to minimize the gamma value.

The third Greek, Theta has different formulas for both call and put options. These are given below: In the first table on the LHS, there are 30 days remaining for the options contract to expire. We have a negative theta value of He has to be sure about his analysis in order to profit from trade as time decay will affect this position.

This impact of time decay is evident in the table on the RHS where the time left to expiry is now 21 days with other factors remaining the same.

As a result, the value of the turbo options for beginners option has fallen from If an options trader wants to profit from the time decay property, he can sell options instead of going long which will result in a positive theta.

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We have just discussed how some of the individual Greeks in options impact option pricing. However, it is very essential to understand the combined behaviour of Greeks in an options position to truly profit from your options position. Let us now look at a Python package which is used to implement the Black Scholes Model.

Python Library - Mibian What is Mibian? Mibian is an options pricing Python library implementing the Black-Scholes along with a couple other models for European options on currencies and stocks. In the context of this article, we are going to look at the Black-Scholes part of this library.

Mibian is compatible with python 2. This library requires scipy to work properly. How to use Mibian for BS Model? The function which builds the Black-Scholes model in this library is the BS function. This list has to be specified each time the function is being called.

Next, we input the volatility, if we are interested in computing the price of options and the option greeks. The BS function will only contain two arguments. If we are interested in computing the implied volatilitywe will not input the volatility but instead will input either the call price or the put price. In case we are interested in computing the put-call parity, we will enter both the put price and call price after the list.

BS [1. We will learn more about this as we move to the next pricing model. Derman Kani Model The Derman Kani model was developed to overcome the long-standing issue with the Black Scholes modelwhich is the volatility smile. One of the underlying assumptions of Black Scholes model is that the underlying follows a random walk the option is based on the right constant volatility.

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However, on calculating the implied volatility for different strikes, it is seen that the volatility curve is not a constant straight line as we would expect, but instead has the shape of a smile. This curve of implied volatility against the strike price is known as the volatility smile.

If the Black Scholes model is correct, it would mean that the underlying follows a lognormal distribution and the implied volatility curve would have been flat, but a volatility smile indicates that traders are implicitly attributing a unique non-lognormal distribution to the underlying. This non-lognormal distribution can be attributed to the underlying following a modified random walk, in the sense that the volatility is not constant and changes with both stock price and time.

In order to correctly value the options, we would need to know the exact form of the modified random walk. More specifically a unique binomial tree is extracted from the smile corresponding to the random walk of the underlying, this tree is called the implied tree.

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This tree can be used to value other derivatives whose prices are not readily available from the market - for example, it can be used in standard but illiquid European options, American options, and exotic options.

What is the Heston Model? Steven Heston provided a closed-form solution for the price of a European call option on an asset with stochastic volatility. This model was also developed to take into consideration the volatility smile, which could not be explained using the Black Scholes model.

The basic assumption of the Heston model is that volatility is a random variable. Therefore there are two random variables, one for the underlying and one for the volatility. Generally, when the variance of the underlying has been made stochastic, closed-form solutions will no longer exist. But this is a major advantage of the Heston model, that closed-form solutions do exist for European plain vanilla options. This feature also makes calibration of the model feasible.