## European vanilla call-put option pricing with python quantstart exchange rate euro to dollar today

This post is part of a larger series on Option Pricing with Python __exchange rate myr to usd__. In order to get the best out of this article, you should be able to tick the following boxes:

• The derivation of the Black-Scholes equation and the Black-Scholes formula for the price of a European Vanilla Call/Put Option (this will be the subject of a later article)

Python has a reputation primarily as a scripting language, functioning as "glue" between other codebases. I would like to augment that reputation and show that with tools like Numpy and SciPy it is perfectly capable of being utilised to price options **usd to aud** exchange rate. However, it is yet another language to learn – so why should you invest the time? I have listed the primary benefits below:

The articles to follow will concentrate on simplicity, rather than optimisation. We will follow Daniel Duffy’s philosophy of "First we get it working, then we get it right, then we optimise it".

To this end, we will not initially be using any complicated syntax – it should be obvious how we have gone from an algorithm to its implementation, with the minimum of head-scratching machine learning artificial intelligence. After we have code working, we can make sure it is producing the correct values. Finally, we can optimise it and put it into production. Black-Scholes Pricing Formula

It has already been outlined that the reader is to be familiar with the Black-Scholes formula for the pricing of European Vanilla Calls and Puts html to word converter. For completeness, the price of a European Vanilla Call, \(C(S,t)\) is given below, where \(S\) is the underlying asset, \(K\) is the strike price, \(r\) is the risk-free rate, \( T \) is the time to maturity and \(\sigma\) is the (constant) volatility of the underlying \(S\) (\(N\) is described below):

The remaining function we have yet to describe is \(N\). This is the cumulative distribution function of the standard normal distribution convert __binary to text__. The formula for \( N \) is given by:

In addition, we would like to have closed form solutions for the "Greeks", i.e. the option price sensitivities to the underlying variables and parameters __fraction worksheets free__. For this reason we also need the formula for the probability density function of the standard normal distribution which is given below:

Our task is now to utilise Python to implement these functions and provide us with values for the closed-form solution to the price of a European Vanilla Call or Put with their associated sensitivities. Python Implementation of Statistical Functions

Open a new Python file in your favourite IDE and name it statistics.py **usa today** crossword. We require some mathematical functions so we will use the math library and import the natural logarithm, the exponential function and the mathematical constant \(\pi\):

This function is fairly self-explanatory. The next function to code up is the CDF of the normal distribution. However, it contains an integral with \(-\infty\) as one of its limits – so how can we appoximate this on the computer? The method utilised (26.2.17) comes from the famous Abramovitz & Stegun textbook on mathematical functions texas baseball. The Wikipedia article on the Normal Distribution sheds more light on this and other methods. Here is the Python listing for the algorithm:

We now have the two statistical functions necessary for calculating the closed-form options prices for European vanilla calls and puts usd to inr today. Python Implementation of Closed-Form European Vanilla Call-Put Prices

We need to create a second file, which we will call closed_form.py. It should reside in the same file directory as the statistics.py file in order for the following import syntax to work correctly:

Referring back to the Black-Scholes formula for the price of a call, \(C(S,t)\), we can see that it requires two separate functions named \(d_j\), \( j \in \{1,2\} \). The \(d_j\) take \(S\), \(T\), \(K\), \(r\) and \(\sigma\) (v in the code) as arguments. It is possible to write the \(d_j\) as a single function which takes an additional argument \(j\). The code for this function is provided below:

"""Price of a European call option struck at K, with spot S, constant rate r, constant vol v (over the life of the option) and time to maturity T"""

"""Price of a European put option struck at K, with spot S, constant rate r, constant vol v (over the life of the option) and time to maturity T"""

This concludes the coding of formulae for the statistical distribution functions in statistics.py and the vanilla __call option__ prices in closed_form.py rm to usd. At this stage it is prudent to check that the formulae produce the correct results and satisfy the known bounds on the prices (such as put-call parity). That will be the subject of the next article.