In previous articles (see Options, Greeks and P&L Decomposition (Part 1) and Options, Greeks and P&L Decomposition (Part 2)) we decomposed the P&L of simple option strategies in different time horizons with the major first and second order greeks. In this article, we analyse the P&L of a short iron butterfly option strategy, which is […]
In this article, we will present the Newton-Raphson method for calculating the implied volatility from option prices. Black-Scholes Price vs Volatility In Black-Scholes model, the price of an option is a function of five variables: The chart below shows the price of a European call option when changing the volatility, all other parameters being fixed. […]
The valuation and the risk management of options can be quickly complex. It depends on the option’s feature and the pricing model. There is often no closed-form solution for the pricing of the derivatives and it involves multiple dimensions. There is a vaste litterature on numerical methods such as binomial / trinomial tree, finite difference, […]
In a previous article (see Options, Greeks and P&L Decomposition (Part 1)) we analysed the decomposition of the P&L of an option strategy in a short period of time with the major first and second order greeks. In this new article, we assume that the option is kept for a certain period of time, delta-hedged […]
The Black-Scholes model is a pricing model used to determine the theoretical price of options contracts. It was developed by Fischer Black and Myron Scholes in 1973 and later refined by Robert Merton. It revolutionized the options market by providing a fair and consistent framework to calculate the fair value and to manage the risk […]
Understanding option sensitivities and greeks is crucial to be successful in trading and risk management of options. In this post we will see how to decompose the P&L of an option strategy in a short time interval with the major first and second order greeks and analyse it with several case studies. P&L Attribution and […]
The Volatility is Constant in the Black-Scholes Model… In the Black-Scholes model, the volatility of a stock price is assumed to be constant, independent of the strike or time-to-maturity. So if the model was correct, a plot of the Black Scholes volatility implied from option prices with a constant time to expiry, would be a […]
Foundations of Stochastic Calculus Stochastic Calculus is a branch of mathematics that deals with random processes. Beyond probabilities it also has links with differential equations, and is widely used in finance particularly for option modelling. Kiyosi Itô (1915-2008) pioneered the field inventing the concept of stochastic integral and stochastic differential equations. Itô Stochastic Integral In […]
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