Listed Volatility and Variance Derivatives
Leverage Python for expert-level volatility and variance derivative trading
Listed Volatility and Variance Derivatives is a comprehensive treatment of all aspects of these increasingly popular derivatives products, and has the distinction of being both the first to cover European volatility and variance products provided by Eurex and the first to offer Python code for implementing comprehensive quantitative analyses of these financial products. For those who want to get started right away, the book is accompanied by a dedicated Web page and a Github repository that includes all the code from the book for easy replication and use, as well as a hosted version of all the code for immediate execution.
Python is fast making inroads into financial modelling and derivatives analytics, and recent developments allow Python to be as fast as pure C++ or C while consisting generally of only 10% of the code lines associated with the compiled languages. This complete guide offers rare insight into the use of Python to undertake complex quantitative analyses of listed volatility and variance derivatives.
- Learn how to use Python for data and financial analysis, and reproduce stylised facts on volatility and variance markets
- Gain an understanding of the fundamental techniques of modelling volatility and variance and the model-free replication of variance
- Familiarise yourself with micro structure elements of the markets for listed volatility and variance derivatives
- Reproduce all results and graphics with IPython/Jupyter Notebooks and Python codes that accompany the book
Listed Volatility and Variance Derivatives is the complete guide to Python-based quantitative analysis of these Eurex derivatives products.
Listed Volatility and Variance Derivatives
Derivatives, Volatility and Variance
The first chapter provides some background information for the rest of the book. It mainly covers concepts and notions of importance for later chapters. In particular, it shows how the delta hedging of options is connected with variance swaps and futures. It also discusses different notions of volatility and variance, the history of traded volatility and variance derivatives as well as why Python is a good choice for the analysis of such instruments.1.1 Option Pricing and Hedging
In the Black-Scholes-Merton (1973) benchmark model for option pricing, uncertainty with regard to the single underlying risk factor S (stock price, index level, etc.) is driven by a geometric Brownian motion with stochastic differential equation (SDE)
Throughout we may think of the risk factor as being a stock index paying no dividends. St is then the level of the index at time t, mi the constant drift, s the instantaneous volatility and Zt is a standard Brownian motion. In a risk-neutral setting, the drift mi is replaced by the (constant) risk-less short rate r
In addition to the index which is assumed to be directly tradable, there is also a risk-less bond B available for trading. It satisfies the differential equation
In this model, it is possible to derive a closed pricing formula for a vanilla European call option C maturing at some future date T with payoff max [ST - K, 0], K being the fixed strike price. It is
The price of a vanilla European put option P with payoff max [K - ST, 0] is determined by put-call parity as
There are multiple ways to derive this famous Black-Scholes-Merton formula. One way relies on the construction of a portfolio comprised of the index and the risk-less bond that perfectly replicates the option payoff at maturity. To avoid risk-less arbitrage, the value of the option must equal the payoff of the replicating portfolio. Another method relies on calculating the risk-neutral expectation of the option payoff at maturity and discounting it back to the present by the risk-neutral short rate. For detailed explanations of these approaches refer, for example, to Björk (2009).
Yet another way, which we want to look at in a bit more detail, is to perfectly hedge the risk resulting from an option (e.g. from the point of view of a seller of the option) by dynamically trading the index and the risk-less bond. This approach is usually called delta hedging (see Sinclair (2008), ch. 1). The delta of a European call option is given by the first partial derivative of the pricing formula with respect to the value of the risk factor, i.e. . More specifically, we get
When trading takes place continuously, the European call option position hedged by dt index units short is risk-less:
This is due to the fact that the only (instantaneous) risk results from changes in the index level and all such (marginal) changes are compensated for by the delta short index position.
Continuous models and trading are a mathematically convenient description of the real world. However, in practice trading and therefore hedging can only take place at discrete points in time. This does not lead to a complete breakdown of the delta hedging approach, but it introduces hedge errors. If hedging takes place at every discrete time interval of length Deltat, the Profit-Loss (PL) for such a time interval is roughly (see Bossu (2014), p. 59)
_ is the gamma of the option and measures how the delta (marginally) changes with the changing index level. DeltaS is the change in the index level over the time interval Deltat. It is given by
_ is the theta of the option and measures how the option value change