CHAPTER 1
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, µ the constant drift, s the instantaneous volatility and Zt is a standard Brownian motion. In a risk-neutral setting, the drift µ 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

where

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 ?t, the Profit-Loss (PL) for such a time interval is roughly (see Bossu (2014), p. 59)

G is the gamma of the option and measures how the delta (marginally) changes with the changing index level. ?S is the change in the index level over the time interval ?t. It is given by

Ø is the theta of the option and measures how the option value changes with the passage of time. It is given approximately by (see Bossu (2014), p. 60)

With this we get

The quantity is called the dollar gamma of the option and gives the second order change in the option price induced by a (marginal) change in the index level. is the squared realized return over the time interval ?t - it might be interpreted as the (instantaneously) realized variance if the time interval is short enough and the drift is close to zero. Finally, is the fixed, "theoretical" variance in the model for the time interval.

The above reasoning illustrates that the PL of a discretely delta hedged option position is determined by the difference between the realized variance during the discrete hedge interval and the theoretically expected variance given the model parameter for the volatility. The total hedge error over intervals is given by

This little exercise in option hedging leads us to a result which is already quite close to a product intensively discussed in this book: listed variance futures. Variance futures, and their Over-the-Counter (OTC) relatives variance swaps, pay to the holder the difference between realized variance over a certain period of time and a fixed variance strike.

1.2 Notions of Volatility and Variance

The previous section already touches on different notions of volatility and variance. This section provides formal definitions for these and other quantities of importance. For a more detailed exposition refer to Sinclair (2008). In what follows we assume that a time series is given with quotes Sn, n ? {0, ., N} (see Hilpisch (2015, ch. 3)). We do not assume any specific model that might generate the time series data. The log return for n > 0 is defined by

• realized or historical volatility: this refers to the standard deviation of the log returns of a financial time series; suppose we observe N (past) log returns Rn, n ? {1, ., N}, with mean return ; the realized or historical volatility is then given by

• instantaneous volatility: this refers to the volatility factor of a diffusion process; for example, in the Black-Scholes-Merton model the instantaneous volatility s is found in the respective (risk-neutral) stochastic differential equation (SDE)

• implied volatility: this is the volatility that, if put into the Black-Scholes-Merton option pricing formula, gives the market-observed price of an option; suppose we observe today a price of C*0 for a European call option; the implied volatility simp is the quantity that solves ceteris paribus the implicit equation

These volatilities all have squared counterparts which are then named variance, such as realized variance, instantenous variance or implied variance. We have already encountered realized variance in the previous section. Let us revisit this quantity for a moment. Simply applying the above definition of realized volatility and squaring it we get

In practice, however, this definition usually gets adjusted to

The drift of the process is assumed to be zero and only the log return terms get squared. It is also common practice to use the definition for the uncorrected (biased) standard deviation with factor instead of the definition for the corrected (unbiased) standard deviation with factor . This explains why we call the term from the delta hedge PL in the previous section realized variance over the time interval ?t. In that case, however, the return is the simple return instead of the log return.

Other adjustments in practice are to scale the value to an annual quantity by multiplying it by 252 (trading days) and to introduce an additional scaling term (to get percent values instead of decimal ones). One then usually ends up with (see chapter 9, Realized Variance and Variance Swaps)

Later on we will also drop the hat notation when there is no ambiguity.

1.3 Listed Volatility and Variance Derivatives

Volatility is one of the most important notions and concepts in derivatives pricing and analytics. Early research and financial practice considered volatility as a major input for pricing and hedging. It is not that long ago that the market started thinking of volatility as an asset class of its own and designed products to make it directly tradable.

The idea for a volatility index was conceived by Brenner and Galai in 1987 and published in the note Brenner and Galai (1989) in the Financial Analysts Journal. They write in their note:

''While there are efficient tools for hedging against general changes in overall market directions, so far there are no effective tools available for hedging against changes in volatility. . We therefore propose the construction of three volatility indexes on which cash-settled options and futures can be traded."

In what follows, we focus on the US and European markets.

1.3.1 The US History

The Chicago Board Options Exchange (CBOE) introduced an equity volatility index, called VIX, in 1993. It was based on a methodology developed by Fleming, Ostdiek and Whaley (1995) - a working paper version of which was circulated in 1993 - and data from S&P 100 index options. The methodology was changed in 2003 to the now standard practice which uses the robust, model free replication results for variance (see chapter 3 Model-Free Replication of...