Equity Derivatives and Hybrids

Frontmatter

1. Empirical Evidence

Abstract

When studying derivatives it is useful to be aware of the empirical properties of the underlying. Those properties should be taken into account when attempting to model the underlying for the risk management of derivatives. Derivatives often depend on one or more closing prices within a time period ranging from few months up to several years. Thus the focus in this chapter is a time series of closing prices, although the methods can also be applied to higher frequency data.

Oliver Brockhaus

2. Equity Derivatives Market

Abstract

In fixed income markets the relevant underlyings are yield curves and there is basically one of those per currency. With the end of risk neutral pricing the use of single curve modelling has become more complicated since effects such as liquidity, funding costs, credit risk and collateral agreements can no longer be ignored. However, arguably, this complicates the nature of fixed income underlyings rather than creates new ones.

Oliver Brockhaus

3. Exotic Equity Derivatives

Abstract

Introducing a knock-out barrier reduces the price of a Vanilla option. An investor may be willing to take the risk of the underlying trading at the barrier level against this sure profit. This will be the case especially if gains on the remaining portfolio offset any losses due a barrier event.

Oliver Brockhaus

4. Implied Volatility

Abstract

In equities markets one has a set of bid and offer market prices for out-of-the-money call and put options on a given underlying. These options may be European or American.

Oliver Brockhaus

5. Dividends

Abstract

One of the attractive features of stocks is that the holder is entitled to dividends. Dividend levels, payment and ex-dates are typically stable and known in advance but are sometimes adjusted by the board of management.

Oliver Brockhaus

6. Short Volatility Models

Abstract

Persistent volatility can be generated through volatility functionally, depending on the stock level, through a stochastic process correlated with equity returns or through a combination of both. Additionally, the volatility process and the return distributions may exhibit jumps. Stochastic volatility models are incomplete and not Markovian in the filtration generated by the stock process. Calibration of stochastic volatility (or indeed any) model to the Vanilla market has an impact on hedging.

Oliver Brockhaus

7. Implied Volatility Dynamics

Abstract

There is a natural order of market data speed, with spot levels changing faster than at-the-money volatility, at-the-money volatility changing more rapidly than volatility skew and volatilities being more volatile than dividend forecasts. Hedging performance can be improved by assuming a link between different market parameters, see Andreasen and Huge (2014). For example, when calculating a price with a new spot, or computing the sensitivity to a stock move (delta) using a spot shift, one may assume that this move is accompanied by a volatility move in the opposite direction or a change in expected dividends in the same direction. Thus, a delta hedge also hedges part of vega if stock and volatility are correlated. If delta and vega are hedged separately one has to be careful not to double count vega exposure. This section discusses pricing approaches assuming a spot move but no new volatility information.

Oliver Brockhaus

8. Correlation

Abstract

There exists a liquid options market for stock indices and, since the components are often stocks of important companies, there are also options on those stocks. In many financial institutions the responsibility for market data (forward, volatility) is separated between stock and index trading desks. Index exposure will be hedged with indices rather than component stocks. If only Vanilla options are traded this is a valid approach and there is no correlation exposure. Forward discrepancies between index and sum of stocks will typically be small and can be attributed to market inefficiencies.

Oliver Brockhaus

9. Copulas

Abstract

Copulas have many applications within financial modelling, including the repre-sentation of the joint distribution of

• several default times, as discussed in Section 12.3,
• various assets at a given future time horizon,
• one asset at several future time horizons (time copula), see Section 9.9.

Oliver Brockhaus

10. Fixed Income

Abstract

Fixed income is at the core of financial modelling. Before considering assets such as stocks, currencies, commodities or credit events it is necessary to value future cash flows.

Oliver Brockhaus

11. Equity-interest Rate Hybrids

Abstract

The problem of modelling assets with stochastic drift has been studied in order to risk manage long dated foreign exchange derivatives. In that context advanced yield curve dynamics is often combined with simplified foreign exchange smile dynamics. In the equity context the aim is to extend smile consistent models to the stochastic interest rate case. Interest rates dynamics is often limited to single-factor short rate models, such as Hull-White’s extension of the Vasicek model or the Cox-Ingersoll-Ross square root process, discussed in Sections 10.4 and 10.5.

Oliver Brockhaus

12. Credit

Abstract

Default can impact a specific trade between two counterparties in three ways, through default of either counterparty as well as default of a firm related to the underlyings. In this section the focus is on modelling default times in the context of credit derivatives. Default of the underlying in the case of equity derivatives is discussed in Chapter 13, while counterparty default is covered in Chapter 14.

Oliver Brockhaus

13. Defaultable Equity

Abstract

Derivatives that require incorporation of default of the underlying’s issuer include derivatives on single stock, in particular convertible bonds.

Oliver Brockhaus

14. Counterparty Credit Risk

Abstract

Since the Lehman default on September 15, 2008 the credit quality of issuers of retail products has received much attention. Arguably, the largest losses to institutions during the crisis were due to credit value adjustment (CVA) rather than to actual default.

Oliver Brockhaus

15. Foreign Exchange

Abstract

Foreign exchange rates are defined by both domestic and foreign currency. These concepts are symmetric in the sense that for a foreign investor my domestic currency is her foreign currency. For example, a European investor may buy 1$ at today’s exchange rate of 0.75€.

Oliver Brockhaus

16. Affine Processes

Abstract

Closed form pricing via Fourier transform methods is the basis for many stochastic volatility models. A general framework has been formulated by Duffie, Pan and Singleton (2000).

Oliver Brockhaus

17. Monte Carlo

Abstract

Equity derivatives contracts can be valued using closed form solutions, finite difference methods and Monte Carlo simulation. The former two approaches are limited to specific products and models. Closed form solutions are available primarily for European options and affine models as discussed in Chapters 11 and 16. For log-normal assets (approximate) formulae are also known for barrier products, as well as basket and Asian products, see Sections 3.1, 3.3 and 3.8. Finite difference, including tree methods, are important for products with early exercise features, such as American options and convertible bonds. For path-dependent and multi-asset products these methods are of limited use due to the high dimensionality of the state space.

Oliver Brockhaus

18. Gauss

Abstract

This chapter collects (and derives) closed form formulae for integrals related to Brownian motion for reference elsewhere in the book. It includes Black-Scholes and Bachelier formulae, as well as a range of barrier products.

Oliver Brockhaus

Backmatter