A survey of Equity pricing models

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A survey of Equity pricing models

Equity linked products remain dominant in most structured products markets around the world. This is because equities are considered to be the natural long term asset class for growth. Equity markets, through indices and headline stocks are well known to all investors. In some markets, credit, FX and fixed income have a strong niche following due to market conditions or local preferences, but equities are in general the asset class of choice.

Equity derivatives (calls, puts, complex or “exotic” options and their combinations) are the building blocks that investment banks use to create an equity-linked structured product.

Different product constructions

The simplest products will only need one derivative component. Examples include participation products which consists of a zero-coupon bond (capital payment) and a long position in a call option on the underlying to provide growth. A reverse convertible Is made up of a zero-coupon bond, an income stream of coupons plus a short position of a put option on the underlying. For the investor this put option translates to capital at risk.

More complex products can combine several options. For example, a dual directional has multiple call and put options. Another common mechanism is to add a barrier or averaging feature. Further up the complexity scale from a pricing perspective are products such as auto-calls because they cannot be easily decomposed into simpler options and those involving multiple underlyings such as a worst-of.

A product may be simple in terms of option construction but still present significant trading or hedging problems. The most common example of this is a hard to hedge underlying such as a custom index or illiquid stock or a very long dated product which would cause challenges for even the most widely traded underlyings such as mainstream benchmark indices.

Selecting a pricing model suite

A reliable suite of equity pricing models is necessary for any trading desk, valuation provider or end user of structured products such as an asset manager or hedge fund. These models are most important to help price complex options which will include elements of path dependency, multiple assets and complex payoff formulae.

Building pricing curves is a crucial first step to choosing and using a pricing model. The main elements of equity pricing curves are interest rates, dividend yields, volatility curves and correlations.

Interest rates and dividend yields present the fewest challenges although marking dividend yields will often be at levels different to economic forecasts because of demand for long delta (bullish) products.

Curves can be created from analysing option prices. This will be from a combination of exchange traded options for the most common underlyings such as indices and stocks augmented by broker markets and direct inter-bank trading. These options and the pricing information from them form the foundation of derivatives and structured product markets.

Any choice of model needs to be calibrated back to these prices so that if they were used to price these options they would get the same answer. If not, arbitrage violations would occur which would lead to mispricing or potential hedging losses.

Choosing a volatility treatment

An equity model implementation requires two choices to be made. Firstly a volatility treatment must be selected. The three most common modelling approaches are constant volatility, local volatility and stochastic volatility. The second choice governs modelling the underlying and performing the payoff calculation. The two main candidates here are tree methods (also known as lattices) and Monte Carlo simulations.

Constant volatility models are very simple to set up and work well with both trees and Monte Carlo. This model is equivalent extending the Black Scholes method to cover a wider set of payoffs. The approach has seen a growth in usage in recent years outside market pricing to satisfy regulatory testing such as FCA stress testing, PRIIPs and economic scenario generation.

A constant volatility model is useful for testing purposes or to provide approximate values but is almost useless for pricing and risk management. The main reason for this is the presence of volatility “skew” and “smile” in equities markets. Skew is caused by the presence of relatively higher prices (and therefore volatilities) at lower strikes compared to at the money. It exists because of the demand for lower strike options to buy crash protection. Smile refers to higher volatilities on the upside too, it is less common and tends to only be seen for higher volatility stocks. Smile is also common in FX markets because of the natural symmetry of the two ways of defining a currency pair.

Since these low-strike options form an important part of the hedging strategy a credible model framework needs to match their pricing and a constant volatility model will not be able to do that.

Addressing skew: local and stochastic volatility

The importance of capturing skew effects has stimulated much academic debate over the years and it has given rise to more sophisticated models.

Local volatility was the first of these and its core purpose is to replicate any volatility by strike and maturity. The local volatility approach requires a mathematically complex methodology to calculate the entire volatility surface. It takes what can be thought of as a look up approach at a given time point and underlying level and fits immediately into a Monte Carlo or tree calculation. Of the two approaches, the binomial (or trinomial) tree is the simpler method but is somewhat limited in its application. It is best suited for single asset European, callable and auto-callable products and it can also deal with American barriers. The Monte Carlo method is more flexible and is also much better suited to multi asset products. Monte Carlo has long become the more popular choice especially since increased computing power has long eliminated disadvantages of speed of calculation.

Our survey of equity models is completed by considering the stochastic volatility model. The local volatility model is popular because it does not require any further parameters other than the volatility surface. However, it is a static model and has serious limitations when calculating hedge ratios and dealing with products dependent on forward volatilities such as cliquets.

The stochastic volatility model introduces an extra degree of freedom to model the volatility process directly and this gives a better treatment in the areas in which the local volatility model suffers. It comes at a cost of extra parameters that need to be fitted and a certain extra complexity in modelling.

Both local and stochastic volatility models try to replicate the volatilities observed at different strikes and maturities by adjusting the volatility dynamic within the model itself. This is logical and generally successful however for some underlying assets such as mainstream indices the skew effect can be very pronounced, and it is necessary to fix parameters at unrealistic levels to fit to the volatility surface. Volatility skew comes about primarily because of supply and demand and not future volatility expectations but as a practical approach nothing better has been proposed.

Equity products remain popular and therefore their pricing and risk management is very important. In order to price, hedge and analyse them accurate data and a flexible modelling approach is required using a combination of models to cope with the challenges this asset class presents.

Tags: Valuations

A version of this article has also appeared on www.structuredretailproducts.com

Image courtesy of:     Daniel Sessler / unsplash.com

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