Understanding option Greeks: part 1

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Understanding option Greeks: part 1

This article is the first of two pieces to look at the importance and role of “Greeks” in Structured Products and Derivatives. In this first part, we will consider Delta and Gamma. These quantities are both directly connected to movements in the underlying asset. The second article will examine Vega (Volatility), Rho (Interest Rates) plus sensitivities to dividend yield, correlation and credit spread.

The terminology of the Greeks was first adopted after the first usage of the Black Scholes model in the 1970s. This advance was so important to pricing and trading vanilla (call and put) options on single stocks and other underlyings on exchanges. Since that time, there has been an explosion of academic literature, expertise within investment banks and increasingly sophisticated trading systems.

Starting with Delta and Gamma

Delta and Gamma are the most fundamental of these quantities and were the first names to be adopted because Alpha (outperformance) and Beta (ratio to the market) already had meaning within investment circles, applied to the measurement of underlying performance. This convention was continued through the other related quantities (not always with genuine Greek letters!) and so the mystical sounding nickname of the “Greeks” has stuck ever since.

All Greeks measure the sensitivity of option or structured product prices to the various parameters that affect them. The collective name of sensitivities is used as an alternative to “Greeks”.

Traders use a variety of models for different asset classes, product types and underlying market assumptions to price and trade derivatives. Since structured products are a combination of derivatives, such models can also be combined or extended to price them.

Price and sensitivities

Most people in the industry take for granted the existence of a price of a structured product. This price is understood to represent the (discounted) average payoff it is clear that this price exists at the start of the product, moving through its lifetime, converging at maturity to the final payoff outcome whatever that may be.

In most cases, the product sensitivities are calculated as the change in price when one parameter is changed by a standard prescribed amount, usually in isolation. These sensitivities are in fact themselves the derivative of the price (mathematical use of the word, not financial) with respect to each parameter.

Applications in risk management

Because the price of a structured product moves constantly through its lifetime, the trader needs to calculate and act on the various Greeks to hedge the position and to execute the bank’s risk management.

A simple example demonstrates why this is necessary. Consider a structured product with initial price 100. Generally, the final payoff could be anything above or below its initial value with a minimum return of zero, such as 50, 100 or 150. The hedging process involves the trader running a series of hedges so that the initial purchase price plus the accumulated profit or loss from the hedging process equals the required final payout as closely as possible, irrespective of market movement. In fact, many trading desks can also make a significant profit by understanding pricing parameters and their movement, or by achieving favourable offsetting positions.

Worked example for Delta

The principle of how Greeks work can be seen from a simple example involving Delta, the sensitivity to the underlying asset. Suppose a structured product is issued with the underlying value at 1000 and has an initial price of 100% as calculated by an option model. The same model can reprice the product if the underlying increases in value by 1%, i.e. it moves to 1010. The value of the structured product might in this instance now be 100.7%.

The Delta is generally defined as the change in structured product price for a 1% move in the underlying asset, giving in this case a value of 0.7, or 70%. This means that the sensitivity of this structured product to the underlying is positive but less than 100% which is what a direct delta one investment would have. This figure is typical for an at-risk product at outset, rising to 100% towards the end of the product lifetime if markets rise and falling to zero if the product is “out-of-the money” and will just return a fixed amount.

Delta can sometimes be in excess of 100% in special situations where the price is very sensitive to moves (e.g. for a Digital or Auto-call near their levels close to maturity), or for a leveraged product. Delta can also be negative for any product that has a “bear” style payoff.

The level of the Delta tells the trader how much of the underlying asset to hold. In the example above, holding 70% of the product notional amount (through stocks or futures) would mean that, if the underlying increases, then the profit from the hedge would offset exactly the increase in product value that the investor holds and the bank books as a liability. If instead the underlying falls then the hedge will lose money but the liability decreases by the same amount again leaving the trader with zero net change.

Gamma – reacting to moves

The rate of change of the value of Delta is known as the “Gamma”. The sign and level of Gamma tells the trader how quickly and in which direction the Delta will change when the underlying asset moves. Gamma is very important since the trader’s main concern is when to rebalance a hedge. Volatile markets, sudden price moves or a lack of liquidity can be very challenging for a trader to avoid losses when the hedge cannot be updated in time.

Delta and Gamma are unique amongst the Greeks since these are driven by underlying asset movement in a way anticipated by standard models such as Black Scholes. The other Greeks that will be discussed in the next article are also important but arise when changes in market parameters such as volatility occur that were not directly accounted for by the model.

Although it is customary to take a pricing model as the driver and then “bump” or change parameters in order to calculate the Greeks it should be remembered that there is a duality between pricing and hedging, one cannot exist without the other. The price formula is only “correct” because the theoretically feasible hedging strategy can be set up and funded. The original Black Scholes paper explains the relationship between the two.

Tags: Valuations

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

Image courtesy of:     Patrick P / unsplash.com

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