Understanding Greeks
The role of “Greeks” in Structured Products and Derivatives is very important. The terminology of the Greeks was first adopted after the initial usage of the Black-Scholes model in the 1970s. This advance was 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.
Most investors and buy-side firms tend to focus on product valuation and likely outcomes. This is rightly the most important consideration, but the “Greeks” play a crucial role in measuring risk, exposures and likely future behaviour.
Fundamental Greeks: Delta and Gamma
The most fundamental Greeks are Delta and Gamma because these quantities are both directly connected to movements in the underlying asset. 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.
Because the price of a structured product moves constantly throughout 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.
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.
The rate of change of the value of Delta is known as the “Gamma”. The sign and level of Gamma tell 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. It is customary to take a pricing model as the driver and then “bump” or change parameters in order to calculate the Greeks.
The use of the Greeks serves two important complementary functions. The first is for the trading desk of the issuing bank to understand how to risk manage the structured product through its lifetime. Any structured product is a contractual obligation to deliver a certain defined return depending on the performance of the underlying assets. In order to meet that requirement, risk management performs a precise role akin to that of an investment manager tracking an index or algorithm by taking into account the changes in the product’s price when market conditions change. The second group that finds the Greeks useful will be those who have bought structured products, such as a private bank or financial adviser. Greeks help them understand how the product’s value will likely move in different scenarios and will indicate aggregate risk and concentrations in a portfolio of such instruments.
Other important Greeks and their applications
Other Greeks are also important, including Vega (Volatility), Rho (Interest Rates) plus sensitivities to dividend yield and correlation.
For all structured products, the payoff is usually defined in terms of direct performance of the underlying assets, such as upside return or fixed returns that come from satisfying Auto-call or barrier conditions. It is clear, therefore, that the secondary market price or fair value of the structured product will move when the underlying moves. This is because the price reflects the previous performance of the underlying asset together with the range of likely outcomes that may happen during the remaining life of the product. As the product approaches maturity, the in-life price will converge to the final payoff of the product.
However, many other variables will contribute to the array of Greeks even though they are not explicitly in the product’s payoff formula. These include the parameters we will consider of volatility, dividend yield, interest rates and correlation. Option pricing models as used by investment banks take into account all relevant market variables that affect the cost of the required hedging strategy. This, in turn will change the product price because of the dual relationship between the price and hedging.
In most standard equity option models, these parameters are assumed constant or deterministic. Volatility is the main exception and can have different treatments depending on the modelling approach of a bank and the complexity of the product being valued. However, despite this assumption, parameters constantly change in value and so the sensitivity to them must be considered.
One of the most important Greeks is the sensitivity to volatility (“Vega”). Any product which has a long option position such as a protected growth product will have positive sensitivity to volatility whereas those where the investor has capital at risk (for example, a reverse convertible) will generally have negative sensitivity. The value of Vega reflects the direction that the product price will take if volatility changes. From a trader’s perspective, these sensitivities translate into practical consequences because as volatility increases, the trader will generally have to do more rebalancing of the underlying as it moves around more. If those rebalances happen in a way that the trader will incur cost (because of being forced to buy the underlying when it goes up and selling when it goes down) then the trader is exposed to volatility increasing and the value of the Vega will indicate that. This is known as being short Gamma and short Vega - two related concepts. The reverse would be true for a long Vega position. Volatility levels in the market can change significantly and quickly, as measured by quantities such as the VIX series and other indicators of both implied and historical volatility.
Rho is the usual name given to price sensitivity to interest rates, and this tends to give the fewest problems to traders because of the easy availability of hedging instruments such as bonds and the fact that the value of Rho does not usually change as quickly as other Greeks.
Challenges in hedging and market dynamics
The two final sensitivities that we will examine are those relating to dividend yields and correlation. Nearly all structured products are long their respective underlyings in some way, whether they be capital protected or capital at risk. Part of the associated long Delta hedge involves dividend exposure if the product is linked to a stock or an index on a price return basis. Historically, the investment banks in the market have always been very exposed to falls in dividend yields since almost all products have the risk in the same direction and external hedging instruments such as dividend futures are not always feasible. The usual phenomenon that is seen, therefore, is that implied dividend yields settle at a lower level than most future consensus estimates. This is intended to create a risk buffer, but in 2020 after the Covid-19 market crash and cutting of company dividends, this proved to be not nearly enough. Other solutions have been sought, most notably the use of fixed dividend indices which eliminate dividend risk from a product.
Correlation sensitivity is also important for any product that features more than one underlying asset. Correlation is even harder to hedge and the market also suffers from the one-way position of risk given that the two most popular product types tend to have long basket options or short worst-of puts which line up the risk in the same direction. Because of these factors, the main approach is to also mark the correlation values higher than most estimates. This also failed to prevent significant losses in 2020. Investors would have been able to measure both these effects which were in their favour this year.
In summary, we see that structured product pricing, hedging and monitoring depends on many factors and it is very important to give the Greeks full attention in any portfolio, particularly during times of market turmoil.
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