## Volatility in structured products

Volatility is a standard for measuring uncertainty about future price trends. Rising volatility leads to a greater likelyhood that the price for an asset will change considerably in the future. Volatility is one of the greatest price factor for structured products.

The two charts below illustrate the price and volatility behavior of the Swiss Market Index (SMI) and a Swiss government bond that expired in December 2006. The left-hand chart shows that the price variations of the SMI were much larger than the bond's. The right-hand chart shows the corresponding volatility of both instruments. Clearly, volatility is anything but constant. While the volatility of the SMI index rose and fell, the bond's volatility slowly decreased until it reached zero at maturity. In fact, the closer it approached its maturity, the smaller the price movements and its volatility became. Because the bond's redemption price was known to be 100%, whatever incertitude remained, slowly disappeared as time elapsed. The SMI has no expiry date; the index is calculated for an indefinite time. Its volatility rises in uncertain times (like when the dot.com bubble burst and 9/11) and decreases when the economy smoothly runs along. One can observe that volatility is negatively correlated with the index: it often rises when the stock market falls and falls down to a minimum level when stock prices increase. The negative *correlation coefficient* demonstrates the negative relationship between the price trend and volatility.

It is also worth noting that volatility has a certain tendency to return to a mean value. This phenomenon is referred to as the “mean-reverting process.” The volatility of an index never falls to zero and does not rise beyond a certain limit, despite the fact that it can experience "spikes", like in the subprime crisis when Lehman Brothers declared bankruptcy.

Volatility is central in the valuation of options. Rising volatility has a positive effect on the value of call and put options. This means that when volatility rises, options on an underlying asset tend to be “expensive,” and investors should consider selling them. Conversely, when volatility is low and “cheap,” buying is recommended. However, it is the forecast about the evolution of volatility an investor makes that should determine if the current volatiltiy is cheap or expensive.

Since the future volatility of an underlying asset cannot be observed, historical data has to be used for option valuation. The *historical volatility* can be calculated by means of the standard deviation, which mathematically measures the distribution of the assets around their mean.

Alternatively, the underlying volatility can be deduced from known option prices. The *implied volatility *calculated in this way makes it possible to assess whether an option is too expensive or too cheap against the background of one’s own volatility expectations.

### Volatility structure

Experience has shown that the volatility of an option can take on different values at the same time, depending on the *strike price* and the *maturity* of an option.

###### Dependence on the strike price (skew)

The chart to the right illustrates the typical relationship between the implied volatility of a call or put option and the strike price of the underlying stock or stock index (also *referred to volatility smile* or *skew*). In this example of the Eurostoxx50^{®} index, the spot price is set at 3'300. The volatility applied in pricing an in-the-money call option is much higher than the volatility of an out-of-the-money call option. The opposite is true for put options.

How does this come about in the case of equities? One possible explanation is the imbalance between supply and demand. Many institutional investors (pension funds or insurances) like to write calls on existing equity portfolios and hedge downside risks with out-of-the-money puts. In other words, there is chronic oversupply of out-of-the-money call options and chronic excess demand for out-of-the-money put options.

Another explanation is the historically asymmetrical price trend of equities. Prices fall faster and harder in crashes than they rise in bull markets. This is why it appears more likely for an out-of-the-money put option to ultimately end up in-the-money than for an out-of-the-money call option to end up in-the-money as a result of strong price gains.

###### Dependence on maturity (term structure)

In addition to the strike price, the maturity also plays a decisive role in option pricing (*volatility term structure*). The chart to the right illustrates a typical term structure for equities (Eurostoxx50^{®}):

The positive, “normal” term structure expresses the opinion that uncertainty gradually rises with time to maturity. The longer the residual maturity, the greater the likelihood that an event (war, weak corporate results, takeovers, etc.) will cause volatility to rise. The opposite case, an inverted term structure, when shorter dated strikes have a higher volatility than longer dated ones, primarily come about when short-term uncertainty is high, but is expected to normalize again in the future.

###### The Greeks

The sensitivity of volatility in calculating options is expressed with a Greek letter: *vega* measures the sensitivity of the option price to changes in the volatility of the underlying asset. Vega is highest when an option is at the money and falls as the maturity approaches.

### Influence of volatility on selected structured products

###### Certificates

Variations in volatility have no influence on certificates since there are no options associated.

###### Capital guaranteed products

In their basic form, capital guaranteed products consist of a zero-coupon bond and long call options. This means that the client is buying volatility. Accordingly, at the inception of the product, the participation rate will be lower when the volatility is high (assuming everything else remains constant) because fewer calls can be bought for the same amount.

The following table uses an example to illustrate how participation decreases as volatility rises.

Example: Eurostoxx50^{®}, 3-year term, interest rate of 4%, capital protection 100%:

Volatility |
15% |
20% |
25% |

Participation |
108% |
83% |
68% |

###### Reverse convertibles

These instruments consist of a bond part and a short put option on an underlying asset, such as a stock, an index, a currency pair, or a commodity. The investor is selling volatility, so the reverse reasoning applies than for capital guaranteed products. Ideally, one invests in such instruments at times of high implied volatility, because a high coupon can be generated.

Reverse convertibles are also used to optimize returns and can be selected in prospects of sideways markets. However, the compensation for the risk of having the security tendered is relatively low in that case. The following table shows an example of various coupons of a reverse convertible with different volatilities in the underlying asset:

Example: Siemens, 1-year term, strike 100%, interest rate of 2%, dividend yield 0.75%, no knock-in:

Volatility |
20% |
25% |
30% |

Coupon |
9.5% |
11.5% |
13.3% |

###### Bonus certificates

As bonus certificates are structured through the purchase of a down-and-out put option, the situation is less clear and depends on circumstances. A down-and-out put option “lives” as long as the barrier is not touched and “dies” as soon as it is reached.

*High* volatility is usually advantageous when investing in bonus certificates. At first sight, this may be surprising because the investor is buying the option embedded in the product. The fact is that when volatility is high the barrier is more likely to be triggered, which means that the option is then more likely to become worthless. The skew plays an important role here too, since the volatility of both the strike and the barrier have to be taken into consideration. The following example shows how the barrier of a bonus certificate changes with changing volatility.

Example: Eurostoxx50^{®}, 3-year term, interest rate of 4%, strike 100%, participation 100%, no bonus:

Volatility |
15% |
20% |
25% |

Barrier (in % of spot) |
68% |
64% |
60% |