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- Volatility Derivatives 1
- The world of Structured Products 4
- Library of Structured Products 0
- Table of Contents
- Vanilla Options
- Volatility, Skew and Term Stru
- Option Sensitivies: Greeks
- Option Strategies
- Correlation
- Dispersion Options
- Barrier Options
- Digitals
- Autocallable Structures
- The Cliquet Family

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- The Cliquet Family

# The Cliquet Family

__13. The Cliquet Family__

Cliquet Options are **appealing retail products** because they provide downside protection while at the same time offer significant upside potential. Cliquet structures are most popular on single indices, but are also structured on stocks, and even baskets of stocks or indices. By introducing various caps and floors, local and global, one is sure to find an attractive yet reasonably priced derivative.

Cliquets are also known as **ratchet options** because they are based on **resetting the strike** of a derivative structure to the last fixing of the underlying asset. The resetting feature is what makes cliquets unique and **introduces the forward skew exposure**.

These are truly beautiful derivatives but must be handled with caution.

**13.1. Forward Starting Options**

A forward starting option is an option that starts at some time in the future and has a maturity after that date.

where t_{0 }< t_{1} < t_{2}. t_{1} is the strike date, or the date on which the option’s strike becomes set but the option, however, is **priced and the premium is fixed and paid at t _{0}**.

It is true that one can apply a Black–Scholes formula to price such an option, but caution must be taken.

Which volatility does one use to price a forward starting option? We know the implied volatilities of vanilla options, but what about forward starting options?

The answer is that one must use the **implied forward volatility**, and this may not be available with the same liquidity we see in regular vanillas. Much like the existence of the skew we see in vanilla options’ implied volatilities across strikes, we have what is known as a **forward skew**. Buying or selling forward starting calls or puts gives the investor exposure to this forward skew.

As we shall see, the simplest of cliquet products are built as just a series of forward starting calls, that is, a series of forward starting calls with the same initial date t_{0}, but where the strike date of the second (or the jth) call is the maturity date of the first (or ( j − 1)th) call. Not all hope is lost in regard to getting the right forward skews as there exist standardized cliquets for which one can obtain some market consensus data from which one can extrapolate a fairly accurate idea of where the market is pricing forward skew.

**13.2. Cliquets with local floors and caps**

Let’s look at a sum of forward starting calls and build what is known as a cliquet structure.

**13.2.1. Payoff Mechanism**

We call this a **symmetric cliquet** when F = −C, that is, the cap and floor are symmetric around zero.

To be exact, it is in fact the **sum of forward starting call spreads**. Setting the floor F = 0 the holder of the cliquet is long the ATM forward starting calls and short the forward starting OTM calls of strike C (the local cap), each with maturity equal to the resetting period.

In conclusion so far, the locally floored cliquet allows the investor to pick up positive annual returns and lock in such profit. This will fit quite naturally into a note structure with capital guarantee, an **easily marketable retail product**: you collect all positive annual returns capped at 10% but none of the negative returns, and of course your money back at maturity.

**13.2.2. Forward Skew and Other Risks **

Cliquet structures have **caps and floors, which immediately implies skew dependency**. Because cliquets are a series of skew-dependent options, the overall structure will itself be quite skew sensitive, and **due to the reset features it is forward skew to which the cliquet is exposed**.

In fact, there is **exposure to more than one forward skew**; taking the example in Table 13.2, the first call spread has exposure to the usual skew, the volatility skew given by the vanilla options surface taken at the required strikes. The second call spread is sensitive to the 1- to 2-year forward skew, and, likewise, the third is sensitive to the 2- to 3-year skew. Obviously, an increase in any of these will increase the price of the cliquet, so the seller of the derivative is short forward skew.

Many of the derivatives we have seen so far can, with some caution, be evaluated using **local volatility models**.

Let us assume that we have calibrated a local volatility model to a set of vanilla options. If we were to simulate the process forward in time, we find that **the forward skews it generates begin to flatten out**.

This can be explained by the fact that, in local volatility models, the volatility is a deterministic function of the underlying price. This dependency of the volatility on the spot results in higher probabilities of the spot moving higher, so as time goes by (or as we simulate forward in time), we find that volatilities and skew go down (thus the flattening out effect). This is cause for serious concern as anyone attempting to price cliquets or any forward skew-dependent derivative with local volatility will almost surely misprice it.

**13.3. Cliquets with global floors and caps**

The above payoff is the same locally floored and capped cliquet we saw above, only the total sum of all the cliquets is now capped and floored (that is, globally capped and floored).

The first thing to note is that, in the case of a local floor F set to zero, a global floor of zero is meaningless as the payoff will already be non-negative. Introducing a positive global floor will act as a minimum guarantee for the option but this makes the option more expensive.

Two popular cliquets are the **locally floored globally capped cliquet** and the **locally capped and globally floored cliquet**.

One thing to note is that we must enforce the restriction that **n × F must be strictly less than GC**, or else the payoff makes no sense as it will always be equal to GC.

*Forward Skew Risk *

If one looks inside the payoff of the LFGC cliquet above, the term inside the sum can be written as:

which is nothing but a call (for each i), plus a minimum guarantee of F. How far in- or out-of-the-money the calls are clearly depends on the level of the floor F .

Next we manipulate the overall payoff as follows:

The second equality makes use of the identity **min(A, B) + max(A, B) = A + B**. In the third we have **taken the term GC outside of the last term** and it cancels with the first term. In the last equality we have used the derivation done above and removed the floor from the payoffs of the floored call so that we can see each as a call struck at F.

**Since GC − nF > 0, the second term of equation minus the third term is always positive**. The term nF makes perfect sense in the last equality as from the payoff definition we can directly see that the option will have to pay at least as much as the sum of all the floors, which is n × F.

Having split the payoff as such we can now see two clear things that will allow us to see the existence of the two key cliquet risks separately.

The **second term is just a series of forward starting call options**.

The **third term is a compound option (an option on an option) and here it is an OTM call on a series of forward starting call options**. It is OTM because we have the constraint that GC−n F > 0 in order that the initial payoff of this cliquet makes sense.

The first effect to consider is **Vega**; the second term is a call option and thus obviously has positive Vega; and the third term, although a compound option, also has positive Vega. **If volatility goes up, both of these increase in value. However, since these two have different signs it is not clear which has the greater effect to determine the volatility position.** This is compounded by the fact that we do not have a prespecified value for F, different values of F set the first set of call options at different levels of moneyness, which also impacts their Vega sensitivity. An ATM call option has a higher Vega than ITM and OTM call options. This effect is also not clear cut on the compound option.

The appearance of the second term, which involves the sum of forward starting call options, will have **forward skew exposure**. **Whether this is a cost or a benefit is not clear until one specifies the value of F. The reason is that if F is negative, then the call options are ITM, and in the presence of skew this increases the ITM call option volatility, in this case forward skew, thus raising their prices. However, if F is positive, the opposite occurs as these calls will be OTM and the OTM vol is lower in the presence of skew. The third term will also have forward skew exposure, but likewise the position is not clear.** It is also the fact that the call options are embedded into another call option in the third term that means the forward skew risk cannot be captured solely by using the forward implied volatility at the correct strike, and here we must apply a model that knows about forward skew.

Another risk to be aware of here is **Vega convexity**. It comes primarily from the third term and is due to the nature of any compound option. The meaning and interpretations of this risk are discussed below once we have introduced the second cliquet in this family.

Looking at the cliquet with local caps and a global floor, the payoff is given by:

The method of using local caps and global floors is an effective method of reducing the price of such a derivative. Here the local caps limit the upside returns, and the global floor acts as a guarantee against a negative overall payout.

**13.3.1. Vega Convexity **

In addition to the forward skew exposure, we have also to worry about Vega convexity.

At each interval [t_{i −1 }, t_{i} ] between each two reset dates t_{i−1 }and t_{i} we are capping the returns, that is, the seller of the option is buying volatility at each of these caps. The seller of the option is selling skew, only here, due to the reset feature, it is the forward skew of the period [t_{i −1 }, t_{i }] to which the seller is exposed.

What is Vega convexity? Or convexity in general firstly?

This general term convexity, or price convexity, is often used to refer to the second-order effect, Gamma. We now want to understand Vega convexity. We know Vega is the sensitivity of our option price to a movement in the volatility of the underlying. **Vega convexity, also known as Volga, is the second-order sensitivity, or convexity, of our option price to a movement in the underlying’s volatility. Mathematically it is the second derivative of the option price w.r.t volatility. **

Under Black–Scholes, the Vega of a call (and a put) is given by:

Under Black–Scholes, the Volga of a call (also a put) option is given by

As we can see in Figure 13.1, **the ATM call has no Vega convexity. As we move away from the money, the OTM and ITM calls start to pick up convexity; however, if we go sufficiently deep in- or out-of-the-money, we see that convexity fades to zero**. In these cases the moneyness of the options is so extreme that in the OTM call it drives the price down towards zero, rendering sensitivities negligible, and in the case of the deep ITM call, the price becomes linear in the underlying and has thus lost its convexities.

Now given that we know the prices of call options from the market, we do not need to worry about this when pricing them. Obviously these are driven by supply and demand, and **these convexities are already priced into the call option values**.

However, our beloved cliquets do have Vega convexity that will need to be priced. **Much the same way that one assumes an underlying stock price to be a stochastic (random) process in order to price options, we will need to introduce the concept of stochastic volatility here**.

Why? When we model a stock price as a random process it allows us to see the convexity to this stock price (Gamma); if we had assumed it to be deterministic (no randomness in the stock price) our option would have no value (as we would not see the convexity). The same thing applies in the case of **Vega convex options, only here we need to allow our volatility (or possibly the variance) to be a random process of its own to enable us to see Vega convexity**.

Using a stochastic volatility model for such product allows us to look at a **vol-of-vol parameter** (the volatility of volatility). Recall, for the volatility to be itself a random process, it must have its own volatility which is known as vol-of-vol. This term is the coefficient of the second derivative w.r.t. volatility (Volga) in a pricing equation.

**13.3.2 Levels of These Risks **

So now we are faced with two risks: **forward skew** and **Vega convexity**. The degree of sensitivity to each of these differs however from one product to the next. Understanding this is the key to pricing these structures correctly.

Starting with the locally floored and capped cliquet, with no global floors or caps, we look at the special case of a **symmetric cliquet**. What is special about this cliquet is that at the ATM point it does not have Vega convexity. This does not mean that its Vega is not sensitive to volatility movements; on the contrary, Vega even changes sign at this point, but **ATM Volga is zero**.

In this case, from a pricing standpoint, we are left with the primary concern of getting the forward skews correct and are not too concerned with the Vega convexity, given it is zero (or very close to it). **However, as the market moves the product will begin to exhibit varying levels of Vega convexity.**

**When the local floor or the cap are positioned differently**, introducing asymmetry, as in the case of local floor set to zero and local cap at 5%, **we begin to pick up Volga**. This happens in much the same way that the Vegas of the two call options in a call spread cancel each other less as the caps and floors become more asymmetric. So, in the case of the 0% locally floored 5% locally capped cliquet, as in the above example, **the sellers will be mainly exposure to forward skew, with the Vega convexity, which the seller of the option is short, gradually increasing**.

**Much like the Vega changes sign for a cliquet, so does the Gamma**. If there is a time for one to look at a higher order it is for an option like this for which the Gamma changes sign. **The third-order effect, the sensitivity of Gamma to a movement in the underlying’s price**, is known as **Speed** and defined as:

The essential idea of this graph will be to notice the point at which the Gamma changes sign to see how sensitive it would be to a movement in either direction of the underlying. As we saw before with Gamma, one will need to compute the two-sided Gamma, that is, the sensitivity to the underlying’s price going up, and also (but computed separately) the case when its price goes down.

**In conclusion, generally, given the nature of these profiles – especially that the Vega profile can change – we will have to use a stochastic volatility model for such options. **

**13.4. Reverse Cliquets **

In this variant on the standard cliquet, the reverse cliquet begins with a headline coupon, and instead of accumulating positive performances in an underlying asset, negative returns are deducted from said coupon. At maturity, the holder of the reverse cliquet receives the part of the coupon that is left after the deduction of the (possibly floored) negative period to period performances:

We should note that the **global floor here plays an important role** as there is a very large potential downside should there be continuously large down movements in the index. This floor will be absolutely necessary if one is to fit this option into a principal-protected structure.

It is clear that the seller of the reverse cliquet is effectively long the puts. **Since they are ATM, we find low skew sensitivity, which can potentially increase**; however, it is **highly convex in Vega** and we must tread with extreme caution on this risk from the onset.

*Reverse Cliquet Risks *

The reason **the structure appeals to investors in an environment of high volatility** is that the seller of the reverse cliquet will be buying volatility at this high level, and is thus able to offer a **higher headline coupon to the investor**. **As volatility goes down, the price goes up, meaning that this would be perfect for an investor wanting to buy an option during a bear market (high volatility) with the view that it is almost over. **

With respect to forward skews we can think of this structure as follows. The investor is short ATM puts, but he is also long the global floor that acts like a strip of OTM puts. **Thinking about this as a put spread, the seller of the reverse cliquet, who is long the put spreads, is short skew, and forward skew in this case owing to the reset feature.** **Since this option is highly convex in Vega (and the seller is short Vega convexity) we expect the reverse cliquet to be worth much more under stochastic volatility than local volatility, reflecting not only the flatter skew the local vol generates but, more importantly, the fact that local volatility does not know about Vega convexity**. Increasing the global floor up from zero can only increase this Vega convexity further.

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