Cliquets

Pension providers that add an equity-linked guarantee to a product, and dealers that sell cliquets, deal with the shape of the payoff and the difficulty of valuing it.

Adding capped and floored returns for each period

A cliquet measures the return on a stock or stock index period by period, applies a cap and a floor to each period, and adds up the results. Some sources name ratchets alongside cliquets¹. A cliquet is a path-dependent option whose payoff is set by the values on discrete observation dates². Overview of Exotic Equity Products covers where cliquets sit among equity exotics.

Let the observation times be $t_0, t_1, \dots, t_N$, and let $S(t_i)$ be the price of the underlying at time $t_i$. The return in period $i$ (from $t_{i-1}$ to $t_i$) is the change relative to the price at the start of the period.

$$R_i = \frac{S(t_i) - S(t_{i-1})}{S(t_{i-1})}$$

Apply a per-period cap (the local cap $C_l$) and floor (the local floor $F_l$), add the results over all periods. Then apply an overall cap (the global cap $C_g$) and floor (the global floor $F_g$) to the sum. The resulting rate multiplies the notional amount³.

$$\text{Payoff} = \text{Notional} \times \max\left(F_g,\ \min\left(C_g,\ \sum_{i=1}^{N} \max\{F_l, \min\{C_l, R_i\}\}\right)\right)$$

The local cap and floor apply to each period, and the global cap and floor apply to the sum.

The buyer pays the premium to the dealer and receives the payoff at maturity. The payoff depends on the returns observed in each period, so its amount is not known when the trade starts.

Deep divePayoff of a reverse cliquet

A variation on the formula above sets a nominal rate $\hat{R}$ and subtracts the amounts from the periods in which the price fell.

$$\text{Payoff} = \text{Notional} \times \max\left(F_g,\ \hat{R} + \min\left(0,\ \sum_{i=1}^{N} \max\{F_l, \min\{0, R_i\}\}\right)\right)$$

When $F_l$ is negative, this is called a reverse cliquet. The buyer can receive a high nominal rate $\hat{R}$, but the payoff falls by the amount of each period's decline, so the buyer bears the downside risk⁴.

Comparing payoffs in a hypothetical example

Consider a hypothetical example in which a buyer buys, from a dealer, a cliquet on a hypothetical stock index calculated by Company Q (the Q index). The notional is 100 million yen, the term is 4 years, and the index is observed once a year. The local cap is 5%, the local floor is 0%, the global cap is 10%, and the global floor is 2%. The premium is 7% of the notional (this is not a model price, and interest and discounting are ignored). With a notional of 100 million yen, 1% is 1 million yen.

Figure description

The curve shows one period's return after the example's local floor and local cap are applied. Every return below the floor is set to the floor value, and every return above the cap is set to the cap value. The return passes through unchanged only between the two. The global cap and floor apply to the sum, so this figure does not include them.

The table compares the payoff on four paths.

On path A, the index rises in every period, but the local cap cuts the 6% and 8% periods to 5%, and the global cap then cuts the 17% sum to 10%. The index rises 22.6% over the whole term, yet the payoff is 10%, so the buyer does not receive most of the rise.

On path B, the index falls in every period, and the local floor sets each period to 0%. The global floor lifts the sum of 0% to 2%, so the buyer receives 2% even though the index fell 23.7% over the whole term. Because the buyer paid a 7% premium, however, the result is a 5% loss.

On path C, the index falls 5.5% over the whole term, but the periods in which it rose still count, so the buyer receives 8%. The amounts from the periods in which the index fell are not subtracted. That is the difference from a vanilla option, whose payoff depends only on the prices at the start and at the end. On path D, the 7% payoff equals the 7% premium, so the profit or loss is zero.

The buyer gains when the sum after the local caps and floors exceeds the 7% premium, as on paths A and C. The buyer loses when the sum falls short of 7%. Path B is an example of the lowest payoff, 2%, within that group.

Use in guaranteed products

A pension provider that adds an equity-linked guarantee to a product can limit the upside to reduce the cost of the guarantee, and products of this kind are cliquet contracts⁵. Large pension providers and individual investors have chosen them as insurance against stock price movements⁶.

Viewing a cliquet as a chain of forward-start options

A forward-start option is a right to receive, on a future reset date and at no additional cost, an option whose strike is proportional to the price of the underlying on that date. Rubinstein introduced it, and on the reset date a standard call is delivered⁷. Let the reset date be $T_1$, the expiry $T_2$, and the ratio $\kappa$. The payoff of the call at expiry is as follows⁸.

$$(S_{T_2} - \kappa S_{T_1})^+$$

$(x)^+$ means $x$ if $x$ is positive and 0 otherwise. One period of a cliquet can be seen as a forward-start option whose strike is the price at the start of the period. The forward-start payoff is the building block of cliquets and ratchets¹.

Looking at the local cap and floor alone, when $C_l \ge F_l$, each period's payoff splits as follows⁹.

$$\max\{F_l, \min\{C_l, R_i\}\} = F_l + (R_i - F_l)^+ - (R_i - C_l)^+$$

$(R_i - F_l)^+$ is the payoff of a call measured as a ratio to the price at the start of the period, with a strike of $S(t_{i-1})(1 + F_l)$. $(R_i - C_l)^+$ is the same with a strike of $S(t_{i-1})(1 + C_l)$. So one period is a forward-start call spread, struck off the price at the start of the period, plus the floor amount. Without the global cap and floor, the whole cliquet would be the sum of the call spreads for the individual periods. The global cap and floor apply to the sum, so the periods can no longer be valued one at a time.

In the Black-Scholes model, the price of a forward-start option corresponds to that of a standard call whose term equals the period from the reset date to expiry. In more general models, such as stochastic volatility models, this correspondence does not hold¹⁰.

Why the valuation depends on the future smile

The distribution of each period's return $R_i$ cannot be read directly from today's vanilla options in the market (those that expire on each observation date and are struck relative to today's price). Today's vanilla options show the distribution of the price on each observation date. One period of a cliquet needs the distribution of the ratio to the price at the start of that period, and the volatility (the size of price movements) relevant here can be called forward volatility¹¹.

Because of the reset, the implied volatility of a forward-start option differs greatly from an ordinary vanilla smile¹². The value of a cliquet depends on the shape of this future smile (the forward smile). Here are three points on the volatility smile that are specific to cliquets.

• The Local Volatility Models page covers models that set volatility as a function of the stock price and time and fit it to vanilla option prices. In them, the forward skew is usually flat. It has been pointed out that the price of a digital cliquet can then fall far below the price in a stochastic volatility model¹³.
• The Heston model and exponential Lévy models are also said to carry large model risk in valuing cliquets¹³.
• In an experiment, a synthetic market was built with a jump-diffusion model, and a local volatility model was fitted to the vanilla option prices it produces. There was no guarantee that the cliquet price would come close to the price in the synthetic market¹⁴.

In that last experiment, adding two corrections used in practice brought the values at the initial price fairly close. One correction makes volatility a function of the ratio of the current price to the price at the previous reset. The other moves the volatility surface (how volatility spreads across maturities and strikes) forward in time, so that the slope on the short-maturity side does not flatten. Away from the initial price, however, the gap widened, and the deltas used for hedging differed greatly¹⁵.

Model risk that remains with the dealer

The selling dealer receives the premium and pays the payoff at maturity. It has been pointed out that the volatility risk of a cliquet is easily underestimated¹⁶. The valuation depends on the model of the underlying's movement, and even a model fitted to vanilla option prices does not necessarily give a value close to that of another model¹⁴.

The contract used in the previous experiment shows how wide the range across models can be. The contract runs 5 years with annual observations, a local cap of 8%, a local floor of 0%, a global floor of 16%, and no global cap. An uncertain volatility model assumes only that the size of price movements lies somewhere between two levels (23.59% and 31.67% per year). Under it, the worst and best prices from the buyer's side are about 15.0% to about 17.5% of the notional, a wide range¹⁷. The paper says this range shows the seller of a cliquet the actual volatility risk¹⁵.

The sensitivities the dealer uses for hedging (delta, vega, and others) are computed from the valuation model. Models give very different deltas, so the same problem as in the valuation appears in the hedge. The paper describes the idea of taking a static position in exchange-traded options that expire on the observation dates to reduce model risk¹⁸. Some papers state that pricing problems with standard cliquets led to losses at several financial institutions¹⁹.

Notes

1. Andrea Mazzon and Andrea Pascucci, "The forward smile in stochastic local volatility models", section 1, "Introduction" (page 2). It says the forward-start call payoff is the building block of cliquets and ratchets, and does not say they are two names for one product. ↩ ↩²

2. H. A. Windcliff, P. A. Forsyth, and K. R. Vetzal, "Numerical Methods and Volatility Models for Valuing Cliquet Options" (revised version of February 14, 2006), page 2, section 1. ↩

3. Windcliff et al., cited above, section 4.1 (pages 11–12), equations (4.1) and (4.2). The symbols follow the source's $C_l$, $F_l$, $C_g$, and $F_g$, and the summation index runs to $N$ (the source's $N_{obs}$). ↩

4. Windcliff et al., cited above, section 4.1 (page 12), equation (4.3) and the sentence below it. The source calls the equation with $F_l < 0$ a reverse cliquet. The contract values in the example are in Table 12 of section 7.4 (page 31). ↩

5. Windcliff et al., cited above, pages 1–2, section 1, "Introduction". It describes the exchange of a guaranteed minimum yearly return for a cap, and guarantees on pension products. ↩

6. The same source, section 8, "Conclusions" (page 32), the opening. It says cliquets became popular as insurance for large pension funds and individual investors against market volatility. ↩

7. Elisa Alòs, Antoine Jacquier, and Jorge A. León, "The implied volatility of Forward-Start options", pages 1–2, section 1, "Introduction". It names Rubinstein as the originator and gives employee stock options and cliquets as examples. ↩

8. Aleksandar Mijatović and Martijn Pistorius, "Exotic derivatives under stochastic volatility models with jumps", section 1.4.3 (page 23). It writes the payoff as $(S_{T_2} - \kappa S_{T_1})^+$. ↩

9. A formula derived in the text, not found in the sources. When $C_l \ge F_l$, splitting into the cases $R_i < F_l$, $F_l \le R_i \le C_l$, and $R_i > C_l$ shows that the two sides agree. ↩

10. Alòs et al., cited above, page 2. In the Black-Scholes model, increments follow the same distribution, so a forward-start option corresponds to an ordinary call with time to maturity $T-s$. In general stochastic volatility models it does not. ↩

11. An explanation derived in the text. The term "forward volatility" is not the sources' own, and follows Mazzon and Pascucci's "forward implied volatility" and Windcliff et al.'s "forward starting volatility skew". ↩

12. Alòs et al., cited above, page 2. Because of the reset feature, the implied volatility of a forward-start option differs greatly from the ordinary vanilla smile. ↩

13. Mazzon and Pascucci, cited above, page 2. It describes the usually flat forward skew, the digital cliquet price that can fall far below the stochastic volatility price, and the model risk of the Heston and exponential Lévy models. ↩ ↩²

14. Windcliff et al., cited above, page 2 (section 1) and section 8 (page 32). Even when a local volatility model is fitted to the vanilla prices of a jump-diffusion model, there is no guarantee that the cliquet prices will be close. ↩ ↩²

15. The same source, section 1 (page 2), section 7.1 (pages 26–27), and section 8 (pages 32–33). It describes the effect of the two practical corrections and says the worst-to-best gap in the uncertain volatility model shows the seller the actual volatility risk. ↩ ↩²

16. The same source, page 2, footnote 2. It says Wilmott (2002) pointed out that the volatility risk of a cliquet is easily underestimated. ↩

17. The same source, Table 11 in section 7.3 (page 29), the contract terms in Table 4 (page 22), and the volatility levels in Table 1 (page 10). Values for a notional of 1 are shown as percentages in the text. ↩

18. The same source, section 7.1 (page 26). It says that, when hedging, one may want to take a static position in exchange-traded options that expire on the observation dates in order to reduce model risk. ↩

19. Mazzon and Pascucci, cited above, page 2. It says the pricing problems of standard cliquets are well documented in the literature and led to losses at several financial institutions, citing Jeffery (2004, Risk magazine). ↩