Cliquet Options

Overview

Cliquet options are financial derivative contracts that provide a guaranteed minimum return in exchange for capping the maximum return over the life of the contract.  Cliquets are appealing to investors because they can protect against downside risk.  FINCAD provides functions to value several different types of cliquet options.  The basic assumption is that the underlying index follows a Black-Scholes lognormal model with a constant volatility or a time-dependent deterministic volatility curve.

Formulas & Technical Details

Let  be the price of an asset or an index and  be  time points. The return of  at time  is defined as:

Most cliquet options have payoffs of one of the following four types:

1.       Cliquets with both local and global floors

2.       Cliquets with global floors but no local floor

 

3.       Cliquets with local floors but no global floor

 

4.       Cliquets with neither local nor global floor

where:  

 is a strike price,

 is a fixed coupon,

 is a scaling factor and

 for a call option and –1 for a put.  Most of the cliquets are call options with a strike price of 0.

Note that at any time ,  the historical coupon of the above four types of cliquets are, respectively:

Historical Coupon 1

 

Historical Coupon 2

 

Historical Coupon 3

 

Historical Coupon 4

 

FINCAD Functions

aaCliquet(d_v, d_m, d_e, princ, prices, ex, local_cap, local_floor, cliquet_type, global_floor, cpn_fix, scaling, ret_type, cpn_his, vol_crv, df_crv, intrp, div_obj, option_type, sam_freq, hl, MC_para, output_type, stat, ReturnStat)

Calculates by Monte Carlo simulation fair value and risk statistics of a cliquet option with periodic sampling dates.  The accuracy of the fair value is also provided.

 

aaCliquet_fs(d_v, d_m, d_e, princ, prices, ex, local_cap, local_floor, cliquet_type, global_floor, cpn_fix, scaling, ret_type, cpn_his, vol_crv, df_crv, intrp, div_obj, option_type, sam_seq, MC_para, output_type, stat, ReturnStat)

Calculates by Monte Carlo simulation fair value and risk statistics of a cliquet option with free-style sampling dates.  The accuracy of the fair value is also provided.

 

Description of Inputs

Input Argument

Type

Description

d_v

Date

Value date

d_m

Date

Maturity date

d_e

Date

Effective date

npa

Double

Notional amount

prices

Double

Underlying prices.  It is an array of two entries (spot price, price at last sampling date <= d_v)

ex

Double

Exercise price.  It is a threshold of the underlying return.  In determining the payoff component of a cliquet at a sampling date the difference of the underlying return and the exercise price is used.  For most cliquets the exercise price is 0.

local_cap

Double

Local cap

local_floor

Double

Local floor

cliquet_type

Integer

Type of cliquets, a switch.  The four values of the switch correspond to the four types of cliquets given above.

global_floor

Double

Global floor

cpn_fix

Double

Fixed coupon:  a fixed coupon that is added to the performance of the underlying.  See above formulas.

scaling

Rate

Scaling factor of underlying return

ret_type

Integer

Return calculation type, a switch:

1 = relative return,

2 = absolute return.  

See above for definitions.

cpn_his

Double

Historical price list or price return table.  It can be of two types:  A) a single entry of a historical coupon.  This is the coupon calculated from historical asset returns up to, including, the valuation date.  See above.  B) an historical price table which will be used to calculate a cumulative historical coupon.  Note that since missing data are not handled in the function, the price list must be complete – one positive price at each historical sampling date.  In function aaCliquet, sampling dates are not adjusted for weekends.  If a sampling day falls on a weekend, the price on the closest prior business day should be used.

vlt_tbl

Double

A constant volatility or a time-dependent volatility table

df_crv

Double

Discount factor curve.  It can be a single-entry of risk-free rate or a discount factor curve.  See note 210 in the function reference page for details.

intrp

Integer

Interpolation method, a switch.

div_obj

Double

Dividend (yield) table.

option_type

Integer

Option type, a switch:

1 = call,

2 = put.  

For a call the underlying price return minus the exercise price is used to determine the payoff component at a sampling date; for a put the exercise price minus the underlying price return is used.

sam_freq

Double

Sampling frequency, a switch.

hl

Date

Holiday list.

MC_para

Double

A single-entry array of number of random trials.  If this parameter is an integer then a different random number seed is used each time the function is called.  If this parameter is not an integer then the same random number seed is used each time, and the results will be the same each time.

output_type

Integer

Output type, a switch.  There are two output types.  The difference is on the risk statistics.  If output type is 1, the delta and other risk statistics are calculated with respect to one unit of a cliquet (npa = 1).  Otherwise, the risk statistics are the so-called dollar risk statistics – no assumption is made on the notional amount.

stat

Integer

Statistic, a switch.

 

Description of Outputs

Output Statistic

Type

Description

1

Double

Fair value

2

Double

Delta

3

Double

Gamma

4

Double

Theta

5

Double

 Vega

6

Double

rho of rate

7

Double

rho of dividend

8

Double

accuracy. This is the accuracy value corresponding to a 95% confidence interval.  For example, if the option value is 3.5 and the accuracy is 0.01 then the 95% confidence interval is (3.5 – 0.1, 3.5 + 0.1) = (3.4, 3.6).

For details about the calculation of Greeks, see the Greeks of Options on non-Interest Instruments FINCAD Math Reference document.

Examples

Example 1

Consider a five-year cliquet with an annual sampling frequency, a global floor of 14% and a local cap of 5%, and with no fixed coupon.  Today’s date is Dec. 1, 2004.  The cliquet was effective on Jan. 24, 2004, expires at Jan. 24, 2009.  Suppose the notional amount is $100, the relevant annual risk free rate is 5% and the annual volatility of the underlying index is 20%.  Call aaCliquet with a random trial number of 10000.5 to get the following results.  (Using a decimal value for the number of trials means that the same random number seed is used each time, so the results will always be as given below.)

aaCliquet

Argument

Description

Example Data

Switch

d_v

Value date

1-Dec-2004

 

d_m

Maturity date

21-Jan-2009

 

d_e

Effective date

21-Jan-2004

 

npa

Notional amount

100

 

prices

Underlying prices

See below

 

ex

Exercise price

0

 

local_cap

Local cap

5%

 

local_floor

Local floor

0

 

cliquet_type

Type of cliquets

1

with both local and global floors

global_floor

Global floor

14%

 

cpn_fix

Fixed coupon

0

 

scaling

Scaling factor of underlying return

100%

 

ret_type

Return calculation type

1

relative return

cpn_his

Historical price list or price return table

0%

 

vlt_tbl

Volatility or volatility table

20%

 

df_crv

Discount factor curve

5%

 

Intrp

Interpolation method

1

 

div_obj

Dividend (yield) table.

0

 

option_type

Option type

1

call

sam_freq

Sampling frequency

1

 

hl

Holiday list

0

 

MC_para

A single-entry array of number of random trials

10000.5

 

output_type

Output type

fair value and risk statistic

1

Underlying prices

Spot price

Price at last sampling date

100

99

Results

Statistics

Description

Value

1

fair value

12.65326

2

Delta

0.000913

3

Gamma

1.61E-05

4

Theta

-3.14E-06

5

Vega

-0.00019

6

rho of rate

-0.00363

7

rho of dividend

-0.0013

8

Accuracy

0.002513

 

Example 2

Suppose in Example 1 the effective date is Jan. 21, 2002 and there is a fixed coupon of 0.03.  To value the cliquet historical performances of the underlying index must be considered.  Suppose the historical prices on the three past sampling dates are given as in the following table:

Historical Price List (cpn_his)

date

price

21-Jan-2002

87

21-Jan-2003

95

21-Jan-2004

99

Then call function aaCliquet with the following inputs

aaCliquet

Argument

Description

Example Data

Switch

d_v

Value date

1-Dec-2004

 

d_m

Maturity date

21-Jan-2009

 

d_e

Effective date

21-Jan-2002

 

npa

Notional amount

100

 

prices

Underlying prices

See Example 1

 

ex

Exercise price

0

 

local_cap

Local cap

5%

 

local_floor

Local floor

0

 

cliquet_type

Type of cliquets

with both local and global floors

1

global_floor

Global floor

14%

 

cpn_fix

Fixed coupon

3%

 

scaling

Scaling factor of underlying return

100%

 

ret_type

Return calculation type

1

relative return

cpn_his

Historical price list or price return table

See below

 

vlt_tbl

Volatility or volatility table

20%

 

df_crv

Discount factor curve

5%

 

Intrp

Interpolation method

1

 

div_obj

Dividend (yield) table.

0

 

option_type

Option type

1

call

sam_freq

Sampling frequency

1

 

hl

Holiday list

0

 

MC_para

A single-entry array of number of random trials

10000.5

 

output_type

Output type

fair value and risk statistic

1

to get the valuation results:

Results

Statistics

Description

Value

1

fair value

20.12432

2

Delta

0.002163

3

Gamma

-1.2E-05

4

Theta

-1.27E-05

5

Vega

-0.00063

6

rho of rate

-0.00458

7

rho of dividend

-0.00345

8

Accuracy

0.004036

Instead of using a historical price list as the input of the parameter cpn_his one can calculate the cumulative historical coupon of the underlying using the formulas given in the section Formulas & Technical Details and use this as an input of the parameter.  The details of the calculation is given in the following table.

Calculation of Cumulative Historical Coupon

date

price

return

local cap

local floor

performance

 

21-Jan-2002

87

 

 

 

 

 

21-Jan-2003

95

0.091954

0.05

0

0.05

 

21-Jan-2004

99

0.042105

0.05

0

0.042105

 

 

 

 

 

 

0.092105

total

The cumulative historical coupon is 0.092105.  It can be checked easily that function aaCliquet gives the same results as above if the parameter cpn_his is set to this value.

References

[1]          Wilmott, P. (Dec. 2002), ‘Cliquet Options and Volatility Models’, Wilmott Magazine.

 

 

Disclaimer

 

With respect to this document, FinancialCAD Corporation (“FINCAD”) makes no warranty either express or implied, including, but not limited to, any implied warranty of merchantability or fitness for a particular purpose. In no event shall FINCAD be liable to anyone for special, collateral, incidental, or consequential damages in connection with or arising out of the use of this document or the information contained in it. This document should not be relied on as a substitute for your own independent research or the advice of your professional financial, accounting or other advisors.

 

This information is subject to change without notice. FINCAD assumes no responsibility for any errors in this document or their consequences and reserves the right to make changes to this document without notice.

 

Copyright

 

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