Bermudan Options

Overview

A Bermudan option is a call or put option which can be exercised on prespecified days during the life of the option. It is reasonable to say that Bermudan options are a hybrid of European options, which can only be exercised on the option expiry date, and American options, which can be exercised at any time during the option life time. As a consequence, under same conditions, the value of a Bermudan option is greater than (or equal to) a European option but less than (or equal to) an American option.

A modification of the Cox-Rubinstein’s binomial model, a popular model in pricing American style options, also provides a satisfactory solution to price Bermudan options. The modification is as follows: since early exercise is not allowed at all times, early exercise is only checked at nodes corresponding to a Bermudan exercise date. For more details see the Standard American Style Options (Cox-Rubinstein Binomial) FINCAD Math Reference document.

FINCAD Functions

Functions aaBERM() and aaBERMdcf() calculate fair values and risk statistics for Bermudan style call/put options:

aaBERM():

For options on futures, FX, indices and stocks modeled using continuous dividends.

aaBERMdcf():

For options on stocks that pay discrete dividends.

aaBERM2dcf():

For options with discrete dividends and a repo-rate (a cost associated with short-selling the underlying asset).

The functions aaBERM_iv() and aaBERMdcf_iv() are used for calculating implied volatility given option prices.

There are several other types of Bermudan option functions available:

Quanto Options: aaQuanto_Berm()

Options with Varying Strikes and Variable Rates: there are several functions available that allow variable strike prices and variable rates.

Description of Inputs

Input Argument	Description
price_u	Current value of an underlying asset
ex	Strike price
d_exp	Expiry date of the option
d_v	Value date
d_berm_list	A list of dates at which the option can be exercised
vlt	The annualized volatility of the underlying asset.
rate_ann	rate - annual - Actual/365
cost_hldg	Also denoted rate1 and rate2, respectively. These rates are quoted on an annually compounded, Act / 365 (fixed) basis. If the underlying is an equity, rate1 is the risk-free rate and rate2 is the annualized dividend yield. If the underlying is a forward or futures price, rate2 should be set equal to the risk-free rate1. If the underlying is an FX (foreign exchange) rate, and quoted on a domestic per foreign basis, rate1 should be the risk-free domestic rate and rate2 the risk-free foreign rate. If the underlying is an FX rate, and quoted on a foreign per domestic basis, rate1 should be the risk-free foreign rate and rate2 the risk-free domestic rate. If the underlying is a commodity, then rate2 should be set to the annualized holding cost of the commodity, including storage and insurance costs as well as marginal convenience value.
risk_free_rate	Risk free rate of interest for aaBERM2dcf(). This can be entered as a single rate, assumed to be annual compounding with accrual method Act365(fixed), a single rate and a compounding frequency (switch 43), a single rate, a compounding frequency and an accrual method (switch 331), or as a discount factor curve.
repo_spread	This rate represents the difference between the risk-free interest rate and the repo rate. In the presence of a repo spread, the underlying asset grows at the risk-free rate minus the repo spread. aaBERM2dcf() only.
interp	Interpolation technique for the discount factor curves. aaBERM2dcf() only.
iter	The number of steps of the binomial tree.
option_type	The type of option: 1=call, 2=put.
stat	See the description of the output.
div_obj	Dividend payment table. The table has two columns, the dividend payment dates (left column) and the corresponding dividend payment amounts (right column). Used in aaBERMdcf() and aaBERMdcf_iv().
price_opt	Given option price. Used in the functions which calculate implied volatilities of options.

Description of Outputs

Output Statistic	Description
fair value	The fair value of the option.
delta	The rate of change in the fair value of the option per 1% change in the current value of the underlying asset. This is the derivative of the option price with respect to the underlying current value.
gamma	The rate of change in the value of delta per 1% change in the current value of the underlying asset. This is the second derivative of the option price with respect to the underlying current value.
theta	The rate of change in the fair value of the option per one day decrease of the option time. This is the negative of the derivative of the option price with respect to the option time (in years), divided by 365.
vega	The rate of change in the fair value of the option per 1% change in volatility. This is the derivative of the option price with respect to volatility.
rho of rate	The rate of change in the fair value of the option per 1% change in the risk-free rate, rate_ann. This is the derivative of the option price with respect to rate_ann.
rho of holding cost rate, rho of repo spread	The rate of change in the fair value of the option per 1% change in the holding cost, cost_hldg (or 1% change in the repo spread for aaBERM2dcf().) This is the derivative of the option price with respect to cost_hldg. If the underlying is futures, this statistic is not available.

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

Examples

Context specific examples are presented for Bermudan-style options on indices, commodity futures and foreign exchange rates. For options involving different underlyings, see the remarks following these examples.

Example 1: Indices

Consider a Bermudan style call option on an index. Early exercise is allowed on the 25th of every month. Suppose the index has a current value of 910 and the strike price of the option is 920. Today’s date is Aug. 1, 1997. The expiration date of the option is Feb. 1, 1998. Suppose the risk-free interest rate (annually compounded, Actual/365 (fixed)) is 7% and the dividend payout rate over the life of the option (annually compounded, Actual/365 (fixed)) is 5%. Suppose the annual volatility of the stock is 12%. Using a 200 step binomial model and the function aaBERM() we obtain the following results:

aaBERM

Argument	Description	Example Data	Switch
price_u	underlying price	910
ex	exercise price	920
d_exp	expiry date	1-Feb-1998
d_v	value (settlement) date	1-Aug-1997
d_berm_list	list of Bermudan exercise dates	see below	t_66
vlt	volatility	0.12
rate_ann	rate - annual - Actual/365	0.07
cost_hldg	holding cost - annual - Actual/365	0.05
option_type	option type	1	call
iter	number of time steps	200
stat	stat list	1…7

Bermudan Dates

25-Aug-1997

25-Sep-1997

25-Oct-1997

25-Nov-1997

25-Dec-1997

25-Jan-1998

Results

Statistics	Description	Value
1	fair value	29.55308
2	delta	0.497785
3	gamma	0.005034
4	theta	-0.10019
5	vega	252.9981
6	rho of rate	202.6499
7	rho of cost of holding	-210.775

Note: The dates in the list of Bermudan exercise dates which are outside the life time of the option are ignored.

Suppose in the above example the volatility of the stock is not known but the quoted price of the option is 29.55308. Calling aaBERM_iv(), with this price and the parameters as above, we obtain an implied volatility of 0.12, as expected.

Example 2: Stocks paying discrete dividends

Consider a Bermudan style call option on a stock. Suppose the stock has a spot price of 100 and the strike price is 105. Today is Aug. 1, 1997. The expiration date of the option is Feb. 1, 1998. The option can be exercised on the 25^th of any month during the life of the option. Suppose the relevant risk-free interest rate (annually compounded, Actual/365 (fixed)) is 7% and the stock pays dividend of 0.5 on the 20^th of March, June, September and December during its life. Suppose the annual volatility of the stock is 12%. Using a 200 step binomial model and the function aaBERMdcf() we can obtain the following results.

aaBERMdcf

Argument	Description	Example Data	Switch
price_u	underlying price	100
ex	exercise price	105
d_exp	expiry date	1-Feb-1998
d_v	value (settlement) date	1-Aug-1997
d_berm_list	list of Bermudan exercise dates	see below	t_66
vlt	volatility	0.12
rate_ann	rate - annual - Actual/365	0.07
div_obj	dividend payment list	see below
option_type	option type	1	call
iter	number of time steps	200
stat	stat list	1…7

Bermudan Dates

25-Aug-97

25-Sep-97

25-Oct-97

25-Nov-97

25-Dec-97

25-Jan-98

Dividend Payment

date	payment
20-Sep-1997	0.5
20-Dec-1997	0.5
20-Mar-1998	0.5
20-Jun-1998	0.5

Results

Statistics	Description	Value
1	fair value	2.313675
2	delta	0.402271
3	gamma	0.045994
4	theta	-0.015844
5	vega	27.790364
6	rho of rate	18.153204

Note: The dividend dates which are outside the life time of the option are ignored.

Example 3: Commodity Futures

Consider a Bermudan-style call option on 1000 barrels of 6 month's crude oil futures with strike price of $25 per barrel. The current futures price of one year crude oil futures per barrel is $24. Today's date is Aug. 1, 1997. The expiration date of the option is Sept. 28, 1997. The option can be exercised on 25^th of any month during the life time of the option. Suppose the relevant risk-free interest rate over the life of the option is 3%, (annually compounded, Actual/365 (fixed)), and annual volatility of the futures price is 20%. Using the function aaBERM() we obtain the following results:

aaBERM

Argument	Description	Example Data	Switch
price_u	underlying price	24
ex	exercise price	25
d_exp	expiry date	1-Feb-1998
d_v	value (settlement) date	1-Aug-1997
d_berm_list	list of Bermudan exercise dates	see below	t_66
vlt	volatility	0.2
rate_ann	rate - annual - Actual/365	0.03
cost_hldg	holding cost - annual - Actual/365	0.03
option_type	option type	1	call
iter	number of time steps	200
stat	stat list	1…6

Bermudan Dates

25-Aug-1997

25-Sep-1997

25-Oct-1997

25-Nov-1997

25-Dec-1997

25-Jan-1998

Results

Statistics	Description	Value
1	fair value	2.313675
2	delta	0.402271
3	gamma	0.045994
4	theta	-0.015844
5	vega	27.790364
6	rho of rate	18.153204

Fair value of the option = 1000* fair value per barrel = 9298.36 ($).

Example 4: Foreign Exchange Rates

Consider a Bermudan style put option on the exchange rate of £/$, sterling pounds per US dollar. The current exchange rate is 0.61 and the strike price of the option is 0.62. The option can be exercised on 25^th of every month during the life of the option. Suppose the current risk-free interest rate of sterling (annually compounded, Actual/365 (fixed)) is 7% and that of the U.S. dollar is 5%. Today’s date is Aug. 1, 1997. The expiration date of the option is Aug. 1, 1998. Suppose the annual volatility of the exchange rate is 12%. Using a 200 step binomial model and the function aaBERM() we obtain the following results:

aaBERM

Argument	Description	Example Data	Switch
price_u	underlying price	0.61
ex	exercise price	0.62
d_exp	expiry date	1-Feb-1998
d_v	value (settlement) date	1-Aug-1997
d_berm_list	list of Bermudan exercise dates	see below	t_66
vlt	volatility	0.12
rate_ann	rate - annual - Actual/365	0.07
cost_hldg	holding cost - annual - Actual/365	0.05
option_type	option type	2	put
iter	number of time steps	200
stat	stat list	1…7

Bermudan Dates

25-Aug-1997

25-Sep-1997

25-Oct-1997

25-Nov-1997

25-Dec-1997

25-Jan-1998

25-Feb-1998

Results

Statistics	Description	Value
1	fair value	0.023137
2	delta	-0.52874
3	gamma	8.329598
4	theta	-4E-05
5	vega	0.166007
6	rho of rate	-0.11312
7	rho of cost of holding	0.117341

If the option is the right to sell $100,000, then the cost will be:

100000 ´ 0.023137 = 2313.7(£)

Remark: Equivalently, one can value the option with respect to the exchange rate in $/£ by switching the values of rate_ann and cost_hldg and change the option type from a put to a call. In more detail, the current price is 1.639344 ($/£), the strike price is 1.612903 ($/£), rate_ann = 0.05 and cost_hldg = 0.07. The function aaBERM() gives the fair value of the call option as 0.061176. If the option is to sell $100,000, or to buy 100000 ´ 0.62 (£), the cost is:
100000 * 0.62 (strike price) * 0.061176 = 3792.9 ($) = 2313.7(£).

Tip: The selection of the appropriate FX rate in valuing an option on FX rates can be confusing. The following table lists all of the different scenarios in the sterling/dollar FX market. One can simply follow this example in his/her modeling.

buy/sell	amount	option type	FX rate	cost of option
sell	100 $	put	£/$	100´option value (£)
buy	100 $	call	£/$	100´option value (£)
buy	50 £	call	$/£	50´option value ($)
sell	50 £	put	$/£	50´option value ($)

Remarks

Commodities

Most options on commodities are options on commodity futures. However, one can also use aaBERM() or aaBERMdcf() to value options on commodity spot prices. To use this function one should identify first the rates of storage cost, insurance cost and convenience yield of the underlying commodity and then combine these rates to define the rate of the cost of holding of the commodity. This value is used as the value of the parameter cost_hldg.

Repo Spread and Continuous Dividend Yield

The repo agreement is a contract whereupon two parties exchange collateral (in this case the underlying asset) for cash, with an agreement to perform the reverse exchange at a predetermined time in the future when the cash has accumulated interest at a predetermined rate: the “repo rate” [2]. The repo rate represents the rate of a “repurchase agreement” that is usually necessary when short selling the underlying asset. Therefore, the repo spread input need only be used when hedging the option requires a short position in the underlying (that is for long calls or short puts); otherwise, enter a zero repo spread. The repo spread is analogous to a continuous dividend yield, in that in the risk-neutral measure the underlying grows at the risk-free rate minus the repo spread (or continuous dividend yield).

The repo spread input to the function aaBERM2dcf() can be entered as a single spread or a discount factor curve. If entering a single spread, this spread equals the risk-free rate of interest minus the repo rate quoted in the market when both rates are quoted as annual, act365(fixed) (conversion to this rate basis and accrual factor can be done with the FINCAD function aaConvert_cmpd2()). A positive repo spread describes the case where the repo rate is greater than the risk-free rate. If you wish to enter the repo spread as a “discount factor” curve, create this curve by the following procedure: First construct the risk-free discount factor curve from money market rates (using, for example, one of FINCAD’s curve bootstrapping functions such as aaSwapCrv()) and the repo rate “discount factor” curve in a similar way from quoted repo rates of different tenors. If the dates in these two discount factor curves do not match, find the missing discount factors by interpolation, using aaInterp(). The repo spread “discount factor” curve can then be created by taking the ratio of the risk-free to the repo rate discount factors on each date (more explicitly the risk-free discount factor divided by the repo rate discount factor, this ratio will be greater than one if the repo rate is greater than the risk-free rate).

This input also provides the ability to enter a term structure of dividend yields. For example, the case where discrete dividends are known up to a particular date and assumed continuous after this date can be handled by using a combination of discrete dividends and the discount factor input. Simply set the continuous dividend yield discount factor equal to one on all dates prior to the last known dividend date to achieve the desired result.

Stocks modeled using continuous dividend payout rates

Valuation of options on stocks modeled using continuous dividend payout rates is similar to the valuation of options on indices.

References

[1] Choudry, Moorad, (2006), An Introduction to Repo Markets, 3rd ed., Chichester, John Wiley & Sons.

[2] Hull, John, (1997), Options, Futures, and Other Derivatives, 3rd ed., Upper Saddle River, Prentice Hall.

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