Single Barrier Options
A single barrier option has an underlying European option that is either knocked in ot knocked out when the underlying asset price touches the barrier level . At the expiry date, if the option has not been knocked in then a rebate is paid.
Consider a generalized European option that pays
where and are constants, is the strike and for call options ( for put options). Standard options have and with binary options having and the payoff amount.
Single barrier options are priced with the formulas from Merton, Reiner and Rubinstein. They are a combination of the following formulas:
Barrier Type | Condition | Parameters | Option Value |
---|---|---|---|
Down-and-Out Call | |||
Down-and-Out Call | |||
Up-and-Out Call | |||
Up-and-Out Call | |||
Down-and-Out Put | |||
Down-and-Out Put | |||
Up-and-Out Put | |||
Up-and-Out Put |
Where for down options or for up options, and
One touch and No touch Binary Options and Rebates
A one touch binary option pays a fixed amount if the barrier is touched during the life of the option, otherwise nothing. The two types are:
- Pay the fixed amount when the barrier is touched
- Pay the fixed amount at expiry
A no-touch binary option pays a fixed amount if the barrier is not touched during its life (equivalent to a fixed amount less a one touch binary option with payment at expiry).
The value of a no touch option is
where for down options or for up options
The value of the one touch option that pays 1 at expiry if S touches the barrier is minus the value of the no-touch option. If it pays 1 when touches the barrier, then its value is
\Phi(\eta z-2\eta\lambda\sigma\sqrt\tau))$$
with
Note that should be floored at for to be defined. One touch and no touch options have strike .
Discontinuous Barrier Sampling
Assuming that barriers are not monitored continuously, an adjustment needs to be made to compensate for discrete sampling. The adjustment is that of Broadie, Glasserman and Kou.
If the barrier is above the asset, the adjusted barrier rate is where is the period between observations and is the constant defined by:
where is the Riemann zeta function. If the barrier is below the asset, the adjusted barrier is
Discrete Asian Options
The payoff of a discrete asian option is max at expiry with:
are the dates defined Sampling Data, is the weight assigned to , , is the strike, and is either for a call or for a put.
If , then is the known price assigned to in the sampling data list otherwise it's the initial price factor value.
For valuation date , let be the smallest index for which (). Then , with
, and is the adjusted strike:
The expectation of under the risk-neutral measure is the forward price with the expectation of given by:
Define:
- to mean
Assuming the moment matching approach (the idea that a sum of lognormal variables may be represented by another lognormal variable with the same first and second moments as the sum), is lognormal and where is given by:
and the option value is
Assuming a constant carry rate and constant volatility, we can approximate and where is the implied volatility at time for expiry and strike and,
Equity Swaps
Equity swap legs consist of a series of equity swaplets.
Equity Swaplet
An equity swaplet has:
- a cashflow
- start date
- end date
- payment date
- a start multiplier
- an end multiplier
- a dividend multiplier
For cashflow dates , the equity swaplet pays:
at time . where
- is the equity price
- is the time- value of dividends per share with ex-dividend date after but on or before
- if the amount type is Principle and if the amount type is Shares.
The swaplets have and either if Include Dividends is Yes or otherwise.
The formula used to value the swaplet at is
where:
- is the forward equity price
- is the usual discount factor
- is the discount factor from the equity's repo rate
The value of the swaplet at is
The value of the swaplet for is the payoff multiplied by
When , the dividend payoff is calculated from Known Dividends and/or the simulated spot and forward prices along the current scenario path
Dividend Timing is assumed to be Terminal i.e. the swaplet pays at time . Dividend timing could also be continuous (i.e. the dividends are settled on their dividend dates) but that is not currently implemented.
EquityBarrierOption
A path dependent option described here
EquityDiscreteExplicitAsianOption
A path independent option described here
EquityForwardDeal
Described here
EquityOptionDeal
A vanilla option described here
EquitySwapLeg
Described here
EquitySwapletListDeal
Described here
FXBarrierOption
A path dependent FX Option described here
FXDiscreteExplicitAsianOption
A path independent option described here
FXForwardDeal
An FX forward is an agreement to buy an amount of one currency in exchange for an amount of another currency at settlement date . The value of the deal in base currency at time , (), is
where
- is the sell currency discount factor
- is the buy currency discount factor
- is the price of the sell currency in base currency
- is the price of the buy currency in base currency
FXNonDeliverableForward
An FX non-deliverable forward effectively an FX Forward deal that is cash settled in a (potentially) third currency. The deal pays
in settlement currency at the settlement date where:
- is the buy currency
- is the sell currency
- is the price of the buy currency in settlement currency
- is the price of the sell currency in settlement currency
The value of the deal in settlement currency at time , (), is
where:
- is the forward price of the buy currency in settlement currency
- is the forward price of the sell currency in settlement currency
FXOneTouchOption
A path dependent FX Option described here
FXOptionDeal
A path independent vanilla FX Option described here
FXSwapDeal
An FX swap is a combination of an FX forward deal with near settlement date and an FX forward deal in the opposite direction with far settlement date , where . The base-currency value of the FX swap at time , , is
where () is amount of the buy currency bought (sold) at , (), and () is amount of the sell currency sold (bought) at (). Typically, is the spot settlement date and