Pricing european digital options
FinancialDerivative [ instrumentparamsambientparamsprop ] computes the specified property prop. FinancialDerivative can compute the values and partial derivatives for many common types of financial derivative contracts.
FinancialDerivative [ ] gives a list of available contracts. FinancialDerivative [ instrument ] lists the names of the contract and ambient parameters. FinancialDerivative [ instrumentparamsambientparams ] computes the value of instrument for the specified ambient parameters. FinancialDerivative [ instrumentsparamsambientparams"Rules" ] computes all available properties and returns the result as a list of rules. Typical contract parameters include: Ambient parameters common to all option types: Volatilities, interest rates, dividends, and the maturity time are assumed to refer to the same time unit, typically years.
The reference time can be given as a date or as a time difference. If it is not specified, the value is assumed to be today's date. Ambient parameter required for basket options, which involve multiple underlying assets: All correlation coefficients are pricing european digital options the range to.
The correlation matrix can also be given as a flat list pricing european digital options the coefficients above the diagonal. Ambient currency parameters relevant to Quanto options: FinancialDerivative can price the following types of Asian options: Barrier Options Barrier options have a payoff that depends on whether or not the price reaches a predetermined barrier pricing european digital options.
The following types of barrier options are supported: Down-and-out options become void if the price decreases to the barrier.
Up-and-in options become valid as the price rises to the barrier level. Down-and-in options become valid when the price falls to or below the barrier level. All single-barrier options require the following parameters: All rebates are paid at option expiration.
Binary Options Binary or digital options have a payoff at maturity that is either a fixed amount or nothing. Binary asset-or-nothing options pay the current price of the asset at expiration if the price of the asset is above the strike price. Binary options require the following parameters: The payoff is taken to be 1 for the binary cash option. Chooser Options A chooser option is a compound contract that requires the buyer to choose between a call and a put option on the same underlying asset at a predetermined expiration time.
Compound Options A compound contract is an option on a European vanilla option. Double-Barrier Options Double-barrier options have a payoff that depends on whether or not the price of the pricing european digital options asset reaches either of the two barrier levels at any time prior to exercise.
The following types of double-barrier options are supported: A double-barrier knock-in option becomes valid as soon as the pricing european digital options breaks out of the interval. Extendible Options Extendible options can be exercised at the time of maturity, or they can be extended by a predetermined period.
The following types of extendible options are supported: A writer-extendible option is extended automatically if the option is out of the money upon initial expiration. Lookback Options A lookback contract is a path-dependent option whose value at exercise depends on the optimal price of the underlying asset over the lifetime of the contract.
The following lookback options are supported: For put options, the highest underlying price is used. In a pricing european digital options lookback call option, the value of the underlying asset at exercise is taken to be the highest price of the underlying asset over the option's lifetime. For put options, the lowest underlying price is used. In addition, it supports American exercise for floating-strike lookback options. One-Touch Options A one-touch contract is a binary option with an American exercise style.
The payoff is taken to be 1. Option on Future An option on a future has a forward contract, rather than a stock security, as its underlying asset. Perpetual Vanilla Options A perpetual vanilla contract is an American vanilla option without an expiration date. Perpetual Lookback Options A perpetual lookback contract is a lookback contract without an expiration date. For put options, the strike price is taken pricing european digital options be the highest underlying price so far. Power Options A power option is a contract for which the payoff is raised to a power.
A powered option raises the difference between the price of the underlying asset and the strike price to the power specified. A capped power option caps the payoff on a power option. The "Vanilla" name specification can be omitted. By default, FinancialDerivative prices American vanilla options by numerically solving the Black — Scholes partial differential equation.
A binomial tree solution method can be specified by setting the Method option to "Binomial". Rainbow Options A rainbow contract entitles the option holder to the maximum of the payouts generated by the individual components of a basket of assets.
American exercise is supported for pricing european digital options maximum of two non-cash assets in a basket. Rainbow Minimum and Maximum Options A rainbow minimum contract has a payout that is the value of an option on the worst-performing asset for a call, and on the best-performing asset for a put. A rainbow maximum contract has a payout that is the value of an option on the best-performing asset for a call, and on the worst-performing asset for a put.
American exercise is supported for a maximum of two assets in a basket. Mountain Pricing european digital options Options Mountain range options are a class of contracts that entitle the holder to a payout that is based on the performance of a basket of assets, with certain time constraints on asset performance. An Altiplano contract is a type of mountain range pricing european digital options that pays out a fixed coupon amount if none of the basket assets have reached their respective barriers, and nothing otherwise.
Parameters for Altiplano contracts: Parameters for Annapurna options: Parameters for Atlas contracts: Parameters pricing european digital options Everest contracts: The time to expiration is divided into subperiods, with the yield for each subperiod determined by the return on the best-performing asset over that period, and with each asset being used to determine a subperiod return exactly once.
Parameters for Himalaya contracts: Quanto Vanilla Options A Quanto vanilla contract is an option whose value at exercise depends on the performance of the underlying pricing european digital options, as well as on the performance of the currency in which the asset is denominated. A vanilla fixed-strike Quanto option is settled at a fixed strike price in a foreign currency at the prevailing exchange rate. Give Feedback Top Thank you for your feedback! Please complete this field.
This post is based on problems 2. I was asked how to price a digital option in a job interview - and had no idea what to do! A call is only worth exercising using if the underlying price,is greater than atas the payoff from exercising is. A digital call option with is similar - it pays off one dollar if at expiration, and pays off zero otherwise:.
Suppose you have a model for pricing regular call options. How can you use to price the digital option? As a starting point, consider buying a call with and selling a call with:.
This is close to the digital option, but not exactly right. Pricing european digital options want to make the slope at steeper, so we need to buy more options. Consider buying two calls with and selling two calls at:.
As opposed to a slope of 1 between andnow we have a slope of two between and Generalizing this idea - consider a number. To get a slope ofyou buy calls at and you sell calls at. How much will the above portfolio cost? You earn from selling the calls, and pay for the calls.
The net cost is: Many complicated payoffs can be re-created as combinations of vanilla puts and calls. Digital Call Options A pricing european digital options call option with is similar - it pays off one dollar if at expiration, and pays off pricing european digital options otherwise: As a starting point, consider buying a call with and selling a call with: Consider buying two calls with and selling two calls at: Given that the slope isto get an infinite slope, we take the limit as goes to zero.
It might look more familiar if I re-wrote it as: Conclusion Many complicated payoffs can be re-created as combinations of vanilla pricing european digital options and calls.