Digital option pricing with C++ via Monte Carlo methods

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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 pricing digital options payoff from exercising is.

A digital call option with is similar - it pays off one dollar if at expiration, and pricing digital options off zero otherwise:. Suppose you have a model for pricing regular call options. How can you use to price pricing digital options 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. We 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 andpricing digital options 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 digital call option with is similar - it pays off one dollar if at expiration, and pays off zero 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 pricing digital options, 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 puts and calls.

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FINCAD offers the most transparent solutions in the industry, providing extensive documentation with every product. This is complemented by an extensive library of white papers, articles and case studies. A Binary Option is an option with a fixed, pre-determined payoff if the underlying instrument or index is at or above the strike at expiration.

The value of the payoff is not affected by the size of the difference between the underlying and the strike price, and can be in the form of a cash payment or delivery of the underlying. The options described here are path independent, which means that the payout profile depends only on the value of the underlying asset on the expiration date of the option. For a call, the payout is received if the underlying asset price is greater than the strike price, and for a put, the payout is received if the strike is greater than the underlying asset price.

The simplest Binary also known as Digital options are cash-or-nothing and asset-or-nothing options. In a cash-or-nothing option, a predetermined amount is paid if the asset is, at option expiry, above call or below put some strike level, independent of the path taken.

An asset-or-nothing option is similar to a cash-or-nothing option except that the holder of the option is entitled to either the full asset value - at option expiry - or nothing. A simple generalization of the asset-or-nothing option is a digital-gap-option. A digital gap option has a payout profile equal to the asset value, less the gap value, depending on whether the asset finishes above or below the strike price.

It is clear that a digital gap option is simply the difference between an asset-or-nothing and a cash-or-nothing digital option with the cash amount set to the gap value. Introduction A Binary Option is an option with a fixed, pre-determined payoff if the underlying instrument or index is at or above the strike at expiration.

Technical Details The simplest Binary also known as Digital options are cash-or-nothing and asset-or-nothing options. Calculate the fair value and delta for a path independent digital or binary gap option. The payout at expiry, if the option is in-the-money, is equal to the asset value, less the gap value.

Calculate the fair value and risk statistics for a path independent digital or binary all-or-nothing option. The payout at expiry, if the option is in-the-money, requires the delivery of either a specified cash amount or the underlying asset. The next generation of powerful valuation and risk solutions is here.

Portfolio valuation and risk analytics for multi-asset derivatives and fixed income.