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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 Barrier Option is a type of digital option for which an option's payout depends on whether or not the asset touched a barrier level at some time during the life of the option. 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 dependent, which means that the payout profile depends on the asset value during the life of the option and the value of the underlying asset when the barrier is hit or on the expiry 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. There are two classes of binary barrier options. The first are options where a payout of cash or the asset is made if the barrier is hit or not hit during the life of the option. The payout is made either when the barrier is hit, or at option expiry. For cash payouts, this distinction will only affect the period of time over which the payment is discounted.

For asset payouts, however, the distinction is more subtle. Binary call option delta and time to expiry date the payout is made when the barrier is touched, then the present value of the payout is equal to the discounted barrier value — since this is the asset value when the barrier is touched. On the other hand, if the payout is made at option expiry, then the present value of the payout is equal to whatever the asset value happens to be at the expiry date, discounted back to the valuation date.

The second class includes options where a payout of cash or the asset is made if the barrier is hit or not hit during the life of the option and if the option is in-the-money at expiry. These are types of knock-in and knock-out binary barrier options.

There are other types of digital options available within the FINCAD library, including various flavors of double barrier binary options. Introduction A Binary Barrier Option is a type of digital option for which an option's payout depends on whether or not the asset touched a barrier level at some time during the life of the option.

Technical Details There are two classes binary call option delta and time to expiry date binary barrier options. Calculate the fair value, risk statistics and probability of hitting the barrier for a binary barrier option with a payoff equal to the asset value if the barrier is touched, or nothing if the barrier is never touched.

Calculate the fair value, risk statistics and probability of hitting the barrier for a binary barrier option with a payoff of a fixed amount of cash if the barrier is touched, or nothing if the barrier is never touched.

Calculate the fair value, risk statistics and probability of hitting the barrier for binary call option delta and time to expiry date knock-in binary barrier call or put option with a payoff equal to the value of the asset if the barrier is touched and the option is in the money. Calculate the fair value, risk statistics and probability of hitting the barrier for a knock-in binary barrier call or put option with a payoff of a fixed amount of cash if the barrier is touched and the option is in-the-money.

Calculate the fair value, risk statistics and probability of hitting the barrier for a binary barrier option with a payoff equal to the value of the asset if the barrier is not touched, or nothing if the barrier is touched.

Calculate the fair value, risk statistics and probability of hitting the barrier for a binary barrier option with a payoff of a fixed amount of cash if the barrier is not touched, or nothing if the barrier is touched.

Calculate the fair value, risk statistics and probability of hitting the barrier for a knock-out binary barrier call or put option with a payoff equal to the value of the asset if the barrier is not touched and the option is in the money at expiry, or nothing if the barrier is touched. Calculate the fair value, risk statistics and probability of hitting the barrier for a knock-out binary barrier binary call option delta and time to expiry date or put option with a payoff of a fixed amount of cash if the barrier is not touched and the option is in the money at expiry, or nothing if the barrier is touched.

Calculate the fair value, delta, and probability of hitting the barrier for a path dependent digital option where the binary call option delta and time to expiry date is on the expiration date. Calculate the fair value, delta, and probability of hitting the barrier for a path dependent digital option where the payoff is made at the time the barrier is touched.

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A European option gives us the right to buy or sell an asset at a fixed price, but only on a particular expiry date. Surprisingly for the case of vanilla options, despite the apparent extra utility of American options, it turns out that the price of American and European options is almost always the same! In general, American options are MUCH harder to price than European options, since they depend in detail on the path that the underlying takes on its way to the expiry date, unlike Europeans which just depend on the terminal value, and no closed form solution exists.

So we can always take the European price to be a lower bound on American prices. Also note that Put-Call Parity no longer holds for Americans, and becomes instead an inequality. How can we go any further? This is the volatility-dependent part of the price, since we are shielded by the optionality from price swings in the wrong direction, but are still exposed to upside from swings in our favour.

Consider the graph above, which shows the BS value of a simple European call under typical parameters. Time value is maximal at-the-money, since this is the point where the implicit insurance that the option provides is most useful to us far in- or out-of-the-money, the option is only useful if there are large price swings, which are unlikely.

What is the extra value that we should assign to an American call relative to a European call due to the extra optionality it gives us? In the case of an American option, at any point before expiry we can exercise and take the intrinsic value there and then. This means that we can sell the option on the market for more than the price that would be received by exercising an American option before expiry — so a rational investor should never do this, and the price of a European and American vanilla call should be identical.

It seems initially as though the same should be true for put options, but actually this turns out not quite to be right. Consider the graph below, showing the same values for a European vanilla put option, under the same parameters. Notice that here, unlike before, when the put is far in-the-money the option value becomes smaller than the intrinsic value — the time value of the option is negative!

What is it that causes this effect for in-the-money puts? It turns out that it comes down to interest rates. Roughly what is happening is this — if we exercise an in-the-money American put to receive the intrinsic value, we receive cash straight away. But if we left the option until expiry, our expected payoff is roughly , where is the forward value. For vanilla options, this is given by.

The plot below shows Theta for the two options shown in the graphs above, and sure enough where the time value of the European put goes negative, Theta becomes positive — the true option value is increasing with time instead of decreasing as usual, as the true value converges to the intrinsic value from below.

In between European and American options lie Bermudan options, a class of options that can be exercised early but only at one of a specific set of times. Since we have an analytical price, we can also calculate an expression for the GREEKS of this option by differentiating by the various parameters that appear in the price. Moreover, differentiating equation [1] above shows that the greeks of a digital put are simply the negative of the greeks of a digital call with the same strike.

Graphs of these are shown for a typical binary option in the following graphs. One final point on pricing, note that the payoff of a digital call is the negative of the derivative of a vanilla call payoff wrt. This means that any binary greek can be calculated from the corresponding vanilla greek as follows.

Price and first-order greeks for a digital call option. Second-order greeks for a digital call option. Greeks for digital puts are simply the negative of these values One final point on pricing, note that the payoff of a digital call is the negative of the derivative of a vanilla call payoff wrt.