# Paradigm Insights | Exotic Options Part 2 - Barriers

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December 22, 2022

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In this piece we’ll foray further into exotic options and take a look at barrier products. These are a step up from vanillas but are still simple in nature. As we previously touched upon, barriers are essentially a vanilla option with an added condition, a trigger. In its simplest form this trigger will either bring the vanilla into play or terminate it.

### Barrier Options

In keeping with tradition, the best place to start is at the beginning so let’s start with a digestible and homely description of a barrier option; remember when you were but a small child. Back in the days before acronyms like FUD and SBF had meaning. Long before your innocence was shattered by the machinations of 10x leverage? Well remember when your screams and demands for dessert were met by ‘Not until you’ve eaten your vegetables!’. Well imagine the desert to be a smooth, rich vanilla ice cream aka the option, leaving the vegetables to be the barrier. Only after action A has been achieved (eat vegetables in this case) can you indulge in action B (engorging your little chubby face with sweet, sweet vanilla goodness). Repressed childhood memories aside, this should prove a good jump off point.

### Vanilla pricing

Let’s compare the vanilla option to a barrier option and try to understand how its existence and placement influence pricing. We’ll begin with a simple world state with zero rates and a sensible vol surface.

You should note that all pricing here will be based on a Local Volatility model which means that we can price any option that is not path dependent i.e. we only care about where the spot rate fixes at expiry and have no interest in the path it took to arrive there.

Inputting the above surface into a Monte Carlo generator spits out the following paths. Despite the limited number of paths displayed, a total of 250,000 paths were generated with 365 time steps.

We’ll start with pricing ATM vanillas. As pictured, Strike and spot will be 100.

###### 3.9775336760898825

Meaning that if you were to trade a call struck at 100 1yr out, you would pay 3.977%. The avid readers amongst you will remember the table below. The remainder of this article will focus on the European barrier style (the barrier is only observed at expiry). For those of you who have committed the unforgivable sin of not having read the first piece, here’s the link.

Now that we have established the cost of outright vanillas, we can assess the impact of adding a barrier.

### Up and In Barrier

As a refresher, a European barrier is a trigger that is only observed at a fixed point in time. More often than not, this will be at the option’s expiry. For a European Call option we will delay the ITM nature of the vanilla until the spot rate fixes at or beyond the barrier level. Beyond the barrier level your pnl will be indistinguishable from a vanilla option, however beneath it, you will not be ITM despite having moved through the strike rate. This gives the below payoff diagram.

I have included 2 barrier levels below; at 110 and 125. Evidently there are fewer paths that fix beyond these levels therefore we can intuit that the price of an Up and In call will be less than that of a vanilla all other things held equal.

The 110 Up and In barrier costs 2.522% whereas the 125 costs 0.335%. As we all know, there’s no such thing as a free lunch. Whilst the 125 barrier option is considerably cheaper, you can only exercise it if the market fixes at or beyond that level at expiry. Note that if the market breaches this level but then fixes beneath it, your option will expire worthless. Buying an Up and In Call option would be particularly prudent in a situation in which you believe that the market would rally considerably.

In addition to triggering options into effect, barrier options can also terminate them. Let us look at a ‘Knock Out’ option. We will use an Up and Out Call for example. Placing the barrier at 110 means that between 100 and 110, you will have a normal vanilla payoff but if the market fixes beyond the barrier of 110 or beneath the strike of 100, the option will expire worthless. This option would cost 1.455%.

Conveniently, 1.4552% + 2.522% = 3.977%

That is to say that-

Up&Out + Up&In = Vanilla (assuming barrier, strike, expiry are identical)

Barrier options have many uses, they can be traded in combination with other options or as a stand alone. Furthermore there are several varieties, some barriers that are triggered upon being touched rather than waiting for expiry. Triggers can be simultaneously placed on both sides of the Strike. In short, they are extremely customisable and can be tailored to your hedging needs.

For further information or indicative pricing on Barrier options as well as other exotics feel free to reach out to our team. In a coming piece we will look at how barriers impact Greeks and the associated hedging implications. Make sure to tune in over the coming weeks.

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