# Digitals

11. Digitals

Digital options pays a specific coupon when a barrier event occurs.

11.1. European Digitals

11.1.1. Digital Payoffs and Pricing

The European cash-or-nothing call is an option paying a fixed coupon C if the spot of the underlying at maturity T is higher than the predetermined barrier level H.

Under Black–Scholes, the price of such an option is given by the following formula:  N(d2) is the probability that the spot at time T is higher than the trigger. That is, the derivative of the call price with respect to K is the price of the digital. In much the same way we saw in Black–Scholes that the Delta sensitivity to underlying S is given by N(d1); in the case of the digital the price is given by N(d2) which is the negative of the derivative with respect to the strike.

Having a long position in a European asset-or-nothing call, and being short a European cash- or-nothing call for which the coupon is equal to the barrier level, is equivalent to a long position in a vanilla call for which strike price is equal to digital trigger. We can then deduct the price of an asset-or-nothing call from a vanilla call and cash-or-nothing call.    In the world of structured products, asset-or-nothing digital options are the less popular of the two. Cash-or-nothing digitals are used more often because they allow holders to receive fixed coupons conditionally on the spot price of a specific underlying crossing a predetermined level.

11.1.2. Replicating a European Digital

The digital can be thought of as a limit of a call spread.  As the distance between the call option strikes and the digital strikes, ε, gets smaller, we need 1/ε call spreads of width 2ε to replicate the digital. In the limit, meaning as ε approaches zero, the call spread replicates the digital exactly.

11.1.3. Hedging a Digital

In practice, one can price and hedge a digital as a call spread. The question is which call spread? The gearing of the call spread required to super replicate the digital depends on how wide we chose the strikes. The term super replicate comes from the fact that the call spread over-replicates the digital.

The smaller the call spread, the more aggressive the price but the more difficult the hedging.

For a digital option, Gamma can be quite large near the barrier, and using a call spread means that we obtain less extreme Greeks. The smaller the call spread the larger Gamma, and Vega can get near the barrier. At the barrier they both shoot up and then shoot down while changing sign.  Here above is the Gamma of a call spread; it, too, changes sign, but is better behaved than that of a digital. Since the plan is to Delta hedge this call spread, we need to make sure that as the underlying approaches the barrier we can still manage to Delta hedge. A large Gamma means that Delta is very sensitive to a movement in the underlying, and if we approach the barrier and our call spread is too tight, then we will need to buy a large Delta of the underlying, which might be difficult in the market. To this end it makes sense that we select the width of our replicating call spreads based on the liquidity of the underlying.

Barrier Shifts

The barrier shift is an amount by which the seller of a digital shifts the barrier, while pricing, so that in fact they are really pricing a new digital whose replicating call spread is the hedge of the actual digital. The barrier shift will be chosen so that the resulting shifted payoff over-replicates the payoff of the digital by the least amount, but such that the Greeks of the new payoff are manageable near the barrier.

We find that the optimal call spread, given MaxDelta, we can trade and to minimize the PV, is given by the call spread around the shifted barrier where:  A smaller shift will mean that we might exceed the MaxDelta, and a larger shift will mean we have increased the option price above the necessary amount. Using this formula we know that the gearing on the replicating call spread is given by Digital Size/Barrier Shift, and by using this shift we have over-replicated the digital by the call spread that can still be hedged.

In practice some traders prefer to just take a constant shift of the barrier, and this is again essentially just an additional margin charged for managing the risks if the spot were to approach the barrier. This can also be the more efficient method to use when risk managing a large book.

11.2. American Digitals

Reflection Principle

The reflection principle gives us an approximate link between the price of a European digital and an American digital with the same maturity and strike. Let’s assume that the log-returns of the underlying are normally distributed, but with mean zero. The symmetry that this introduces in the paths of the underlying means that the paths where the barrier is hit and reflected has the same probability as the path that crosses through the barrier. This implies that the probability of hitting the barrier is exactly twice the probability of ending up above the barrier at expiry. The latter of these two gives us the price of the European digital of the same maturity, and we deduce that the value of the American digital is twice that of the European equivalent.

Exercise:

You are a structurer visiting a client who wants to invest in a volatile market. He believes KOS company is going to perform quite well in the coming years. The client wants to buy from you a digital call option that pays \$100 whenever KOS stock price reaches \$1. KOS actual spot price is equal to \$0.7. You need to give the client an offer price for this option straight away. At what price would you sell it?

The real problem in pricing this option comes from the fact that there is no specified maturity. So it is impossible to compute the probability of striking and then just discounting it. Now remember, the price of an option is the cost of hedging it. This American digital option is equivalent to 100 American digital options paying a coupon of \$1 if KOS spot reaches \$1. If the trader selling these options wants to perfectly hedge them, he just needs to buy 1 KOS share for each individual option, so that if the spot reaches \$1, he can just sell 1 KOS stock for \$1 and pays the coupon to the client. Then an upper bound price for this option is 100 × \$0.7 = \$70.

So there is no easy way to compute the fair value of this option but you know that you should sell it for less than 70\$. The offer price will then mainly depend on the relationship you have with your client and how confident you feel trading on the required market.

Upper Bound Price for American Digitals

Assume that you need to give an upper limit to an American digital option paying a coupon C if the spot of an underlying stock reaches K. This option is equivalent to C/K options paying a coupon equal to K if the spot reaches K. The trader selling this option needs to buy C/K underlying shares to be completely hedged. The fair price of this option should then be lower or equal to C/K × S(0).

11.3. Risk Analysis

11.3.1. Single Asset Digitals

Digital Call option

The holder of such an option is long the forward price and therefore short dividends, long interest rate and short borrow costs of the underlying. This option having a positive delta, the trader selling it will have to buy delta of the underlying. Therefore he will be long dividends, short interest rates and long borrow costs of the underlying.

The vega of a digital depends about the position of the forward price regarding the barrier. In the case of a digital call, the owner will be long volatility if the forward price is lower than the barrier level since a higher volatility will increase the probability of the barrier being reached.

Time to maturity has the same effect on a digital option’s fair value as volatility. Note that time also has a second effect as we must discount it when pricing, although the previous effect is generally larger. (a developper)

Skew risk is a critical consideration for the seller of a digital. When hedging a short position in a digital, the trader takes an opposite position in a call spread. Skew makes a call spread more expensive. If we consider the digital as a call spread, we can immediately see what is happening. The seller of the digital is clearly selling skew.

One can even write down the price of a digital, using the limit defined in section 11.1.2, and combine it with a parameterization of the skew. Then the price of the digital is given by  Exercise:

Consider two European digitals A and B on the same underlying stock. The options pay the same coupon if the underlying’s spot reaches a trigger equal to 160%. Option A expires in 1 year; the other option expires in 3 years. Which one is cheaper and why?

For issues concerning Theta, it is always more convenient to think about it in terms of volatility first. Both options A and B are deep OTM. Therefore volatility increases the price of the digital options. Time to maturity having the same effect on digital prices as volatility, we can say that option B is more expensive than option A.

11.3.2. Digital Options with Dispersion

Worst Of Digital Call option

With respect to forward price sensitivity, higher interest rates, lower dividends and borrow costs will increase the forward prices of the different underlying shares. These will increase the forward price of the worst performing stock and thus make the option more expensive.

For the Vega of a worst-of digital, things get more interesting. Recall that the seller of a worst-of call is short volatility because of the call option feature, but is buying volatility because it is long dispersion, and the overall position for the worst-of call was a function of the levels of volatility, correlation and the forwards, and can be either long or short in volatility.

Since this is a worst-of style multi-asset product, a higher dispersion decreases the price of the worst-of digital and our correlation exposure comes from this dispersion effect. The seller of the worst-of digital is short correlation.

As for the skew, we previously saw the skew effect on single asset digitals as well as the skew effect on worst-of products. In the case of worst-of digitals, skew generally makes their price more expensive.

In the multi-asset case, which digital barrier shift should be applied? One can be conservative and apply a unique shift taking into account the lowest liquidity (hedged against the highest risk around the barrier). Otherwise, one can apply individual shifts depending on the different stocks’ average daily traded volumes.

11.3.3. Volatility Models for Digitals

To correctly price a digital one must use a model that knows about skew.

If the option’s payoff is only a function of the returns of the underlying at maturity, then it is imperative to get that particular skew correct in the calibration and we would use the exact date-fitting model.

If there is path dependency then we need to use a form of smooth surface calibration in order to capture the effect of surface through time. Specifically, because the digital can be triggered at any time prior to maturity, the volatility hedge will need to consist of a set of European options with different maturities, i.e. Vega buckets. The American digital will have Vega sensitivity to the volatilities of these Europeans and we need the model to be calibrated to them in order to show risk against them. As such they can serve as valid hedging instruments and the model price will reflect this.

In the multi-asset cases the calibration will need to be done to each of the respective implied volatility surfaces of the individual underlyings. In the multi-asset case, all payoffs discussed are sensitive to the correlations between the various underlyings. The effects on the digital again depend on the nature in which the multi-asset feature enters. These correlations must be correctly specified by the criteria laid out in Chapter 7 and also by observing the correlation sensitivity of the seller of such options.

11.4. Structured Products involving European Digitals

11.4.1. Strip of Digitals Note

Payoff Mechanism

At each observation date t, the holder receives a conditional coupon equal to:  This note is capital guaranteed, and the holder receives the following payoff at maturity T :

Notepayoff  = 100% + Coupon(T)

The non-risky part of this product is a zero coupon bond that redeems 100% of invested capital at maturity. The risky part is in fact a set of different European digitals starting at the start date of the note and having maturities corresponding to the different observation dates.

To price a strip of digitals, we have to price the individual digitals separately since they are all independent. To do so, we should compute the sum of discounted probabilities of being ITM for the different digitals and multiply these probabilities by the coupon paid.

The seller of a strip of digitals note is short the digitals. Therefore, the trader taking a short position in this structure is short the forward price of the underlying share and will need to buy Delta in the underlying asset on day 1, and adjust dynamically through the life of the trade to remain Delta neutral. The seller will be short interest rates, long borrow costs and long dividends, short skew and long the barrier.

The position in volatility is not obvious. In order to check his assumptions on his position in volatility, a trader selling this structure can have a look at the different undiscounted probabilities of being ITM and make sure that maturity decreases these probabilities in case he is long volatility or increases the probabilities if he is short volatility.

11.4.2. Growth and Income

Payoff Mechanism

At each observation date t, the holder receives a conditional coupon equal to:  This note is capital guaranteed, and the holder receives the following payoff at maturity T :  The idea behind this product is to add an additional opportunity to capture the final performance of the underlying in case the holder did not receive very much in previous coupons.

The Growth and Income note is capital protected since the buyer is long a zero coupon bond having the same maturity as the note plus an option structure. The risky part is composed of a series of digitals paying a predetermined periodic coupon when the underlying stock price is higher than a specific trigger. At maturity, the holder is also long an out-of-the-money European call option with strike level equal to 100% plus the sum of coupons already paid by the digitals.

To price the Growth and Income option, we have to price the set of individual digitals as well as the out-of-the-money European call. Concerning digital pricing, the risks are identical to those associated with the strip of digitals, trader selling these digitals will need to buy Delta in each of the underlying assets on day 1, and adjust dynamically through the life of the trade to remain Delta neutral. The seller is short interest rates, long borrow costs, long dividends. He also has a short skew position and a long barrier position. The volatility position depends on the underlying stock forward price and the digitals’ trigger.

Concerning the out-of-the-money call option pricing, one should note that it is not a simple European call since it is now a path-dependent option. The trader selling this call will need to buy Delta in the underlying asset on day 1, and adjust dynamically through the life of the trade to remain Delta neutral. The seller is short interest rates, long borrow costs, long dividends. He is also short volatility since he is selling a call option. The trader selling a Growth and Income note should check the overall position in volatility.

The skew effect is interesting in this case. In fact, skew will decrease the price of the out- of-the-money call, so there is a skew benefit for the buyer. Moreover, since skew increases the price of digitals, this means that skew increases the probabilities of coupons paid. This effect enhances the probability of a higher strike which, in turn, increases the skew benefit on the call since it makes it more out-of-the-money. This is called the second skew effect. Therefore, one should be cautious with respect to the overall effect of the skew but, generally, the skew sensitivity of the digitals is higher in absolute value than that of the out-of-the-money call. Thus, skew usually increases the price of the Growth and Income note.

Extending the discussion of volatility models, there is a need for caution. Although the digitals are not path dependent and can be priced by an exact fitting of the skew at the correct dates, the call option is path dependent and needs a smooth fitting across all these dates.

11.5. Structured Products involving American Digitals

11.5.1. Wedding Cake

The wedding cake is an option that pays a fixed payout based on the movement of the underlying reference rate within certain predefined barriers. It will typically pay a lower coupon where the reference rate moves within the wider range, or no coupon if it touches the outside barrier levels.

The structure can be thought of as a set of two-sided no-touch digitals. In the example in Figure 11.8, the wedding cake structure pays a 15% coupon at maturity, provided the underlying never went outside the range [95%, 105%]. It pays a coupon of 10% if the underlying goes outside the first range but does not exit from the second range [90%, 110%]. It pays a coupon of 5% if the second range has ever been breached but not the third range [85%, 115%], and it pays zero if the third range has ever been breached.

This can be broken down into three two-sided no-touch digitals in the following sense: start with a digital that pays 5% if the underlying is never outside the range [85%, 115%], add to this a second digital that also pays 5% if the range [90%, 110%] is never breached, and similarly another no-touch of coupon 5% and range [95%, 105%]. Pricing each of these separately and adding them together gives us the price of the structure.

This product is an example of the curvature effect of the skew. That is, the skew sensitivities on each side of the digitals cancel each other out somewhat, assuming the skew is relatively non-convex. Here the more positive the curvature, the more the skew begins to impact the price. 11.5.2. Range Accrual

The range accrual pays a coupon at maturity based on the amount of time, typically the number of days, that the underlying has spent within a given range. This can be used to obtain an above market coupon by taking a view on the path of an underlying, or can be used to hedge other risks.  We can specify this option with a minimum payoff at maturity and structure it into a note. Since the payoff is positive in all cases, a minimum value of zero (a global floor) makes no sense. This can also be structured as a swap, where, for example, the investor pays LIBOR plus a specific spread, and the other party pays an equity contingent coupon according to the range accrual.

Obviously the analysis for a range accrual depends on where the range is specified. But generally, with a range similar to the one specified, it is tailored towards the view that one expects relatively flat volatility in the underlying.

One can think of a range accrual as a set of daily no-touch digitals. To obtain a higher coupon one can introduce a multi-asset component and bring dispersion into the picture. Let’s assume that we have N assets and we specify the range accrual as  The seller of the range accrual in the multi-asset case specified as above is long dispersion: as dispersion goes up, one of the assets will probably leave the range. The seller is short correlation and long volatility, but caution must be taken as the size of the range and the position of the range can change the effects of these.

If we think about a range accrual as a set of forward starting daily digitals, we can see that because of its path-dependent nature, the dynamics of the model specified will have an impact on the price of the range accrual. If we are concerned with the effect of future volatility implied by the model, we will need to use a stochastic volatility model to capture this. Whichever model is chosen must be correctly calibrated to obtain as smooth a calibration through time as possible to European options of different maturities, the Vega buckets.

11.6. Outperformance Digital

11.6.1. Payoff Mechanism  11.6.2 Correlation Skew and Other Risks

When pricing and hedging this option, one should be careful because the payoff of an outperformance digital looks similar to that of an outperformance option, but the volatility and correlation sensitivities are quite different and actually depend on the difference between the underlying’s forwards.

The trader taking a short position in the outperformance digital option described above is obviously short the EuroStoxx index forward price and long the S&P forward price.

If the forward performance of EuroStoxx is higher than the forward performance of S&P, then a higher dispersion decreases the price of the outperformance digital. Otherwise, dispersion increases its price.

Correlation, like volatility, has a relationship with the underlying price. In equity markets, for large moves down, we see an increase in correlation. This is what we call correlation skew. Since the position in correlation can potentially change during the life of the option, one should use a stochastic correlation model to capture this effect or to at least be able to notice the existence of such an effect and be able to quantify it.

The issue is linked to convexity – that is a second-order effect. In this case it is insufficient to measure a correlation sensitivity by computing a first-order derivative w.r.t. correlation. Since the price is no longer linear in correlation, and this sensitivity may change, the seller should see the second-order effect. The correlation convexity is not something that can be directly or easily hedged, and thus seeing its effect through the use of a correlation skew model is purely to include its effect in the price, if nothing else as a fixed charge for bearing the additional risk. If one knows the magnitude of this effect, then a simple edge can be taken onto the price without the added modeling complexity.

Although this effect is present in the outperformance digital, one must first be sure to take the volatility skew sensitivity of the product into account. The product is a digital and thus carries skew risk and is sensitive to the implied volatility skews of both underlyings.