Mini Chapter Sixteen
Digitals
- Are a binary payoff strategy – i.e. payoff is either zero or the notional of the trade – conditional on the underlying’s price trading at or above/below a specific strike. We can have either upside digitals or downside digitals. A zero payoff implies just the loss of the option premium in case the underlying fails to trade the payoff condition.
- As an example of an upside digital – an investor wishes for a payoff (also known as the notional) of USD 10 mio if USDJPY at expiry trades at 150 or above, current spot being 135. Similar to other strategies option price/premium of the digital is quoted as a percentage of the notional.
- Since the payoff is tied to a specific level of the underlying it is discontinuous in nature which makes hedging of the greeks in this strategy very different from those with continuous payoffs. In other words a digital’s payoff cannot be exactly replicated by any combination of simple calls and or puts but can be approximated by a call or put spread placing the short strike at the strike of the digital. The long strike of this spread is at a distance from the digital’s strike – referred to as ramp in the options world – such that this distance when multiplied by the contracts in the call spread amounts to the final payoff. Effectively the max payout of this call/put spread is equal to the digital’s payoff.
Graph 61 – Long European Call Spread Pay-off over-replicates a Digital
Source: Pandemonium.
Graph 62 – Dealer’s Short Call Spread Delta
Source: Pandemonium.
Graph 63 – Dealer’s Short Call Spread Gamma
Source: Pandemonium.
- From the diagram above consider a USDJPY upside digital for a USD 10 mio notional – current USDJPY at 135 with digital strike at 150. Payoff on this can be replicated by a) 135 to 150 Call Spread b) 145 to 150 Call spread c) 149 to 150 call spread. Referencing above the distance between the strikes would determine the notionals on the respective call spreads to maximise the payout to USD 10 mio at 150. Therefore to attain a max payoff one needs to buy 100 mio, 300 mio and 1.5 bio notionals respectively for a, b and c; notional amounts are computed using 150 as the USDJPY FX level.
- Given the nature of payoff for a call spread a dealer would make money on this approximated risk of the digital before USDJPY trades at 150. Hence the call spread is a conservative approximation of the digital. The area under each of the call spreads denotes the probability of the payoff for it – larger the area, higher the payoff, higher the price of the spread, more conservative the call spread.
- Pricing of a digital therefore is derived from the cost of hedging this short call spread – narrower ramps would mean a more aggressive pricing of the digital.
-  While the digital’s notional is a prime consideration for assessing the width of the ramped call spread, other factors like liquidity of the underlying, peak delta as we approach the barrier/digital strike and the implied volatility around it that impacts the moneyness and prospects of large gamma/delta hedging.
- As mentioned above from a dealer’s point of view selling a digital is equivalent to being short a call spread, greeks for which would have the following behaviours:
- Delta is negative to start with and gets to a peak negative level around the midpoint of the call spread and starts to flip lower towards zero as we approach and go beyond the long/digital strike. Whether the trough appears right at the mid-point or ahead of it is a function of the implied vol of the underlying in that region and or the time to maturity of the option (refer to Delta on call spreads across expiries and implied vols).Â
- Gamma would be negative too but would get to peak negative (much sooner than delta) at the lower/short strike, gets down to 0 when delta peaks and flips back into positive territory as we approach the long/digital strike. The peak and trough positions relative to the call spread strikes and time to maturity (refer to Gamma on call spreads across expiries and implied vols).Â
- Narrower ramps therefore imply higher peaks for both delta and gamma and a faster flipping from negative to positive or vice versa (depending on underlying’s vol), worsened even more by shorter time to expiry. At the same time a wider ramp would imply a much smaller delta for the dealer to monetise in the positive gamma zone.
- Vega’s profile would be like gamma’s (both are sensitive to the placement of barrier vs implied forward) though magnitude would be different depending on the time to maturity of the option. As the underlying’s price approaches the digital (call as per the example) strike, higher volatility in the run-up would increase the probability of the digital payout being delivered implying peak positive vega around the strike.
Note that the gamma and delta profiles for a buyer would be just the mirror image of those for a dealer (seller).
- Cost of managing the greeks closer to the digital strike and  the time to expiry are high for both the buyer and the seller. If the underlying is trading near the strike closer to maturity both the buyer and seller would be keen to unwind to avoid the uncertainty of having a large drawdown for a small movement of the underlying.
