Super-Replicating a Digital Call with a Vanilla Call Spread

# Risk management of options


# Risk management of options












Your client would like to buy a digital call option. the digital call option pays the buyer in one years time (i.e at maturity )

- N=1m SGD, if the SGD USD spot rate at maturity is above a prescribed level k_0 and nothing otherwise

Risk management of this structure is done, via conservative replication, with a pair of call options of strikes $K_1$ and $K_2$, $K_2$ > K_1 and with a spread size $\Delta K =K_2-K_1>0$.

Question: From the seller point of view, how would you choose the strikes $K_1 , K_2$ and notionals $N_1,N_2$ (possibly negative) of these call options as a function of $K_0$, $\Delta K$, and $N$ in such a way that

1) you overestimate the actual amount paid to the client if the spot falls between $K_1$ and $K_2 $

2) and match exactly actual amount otherwise?

Anybody can help me with this, and if possible ,explain what is meant by conservative replication as well as how can the notionals be negative? It does not make sense that i am a seller of a call option and i give a negative payout?

## Answer by Peter Carr (score 2)

https://quant.stackexchange.com/a/10333

In what follows, I assume no default and no frictions. A perfect hedge for your short position in a binary call would be a long position in a digital call with the same notional, strike price, and maturity date. Your goal is to super-replicate the payoff from a long digital call by a static position with a pair of co-terminal vanilla calls, one with strike K1 and the other with strike K2 > K1. The notional of the target digital call is N>0.

Here is a super-replicating portfolio. Set K2 = K0. Set K1 = K2 - ΔK. Buy one call with strike K1 and notional N1 = N/ΔK. Sell one call with strike K2 and notional N2 = N/ΔK. Done. If you wish, you can think of the second call position as Buy one call with strike K2 and with negative notional N2 = -N/ΔK, but this would be an unconventional view. The super-replicating portfolio above would be an example of conservative replication.

## Answer by Peter Carr (score 0)

https://quant.stackexchange.com/a/10424

The convention in the world of finance is that notionals are always nonnegative. In mathematical finance, notional could be real-valued, so let's call it MFNotional to distinguish. If MFnotional is negative, then it means short an option.

Here is an example to indicate how I arrived at N1 = N/ΔK. Suppose K0 = 0.8, ΔK= 0.2, and N = 1m SGD. Then K2 = 0.8 and K1 = 0.6. Let X denote the unknown exchange rate at expiry with X SGD = 1 USD. The goal is to super-replicate the payoff 1( X > 0.8) m SGD. Suppose we scale down our payoff substantially to just 1(X > 0.8) SGD i.e. divide out the 1m SGD notional. The vanillas to be used to super-replicate 1(X > 0.8) SGD pay (X-0.6)^+ SGD and (X - 0.8)^+SGD for each SGD of notional. Consider the payoff from 5 (X - 0.6)^+ - 5 (X - 0.8)^+. Notice 5 = 1/(0.2). Refer to the vanilla position as a call spread. The notional in the long call is 5 SGD and the notional in the short call is also 5 SGD. If X < 0.6, the call spread payoff vanishes as does 1(X > 0.8). If X> 0.8, then call spread payoff = 1 SGD as is 1(X> 0.8). If X inbetween 0.6 and 0.8 then call spread payoff is 5 (X - 0.6) SGD >= 0 SGD so we overestimate/super-replicate the digital call payoff of 0 SGD. When we recognize that the actual notional of the target is 1m SGD rather than 1 SGD, then we have to scale up the notionals in the two calls by a factor of 1m, so that they become 5m SGD each. Notice that N1 = N/ΔK = 1m SGD/(0.2) = 5m SGD.