# 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?

-

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.

-
Hi, thank you for your answer. i like to ask when your notional value is negative, it implies that you short an option? How did you arrive at the numbers N/delta K? why not N/100*delta K? lastly, let me get the 3 scenarios correct case 1: less than k1 - in which case nobody gets anything, so i get 0, clients gets 0 which matches the exact amount case 2 :more than k2, so i have to pay my client N amount (since the notionals from the two call options will cancel out) case 3: in between - clients gets 0, i get N/delta k from K1 call option. But in this case why am i 'overestimating'? –  user7315 Feb 22 at 18:15
after reading up more, i realized certain misconceptions i had. could you explain to me how does a fx option works with respect to notionals? and what will be the payout/situation in each of the free cases? –  user7315 Feb 23 at 4:46