Approximating Digital Call Payoffs with Call Spreads and Hedges

# How to replicate a digital call option


# How to replicate a digital call option












Call Option S=100 K=100 Payoff=1 (option is not available) How can i replicate this (payoff) with calls and puts with strike prices with multiples of 5$

Thanks for help

## Answer by SmallChess (score 13)

https://quant.stackexchange.com/a/18938

A digital call option (cash-or-nothing) can be replicated with two call options with different Strike. When we make the delta infinitely small and assume we have arbitrary strike prices. We get:

$C_{\textsf{bin}}(K,t) = \lim_{h \to 0} \frac{C(K,t)-C(K+h,t)}{h} = -\frac{\partial C(K,t)}{\partial K}$

## Answer by glyphard (score 3)

https://quant.stackexchange.com/a/1465

use a vertical spread and delta hedge it.

http://www.wilmott.com/messageview.cfm?catid=3&threadid=65988

## Answer by Mats Lind (score 0)

https://quant.stackexchange.com/a/49401

Going back to the original question, there is no static replication. This is clear from the first answer above which states that a call spread with an infitesimal difference between their strikes is needed. To arrive at an approximate replication, we need the probability of the underlying fixing in the USD 5 interval between the calls to be small, hence:

- Far away from the binary's strike, at large volatilities and long expiries, the call-spread replication could be good enough.

Going towards the strike at limited volatlity and time to expiry, the delta of the binary rises above that of the replica. Adjusting the positions in the options of the replica would probably not liquidity-wise be feasible - hence the second answer:

- Add a dynamic delta hedge to cover the delta mismatch near the strike of the binary.

## Answer by siou0107 (score 0)

https://quant.stackexchange.com/a/80100

Short answer: you cannot replicate exactly the digital option because its payoff function is discontinuous, making delta-hedging impossible close to the strike. However, you can over-hedge it: buying a static portfolio, payoff of which will dominate the digital payoff function at expiry.

Buy 0.2 call spread with strikes 95\$ and 100\$. When your digital option ends up in the money, the call spread payout will be 1; if it ends up out, you may get some extra money. Because that portfolio pays off more or the same than your digital, its value is theoretically higher than the digital's. It therefore gives you an upper bound on the price you should charge for the digital.

An equivalent lower bound is given by that of 0.2 100\$-105\$ call spread; that portfolio is worth the same as the digital on $\left[105, + \infty\right)$ and on $\left(-\infty, 100\right]$, but less than it on $\left(100, 105\right)$.

In your market conditions, you cannot do better. Any reliable model should give you a price for the digital option within those bounds.