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Topic Title: question regarding static replication of Binary Option/Digital Option
Created On Wed Oct 15, 08 03:47 PM

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AutumnTale
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Wed Oct 15, 08 03:47 PM

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I am trying to replicate a Digital Optino using two call options, by short one call option at strike K, and long another at K-delta.
and I want to minimize the replication error by using smaller delta.
The questions is: no matter how small the delta is, the replication error does not change much, and the replication error remain about 10% of the notional for at the money option.

I am wondering if I am doing it right? are there anyway to reduce the replication error?

Thanks a lot.

daveangel
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Wed Oct 15, 08 04:03 PM

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clearly not - as delta->0 the call spread tends towards the binary.

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PaperCut
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Wed Oct 15, 08 04:10 PM

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Quote
Originally posted by: AutumnTale
I am trying to replicate a Digital Optino using two call options, by short one call option at strike K, and long another at K-delta.
and I want to minimize the replication error by using smaller delta.
The questions is: no matter how small the delta is, the replication error does not change much, and the replication error remain about 10% of the notional for at the money option.

I am wondering if I am doing it right? are there anyway to reduce the replication error?

Thanks a lot.

How are you measuring hedging error?
MC? If so, you may want to include jumps before you proceed.

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"A man should be learned in several sciences, and should have a reasonable, philosophical and in some measure a mathematical head, to be a complete and excellent poet."

BrightDay
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Wed Oct 15, 08 04:12 PM

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You are thinking like a quant in terms of non-existent theoretical prices, instead of thinking like a trader in terms of what you can find on the market to hedge your risk (i.e. on the market you won't be able to hedge the binary with an "infinitesimally" narrow spread).

Edited: Wed Oct 15, 08 at 04:16 PM by BrightDay

list
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Wed Oct 15, 08 04:21 PM

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Be accurate you need to start with the graph of the profit loss at maturity if you use two option with the same T and compare it with the digital payoff. The difference between two payoffs would present a source of theoretical modeling error.

AutumnTale
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Wed Oct 15, 08 04:24 PM

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Thanks for response. Let me clearify it:

The static replication is: [C(k-delta,T)-C(k,T)]/delta
where the C(k,T) is the call option with maturity at T, and strike at k.
The reason I need to divide the call spread by delta is I need to match the pay off of this call spread with that digital option.

Theoretically, when delta approach zero, this portfolio should approach a digital option.
But I did a small experiment using excel and Black-Sholes, the replication error did not go away. Instead, it is very stable.
Lets assume Black-Shole is right. and true price for a digital option is just N(d1)*discounting factor. and the replication error is just the difference between this true digital option price and the value of that call spread portfolio.
I am wondering why this replication error does not approach zero when delta is very small?

daveangel
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Wed Oct 15, 08 04:25 PM

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the digital call is N(d2) not N(d1)

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AutumnTale
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Wed Oct 15, 08 04:42 PM

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ohhhhhhhhhhhhhhhhhhhhhhhh, my god.
yes, you are right.
Thanks, and thanks a lot.

tw
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Wed Oct 15, 08 04:47 PM

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what are you doing about the skew?

Quote
Originally posted by: AutumnTale
Thanks for response. Let me clearify it:

The static replication is: [C(k-delta,T)-C(k,T)]/delta
where the C(k,T) is the call option with maturity at T, and strike at k.
The reason I need to divide the call spread by delta is I need to match the pay off of this call spread with that digital option.

Theoretically, when delta approach zero, this portfolio should approach a digital option.
But I did a small experiment using excel and Black-Sholes, the replication error did not go away. Instead, it is very stable.
Lets assume Black-Shole is right. and true price for a digital option is just N(d1)*discounting factor. and the replication error is just the difference between this true digital option price and the value of that call spread portfolio.
I am wondering why this replication error does not approach zero when delta is very small?

daveangel
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Wed Oct 15, 08 04:49 PM

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Quote
Originally posted by: AutumnTale
ohhhhhhhhhhhhhhhhhhhhhhhh, my god.
yes, you are right.
Thanks, and thanks a lot.

a eureka moment

-------------------------
Knowledge comes, but wisdom lingers.

list
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Wed Oct 15, 08 05:08 PM

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It is clear that using options with K and K- delta you approximate digital payoff. The position payoffs would be different on [ K - delta , K ]. Check if you have digital option the payoff would be 0 or 1 if you have other digital it would be 0, N. So you need to normalize your combination in order to approximate what you need. When you get an approximated payoff of the option the current price theoretical or historical present you approximation of the digital option.

HOOK
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Wed Oct 15, 08 07:19 PM

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Also, if you want to take into account the volatility skew, your binary/digital should be exp(-rt)N(d2)-vega*d(vol)/d(strike). The last term all being evaluated on the strike.

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