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## Tuesday, December 13, 2011

### if given binary + asset-or-nothing call prices...

There's something intuitive (and therefore important) in this scenario -

Suppose we know the current prices of
- binary call (B) on an asset like IBM stock
- asset-or-nothing call (denoted A) on that asset

Q: What can we say about the vanilla call current price?

First, remember binary call must be worth between \$0 and \$1 before maturity, since the
payout is either \$0 or \$1. In contrast, asset-or-nothing call has current value comparable to current asset price.

It's useful to use concrete numbers, like K=\$15, exp(-rT) = 0.8 ...

Consider a portfolio { A - K*B } i.e.
++ long 1 unit of asset-or-nothing call
-- short K (i.e. 15) units of the binary call. Note even if there's a discount factor of 0.87 from maturity to today, we still go short K units.

If expired ... easy. If (both) exercised, we end up with 1 unit of the asset, and we owe someone \$15.

In a way, the (current prices of) A and B reveal the crucial N(d1) and N(d2) values as of today.

This replicating portfolio needs no rebalancing.