american options - How far the spot price is likely to go from the current level in three months if its volatility is 15.7%

On Page 24 of N. Taleb's "Dynamic Hedging" the author gives the following example

Example: Assume that an asset trades at \$100, with interest rates at 6% (annualized) and volatility at 15.7%. Assume also that the 3-month 80 call is worth \$20, at least if it is American. Forgoing early exercise would create an opportunity cost of 20 x 90/360 x .06 = .30 cents, the financing of \$20 premium for 3 months. The time value of the equivalent put is close to zero (by put-call parity), so the intelligent operator can swap the call into the underlying asset and buy the put to replicate the same initial structure at a better cost. He would end up long the put and long the underlying asset.

The possible position of the operator before swap:

• 1 call worth of \$20 a


• \$80 in cash.

The position after the swap:

• 1 asset worth of $100 at the current spot price


• 1 put worth of almost zero

If I could earn 6% both on \$80 in cash and on 1 asset (i.e. if the asset is another currency for example) and the spot price would remain the same \$100 then I would agree with the calculations of the author:

1 asset x 90/360 x .06 - \$80 x 90/360 x .06 = (100 - 80) x 90/360 x .06 = 30 cents

But if the price of the asset will go down to say \$75, then I'd better stay with the call because:

1 asset x 90/360 x .06 - \$80 x 90/360 x .06 = (75 - 80) x 90/360 x .06 = -7.5 cents

So,how far the spot price is likely to go from the current level in three months if its volatility is 15.7%?

Answer

Keep in mind that there is nothing dynamic about this at all...it is only a snapshot and it is only a 1 sigma range.

High side:

([Price] * (1 + ([Vol] * SQRT[days to expiry]/365)))
(100 * (1 + (.157 * sqrt(90/365)))
107.7960

Low side:

([Price] * (1 - ([Vol] * SQRT[days to expiry]/365)))
(100 * (1 - (.157 * sqrt(90/365)))
92.2040