Financial Hacking:Evaluate Risks, Price Derivatives, Structure Trades, and Build Your Intuition Quickly and Easily by Philip Maymin

Author:Philip Maymin
Language: eng
Format: mobi
Tags: General Fiction
Publisher: World Scientific Publishing
Published: 2012-09-30T23:00:00+00:00

Cell Title Formula

A4 Expiry =4/252

B4 Stock 100

C4 Delta =BSCallDelta(100,A4,B4,.2,0,0)

D4 Hedge P&L

And here is what the Excel formulas would look like for the next row:

Cell Title Formula

A5 Expiry =A4-/252

B5 Stock =B4*LOGNORM.INV(RAND(), 0−1/2*(0.2*SQRT(1/252))^2, 0.2*SQRT(1/252))

C5 Delta =BSCallDelta(100,A5,B5,.2,0,0)

D5 Hedge P&L =−C4*(B5−B4)

The new expiry merely decrements by one business day.

The new stock price is the old stock price multiplied by a random number drawn from a lognormal distribution with an associated drift of zero (adjusted for the volatility as described in Section 5.2.3) and a standard deviation of .

The new delta is the same formula as the old delta, but using the new expiry and underlying price.

The hedging P&L calculates the amount of money you made or lost from yesterday's trade.

The formulas for the remaining rows are exactly the same, including the last row. This is an additional benefit of never using information from the future. If we had associated the hedging P&L with the date of entry into the trade, we would have had to be careful to be sure that the last row was blank, otherwise we would accidentally calculate the profit assuming we liquidated our delta hedge tomorrow at a blank price, or zero.

You might wonder why we don't need to keep track of all of our stock trades. After all, if we sold 200 shares on day one, and then our option delta turned out to be 250 on day two, we only need to sell an additional 50 shares, and so it seems we need to compute the P&L from each of our trades individually.

“I can't be expected to keep track of all my wheelings and dealings!”

Homer Simpson

Here's the assumption hidden in our formula: it is true in that example that you only need to sell an additional 50 shares, but, equivalently, you can imagine that you bought back all 200 shares that you sold the day before, and then sold again 250 shares. This way, you can calculate the hedging P&L based only on your net stock position each day, and not have to keep track of each individual trade.

You would, however, have to keep track of each individual trade once you take transactions costs into account, because you would not want to pay commissions to both buy and sell 200 shares with yourself for no economic benefit.

Okay, so now that this is done, we can test Black-Scholes. Remember that according to Black-Scholes, the total profit from hedging should exactly offset the initial cost of the option. In our example, a five-day option on a 20-vol stock should cost $1, a number we can compute either from the Black-Scholes formula or from the approximation in Section 5.2.1:

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