Motivation. Imagine you are an investment bank, selling (which is equivalent to short-selling) a call option. At the end, you have to pay to the holder . When the option is on the money at expiration time , you have to pay . You want to pay less, though; so you try to hedge against small changes in underlyer price.

Delta Hedging for Call/Put Options

At time (day 0)

  • Sell the option , and get the cash
  • Borrow amount of money to buy units of stock. (this is the hedging part)
    • & this is the same as the delta of the call option.

It is obvious that the value of this portfolio is , at time , but to show that this portfolio at any time always has value , we observe the fact that the last three terms (let ) is a self-financing portflio that tracks the call price exactly. Observe from BSM derivation that a portfolio (we already used ) is self-financing and replicates when:

  • In our case, we have:
  • And thus by the above equation.

At time (day 1)

  • Option value changes to
  • Cash holding grows to
  • We must “Re-hedge” or “balance” the loan and underlier such that we maintain units of underlyer. We break it out into three cases:
    • (delta stays same): time value of money, and use dividend to pay back loan
    • (delta increases): Borrow more money to buy more stock:
    • (delta decreases): Sell stock to reimburse loan.
    • In all three cases, we can say:
  • We then purchase or sell units of the security to get As we show above, is self-financing. This means that total value of the portfolio is:

At time , (expiration date)

On expiration and before the call is cashed out, the value of the portfolio is:

We then consider our customer cashing out their loan:

  1. If in the money,
    1. becomes
    2. Thus since synthetic call still holds
      1. i.e., liquidate the stock to pay
    3. Thus
      1. i.e., pay back the loan with remaining and cash invested
  2. If out of the money:
    1. becomes
    2. Thus since synthetic call still holds
    3. Thus
      1. i.e. pay back the loan with cash invested and liquidating the stock

Greek Neutral Portfolio of Derivatives

Any derivative we can modify to make delta-neutral. For a security modeled by geometric brownian motion (with no dividends, ):

Notation

Since there are lots of notaions here, let’s outline them first:

  • is the underlying security
  • is the call option
  • is the delta-neutral portfolio when parameter
  • is the gamma-neutral portflio when parameters ,
  • is the delta-gamma neutral portflio when parameters

let us create a portfolio of a derivative of this stock by having one call option and units of the underlier. The value of this portflio is:

  • have greeks
  • which have greeks , , . Using ito’s formula, we get:

Where from Ito Isometries

(dS_{t})^{2}&=m^{2}S_{t}^{2}\underbrace{ (dt)^{2} }_{ =0 }+\sigma^{2}S_{t}^{2}\underbrace{ d\mathfrak{B}_{t}^{2} }_{ = dt}+2m\sigma S_{t}^{2}\underbrace{ dtd\mathfrak{B}_{t} }_{ =0 } \\ &= \sigma^{2}S_{t}^{2}dt \end{align}

Now, from the above formula we see that there are three components that cause changes in

  • i.e. non-zero delta
  • , i.e. non-zero gamma
  • , but considering it’s impossible for this to be zero.

Delta hedging.

Lets make . We can do this simply by setting :

Delta-Gamma Hedging

After the delta hedging we still have a positive gamma. Lets make . We create a new portfolio of the delta-neutral portfolio, and units of the call:

Setting will make . But this incurs a new problem: is non-zero again. We add more units of to combat this:

  • #task finish off gamma hedging