6. (15 pt) You have a portfolio of derivatives ? that is ?-neutral (Portfolio ?

Question

6. (15 pt) You have a portfolio of derivatives ? that is ?-neutral (Portfolio ? = 0) but has the following Gamma, ?(?) = ?30. (a) (7 pt) Make a prediction about the change in the value of the portfolio in case the underlying changes by $0.2. (b) (8 pt) In the market there is a derivative U1 with ?(U1) = 0.4 and ?(U1) = 0.5. Explain how you should modify your portfolio (using U1 and the underlying asset) to make it both ?- and ?-hedged. Give the precise numerical quantities involved. If the price of U1 is 1.50 and the price of the underlying is $2, what is the total cost of this modification?

6. (15 pt) You have a portfolio of derivatives ? that is ?-neutral (Portfolio ?-0) but has the following Gamma,?(11)--30 (a) (7 pt) Make a prediction about the change in the value of the portfolio in case the under- lying changes by $0.2. (b) (8 pt) În the market there is a derivative U1 with?(UI) = 0.4 and ?(U) = 0.5. Explain how you should modify your portfolio (using Ui and the underlying asset) to make it both and?-hedged. Give the precise numerical quantities involved. If the price of Ui is 1.50 and the price of the underlying is S2, what is the total cost of this modification?

Explanation / Answer

Answer

Part (b)

What is the approach behind it;-

At present the portfolio gamma is -30, to make it gamma neutral we have to add the derivative U1,after it is gamma neutral, the corresponding delta will also change for the portfolio, and hence we have to adjust the underlying asset to keep it delta neutral.

Let’s assume we add P units of U1 for each unit of portfolio

Now portfolio gamma is equal to= P x 0.4 + 1 x (-30)

To be gamma neutral, portfolio gamma now has to be zero (0)

Hence,

                                  P x 0.4 – 30 = 0

Solving for P= 30/0.4= 75 units of U1

Hence we need to add 75 units of U1 to the portfolio to make it gamma neutral.

Now It will change the delta of the portfolio, we need to make it delta neutral again

Now present delta of modified portfolio= previous delta+ delta added due to addition of U1

                                                                        = 0 + 75 x 0.5

                                                                        = 37.5

Assumption :- We add Q units of underlying asset to the modified portfolio to make it delta neutral. (Delta of any underlying asset is 1)

Modified delta of the portfolio now = Q x 1 + 1 x 37.5

For delta neutral modified delta will have to be zero hence

                                                             Q x 1 + 1 x 37.5 = 0

                                                              Q = -37.5 units

Hence we have to short (sell) 37.5 units of the underlying asset. (Please note, adjusting the underlying asset quantity will not now change the gamma of the portfolio as thegamma of any underlying asset is zero)

So to make the overall position delta and gamma neutral, we have to:

Cost of modification:

Net cost of modification= $ 112.5 - $75 = $37.5

We should incurred $37.5 to keep the portfolio both delta and gamma neutral.

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