Question 2 (25 marks)

2.1 The gamma and Vega of a delta-neutral portfolio are 55 and 27, respectively, where vega

is per %. Estimate what happens to the value of the portfolio when there is a shock to the

market causing the underlying asset price to decrease by $5 and its volatility to increase

by 5%. (3)

2.2 Can the Vega of a derivative portfolio be changed by taking a position in the underlying

asset? Explain your answer. (2)

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2.3 A financial institution has the following portfolio of over-the-counter options on Canadian

dollar:

Type Position Delta of

Option

Gamma of

Option

Vega of

Option

Call 1,100 0.55 2.3 1.7

Call 550 0.82 0.7 0.2

Put 2,100 0.43 1.2 0.8

Call 550 0.71 1.7 1.5

A traded option is available with a delta of 0.6, a gamma of 1.6, and a vega of 0.75.

What position in the traded option and in Canadian dollar would make the portfolio both vega

neutral and delta neutral? (5)

2.4 A trader's portfolio is delta neutral and has a gamma of -4,250. The delta and gamma of

a particular traded call option are 0.62 and 1.52, respectively. The trader wants to make

the portfolio gamma neutral as well as delta neutral. What position should the trader take.

Explain to the trader what protection delta and gamma neutrality can provide to his

portfolio. (5)

2.5 Portfolio A consists of a one-year zero-coupon bond with a face value of $2,000 and a 10-

year zero-coupon bond with a face value of $6,000. Portfolio B consists of a 5.95-year

zero-coupon bond with a face value of $5,000. The current yield on all bonds is 10% per

annum (continuously compounded).

(a) Show that both portfolios have the same duration.

(b) Show that the percentage changes in the values of the two portfolios for a 0.1% per annum

increase in yields are the same.

(c) What are the percentage changes in the values of the two portfolios for a 5% per annum

increase in yields? (10)