Chapter 12 - The Greeks

Question 1

The Greeks

Answer 1

The Greeks are a group of mathematical derivatives that can be used to help us to manage or understand the risks in our portfolio.

Question 2

Delta hedging

Answer 2

Taking a mixed portofio of a derivative and the underlying asset to create an instantaneously risk-free portfolio.

Delta-hedged portfolio is a portfolio for which the weighted sum of the deltas of the individual assets is equal to zero.

Question 3

Static vs dynamic hedging

Answer 3

The process of simply constructing an initial portfolio with a total delta of zero, at time 0 say, and not rebalancing to reflect the subsequent changes in delta, is known as static delta hedging.

The process of continuously rebalancing the portfolio in order to maintain a constant total portfolio delta of zero is known as dynamic hedging.

Question 4

What is the value of delta for the underlying asset?

Answer 4

For the underlying asset, whose value is St, Δ=1

Question 5

What is the value of gamma for the underlying asset?

Answer 5

For the underlying asset, St, Γ=0

Question 6

Discuss the use of gamma in portfolio risk managment

Answer 6

Suppose a portfolio is following a delta hedging strategy. If the portfolio has a high value of Γ then it will require more frequent rebalancing or larger trades than one with a low value of gamma.
It is recognised that continuous rebalancing of the portfolio is not feasible and that frequent rebalancing increases transaction costs. The need for rebalancing can, therefore, be minimised by keeping gamma close to 0.

Question 7

What is the value of vega for theunderlying asset?

Answer 7

For the underlying asset, St, V=0.

Question 8

Discuss the value of vega in portfolio risk management

Answer 8

The value of a portfolio with a low value of vega will be relatively insensitive to changes in volatility. Put another way, it is less important to have an accurate estimate of σ if vega is low. Since σ is not directly observable, a low value of vega is important as a risk-managment tool. Furthermore, it is recognised that σ can vary over time. Since many derivative pricing models assume that σ is constant through time, the resulting approximation will be better if V is small.

Question 9

Discuss the use if rho in portfolio risk management

Answer 9

Rho tells us about the sensitivity of the derivative price to changes in the risk free rate of interest. The risk free rate of interest can be determined with a reasonable degree of certainty, can vary by a small amount over the (usually) short term of a derivative contract. As a result, a low value of ρ reduces risk relative to uncertainty in the risk free rate of interest.

Question 10

Discuss the use of Θ in portfolio risk managment

Answer 10

Since time is a variable that advances with certainty, it does not make sense to hedge against changes in t in the same way as we do for unexpected changes in the price of the underlying asset.