Computational Finance

1
Computational Finance

• Lecture 3
• Derivatives Forwards, Futures and Swaps
• Part II Hedging

2
Derivatives as Risk Hedging Tools

• The primary use of futures contracts is to hedge
  against risk. We shall illustrate some of the
  main hedging strategies in the remainder of the
  lecture.

3
Perfect Hedge

• The simplest hedging strategy is the perfect
  hedge, where the risk associated with a future
  commitment to deliver or receive an asset is
  completely eliminated by taking an equal and
  opposite position in the futures market.

4
Perfect Hedge

• Consider the example in the part I.
• The US company has 1M pounds obligation to pay
  on Feb. 23, then it can hedge its foreign
  exchange risk by taking a long position in the
  futures market to buy 1M pounds.

5
Perfect Hedge

• Another example from Part I
• A manufacturer of heavy electrical equipment
  needs copper in 9 months and takes a long
  position of a forward contract for delivery of
  copper in 9 months.

6
The Minimum-Variance Hedge

• It is not always possible to form a perfect hedge
  with futures contracts. For instance, an airlines
  want to hedge the risk of aviation fuels in the
  future. However, there are rarely such futures
  contracts traded in the market.
• One option is to find substitutions. For example,
  crude oil futures.

7
The Minimum-Variance Hedge

• In general, there may be no contract involving
  the exact asset that is to be hedged, the
  delivery dates of the available contracts may not
  match the asset obligation date, the amount of
  the asset obligated may not be an integral
  multiple of the contract size, and so on.
• In these situations, the original risk can not be
  eliminated completely.

8
The Minimum-Variance Hedge

• If the risk can not be eliminated completely, we
  of course want it as small as possible.
• Then, we need a definition of riskiness. What is
  a large risk and what is a small risk?

9
A Popular Risk Measure Variance

• In the probability theory, variance is often used
  as a measure of uncertainty.
• Using variance, we may be able to do risk
  hedging
• 1 unit of asset Y to be hedged at time T
• How many units of hedging asset X need now?

10
The Minimum-Variance Hedge

• The total cash flow is
• spot price of Y at time T,
• spot price of X at time T,
• hedging amount
• spot price of X now

11
The Minimum-Variance Hedge

• We want to minimize
• by choosing a proper .
• Mathematically, it is easy to show that

12
The Minimum-Variance Hedge

• A practical issue is how we get such variance of
  X and covariance of X and Y. Usually they can be
  obtained from statistical methods and historical
  data.

13
Statistical Methods to Estimate Variance and
Covariance

• Suppose that we have observations that the
  historical data of the spot prices of X and Y are
  given by
• , , , ,
• Then, two unbiased estimators