Derivatives

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Derivatives Risk Management

• Lecture 2 Forwards Futures - pricing

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Summary of Lecture 1

• Different types of derivatives
• Different market organizations
• OTC vs. standardized contracts
• Different uses
• asset-liability management
• risk management
• speculation

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Lecture 1 summary

• Futures contract specification
• Hedging with futures (basis risk)

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Overview of this lecture

• A review of continuous compounding
• Some key concepts
• Assumptions of the framework
• Notation
• Pricing forward contracts with no-arbitrage
  methods

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Continuous compounding

• Future of 1 invested at a rate R for n years
  with interest paid yearly

The future value with interest paid m times p.a.
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Examples

• Quarterly compounding for 5 years

• Semi-annual compounding for 3 years

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Continuous compounding

• As you progressively increase the periodicity of
  interest payments you will eventually have
  continuous compounding.
• 1 invested at a continuously compounded rate for
  1 year yields a future value of
• Exercise show that

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Continuous compounding

• Why is continuous compounding useful?
• Effective return is linear
• The exponential functional form is easy to work
  with and as a result extremely common in
  derivatives theory

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Conversion

• Consider the future value of a given investment
  opportunity.
• This quantity should be the same regardless of
  whether the return is expressed in terms of
  quarterly of continuously compounded rates

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Conversion II

• Formally

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Conversion III

• This equation can then be used to convert
  continuous rates to ones with arbitrary interest
  rate periodicity (and vice versa)

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Short sales

• Short selling involves selling securities you do
  not own
• Your broker borrows the securities from another
  client and sells them in the market in the usual
  way

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Short sales II

• At some stage you must buy the securities back so
  they can be replaced in the account of the client
• You must pay dividends other benefits the
  owner of the securities receives

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Repo rates

• The repo rate is the relevant rate of interest
  for many arbitrageurs
• A repo (a.k.a. repurchase agreement) is an
  agreement where one FI SELLS securities to
  another FI agrees to BUY them back later at a
  slightly higher price

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Repo Rates II

• The difference between the SELLING price the
  BUYING price is the interest earned
• It can be viewed as a guaranteed loan
• Think of repo rate as risk-free rate

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Forwards Pricing - Notation

• T maturity / expiration of the forward
• S0 current price of security
• ST future price of security
• K delivery price of the forward
• f value of the forward
• F0 forward price at time 0
• r risk free interest rate for period (0,T)

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Assumptions

• No transaction costs
• All trading profits subject to the same tax rate
• Borrowing and lending at the same constant risk
  free interest rate r

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The simplest case

• A forward on a security paying no dividends
• Method of derivation
• set up two portfolios with identical payoffs
• assume that no arbitrage opportunities exist in
  the market

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The argument

• Portfolio A one forward contract plus an amount
  of cash equal to the present value of the
  delivery price
• Portfolio B one unit of the underlying security

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The argument II

• At time T they will both be worth one unit of the
  underlying security

Hence they must also have the same value today
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The argument III

• The delivery price is set so that the forward has
  no initial value, i.e. f0. The forward price is
  defined as this delivery price
• We can then rewrite

as
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Example

• 1 year forward on a non-dividend paying stock
  S040, r10 p.a.

• Forward price

• Initial value of the forward contract

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Example II

• Let 6 months pass
• The Forward price at t0.5 is

• The value of the forward contract is

• This is the gain that could be locked in if we
  entered into a short forward with the same
  expiration

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When the underlying pays a known cash dividend

• Slight adjustment of the portfolios needed for
  valuation
• Portfolio A remains unchanged one forward
  contract plus an amount of cash equal to the
  present value of the delivery price
• Portfolio B is changed to
• one unit of the underlying security plus
  borrowings equal to the present value of the
  dividends paid during the life of the forward

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When the underlying pays a known cash dividend
(II)

• We use the dividends as they are paid to repay
  the loan in portfolio B
• At expiration / maturity of the forward we are
  still left with one unit of the security
  regardless of whether we own portfolio A or B

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When the underlying pays a known cash dividend
(III)

• Hence

• And remembering that at t0, f0 and F0K

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The case of a known dividend yield

• Portfolio A unchanged
• Portfolio B consists of

• Portfolio B will grow to one unit of the security
  assuming that dividends are reinvested in the
  security

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The case of a known dividend yield (II)

• Hence

and
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Futures vs. forward prices

• Throughout the course we will for valuation
  purposes ignore any differences
• With stochastic interest rates and for commodity
  contracts, this may not be a reasonable
  assumption and may favor more complex models

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Futures vs. forward prices (II)

• Suppose an asset S is strongly positively
  correlated with interest rates.
• When S increases the holder of a long futures
  contract makes an immediate gain which may be
  reinvested at a relatively high interest rate
• conversely when S falls, the loss will be
  financed at a lower than average rate

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Futures vs. forward prices (III)

• Hence with positive correlation one would expect
  a long futures contract to be worth more than a
  corresponding forward
• Ignoring this effect may be reasonable for
  shorter time horizons

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Futures vs. forward prices (IV)

• Other factors that may affect the relative prices
• Tax treatment
• Transaction costs
• Liquidity
• Margin requirements

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Index futures

• 3 month futures contract on the SP 500 index.
  Suppose that
• the annual dividend yield is 3
• the risk free interest rate is 8
• If the index currently stands at 400 then the
  futures price will be

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Index arbitrage

• Suppose that F0gt405.03 then you can go long in
  the index and sell the futures contract and lock
  in an arbitrage profit
• And if F0lt405.03 you short the index and take a
  long futures position

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Hedging with index futures

• The properties of a well diversified portfolio
  may be relatively similar to some index
  portfolio.
• The CAPM tells us that the return on a portfolio
  is

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Hedging with index futures II

• We can thus alter the risk profile of a portfolio
  by changing its beta.
• A simple way of doing this is to take positions
  in index futures
• Notation

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Hedging with index futures III

• We know from portfolio theory that the beta of a
  portfolio is a linear combination of the betas of
  its components

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Hedging with index futures example

• Portfolio worth 2,100,000
• We want to hedge the portfolio with 4 month SP
  500 index futures.
• Hedging can be interpreted as setting the beta
  equal to zero.
• Question What is the expected return of a zero
  beta portfolio?

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Hedging with index futures example

• As a reasonable approximation we can assume that
  the futures contract and the index have the same
  beta
• Assume that the SP 500 index is a close proxy
  for the market portfolio

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Hedging with index futures example
Current Futures Price
Unknown target quantity no. of futures
Current portfolio value
Assumed equal to 1
Given as 1.5
Number of times the index Institutional
feature e.g. 500
We want to set this to 0
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Hedging with index futures example

• The current futures price is 300

or
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More on the use of Index futures

• These arguments are not limited to hedging.
• The same methods will allow you to lever your
  portfolio, or just fine-tune its risk profile

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Changing beta

• Example suppose instead of hedging we would
  like to increase returns by doubling the risk of
  our portfolio

Which yields
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Currency contracts

• Analogous to the case of a forward on a security
  paying a known dividend yield
• Review (forward to buy with )
• Portfolio A long forward cash equal to PV of
  delivery price
• Portfolio B an amount in

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Currency contracts
Local () interest rate
Foreign () interest rate
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Currency forward example

• The 2 month interest rates in Switzerland and the
  U.S. are 3 and 8 p.a. respectively
• The spot price of a SFr is USD 0.6500. The 2
  month forward price is USD 0.6600.
• Is this consistent with the absence of arbitrage?

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Currency forward example II

• We should observe a forward price of

Hence we should sell SFr forward and buy spot
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Currency forward example III

• Buy PV(100) CHF today
• Borrow PV(66) dollars
• At expiration PV(100) CHF interest 100CHF.
• Sell at 66 USD
• Repay loan
• Net effect in two months time zero cash flow and
  today

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Commodity futures

• With commodities part of the fundamental
  arbitrage argument may break down
• Well first consider the case of investment
  commodities (gold, silver, etc)

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Commodity futures II

• If a commodity is held mainly for investment
  purposes we only need to consider the effect of
  storage costs.
• These can be considered either as negative income
  in cash terms or as a rate.

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Commodity futures III

• Non-investment commodities
• Suppose

Then we can short a futures contract, borrow
S0U, buy the commodity and pay the storage costs
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Commodity futures IV

• At time T we sell the commodity for F0 and repay
  the loan

This will generate a profit of
without any outlay at time t
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Commodity futures V

• Suppose instead
• To take advantage of this you would need to
• sell the commodity (and save storage costs) and
  invest proceeds at r
• go long in a futures contract
• If you hold the commodity for consumption in the
  first place you may be reluctant to do this.

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Commodity futures VI

• This reluctance can be captured by the
  convenience yield

Measures market expectations for future demand of
commodity
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Futures prices and the expected spot price

• Question

or
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Futures prices and the expected spot price II

• Keynes in Theory of the Forward Market in
  Treatise on Money 1930
• If hedgers on aggregate hold short positions the
  we should expect normal backwardation in order
  for speculators to be willing to take long
  positions

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Futures prices and the expected spot price III

• If on aggregate they hold long positions we
  should observe the opposite
• Contango

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Risk and Return

• Only systematic (non-diversifiable) risk is
  priced
• Expected return positive if positive correlation
  with market return and vice versa

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Risk and Return II

• Consider a speculator who bets on a rise in the
  asset price
• He takes a long forward position and deposits the
  present value of the delivery price at the risk
  free rate
• Cash flows
• at time t
• at time T

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Risk and return III

• Present value of strategy

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Risk Return IV

• Suppose the relationship between risk and return
  is CAPM-like

Then