D&R lecture 2.pptx

AF5002: Derivatives & Risk
Lecture 2
Pricing Futures & Forwards

Learning Outcomes
By the end of the lecture, you will be able to:
• Explain the assumptions that underlie the process of trying to value
alternative strategies to an existing Futures/Forward agreement ‘mid-
term’.
• Explain what holding costs/benefits are.
• Explain the mechanics of a Long Hedge and a Short Hedge.
• Understand some of the ‘perils’ of hedging using Futures/Forwards.

What have we learnt so far …
• We have looked at a single period Futures/Forwards
contract from the perspective of a HEDGING counterparty.
Time : T0
The time the
contract is made
Time : T
The maturity date
of the contract
Asset
Price
F
The
agreed
Futures
Price
FT0

• We have looked at a single period Futures/Forwards
contract from the perspective of a HEDGING counterparty.
Time : T0
The time the
contract is made
Time : T
The maturity date
of the contract
Asset
Price
The
agreed
Futures
Price
FT0
Margin
Account
/
Cashflow
Margin
SPOTT = FUTUREST
FT0

The SPOT (current) price is
lower than
the futures price agreed in
Jan’ 2018.
The merchant loses out by
this amount.
The farmer benefits by this
amount.
THIS CREATES A
POTENTIAL CREDIT RISK
FOR THE FUTURES
INSTRUMENT!
Note :
The SPOT Price at 12/12/18 and the futures
price for December delivery will have
converged at the delivery date.
Futures

Pricing Futures & Forwards – An Introduction …
» There is a fundamental assumption when seeking to price forwards and futures :
a) No Arbitrage - Markets are efficient & arbitrage cannot persist
» There is also a guiding principle :
b) Replication - The payoffs of the derivative product (in this
case the future or the forward) are
determined by price movements in the
underlying assets.
The derivative can be “replicated” by using
the underlying assets.

Let’s consider some notation :
Time  0 Current Date
T Maturity Date of the Forward (Years)
Price  S0 Current Price of the Underlying Asset
ST The Price of the Underlying Asset at Time = T
This is the Maturity Price of the Forward
F Delivery Price of a Forward (payable at Time T).

An investor needs to ensure that they have a given a unit of asset at a set point in time
(Time =T). They have two alternatives :
a) They could (Long) buy a forward at Delivery Price FT
b) They could buy a unit of Asset now (at Time = 0) and hold it
Either of these strategies deliver equivalent results for the investor … at Time = T the
Investor holds x unit(s) of Asset
In a perfect market with no transaction, holding or delivery costs, the Present Value of
each of the strategies should be the same.

The notation for this would be:
PV(F) = PV(S)
PV(F) Forwards Strategy
This is, at Time = 0, the present value of the single
payment under the futures contract upon delivery of the asset
PV(S) Replication Strategy
This is more complicated and may involve:
a) “Holding Benefits” e.g. dividends or coupons
b) “Holding Costs” e.g. storage or insurance costs

Let’s break down the notation for the Replication Strategy to include Net Holding Costs:
Consequently, the notation for the Replication Strategy becomes:
PV(S) = S0 + M
where S0 is the Spot Price at Time = 0
where M = Net Holding Costs (Replication Strategy)
= PV(Holding Costs) - PV (Holding Benefits)
Therefore:
PV(F) = S0 + M

Pricing Futures & Forwards – The assumptions …
The principal assumptions made using this approach are:
» No transaction costs
» No restrictions on Short Sales
Short Sales are possible and therefore the full proceeds from short sales are available
immediately
BUT, issues arise with this assumption (e.g. catastrophe futures where the underlying
is not traded & electricity where the cost to hold it is prohibitively expensive)
» The rates of interest are the same for borrowing and lending (incl. the default-free or
risk-free rate of interest)

Pricing Futures & Forwards – So What …?
What does this tell us?
Why bother?
Which strategy should the Investor use?
What if we could determine what the Holding Costs and Holding Benefits of buying an
asset now and holding it?
What if we had access to a market information system (e.g. Bloomberg) to determine
Spot Prices?

Pricing Futures & Forwards – So What …?
What does this tell us?
Why bother?
Which strategy should the Investor use?
Pricing Futures in the way enables us to make informed decisions:
» We will look at Hedging Strategies later …
» For now we’ll look at what an Arbitrager might do!!

Pricing Futures & Forwards – Arbitrage I …
We are examining a position at which : PV(F) ≠ S0 + M
In this instance we shall consider the following:
PV(F) > S0 + M The futures is over-valued therefore to Arbitrage the position the
Arbitrageur must:
Short the Futures and Long the Spot Mkt
» Enter a position to sell asset in the future.
» Borrow money for the period of the futures instrument
» Buy x unit(s) of asset on the Spot Mkt
» Repay borrowings when the futures settles

Let’s assume the following, for wheat prices on 01 May 2018:
» The present value of SPOT plus Hold is given by :
S0 = $26,400 (528.00 * $50)
M = $0
S0 + M = $26,400
» The present value of the Futures is given by:
PV(F) = $28,950 (579.00 * $50)
• Short (Sell) the Futures @ 579.00
• Borrow $26,400
• Buy (Long) the SPOT @ 528.00
• Hold and then Settle at the Futures Delivery Date
Pricing Futures & Forwards – Arbitrage Ia…

Pricing Futures & Forwards – Arbitrage Ib…
This generates the following cashflows:

Pricing Futures & Forwards – Arbitrage II …
We are examining a position at which : PV(F) ≠ S0 + M
In this instance we shall consider the following:
PV(F) < S0 + M The futures is under-valued therefore to Arbitrage the position the
Arbitrager must:
Long the Futures and Short the Spot Mkt
» Enter a position to buy asset in the future.
» Sell x unit(s) of asset on the Spot Mkt
» Invest proceeds for the period of the futures instrument
» Buy back x unit(s) of asset on the Spot Mkt
NOTE : This strategy assumes that you hold assets, have spare available and want it back or can borrow and then
return assets!

Let’s assume the following situation for wheat prices:
» The present value of SPOT plus Hold is given by :
S0 = $29,750 (595.00 * $50)
M = $0
S0 + M = $29,750
» The present value of the Futures is given by:
PV(F) = $28,950 (579.00 * $50)
• Long (Buy) the Futures @ 579.00
Borrow 5,000 Bushels of Wheat for Delivery NOW
• Short (Sell) the Spot @ 595.00
• Invest the funds received
• Settle at the Futures Delivery Date
Deliver back the 5,000 bushels of Wheat borrowed.
Pricing Futures & Forwards – Arbitrage IIa…

Pricing Futures & Forwards – Arbitrage IIb…
This generates the following cashflows:
Make a contract to buy the wheat in the future.
Borrow some wheat now and sell it now.
Buy the wheat using the futures instrument and give it
back to the person you borrowed it from!

Pricing Futures & Forwards – Limitations …
This pricing model appears very neat …
BUT REMEMBER!
a) The model makes no allowances for:
» Delivery Options }
» Margining (Daily Marking to Market) }
» Transaction costs
» Restrictions on Short Sales
(e.g. catastrophe futures where the underlying is not traded & electricity where the cost to hold it is
prohibitively expensive)
» Different rates of interest for borrowing and lending
(incl. the default-free or risk-free rate of interest)
Futures

Pricing Futures & Forwards – Applications …
Notwithstanding the limitations of the pricing model, it is extremely useful:
• Given the potential duration of both Futures and Forwards contracts, it is unlikely that
prices will remain static during the period of the contract.
• If such volatility is significant enough, at some point in the lifetime of a forwards (or
futures contract), assessment of the current position may be prudent. Using such a
pricing model would enable a counterparty to determine its current position.

Pricing Futures & Forwards – A Decision Tool …
If we use this tool to determine the value of a position mid-term (at Time = Mid) then it
enables us to make a decision about our position:
If :PV(F) ≠ SMid + M then we can either: a) Examine our Hedges
b) Seek to Arbitrage
If : PV(F) > SMid + M The forward is over-valued therefore either:
Hedge : Examine & Assess
Arbitrage : Short Forward and Long spot (Prior Example)
If : PV(F) < SMid + M The forward is under-valued therefore either:
Hedge : Examine & Assess
Arbitrage : Long Forward and Short spot (Prior Example)

Pricing Futures & Forwards – Hedging Strategy I …
If : PV(F) ≠ SMid + M The delivery payment due under the current forward is not the
same as the original contract so we can:
a) Take advantage of the price mismatch
(using the SPOT market)
(We arbitraged in this way in the previous examples)
b) Seek to trade out of the position, or
c) Hold the original position,

Pricing Futures & Forwards – Hedging Strategy II …
• This all looks very easy …
BUT!!
• Mid-term, the contract cannot be unilaterally unwound!
The obligations required under the contract must be reversed.
Practical Issues that Must be Considered
Investors are now operating at Time = TMid (the market has moved)!
a) There may be an Asset and Delivery mismatch
b) Margining for Futures contracts would have to be considered

Pricing Futures & Forwards – mismatches …
Asset Mismatches
• The assets underlying the original contract may not be available - there may be
similar assets but what is the difference in value?
Delivery Mismatches
• It may be impossible to absolutely match the delivery criteria of the original contract in
the current market.
• Given that Futures contracts generally favour the Short position (Sellers) and therefore
favour supply over demand, it means that Futures contracts will, ceteris paribus, be
cheaper than Forwards contracts.

Pricing Futures & Forwards – Margining …
• Given the daily marking-to-market, initial margin and variation margins for Futures
contracts (that have no counterpart in Forwards contracts) Futures price movements
will differ to those of Forwards.
Derivatives Principles and Practice, Sundaram & Das (2011), McGraw Hill, Singapore

HEDGING STRATEGIES USING FORWARDS

Hedges – What are They …
“a way of protecting oneself against financial loss
or other adverse circumstances. ”
Oxford Dictionaries Online @
http://www.oxforddictionaries.com/definition/english/hedge

Short & Long Hedges (A Review)…
• Long Hedge An attempt to manage risk using an agreement to buy an asset/assets at
a point in the future.
It is used when the Investor knows that they will need an asset/assets at a known point
in the future and/or believes that the price of the asset is likely to rise over time.
• Short Hedge An attempt to manage risk using an agreement to
sell an asset/assets at a point in the future
It is used when the Investor already owns (or has a contract to borrow an asset), wants
to sell it at some point in the future (and, as applicable, return the borrowed asset)
and/or believes that the price of the asset may fall over time.

Pippa & Ronny Pizzas Limited – Long Hedge I …
• It’s mid 2018 …
• Pippa & Ronny Pizzas Limited win a contract to supply a major US Supermarket chain
with frozen pizzas from 2019
• They are concerned about the price per pizza that they have had to negotiate and want
some certainty regarding the costs of some key ingredients. Consequently, they
negotiated a Forwards contract for cheese:
» Start Date : 02-May-18
» Last Price : $1.698 per lb
» Amount : 200,000 lbs of cheese
» Delivery Date : 04-Jan-2019

Towards the end of the period (03-Dec-18) Pippa & Ronny assess their cheese Hedge
… and are faced with the following situation:
» The present value SPOT & Hold is given by :
SMid = $294,000 ($1.47 * 20,000 * 10)
M = $0
SMid + M = $294,000
» The present value of the forward is:
PV(F) Mid = $281,400 ($1.407 * 20,000 * 10)
Key Questions
» What is Pippa & Ronny’s current cash position?
» Can the existing Hedge be unwound?
» Can they re-Hedge?
» Do they want to?
Pippa & Ronny Pizzas Limited – Long Hedge II …
We need 200,000 lbs of cheese
therefore we multiply by 10 :
Each contract is for 20,000lbs
and
200,000lbs = 20,000lbs x 10
Security CHEZ8 Comdty
Contract Size 20,000lbs
Delivery Date 04-Jan-19
Price USD/lb
Exchange
Chicago
Mercantile
Date (Original Forward) 02-May-18
Last Price 1.698
Date 03-Dec-18
Forward Price (FMid) 1.407
Date 03-Dec-18
Spot Price (SMid) 1.47
Holding Costs Zero
Interest Cost/Returns 0%

» Their cash position is zero (at 03-Dec-18)
They negotiated their forwards contract but it has not yet delivered and therefore has
not cash settled.
Hedging, they could:
» Hold their position (wait and see)
» Try to manage the current position which indicates they may
overpay
PV(F) > PV(F)Mid $1.698 > $1.407
Pippa & Ronny Pizzas Limited – Long Hedge III …
Price Fluctuations to Date
(02 May 2018 to 03 December 2018)
Forward Price (02/05) 1.6980
Forward Price (03/12) 1.4070
Maximum Price 1.7330
Minimum Price 1.4070
Average Price 1.6281
St. Dev Price 0.0838

Pippa & Ronny Pizzas Limited – Long Hedge IV …
BUT WHAT DOES “HEDGING” MEAN IN THIS CONTEXT?!
The purpose of the strategies we are examining is to manage Pippa & Ronny Pizzas
Limited issues in the context of their need for cheese and the price they will pay for that
cheese!
Based on this premise, are other strategies simply speculative … ?
Pippa & Ronny Limited could (amongst other strategies):
1) Short Forward (Unwind the current trade) and Long Spot & Hold
2) Short Forward (Unwind the current trade) and Long Forward

Short Forward (Unwind the current trade) and Long Spot & Hold as at 03-Dec-18
generates the following cashflow:
There is little point to this hedging strategy – because the Spot Price is so high Pippa & Ronny Pizzas Limited’s final
cashflow position would deteriorate ($339.6k to $352.2k) not to mention the additional risk of having to hold the
asset until required!
Pippa & Ronny Pizzas Limited – Long Hedge V …
Cashflow Analysis (as at 03 December 2018) Cashflow Final
as @ 03-Dec-2018 Cashflow
Initial Trade @ 02-May-18 Long (Buy) Forward @ 1.698 $ - $ -339,600.00
Hedging Strategy Trade as @ 03-Dec-18 Short (Sell) the Forward @ 1.407 $ - $ 281,400.00
Cost/Profit upon Unwind n/a $ -58,200.00
Hedging Strategy Trade @ 03-Dec-2018 Long (Buy) the Spot @ 1.47 $ -294,000.00 $ -
Holding Costs $ - $ -
Net cashflows of All Trades $ -294,000.00 $ -58,200.00
Final Cashflow (All Trades) n/a $ -352,200.00

Short Forward (Unwind the current trade) and Long Forward as at 03-Dec-18 generates
the following cashflow:
There is little point to this hedging strategy – it hasn’t altered Pippa & Ronny Pizzas Limited’s final cashflow at all and
has arguably increased their risk - delivery and asset mismatches on the additional trades (even assuming transaction
costs of zero)!
Pippa & Ronny Pizzas Limited – Long Hedge VI …
Hedging Strategy Trade @ 03-Dec-2018 Long (Buy) the Forward @ 1.407 $ - $ -281,400.00
Net cashflows of All Trades $ - $ -339,600.00

Pippa & Ronny Pizzas Limited – Long Hedge VII …
We investigated two Hedging strategies
1) Short Forward (Unwind the current trade) and Long Spot & Hold
2) Short Forward (Unwind the current trade) and Long Forward
Given that neither of these strategies improve the position of Pippa & Ronny Pizzas
Limited (and arguably make it worse) …
Key Questions
» What is Pippa & Ronny’s current cash position? $ 0.00
» Can the existing Hedge be unwound? Yes
» Can they re-Hedge? Yes
» Do they want to? No
Holding the Original Position provides the “best” at the current date. (as @ 03-Dec-18)

Pippa & Ronny Pizzas Limited – Long Hedge VIII …
Why in this example is “Holding the Original Position”
the best position for the company?
This is the result of the interaction between:
• The Original Forward Price Fixed
• The Present Value of the Forward Variable
• The Spot Market Price Variable

Pippa & Ronny Pizzas Limited – Long Hedge X …
One strategy would be to examine how much:
a) The Spot Market moves relative to the Present Value of the Forward
Things to Remember :
• Of the Hedging strategies we have examined:
» Short Forward then Long Forward results in the same cashflow and is not therefore a reasonable
trade.
» Short Forward then Long Spot and Hold could be a strategy to follow
• If the original cost position to Pippa & Ronny Limited could be reduced by, say $25,000
from $339,600 to $314,600 the company may be tempted to trade. But how much
would the SPOT price have to move to make this worthwhile?

So what do we do … :
• If we hold the Present value of the Forward constant we can see how much the Spot
Price would have to move to achieve an overall cost of $314,600
Pippa & Ronny Pizzas Limited – Long Hedge X …

Pippa & Ronny Pizzas Limited – Long Hedge XI …
Another strategy would be to examine how much:
b) The Present Value of the Forward moves relative to the Spot Market
Things to Remember :
• Of the Hedging strategies we have examined:
» Short Forward then Long Forward results in the same cashflow and is not therefore a reasonable
trade.
» Short Forward then Long Spot and Hold could be a strategy to follow
• If the original cost position to Pippa & Ronny Limited could be reduced by, say $25,000
from $339,600 to $314,600 the company may be tempted to trade. But how much
would the Forward price (FMID) have to move to make this worthwhile?

So what do we do … :
• If we hold the Present value of the Spot Market Price constant we can see how much
the Present Value of the Forward Price would have to move to achieve an overall cost
of $314,600.
Pippa & Ronny Pizzas Limited – Long Hedge XII …

Pippa & Ronny Pizzas Limited – Long Hedge XIII …
How reasonable is it to assume that only one of the prices will move whilst the other is
held constant?
Remember, the position we are examining is the result of the interaction between:

Pippa & Ronny Pizzas Limited – Long Hedge XIV …
From the perspective of an entity that is Hedging (notwithstanding their attitude to risk), it
is the difference between the :
Relative to :
that determines whether the Pippa & Ronny Limited should:
1) Hold their Original Position
2) Trade their Original Position

Abba, Dean & Angus Limited – Short Hedge I …
• It’s early 2018 …
• Abba Dean & Angus Limited breed cattle. They have a herd of cattle that will be ready
for market in early 2019.
• They are concerned that their breeding costs may not be met if they cannot sell the
cattle. Consequently, they negotiate a Forwards contract to Short (sell) the cattle to a
meat wholesaler:
» Start Date : 01-Feb-18
» Last Price : $1.1835 per lb
» Amount : 40,000lbs of cattle (approx. 60 animals)
» Type : 55% Choice, 45% Select, Yield Grade 3 Live Steers
» Delivery Date : 15-Feb-18

Part way through the period, Abba Dean & Angus Limited assess their cattle Hedge …
and are faced with the following situation:
» The present value of the SPOT & Hold is given by :
SMid = $50,984 ($1.2746 * 40,000)
M = $0 Assume : Holding Benefits = 0 (in this case).
SMid + M = $50,984
» The present value of the Forward is given by :
PV(F) = $46,600 ($1.1650 * 40,000)
Key Questions
» What is Abba Dean’s current cash position?
» Can the existing Hedge be unwound?
» Can they re-Hedge?
» Do they want to?
Abba Dean & Angus Limited – Short Hedge II …

» Their cash position is zero (at 03-Jul-18)
They negotiated their forwards contract but it has not yet delivered and therefore has
not cash settled.
They could:
» Hold their position (wait and see)
» Try to secure the current position and the similar price …!
PV(F0) < PV(FMid ) $1.1835 > $1.1650
Abba Dean & Angus Limited – Short Hedge III …

Abba Dean & Angus Limited – Short Hedge IV …
BUT WHAT DOES “HEDGING” MEAN IN THIS CONTEXT?!
Given the definition of “Hedge” and the fact that the original Forward should deliver a
‘fixed’ price (albeit this is less than the current SPOT price), any strategy here might
arguably be speculative – Abba Dean & Angus have a need to manage the sale of their
cattle and the price they will receive!
Abba Dean & Angus Limited could (amongst other strategies):
1) Long Forward (Unwind the current trade) and Short Spot
2) Long Forward (Unwind the current trade) and Short Forward

Long Forward (Unwind the current trade) and Short Spot as at
03-Jul-18 generates the following cashflow:
This hedging would work – it returns $51,724 as opposed to the $47,340 the original Forward would return.
Remember, however, that Abba Dean & Angus could only use this strategy if their cattle were ‘sellable’ on 03-Jul-18
or could borrow equivalent cattle!!
Abba Dean & Angus Limited – Short Hedge V …

Long Forward (Unwind the current trade) and Short Forward as at
03-Jul-18 generates the following cashflow:
There is little point to this hedging strategy – it hasn’t altered Abba Dean & Angus Limited’s final cashflow and they
may have increased their risk (delivery and asset mismatches on the additional trades)!!
Abba Dean & Angus Limited – Short Hedge VI …

Abba Dean & Angus Limited – Short Hedge VII …
We investigated two Hedging strategies
1) Long Forward (Unwind the current trade) and Short Spot
2) Long Forward (Unwind the current trade) and Short Forward
It is arguable whether either of these strategies improve the position of Abba Dean &
Angus Limited (and potentially they make it worse) …
Key Questions
» What is Abba Dean & Angus’s current cash position? $ 0.00
» Can the existing Hedge be unwound? Yes
» Can they re-Hedge? Yes
» Do they want to? Possibly
Practical issues as well as Abba Dean & Angus’ risk approach will determine the ‘best’
(but possibly risky) outcome. (as @ 03-Jul-18)

Futures & Forwards : Cautionary Tales I …
There are a number of examples where companies have sought to Hedge exposures,
mismanaged their strategies and lost money.
There are two extremely good case studies in :
Sundaram, R & Das, S Derivatives Principles & Practice
McGraw-Hill Irwin, New York, USA
Chapter 2, P.39
• Metallgesellschaft AG Gas, Heating Oil & Diesel
• Amaranth Natural Gas

Futures & Forwards : Cautionary Tales II …
The result for each :
• Metallgesellschaft AG Losses in excess of $1bn Bankruptcy
• Amaranth Losses in excess of $4.3bn Portfolio Unwound
The key lessons to be learnt :
a) Leverage Forwards and Futures are Highly Leveraged
b) Liquidity Market Liquidity is critical – the less liquid a market the greater the impact of a
participant’s own trading
c) Volatility Forwards, Futures (and particularly commodities markets) can be very volatile

Learning Outcomes
By the end of the lecture, you will be able to:
• Explain the assumptions that underlie the process of
trying to value alternative strategies to an existing
Futures/Forward agreement ‘mid-term’.
• Explain what holding costs/benefits are.
• Explain the mechanics of a Long Hedge and a Short
Hedge.
• Understand some of the ‘perils’ of hedging using
Futures/Forwards.

D&R lecture 2.pptx

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D&R lecture 2.pptx