Futures and Options - Futures _ Forwards by RushenChahal


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Futures & Forwards

      Prof. Rushen Chahal   1
Comparisons & Contrasts
   We have said that forward and futures
    contracts contain an essential similarity,
    in the sense that both call for the
    delivery of a specified quantity of the
    underlying asset, on a pre-specified
    date, at a price that is agreed upon at
    the outset.

                   Prof. Rushen Chahal       2
Comparisons & Contrasts
   Besides, both types of contracts,
    impose an obligation on the buyer as
    well as the seller.
   While there are obvious similarities
    between the two types of contracts,
    there are certain vital differences.
   We have already discussed one
    fundamental difference.
                  Prof. Rushen Chahal      3
Comparisons & Contrasts
   Futures contracts are traded on an
    organized exchange whereas forward
    contracts are private or OTC contracts.
   Now we will go on to look at other

                   Prof. Rushen Chahal        4
   Since a forward contract is traded OTC,
    it is customized in nature.
   That is, the core features of the
    agreement to transact, are arrived at by
    a process of consensus between the
    two parties to the contract.
   What exactly are these core features?

                  Prof. Rushen Chahal      5
   Core Features
1. How many units of the underlying
  asset is the short required to deliver to
  the long per contract?
2. What grade of the asset (this is
  applicable mainly for commodities) is
  acceptable for delivery?
  Or can more than one grade be eligible
  for delivery?

                     Prof. Rushen Chahal      6
Core Features (Cont…)
3. Where should delivery be made?
  Or is there a choice of locations for one
  of the parties?
4. When can delivery be made?
  Is it possible only on a fixed date, or is
  there a specified time interval during
  which it can occur?

                  Prof. Rushen Chahal          7
Core Features (Cont…)
   In the case of contracts where multiple
    grade and/or locations are allowed, who
    gets to choose as to what and where to

                  Prof. Rushen Chahal     8
Customized Contracts
   In the case of customized contracts, all
    these issues have to be resolved
    through bilateral negotiations and
    incorporated into the contract.
   This is therefore the case with forward

                   Prof. Rushen Chahal         9
Standardized Contracts
   In the case of a standardized contract,
    like a futures contract, there is a third
    party, which will spell out the
    permissible terms and conditions that
    may be incorporated into the contract.
    This party is the exchange.
   Let us take each of the issues raised
    above from the standpoint of a futures
    contract.       Prof. Rushen Chahal       10
Contract Size
   The contract size or the number of units
    of the underlying asset that is required
    to be delivered by the short to the long,
    is fixed by the exchange.
   While setting the size, two opposing
    forces have to be judiciously balanced.
       While a smaller size will favour small
        traders, larger sizes will reduce the
        average transactions cost.
                       Prof. Rushen Chahal       11
Allowable Grades &
Delivery Locations
   The grade or grades that are eligible for
    delivery are specified by the exchange.
   The place or places where delivery is
    permitted, are specified by the
   In the case of multiple deliverable
    grades and/or delivery locations, one of
    the parties to the contract has to be
    given the right to choose in order to
    avoid a conflict. Rushen Chahal
                    Prof.                   12
Grades & Locations (Cont…)
   Traditionally futures exchanges have
    given this choice to the short.
   In the case of multiple allowable
    grades, one grade is designated as the
    par grade.
   Other grades may be delivered at a
    specified premium to or discount from
    the par grade.
                  Prof. Rushen Chahal        13
Grades & Location (Cont…)
   The premium/discount may be applied
    additively or multiplicatively as we will
    see later.
   What will happen in practice is that one
    of the grades will emerge as the
    Cheapest to Deliver grade.
   Consequently everyone will deliver it.

                   Prof. Rushen Chahal      14
Grades & Location (Cont…)
   If multiple locations are specified the
    short can choose the place of delivery.
   If he decides to deliver at a location
    other than the par location, he will
    receive a premium or discount per unit
    of the underlying asset that is delivered.

                   Prof. Rushen Chahal      15
Delivery Period
   The exchange will also specify a
    delivery period by indicating the first
    date on which delivery can be made,
    and the last possible date on which it
    can occur.
   The decision to deliver is made by the
    short in practice.
   That is, the long cannot call for
    delivery.       Prof. Rushen Chahal       16
Unit Price
   The last remaining issue is the price at
    which the underlying asset is to be
   In the case of both forward as well as
    futures contracts, this is arrived at by
    negotiations between traders on
    opposite sides of the market, and is a
    function of supply and demand forces.
                   Prof. Rushen Chahal         17
Dissimilarities (Cont…)
   The second major difference between
    the two types of contracts is that while
    most of the forward contracts that are
    entered into, result in eventual delivery
    of the underlying asset, only a very
    small percentage of the futures
    contracts that are entered into, actually
    result in delivery.
                   Prof. Rushen Chahal      18
   What happens when a trader who has
    entered into a futures contract, desires
    to get out of his position without having
    to deliver or to accept delivery?
   He can simply offset by taking a
    counter-position in the same contract,
    on the floor of the futures exchange.

                   Prof. Rushen Chahal      19
   What is a counter-position?
   If a trader has taken a long position
    initially, taking a counter-position would
    entail going short subsequently and vice

                   Prof. Rushen Chahal      20
The Advantage of
Standardized Contracts
   Forward contracts are customized.
   Thus each contract will typically have
    features that are unique to it.
   Therefore to abrogate an existing
    contract, it is essential to seek out the
    party with whom the trade was
    originally consummated, and have the
    contract nullified.
                    Prof. Rushen Chahal         21
Standardization (Cont…)
   A futures contract however is
   Thus a contract to trade 100 kg of
    wheat of type Indian Red in New Delhi
    on 31 October, entered into between
    Poonam and Kunal, will be the same as
    a contract to trade 100 kg of wheat of
    type Indian Red in New Delhi on 31
                          into between Aditi and
    October, entered Rushen Chahal
                    Prof.                      22
Standardization (Cont…)
   Let us assume that Poonam has gone
    long to start with.
   If she wants to get out of her position,
    she has to simply call her broker and
    indicate her desire to go short in a
    contract with the same features as the
    one she had traded initially.

                   Prof. Rushen Chahal         23
Standardization (Cont…)
   That is, she should seek to establish a
    short position in an October contract on
    Indian Red wheat.
   It will obviously have a contract size of
    100 kg, because that would have been
    fixed by the exchange.
   So too would be the place of delivery,
    which in this case is New Delhi.

                   Prof. Rushen Chahal      24
Offsetting (Cont…)
   When Poonam’s order goes to the floor
    of the exchange, it will be matched with
    an order to go long that has been
    placed by another trader.
   It is important to note that this trader
    need not be Kunal, the person with
    whom she had originally traded.

                  Prof. Rushen Chahal      25
Offsetting (Cont…)
   Once this trade is put through, Poonam
    will have a net position of zero.
   Of course, in the process she may have
    made a profit or a loss, since the
    prevailing futures price when she had
    initially gone long, would in general be
    different from the price prevailing at the
    time of offsetting.
                   Prof. Rushen Chahal      26
Offsetting (Cont…)
   Is standardization the only reason why
    offsetting is easy?
   The answer is no.
   In the case of a forward contract, there
    is an element of credit risk, because
    each party is exposed to the risk of
    nonperformance on the part of the
                   Prof. Rushen Chahal     27
Credit Risk
   Consequently, each party must verify
    the credit history and antecedents of
    the other.
   It is for this reason that parties to a
    forward contract tend to be large
    institutional players like commercial
    banks, investment banks, investment
    funds, insurance companies, and
    brokerage firms. Rushen Chahal
                      Prof.                   28
Futures & Credit Risk?
   What about futures contracts?
   These contracts are traded on
    organized exchanges.
   And each exchange has an associated
    entity known as a clearinghouse.
   The clearinghouse may be a wing of the
    exchange or may be a separate legal
                  Prof. Rushen Chahal    29
The Clearinghouse
   The clearinghouse takes the stand that
    once a buyer and a seller enter into a
    futures contract, it will come in between
    and provide a performance guarantee
    to both the parties.

                   Prof. Rushen Chahal     30
The Clearinghouse (Cont…)
   That is, as far as the buyer is
    concerned, the clearinghouse will
    become the effective seller, and as far
    as the seller is concerned, it will
    become the effective buyer.
   Notice that neither party actually trades
    with the clearinghouse. It therefore
    enters the picture only after a deal has
    been struck betweenChahallong and a
                     Prof. Rushen            31
The Clearinghouse (Cont…)
   How does the clearinghouse guarantee
    the performance of both the parties?
   It does so by insisting that both the
    parties deposit a performance
    guarantee called Margin with it.
   What is this concept of a Margin?

                  Prof. Rushen Chahal       32
   Let us first see as to why a margin is
   Since a futures contract imposes an
    obligation on both the parties, it is but
    logical that given a chance, one of the
    two parties would like to default on the
    expiration date, since it will not be in
    his interest to go through with the
    contract.       Prof. Rushen Chahal       33
Margins (Cont…)
   The question is, how can the
    clearinghouse guard itself against this
   Remember it has provided a
    performance guarantee to both the
    parties, and unless it takes adequate
    precautions, it could be saddled with a
                   Prof. Rushen Chahal        34
Margins (Cont…)
   The protection technique is fairly
   Estimate the potential loss for either
    party and collect it in advance.
   Since both the parties have an
    obligation, both can lose in principle.
   Consequently it is necessary to collect
    collateral from both the parties.
                   Prof. Rushen Chahal        35
Margins (Cont…)
   Once the potential loss is collected from
    a party, there is no incentive to default.
   And even if the party that ends up on
    the losing side of the transaction were
    to fail to perform its obligations, the
    clearinghouse is in a position to take
    care of the interests of the other party.
   This is the crux of margining.

                   Prof. Rushen Chahal       36
Marking to Market
   Closely related to the system of
    margining is the concept of marking to
   What is marking to market?
   Now, the reason for collecting margins,
    is to protect both the parties against
    default by the other party.

                  Prof. Rushen Chahal     37
Marking to Market (Cont…)
   The potential for default arises because
    a position once opened, can and
    invariably will lead to a loss for one of
    the two parties if that party were to
    comply with the terms of the futures
   Now this loss will not simply arise all of
    a sudden at the time of expiration of
    the contract. Prof. Rushen Chahal          38
Marking to Market (Cont…)
   As the prices in the market fluctuate
    from trade to trade, one of the two
    parties to an existing futures position
    will experience a loss while the other
    will experience a gain.

                   Prof. Rushen Chahal        39
Marking to Market (Cont…)
   The total loss or gain from the time of
    getting into a position, till the time the
    contract expires or is offset by taking a
    counter-position, whichever were to
    happen first, will be the sum of the
    these small losses/profits corresponding
    to each observed futures price in the
                   Prof. Rushen Chahal       40
Marking to Market (Cont…)
   Marking to Market refers to the process
    of calculating the loss for one party,
    which implies a corresponding gain for
    the other, at specified points in time,
    with reference to the futures price that
    was prevailing at the time the contract
    was previously marked to market.

                   Prof. Rushen Chahal     41
Marking to Market (Cont…)
   When a futures contract is entered into,
    it will be marked to market for the first
    time at the end of that day.
   Subsequently, it will be marked to
    market everyday until either the
    position is offset or else the contract

                   Prof. Rushen Chahal      42
Marking to Market (Cont…)
   Remember that both the parties have
    deposited a performance guarantee or
    collateral in their margin accounts.
   The amount of margin that is deposited
    when the contract is first entered into is
    called the Initial Margin.

                   Prof. Rushen Chahal       43
Margin Accounts
   If a profit is made subsequently, the
    margin account will be credited, while if
    a loss is made, it will be debited.
   An increase in the balance in the
    margin account can be withdrawn by
    the investor.
   However the broker has to ensure that
    the balance in the account does not dip
    below a threshold level due to repeated
    losses.          Prof. Rushen Chahal    44
Margin Accounts (Cont…)
   Otherwise the entire purpose of
    depositing margins will be defeated.
   This threshold is known as the
    Maintenance Margin.
   If the balance dips below this level, the
    trader will get a Margin Call asking for
    additional funds to be deposited, to
    take the balance back to the Initial
    Margin level.
                   Prof. Rushen Chahal      45
   The whole purpose of margining is to
    remove the incentive to default by
    collecting an amount equal to the
    potential loss in advance.
   The pertinent question is therefore,
   `how do we calculate the potential loss
    over a given time horizon?’

                  Prof. Rushen Chahal     46
VaR (Cont…)
   This is where the concept of Value at
    Risk or VaR comes in.
   The Value at Risk of a position is a
    statistical measure of the possible loss
    of value of a portfolio of assets over a
    specified time horizon.

                   Prof. Rushen Chahal         47
VaR (Cont…)
   For instance, the 99% VaR of a
    portfolio over a one day horizon,
    signifies that the portfolio is expected to
    suffer a loss exceeding the calculated
    VaR only with a probability of 1% over
    a one day horizon.
   This does not mean that the maximum
    loss that a portfolio can suffer from one
    day to the next, is equal to the VaR. 48
                     Prof. Rushen Chahal
VaR (Cont…)
   The maximum possible loss of value is
    always equal to the initial value of the
    portfolio, since, in principle, the value of
    the assets constituting the portfolio can
    go to a minimum of zero.
   A given measure of VaR is meaningless
    unless the corresponding probability
    level and time horizon are specified.
                    Prof. Rushen Chahal       49
Spot-Futures Convergence
   Before we go on to look at numerical
    illustrations of the concept of
    margining, it is essential to understand
    a fundamental pricing relationship.

                   Prof. Rushen Chahal         50
Convergence (Cont…)
   At the point in time, at which a futures
    contract is about to expire, the
    prevailing price for a futures position
    should be equal to the price of the
    underlying asset in the cash or spot

                   Prof. Rushen Chahal         51
Convergence (Cont…)
   Symbolically, if we are at T, where T
    denotes the time of expiration of the
    futures contract, we require that
             FT = ST
   where FT is the price of a futures
    contract per unit of the underlying
    asset, and ST is the price of the asset in
    the spot market.

                   Prof. Rushen Chahal       52
Convergence (Cont…)
   What is the rationale for this?
   A futures contract that is about to
    expire is nothing but a contract that
    calls for immediate delivery.
   Thus it is no different from a spot
   Consequently the two prices must be
    equal to preclude arbitrage.

                   Prof. Rushen Chahal      53
   Let us see the consequences if this
    relationship were to be violated.
   Case A: FT > ST
   Let FT = $ 405 per unit and ST = $ 400
    per unit.
   Under such circumstances, an
    arbitrageur will do the following.

                  Prof. Rushen Chahal        54
Arbitrage (Cont…)
   He will acquire the asset in the spot
    market and simultaneously go short in
    the futures market.
   Since by assumption the futures
    contract is about to expire, he will have
    to immediately deliver as per the

                   Prof. Rushen Chahal      55
Arbitrage (Cont…)
   So the transaction entails a purchase at
    $ 400 for an immediate delivery at
    $ 405.
   The difference of $ 5 is obviously an
    arbitrage profit.

                   Prof. Rushen Chahal     56
Arbitrage (Cont…)
   Case B: FT < ST
   Let FT = 395 per unit and ST = 400 per
   An arbitrageur will now do the
   He will short sell the asset in the spot
    market and go long in a futures
                   Prof. Rushen Chahal         57
Arbitrage (Cont…)
   Therefore he will receive $ 400 by way
    of the short sale, which he will
    immediately cover by paying $ 395 to
    acquire the asset under the futures
   The difference of $ 5 is once again an
    arbitrage profit.

                  Prof. Rushen Chahal        58
Illustration of Credit Risk
   Assume that Poonam goes long in a
    futures contract to buy the asset five
    days hence at a price of $ 400, and that
    Kunal takes the opposite side of the
   We will assume that no margin is
    collected, and that neither party quits
    the market prior to the expiration of the
    contract by offsetting.
                    Prof. Rushen Chahal     59
Illustration (Cont…)
   We will first assume that the spot price
    of the asset five days hence is $ 425.
   If Kunal does not have the asset, he will
    have to buy it by paying $ 425.
   It will then have to be delivered for
    $ 400.
   Else if he already owns it, he will have
    to forego the opportunity to sell it for
    $ 425 since he is committed to selling
    at $ 400.       Prof. Rushen Chahal      60
Illustration (Cont…)
   Quite obviously Kunal will default unless
    he has an impeccable conscience and
   Now let us see if on the other hand, the
    spot price at the end of five days were
    to be $ 375.

                   Prof. Rushen Chahal      61
Illustration (Cont…)
   In this case, if Kunal already has the
    asset, he would be delighted to deliver
    it for $ 400.
   For, the alternative is to sell it in the
    spot market for $ 375.
   Else, he will be more than happy to
    acquire the asset for $ 375 and deliver
    it at $ 400.
                   Prof. Rushen Chahal          62
Illustration (Cont…)
   The problem now is that Poonam will
    refuse to pay $ 400 for it.
   There are two ways of looking at it.
   If she does not want the asset, and she
    buys it at $ 400, she will have to
    liquidate it at $ 375.
   Even if she wants the asset, she would
    rather buy it in the spot market for
    $ 375.
                  Prof. Rushen Chahal     63
Illustration (Cont…)
   So once again we have a situation that
    is conducive for default.
   Margins are collected to get over this
   Let us assume that the prices at the
    end of each day during the five day
    period are as follows.

                  Prof. Rushen Chahal        64
Price Sequence
 DAY      PRICE             Daily
  0        400                -
  1        415              +15
  2        405               -10
  3        390               -15
  4        360               -30
  5        375              +15
          TOTAL Chahal
           Prof. Rushen      -25      65
Illustration (Cont…)
   Over a period of five days, Poonam will
    have suffered a loss of $ 25 if she were
    to be forced to go through with the
   This loss of $ 25 did not accrue
   If we look at the last column, the
    overall loss (it could have been a profit
    instead), is nothing but the sum of daily
    gains and losses. Rushen Chahal
                     Prof.                    66
Illustration (Cont…)
   At the end of the first day, Poonam has
    gained $ 15.
   Who has lost $ 15.
   Obviously Kunal.
   At the end of the second day, as
    compared to the end of the first day,
    Poonam has lost $10.

                  Prof. Rushen Chahal     67
Illustration (Cont…)
   Who has gained $ 10?
   Obviously Kunal.
   So Kunal’s gains/losses may be depicted
    as follows.

                  Prof. Rushen Chahal    68
Price Sequence
 DAY      PRICE             Daily
  0        400                -
  1        415               -15
  2        405              +10
  3        390              +15
  4        360              +30
  5        375               -15
          TOTAL Chahal
           Prof. Rushen     +25       69
Illustration (Cont…)
   Kunal has thus gained $ 25 over the
    five day period.
   As you can see, one person’s gain/loss
    is exactly equal to the other person’s
   This is why futures contracts are called

                   Prof. Rushen Chahal         70
Illustration (Cont…)
   Now the purpose of collecting the
    margin is to take away the incentive for
   In this case, if both the parties had
    been forced to deposit an amount equal
    to or exceeding $25 at the outset, there
    would be no scope for default.
   But how do we know how much one
    party will lose over five days?
                   Prof. Rushen Chahal     71
Illustration (Cont…)
   In practice, it is always easier to
    forecast the probable loss over a
    shorter period such as one day than
    over a longer period like five days.
   And in real life, futures contracts often
    have expiration dates that are three
    months or more away.

                   Prof. Rushen Chahal          72
Illustration (Cont…)
   Quite obviously, it is not easy to
    estimate the collective loss over such a
    long period.
   Besides most traders may not remain in
    the market till the end.
   Remember that the vast majority of
    contracts are offset prior to maturity.

                   Prof. Rushen Chahal     73
Illustration (Cont…)
   There are two ways to address this
    situation in practice.
   The probable loss over a one day period
    can be calculated and collected in the
    form of an Initial Margin.

                  Prof. Rushen Chahal    74
Illustration (Cont…)
   At the end of each day, the profit/loss
    for a position will be calculated with
    respect to the futures price that was
    prevailing at the end of the previous
   Gains if any are credited to the margin
    account and can be withdrawn by the
                   Prof. Rushen Chahal        75
Illustration (Cont…)
   The loss if any must be settled by the
    losing party by paying the other party
    before the commencement of trading
    on the following day.
   Thus the account will see either a cash
    inflow or an outflow on a daily basis.

                   Prof. Rushen Chahal        76
Illustration (Cont…)
   If the trader fails to deposit the loss,
    the broker will automatically offset his
    position, thereby ensuring that there
    are no further liabilities.
   This is the essence of the margining
    system that is prevalent in India.

                   Prof. Rushen Chahal         77
Illustration (Cont…)
   In India the initial margin is calculated
    on a real-time basis using the value at
    risk principle.

                    Prof. Rushen Chahal         78
Illustration (Cont…)
   There is another approach to margins
    that is followed in many global
   The initial margin is set at a high
    enough level to cover more than one
    day’s anticipated loss, and is collected
    from both parties at the outset.

                   Prof. Rushen Chahal         79
Illustration (Cont…)
   As trades keep occurring and the
    futures price fluctuates, both the parties
    will keep makes gains and losses.
   The broker will specify a threshold level
    for the balance in the margin account,
    which will be below the initial margin

                   Prof. Rushen Chahal      80
Illustration (Cont…)
   This threshold is called the Maintenance
    Margin level.
   As long as the cumulative loss does not
    cause the account balance to breach
    the maintenance margin level, the
    broker will take no action.

                   Prof. Rushen Chahal     81
Illustration (Cont…)
   However if a party were to incur losses
    which cause the account balance to dip
    below the threshold the broker will
    issue a Margin Call.
   A Margin Call is a call for more margin.

                   Prof. Rushen Chahal     82
Illustration (Cont…)
   The trader must respond by putting
    additional collateral to take the balance
    back up to the initial margin level.
   The additional margin deposited in
    response to a margin call is called the
    Variation Margin.

                   Prof. Rushen Chahal      83
Illustration (Cont…)
   Failure to respond to a margin call will
    lead to the offsetting of the position by
    the broker.

                   Prof. Rushen Chahal          84
Multiple Deliverable Grades
   We have already seen that futures
    prices and spot prices ought to
    converge at expiration.
   That is: FT = ST
   But what would happen if more than
    one grade were to be eligible for

                 Prof. Rushen Chahal     85
Multiple Grades (Cont…)
   If n possible grades can be delivered,
    then each will have its own spot price:
   S1,T , S2,T , S3,T , Sn,T
   The question is, which price will FT
    converge to?
   In the case of multiple deliverable
    grades, one grade will be specified as
    the par grade.
                   Prof. Rushen Chahal        86
Multiple Grades (Cont…)
   If any other grade is delivered, the
    short will receive a premium or a
   This premium or discount may be
    determined multiplicatively or
   In the multiplicative system, if grade i is
    delivered, the short will receive aiFT.
                    Prof. Rushen Chahal       87
Multiple Grades (Cont…)
   For the par grade: ai = 1
   For premium grades: ai > 1
   For discount grades: ai < 1
   Which grade will a short prefer to
   The opportunity cost if grade i is delivered is
   The revenue from delivering grade i is aiFT
                      Prof. Rushen Chahal             88
Multiple Grades (Cont…)
   The short will obviously like to
   aiFT – Si,T
   The preferred graded i is the one for
   aiFT – Si,T > ajFT – Sj,T  j

                   Prof. Rushen Chahal      89
Multiple Grades
   In other words:
   ai[ajFT] – Si,T[Sj,T] > ajFT – Sj,T  j
    --------- ----------
       aj         Sj,T

 ai/aj > Si,T/Sj,T
 Si,T/ai < Sj,T/aj

                      Prof. Rushen Chahal     90
Multiple Grades (Cont…)
   S/a is called the delivery adjusted spot
   Thus the preferred grade for delivery
    will be the one with the lowest delivery
    adjusted spot price.
   This is called the Cheapest to Deliver
    (CTD) grade.

                   Prof. Rushen Chahal         91
Multiple Grades (Cont…)
   To rule out arbitrage, we require that:
   aiFT – Si,T = 0 for the CTD grade.
     FT = Si,T
    Thus the futures price will converge to
    the delivery adjusted spot price of the
    cheapest to deliver grade.
                   Prof. Rushen Chahal        92
Multiple Grades (Cont…)
   Since Si,T/ai < Sj,T/aj for any other grade
    ajFT – Sj,T < 0
   Hence it will not be attractive to deliver
    any other grade.

                    Prof. Rushen Chahal       93
The Additive System
   In this system, if grade i is delivered,
    the short will receive:
   FT + ai
   ai = o for the par grade
   ai > 0 for premium grades
   ai < 0 for discount grades

                    Prof. Rushen Chahal        94
Additive (Cont…)
   The short will try to maximize:
   FT + ai – Si,T
   So the cheapest to deliver grade is the
    one for which:
   FT + ai – Si,T > FT + aj – Sj,T  j
     Si,T – ai < Sj,T – aj  j

                   Prof. Rushen Chahal        95
Additive (Cont…)
   S – a is called the delivery adjusted
    spot price.
   Thus the CTD grade is again the one
    with the lowest delivery adjusted spot
   To rule out arbitrage we require that:
   FT + ai – Si,T = 0
     FT = Si,T - ai
                   Prof. Rushen Chahal       96
Additive (Cont…)
   Once again, the futures price will,
    converge to the delivery adjusted spot
    price of the Cheapest to Deliver Grade.
   Notice that irrespective of whether we
    use multiplicative or additive
    adjustment, the cheapest to deliver
    grade need not be the grade with the
    lowest spot price.
                   Prof. Rushen Chahal        97

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