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Unformatted text preview: EC3070 FINANCIAL DERIVATIVES FUTURES: MARKING TO MARKET
The holder of a futures contract will be required to deposit with the brokers
a sum of money described as the margin, which will be calculated at a
percentage of the current spot price S0 of the asset. At the end of each
day of trading, the margin will be adjusted to reﬂect the gains or losses of
the contract holder.
Should the cumulated losses reduce the margin to below a certain threshold level, described as the maintenance margin, then extra funds will be
called for to maintain its level.
The process of adjusting the margin account is described as marking to
market. Its eﬀect is to ensure that, at the end of any day of futures trading,
when the daily settlements have been made, there will be no outstanding
obligations.
This will allow the position of the contract holder to be closed without
further losses or gains, thereby virtually eliminating the risk of a default
on the contract.
1 EC3070 FINANCIAL DERIVATIVES Let Fτ t be the value at time t of a futures contract that is to be settled
at time τ , and let δt be the gains or loses realised on day t, which is a sum
that must be paid to the brokers or paid by them to the contract holder.
Day Futures Price Gain or Loss t Fτ  t δt 0
1
2
.
.
.
τ Fτ 0
Fτ  1
Fτ  2
.
.
.
Fτ  τ = Sτ —
δ 1 = F τ  1 − Fτ  0
δ 2 = F τ  2 − Fτ  1
.
.
.
δτ = Sτ − Fτ τ −1 It can be assumed that Fτ t → Sτ as t → τ , which is to say that the
futures price converges to the spot price as the delivery time approaches. 2 EC3070 FINANCIAL DERIVATIVES The settlement on the ﬁnal day, which is δτ = Sτ − Fτ τ −1 , will be a
negligable amount. At that time, the cumulated total of the adjustments
is
τ δ t = S τ − Fτ  0 ,
t=1 which is the diﬀerence between the contract price and the spot price on
the date of delivery.
Observe, however, that the contract could be closed at any time t prior to
the delivery date. Then the cumulated total of the adjustments would be
t δj = Fτ t − Fτ 0 .
j =1 3 EC3070 FINANCIAL DERIVATIVES OPTIONS
Options give their holders rights to buy (if they are call options) or rights
to sell (if they are put options). The party owning the right to buy in a
call option or the right to sell in a put option is on the long side of the
contract. The party with the corresponding liabilities is on the short side
of the contact.
Call
Short
Long Put liability to sell
right to buy liability to buy
right to sell A party who is long on a call option or short on a put option will proﬁt if
prices rise. (They have bullish expectations.)
A party who is short on a call option or long on a put option will proﬁt if
prices fall. (They have bearish expectations.)
4 EC3070 FINANCIAL DERIVATIVES We denote the contract date by t = 0 and the expiry date by t = τ . The
price agreed in the contract is the exercise price denoted Kτ 0 and the spot
price that prevails at the time of expiry is Sτ . The price of a call option,
or its premium, will be denoted by cτ 0
The call option will be exercised only if
S τ > Kτ  0 ,
when it will be worth more than what will be paid for it. The call option
will have proved proﬁtable to the holder only if
Sτ > Kτ 0 + cτ 0 erτ
which is when the value of what is called for exceeds what is paid for it
plus the cost, up to the present time, of holding the option. 5 EC3070 FINANCIAL DERIVATIVES The price of a put option, or its premium, will be denoted by pτ 0 .
The put option will be exercised only if
S τ < Kτ  0 ,
which is when the asset possessed by the option holder is worth less on the
open market that can be claimed for it under the terms of the contract.
The put option will have prove proﬁtable to the holder only if
Kτ 0 > Sτ + pτ 0 erτ ,
which is when the price that is paid for it exceeds its current value plus
the cost, up to the present time, of holding the option. 6 EC3070 FINANCIAL DERIVATIVES Cτ Sτ Cτ
Kτ  0 Sτ
Kτ  0 Figure 1. The proﬁts of a call option written at time t = 0 and with a date
of expiry of t = τ . The premium evaluted at time t = τ is Cτ = cτ 0 erτ , the
strike price is Kτ 0 and the spot price on expiry is Sτ . On the left are the proﬁts
of the option holder (in the long position) and on the right are the proﬁts of the
writer (in the short position). 7 EC3070 FINANCIAL DERIVATIVES Pτ Sτ Pτ
Kτ  0 Sτ
Kτ  0 Figure 2. The proﬁts of a put option written at time t = 0 and with a date
of expiry of t = τ . The value of the premium is Pτ = pτ 0 erτ , the strike price
is Kτ 0 and the spot price on expiry is Sτ . On the left are the proﬁts of the
option holder (in the long position) and on the right are the proﬁts of the writer
(in the short position). 8 EC3070 FINANCIAL DERIVATIVES UPPER BOUNDS FOR OPTION PREMIA
pτ 0 denotes the price at time t = 0 of a put option eﬀective at time τ .
cτ 0 denotes the current price of a corresponding call option.
Kτ 0 denotes the strike price of the options at the time τ of their expiry,
S0 and Sτ are the spot prices at time t = 0 and t = τ , respectively.
The current spot price is an upper bound on the price of a call
option:
cτ 0 ≤ S0 .
By paying cτ 0 , one acquires the opportunity either of owning the stock
at a later date, if Kτ 0 < Sτ , or of foregoing ownership, if Kτ 0 > Sτ .
If cτ 0 ≥ S0 , then one has the possibility of owning the stock for certain
both now and at time τ at the lesser cost of S0 . Therefore, the cost of the
option cannot exceed S0 .
9 EC3070 FINANCIAL DERIVATIVES The value of a put option cannot exceed the present value of the
strike price:
pτ 0 ≤ Kτ 0 e−τ
Either the put option becomes worthless, if Sτ ≥ Kτ 0 , or else it serves to
secure a payment Kτ at time τ , which has a discounted present value of
Kτ e−τ at time t = 0.
If pτ 0 > Kτ e−τ , it would be better to have Kτ e−τ for certain at time
t = 0 instead of spending it on a put option. Therefore, the cost of the
option cannot exceed Kτ e−τ . 10 EC3070 FINANCIAL DERIVATIVES THE PUTCALL PARITY
Portfolio C consists of one call option on a unit of stock, valued at cτ 0
and with has a strike price of Kτ 0 , together with a cash sum of Kτ 0 e−rτ ,
which will yield Kτ 0 when invested at a riskless compound rate of return
of r.
At time τ , the portfolio C will be worth
Kτ 0 + max(Sτ − Kτ 0 , 0) = max(Sτ , Kτ 0 ).
At worst, the option will not be exercised and the funds Kτ 0 will be
retained, whereas, at best, an asset worth Sτ > Kτ 0 will be acquired at
the cost of the strike price of Kτ 0 . 11 EC3070 FINANCIAL DERIVATIVES Portfolio P consists of one put option valued at pτ 0 and a unit of the
stock with a spot price of S0 .
At time τ , the portfolio P will be worth
Sτ + max(Kτ 0 − Sτ , 0) = max(Sτ , Kτ 0 )
At worst, the option will not be exercised and the asset worth Sτ will be
retained. At best, the sum of Kτ 0 > Sτ will be received for the delivery
of an asset worth Sτ .
The values of these two portfolios, which are equal at time τ , must also
be equal at time t = 0. It follows that
cτ 0 + Kτ 0 e−rτ = pτ 0 + S0 .
This is the formula for the put–call parity of European options. 12 ...
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 Spring '12
 D.S.G.Pollock

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