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percentage of the current spot pricea sum of money described as the margin, which will be calculated at a The holder of a futures contract will be required to deposit with the brokers FUTURES: MARKING TO MARKETEC3070 FINANCIAL DERIVATIVES 0 Sof the asset. At the end of each
on the contract.further losses or gains, thereby virtually eliminating the risk of a defaultThis will allow the position of the contract holder to be closed withoutobligations.when the daily settlements have been made, there will be no outstandingmarket. Its effect is to ensure that, at the end of any day of futures trading,The process of adjusting the margin account is described as marking tocalled for to maintain its level.old level, described as the maintenance margin, then extra funds will beShould the cumulated losses reduce the margin to below a certain thresh-the contract holder.day of trading, the margin will be adjusted to reflect the gains or losses of 1
t at time^ τ^ , and let^ δbe the gains or loses realised on dayτ^ |t^ Let^ Fbe the value at time (^) t (^) of a futures contract that is to be settledEC3070 FINANCIAL DERIVATIVES (^) t, which is a sum
Day (^) Futures Pricethat must be paid to the brokers or paid by them to the contract holder. (^) Gain or Loss τ (^) |t t^ F tδ τ (^) | 0 0 F — τ (^) | 1 1 F τ (^) | (^1) τ (^) | 01 δ=^ F− (^) F τ (^) | 2 2 F τ (^) | (^2) τ (^) | 12 δ=^ F− (^) F
τ (^) |τ τ^ F= (^) S τ (^) τ (^) τ (^) τδ= (^) S− (^) F (^) |τ (^) −| 1
It can be assumed that τ (^) |t (^) τ F→ (^) Sas (^) t (^) → (^) τ (^) , which is to say that the
futures price converges to the spot price as the delivery time approaches. 2
Options give their holders rights to buy (if they are OPTIONSEC3070 FINANCIAL DERIVATIVES (^) 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 Put
Short (^) liability to sell (^) liability to buy
Long (^) right to buy (^) right to sell
prices fall. (They have bearish expectations.)A party who is short on a call option or long on a put option will profit ifprices rise. (They have bullish expectations.) A party who is long on a call option or short on a put option will profit if 4
We denote the contract date by (^) t (^) = 0 and the expiry date byEC3070 FINANCIAL DERIVATIVES
(^) t (^) = (^) τ (^). The
price agreed in the contract is the (^) exercise price (^) denoted τ (^) | 0 Kand the (^) spot
price (^) that prevails at the time of expiry is τ S. The price of a call option,
or its premium, will be denoted by τ (^) | 0 c
The (^) call option (^) will be exercised only if τ (^) | 0 τ^ S> K,
will have proved profitable to the holder only if when it will be worth more than what will be paid for it. The call option τ (^) τ (^) | (^0) τ (^) | 0 S> K+ (^) cerτ
plus the cost, up to the present time, of holding the option. which is when the value of what is called for exceeds what is paid for it 5
EC3070 FINANCIAL DERIVATIVES
τ^ -C
τ | 0^ K τ S
K τ^ C
τ | 0 τ S
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
τ (^) τ (^) | 0 t^ =^ τ^ is^ C= (^) c e, therτ
τ (^) | 0 strike price is^ Kand the spot price on expiry is τ S. On the left are the profits
writer (in the short position).of the option holder (in the long position) and on the right are the profits of the 7
EC3070 FINANCIAL DERIVATIVES
τ^ -P
τ | 0^ K τ S
K^ τ^ P
τ | 0 τ S
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 τ (^) τ (^) | 0 P= (^) pe, the strike pricerτ
τ (^) | 0 is^ Kand the spot price on expiry is τ S. (^) On the left are the profits of the
(in the short position).option holder (in the long position) and on the right are the profits of the writer 8
τ (^) | 0 p≤ strike price:^ The value of a put option cannot exceed the present value of theEC3070 FINANCIAL DERIVATIVES τ (^) | 0 Ke−τ
Either the put option becomes worthless, if τ (^) | 0 τ^ S≥^ K, or else it serves to
secure a payment τ Kat time (^) τ (^) , which has a discounted present value of τ (^) | (^0) τ If^ p> Ke, it would be better to have−τ τ^ Keat time (^) t (^) = 0.−τ τ Kefor certain at time−τ
t (^) = 0 instead of spending it on a put option. (^) Therefore, the cost of the
option cannot exceed 10 τ^ Ke.−τ^
Portfolio (^) C (^) consists of one call option on a unit of stock, valued at THE PUT-CALL PARITYEC3070 FINANCIAL DERIVATIVES
τ (^) | 0 c
and with has a strike price of τ (^) | 0 K, together with a cash sum of τ (^) | 0 Ke,−rτ
which will yield (^) K τ (^) | (^0) when invested at a riskless compound rate of return
At time (^) τ (^) , the portfolioof (^) r. τ (^) | (^0) τ (^) | 0 τ K+ max(S− (^) K, (^) 0) = max( (^) C (^) will be worth τ (^) | 0 τ^ S, K).
At worst, the option will not be exercised and the funds
τ (^) | 0 Kwill be
retained, whereas, at best, an asset worth τ (^) τ (^) | 0 S> Kwill be acquired at
the cost of the strike price of 11 τ^ |^0 K.
