Refresher lectures
Forwards and futures are not options, since they are a binding contract in which the seller of the
contract (short position) agrees to sell the underlying at a pre-determined price and date in the
future and the buyer of the contract (long position) agrees to buy the underlying under those
specifications.
No-arbitrage pricing forwards comes down to forwards being priced such that there are no remaining
arbitrage opportunities. This results in the following theorem, with the 𝑟 being the risk-free rate.
Under no-arbitrage pricing, the value of a forward contract at time 𝑡 = 0 is set in the market at 0, by
definition. This general principle of pricing by no arbitrage relates to the following:
When we want to value forwards we can use the following theorem:
,Put into practice this would result in:
Futures are essentially forwards but a standardized form of a forward. They are characterized by:
,Futures and forward prices should coincide, since they are essentially the same type of products. If
this would not be the case there would be arbitrage opportunities. However there are some
considerations that lead to futures having a different price as opposed to forwards. For example
them marking to market in futures leads to a large difference.
Since with futures there is marking to market and thus the counterparties need to make payments to
each other at each time interval relating to the value of the contract, there are some considerations.
For example if there is a positive correlation between the underlying and the short-term interest rate
then:
However if there is actually a negative correlation between the underlying and the short-term
interest rate we get exactly the opposite situation:
, Options are contracts that give the purchaser a certain option in the future, but not an obligation. On
the other hand the seller of the option always is obliged to conform to the contract if the buyer
exercises the option. In a call option the buyer has the option to buy the underlying in the future at a
pre-determined price and in a put option he has the option to sell the underlying. This information
leads to the following results:
Forwards and futures are not options, since they are a binding contract in which the seller of the
contract (short position) agrees to sell the underlying at a pre-determined price and date in the
future and the buyer of the contract (long position) agrees to buy the underlying under those
specifications.
No-arbitrage pricing forwards comes down to forwards being priced such that there are no remaining
arbitrage opportunities. This results in the following theorem, with the 𝑟 being the risk-free rate.
Under no-arbitrage pricing, the value of a forward contract at time 𝑡 = 0 is set in the market at 0, by
definition. This general principle of pricing by no arbitrage relates to the following:
When we want to value forwards we can use the following theorem:
,Put into practice this would result in:
Futures are essentially forwards but a standardized form of a forward. They are characterized by:
,Futures and forward prices should coincide, since they are essentially the same type of products. If
this would not be the case there would be arbitrage opportunities. However there are some
considerations that lead to futures having a different price as opposed to forwards. For example
them marking to market in futures leads to a large difference.
Since with futures there is marking to market and thus the counterparties need to make payments to
each other at each time interval relating to the value of the contract, there are some considerations.
For example if there is a positive correlation between the underlying and the short-term interest rate
then:
However if there is actually a negative correlation between the underlying and the short-term
interest rate we get exactly the opposite situation:
, Options are contracts that give the purchaser a certain option in the future, but not an obligation. On
the other hand the seller of the option always is obliged to conform to the contract if the buyer
exercises the option. In a call option the buyer has the option to buy the underlying in the future at a
pre-determined price and in a put option he has the option to sell the underlying. This information
leads to the following results:
