Forward
Compute forward price of at . Consider the contract that pays at .
Write for the its no-arbitrage price at , then
The Forward price is
Example: if we want to price
Solution:
then,
where the is the terminal value of quotiant of martingales.
The Quotient of martingales:
Futures
Denote the successive days as .
At : the payment of the future is . If enter at and exit at , the total payment is
After marking to market, the price of the future is 0.
Derivation of Fut(t, T)
Discrete time
For the discount factor, we have
which is measurable.
As for the price at :
Therefore, is a martingale under .
Continuous time
is a martingale under and , therefore,
Check: the price of entering or exiting should be zero.
Price at is equal to
The math meaning is that (the martingale representation)
Thus, the price at is
Forward-futures spread
The forward-futures spread is an indicator of correlation between interest rate and the underlying asset.
For the asset , we have
For the futures, we have
The forward-futures spread is
Example
Let the asset be the short rate, i.e. . Assume the Ho-Lee model
The forward price is
The futures price is
Thus
that is,
Portfolios of futures
Example: replicate a call on by trading in futures.
Usual portfolios:
Futures portfolios:
as for the discount portfolio:
which is a martingale.
Black-Scholes:
Replicate by trading in futures
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