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Business Math Forward Contract Valuation

nickoh

New member
Aug 16, 2013
4
Hi all,

I'm having issues with a question regarding forward contract values.

Basically here is the question:

The risk free rate is 10%
Underlier is currently trading at \$100
It is expected to trade at either \$90 or \$120 at the end of the period.
The forward asset price in the contract is \$110

I need to find the no-arbitrage value of a forward contract on the underlier.

----

I'm stumped for a number of reasons. I can't seem to work out how to deal with the two probabilities of the end of period prices (\$90 and \$120).

I get that 10% x 100 = \$110, which is the risk-free growth expected at the end of the period.

I believe that to find the value of the forward contract I would do this:

Traded value at end of period - Actual value at end of period.

How do I go about doing this question? I literally can't even get a start. I'm looking at theory from my book, but it doesn't seem to deal with multiple trading price probabilities.

Any help would be greatly appreciated.
 

springfan25

New member
Mar 3, 2012
14
the "no arbitrage price" is the price of a hypothetical portfolio which has the same payoffs as the contract you are trying to value.

To work it out you will follow 3 steps:
1) Work out the payout from the forward contract in each of the 2 states ("high underlier price" and "low underlier price"

2) Work out a "replicating portfolio" which has the same payouts as the forward contract in both states

3) Note that the price of the "replicating portfolio" must be the no-arbitrage price of the forwar contract.

Step 1:
You said "The forward gives an asset price of £100". That doesn't mean anything to me, but I assume it gives you an obligation to sell the underlier at £100. The payoff from the forward in the two states is therefore:

"Low" underlier price: Payoff = £100 - £90 = £10
"High" underlier price: Payoff = £100 - £120 = -£20


Step 2
We will construct a portfolio of cash (x) and the underlier (y) which has the same payoffs as above. In 1 period, the cash will accumulate to 1.1x. The underlier will be worth y * price.

Low scenario: 1.1x + 80y = 10
High scenario: 1.1x + 120y = -20


Solve these simultaneously to get
y=0.25
x= -9.0909

T he current price of the underlier is £100. So the value of the replicating portfolio today is 100y + x= 0.25*100 - 9.09 = £15.909

Step 3
Hence the answer is £15.91

(comment: perhaps the forward is an obligation to buy the underlier rather than sell it, then the answer would be positive).