MHB Valuing a Forward Contract with Multiple Trading Price Probabilities

[nickoh]

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]

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 = -£20Step 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 = -20Solve 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).