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# HULL Ch 5 Practice Question 5.20

##### Active Member
In Reference to FIN_PRODS_HULL_CH5_Determination_Of_Forward_and_Futures_Prices_Practice_Question_5.20 :-

I have the following Practice Questions Hull 5.20:-

It states that F0 = S0 * e^(r-q) *T
Should it not be Ft = S0 * e^(r-q) *T instead ..?

Because if F0 = St * e^(r-q) *T , then , the arbitrager makes N* ( F0* e^qT - S0 * e^rT) boils down to
N*[ ( S0 * e^(r-q)T ) * e^qT - S0* e^rT ] => N*[ ( S0 * e^(r -q + q )T - S0* e^rT ] =>
N*[ ( S0* e^r T - S0* e^rT ] = Zero ....

What am I missing here... ? ...

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi @gargi.adhikari I am not sure you are actually missing something: by showing that the would-be arbitrager does not make a profit (or makes a profit of zero), you are demonstrating the "proof" that the F(0) must be equal to S(0)*exp[(r-q)*T], because under this proof the key idea is that F(0) is the price that avoids an arbitrage opportunity; that is, F(0) is the no-arbitrage price because an would-be arbitrage cannot do better than zero profit.

Hull is saying that today she, the arbitrageur, buys one unit (I'll assume N = 1.0 since it drops out anyway) at a cost of -S(0). She can simultaneously sell forward at the forward price, F(0). Please keep in mind that F(0) is a price today that is the guaranteed (predetermined) strike price in the future already; e.g., if F(0) = $13.00 that means that she can enter the forward contract today and guarantee the receipt of$13.00 in the future at the delivery date. For this reason, a more elaborate notation is sometimes used, F(0,T) = \$13.00, to reflect this is a price that is today observed for delivery in the future; i.e., the price today that is guaranteed to be received or paid in (T) years at contract maturity.

... so she can sell forward at F(0); i.e., lock in the cash she will receive in the future. With reinvested income, that means she will cumulative receive F(0)*exp(qT) which, again, is her future cash received. If F(0)*exp(qT) is the future receipt, then in present value terms, this is F(0)*exp(qT)*exp(-rT) = F(0)*exp[(q-r)*T]; discounting at the risk-free rate is justified because there is no uncertainty that she will receive F(0) in the future. So today's S(0) = F(0)*exp[(q-r)*T], or F(0) = S(0)*exp[(r-q)*T]. I hope that helps a little!