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2. forward contract
a. K is the delivery price of underlying assets on the forward, and St is the price of underlying assets at maturity time T.
E[(exp(-r(T-t))(ST-K)]=0, K=E(ST);

b. delivery price=forward price=spot price+cost of fund+storage cost-dividend of holding assets
*cost of fund: the forward writer spend some money in buying the underlying assets at spot price and lose the deposit interest of the money.*cost of carry=cost of fund+storgae cost-dividends

c. value and price of a forward contractf-forward value,F(s,T-t) is the forward price,k is the delivery price, S is the spot price at time 0. B(T-t) is the discount rate.
f=S-kB(T-t)
The forward price equals the delivery price, which makes forward value zero. F(S,T-t)=k=S/B(T-t)=spot price +cost of fund
*S/B(T-t) is the future price of S, which equals to S(1+interest rate)=spot price+cost of fund.
*with time passing by, the f !=0.

d. Take dividend into account
Discrete dividend paying asset
D is the present value of all dividens of forward holding period.
f=S-kB(T-t)-D
k makes f=0, k=F=(s-D)/B(T-t)
e. cost of carry
U is the other cost of carring the assets.
F=(S+U)exp(r(T-t))   F=S*exp[(r+u)*(T-t)]   F=S*exp[b(T-t) , where b=r+u-q    .  q is the dividend yield.
Discrete carrying costs
F=S/d(0,M)+∑c(k)/d(k,M)     discount rate^-1 calculates the future price of  S, c(k)
d(0,k)d(k,M)=d(0,M)  The price at time M discount to time 0 equals the price firstly discount to time k and then to time 0.
Eo[d(k,M)]=B(0,M)/B(0,k)

thx~

f.proportonal carring charge

the carrying charge in period k is qS(k-1).

F=S(0)/[(1-q)^M]*d(0,M)
1/d(0,M) and 1/(1-q)^m can be seen  as e^r(T-t) and e^u(T-t) * get S(0)/(1-q)^M units assets at the first, sell (1-q) portion of assets to pay for the carring cost. After M periods, holder own S(0) units of assets. f+ kd(0,m)=S/(1-q)^M