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Lesson 2: Forward Contracts I Part 2

Forward Contract on Investment Assets with Known Dollar Income (continued)

Formalize: Forward Contracts on Investment Assets with Known Dollar Income

Forward Price:

Assume I = PV0(the income during life of forward contract)

                   F(0,T) = (S0I )erT                          [continuous]

Proof: Consider two portfolios:

  1. Long forward at forward price F(0,T):
    t = 0: no CF
    t = T: pay F(0,T) & receive the asset
  2. Synthetic Forward: Buy the asset spot with borrowed money
    t = 0: CF0 = + S0 (loan) – S0 (pay for stock) = 0
                      + Asset (assume no storage cost)
    t = T: CFT = - S0 · erT (←loan pmt)
                     + I · erT  (←FV at t=T of income)
  3. Since the two portfolios should be equivalent, we get:
    - F(0,T) = - S0 · erT + I · erT    F(0,T) = (S0I )erT
Valuation of an Existing Forward Position:

Present Value of a Long Forward at t

Note: The following refers to the equation below found in section i. Discrete Compunding
The term, D/(1+r) (T-t), is PV of the dividend.
The term, F(0,T)/(1+r) (T-t), is PV of F(0,T).

  1. Discrete compounding:
    At t = t:   Short sell at St
    At t = T:  Close the existing long forward: Pay F(0,T) and receive one share
                Close the short sell: Return the share & pay div (–D)
    → Vt (0,T)   = St – PV{D + F(0,T)}
                       = St – D/(1+r) (T-t) – F(0,T)/(1+r) (T-t)
  2. Continuous compounding:
         Vt (0,T)  = St - De-r(T-t) – F(0,T)e –r(T-t)
         

Forward on Assets with Known Income:
            F(0,T) = (S0I0 )erT
Value of an existing long forward contract:
            Vt (0,T) = St - It – F(0,T)e –r(T-t)
Where:
            I0 = PV of income from the asset during (0,T) period
            I= PV of income from the asset during (t,T) period

 


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