Main Content
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) = (S0 – I )erT [continuous]
Proof: Consider two portfolios:
-
Long forward at forward price F(0,T):
t = 0: no CF
t = T: pay F(0,T) & receive the asset -
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) -
Since the two portfolios should be equivalent, we get:
- F(0,T) = - S0 · erT + I · erT ⇒ F(0,T) = (S0 – I )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).
-
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) -
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) = (S0 – I0 )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
It = PV of income from the asset during (t,T) period