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Lesson 2: Forward Contracts I Part 1

What Should be an Equilibrium Forward Price? (continued)

Forward Price can be determined using an arbitrage argument:
  • If two portfolios have identical cash flows (equivalent), the arbitrage-free condition requires that the two portfolios must be priced the same.
  • Forward Prices reflect Costs and Benefits:
    (Forward Price)
    = (Spot Price) + FV of {(cost - benefits) for deferred transaction}
Example 2.3. continued (Arbitrage-free and Forward Price, F0):

Arbitrage-free condition will require the gold forward price to be equal to the synthetic gold forward cost:
F0 = FSYNTHETIC (= $1,040)

What if a dealer quotes a gold forward price of F0 = $1,100 (higher than $1,040)?

Click on Example 2.3. Solution with gold forward price of F0= $1,100 (higher than $$1,040) to view the solution.

Example 2.3. Solution with gold forward price of F0 = $1,100 (higher than $1,040):

At t = 0: Short forward + Long synthetic forward

  1. Short forward at F0 = $1,100
  2. Borrow S0 = $1,000 at 6%         +$1,000
  3. Buy gold spot at S0 = $1,000     -$1,000
  4. Lease the gold at 2%
    → Net CF0 = zero; no gold in possession
At t = 1 (one year later):
  1. Pay back the loan and accrued interest = - $1,000 · 1.06 = - 1,060
  2. Receive accrued fee on gold lease: 1,000 · 2% = $20
  3. Close the short forward position:
        Deliver gold for F0 = $1,100     +$1,100
  4. Net Profit                                      +$60
    This is a risk-free profit with no initial investment - pure arbitrage profit!
    As long as F0 > FSYNTHETIC, this strategy will yield an arbitrage profit, driving F0 down to FSYNTHETIC

What if the dealer quotes a gold forward price of F0 = $1,020?

Click on Example 2.3. Solution with gold forward price of F0= $1,020 to view the solution.

Example 2.3. Solution with gold forward price of F0 = $1,020:

At t = 0: Long forward + short synthetic forward

  1. Long forward at F0 = $1,020
    → Net CF0 = zero; no gold in possession
  2. Borrow gold at 2%
  3. Sell gold spot at $1,000         +$1,000
  4. Deposit $1,000 at 6%             -$1,000

At t = 1 (one year later):

  1. Close the long forward position:
    Accept delivery of gold for F0 = $1,020              ($1,020)
  2. Return borrowed gold with accrued fee:              ($20)
  3. Get back the deposit with interest, S0 · (1+6%)   +$1,060
  4. Net Profit                                                             +$20
    This is a risk-free profit with no initial investment - pure arbitrage profit!
    As long as F0 < FSYNTHETIC, this strategy will yield an arbitrage profit, driving F0 upward to FSYNTHETIC

Wrap-up: The arbitrage-free condition establishes the forward price as F0 = (spot price) + (interest cost) - (gold lease fee), i.e.

F0 = (spot price) + FV of {(cost - benefits) for deferred transaction}
    = S0 (1 + (R0 - I0)*t)  or = S0 (1 + R0 - I0)t                  (discrete compounding)
     = S0 EXP{(R0 - I0) · t}           (continuous compounding)

 


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