Pricing and Valuation of Forward Commitments

This reading on forward commitment pricing and valuation provides a foundation for understanding how forwards, futures, and swaps are both priced and valued.

Key points include the following:

• The arbitrageur would rather have more money than less and abides by two fundamental rules: Do not use your own money, and do not take any price risk.

• The no-arbitrage approach is used for the pricing and valuation of forward commitments and is built on the key concept of the law of one price, which states that if two investments have the same future cash flows, regardless of what happens in the future, these two investments should have the same current price.

• Throughout this reading, the following key assumptions are made:

  • Replicating and offsetting instruments are identifiable and investable.

  • Market frictions are nil.

  • Short selling is allowed with full use of proceeds.

  • Borrowing and lending are available at a known risk-free rate.

• Carry arbitrage models used for forward commitment pricing and valuation are based on the no-arbitrage approach.

• With forward commitments, there is a distinct difference between pricing and valuation. Pricing involves the determination of the appropriate fixed price or rate, and valuation involves the determination of the contract’s current value expressed in currency units.

• Forward commitment pricing results in determining a price or rate such that the forward contract value is equal to zero.

• Using the carry arbitrage model, the forward contract price (F₀) is:

  F₀ = FV(S₀) = S₀(1 + r)^T (assuming annual compounding, r)

  F 0 = FV ( S 0 ) = S 0 exp r c T (assuming continuous compounding, r_c )

• The key forward commitment pricing equations with carry costs (CC) and carry benefits (CB) are:

  F₀ = FV[S₀ + CC₀ – CB₀] (with discrete compounding)

  F 0 = S 0 exp ( r c + CC − CB ) T (with continuous compounding)

Futures contract pricing in this reading can essentially be treated the same as forward contract pricing.

• The value of a forward commitment is a function of the price of the underlying instrument, financing costs, and other carry costs and benefits.

• The key forward commitment valuation equations are:

  Long     Forward:   V t = PV [ F t − F 0 ] = [ F t − F 0 ] ( 1 + r ) T − t

  and

  Short     Forward:   − V t = PV [ F 0 − F t ] = [ F 0 − F t ] ( 1 + r ) T − t ,

  With the PV of the difference in forward prices adjusted for carry costs and benefits. Alternatively,

  Long     Forward:   V t = S t − P V [ F 0 ] = S t − F 0 ( 1 + r ) T − t

  and

  Short   Forward: − V t = PV [ F 0 ] − S t = F 0 ( 1 + r ) T − t − S t

• With equities and fixed-income securities, the forward price is determined such that the initial forward value is zero.

• A forward rate agreement (FRA) is a forward contract on interest rates. The FRA’s fixed interest rate is determined such that the initial value of the FRA is zero.

• FRA settlements amounts at Time h are:

  Pay-fixed (Long): NA × {[L_m – FRA₀] t_m}/[1 + D_mt_m] and

  Receive-fixed (Short): NA × {FRA₀ – L_m] t_m}/[1 + D_mt_m].

• The FRA’s fixed interest rate (annualized) at contract initiation is:

  FRA₀ = {[1 + L_Tt_T]/[1 + L_ht_h] – 1}/t_m.

• The Time g value of an FRA initiated at Time 0 is:

  Long FRA: V_g = NA × {[FRA_g – FRA₀] t_m}/[1+ D_(T–g) t_(T–g)] and

  Short FRA: –V_g = NA × {[FRA₀ – FRA_g] t_m}/[1+ D_(T–g) t_(T–g)].

• The fixed-income forward (or futures) price including conversion factor (i.e., adjusted price) is:

  F₀ = Q₀ × CF = FV[S₀ + CC₀ – CB₀] = FV[B₀ + AI₀ – PVCI],

  and the conversion factor adjusted futures price (i.e., quoted futures price) is:

  Q₀ = [1/CF] {FV [B₀ + AI₀] – AI_T – FVCI}.

• The general approach to pricing and valuing swaps as covered here is using a replicating portfolio or offsetting portfolio of comparable instruments, typically bonds for interest rate and currency swaps and equities plus bonds for equity swaps.

• The swap pricing equation, which sets r_FIX for the implied fixed bond in an interest rate swap, is:

  r F I X = 1 − PV n ( 1 ) ∑ i = 1 n PV i ( 1 ) .

• The value of an interest rate swap at a point in Time t after initiation is the sum of the present values of the difference in fixed swap rates times the stated notional amount, or:

  V S W A P , t = NA   ×   ( FS 0 − FS t )   × ∑ i = 1 n PV i (Value of receive-fixed swap)

  and

  − V S W A P , t = NA × ( FS t − FS 0 ) × ∑ i = 1 n PV i (Value of pay-fixed swap).

• With a basic understanding of pricing and valuing a simple interest rate swap, it is a straightforward extension to pricing and valuing currency swaps and equity swaps.

• The solution for each of the three variables, one notional amount (NA_a) and two fixed rates (one for each currency, a and b), needed to price a fixed-for-fixed currency swap are :

  NA a = S 0 × NA b;   r a = 1 − PV n , a ( 1 ) ∑ i = 1 n PV i , a ( 1 )  and  r b = 1 − PV n , b ( 1 ) ∑ i = 1 n PV i , b ( 1 ) .

• The currency swap valuation equation, for valuing the swap at time t (after initiation), can be expressed as:

  V C S = NA a ( r F i x , a ∑ i = 1 n PV i ( 1 ) + PV n ( 1 ) ) − S t NA b ( r F i x , b ∑ i = 1 n PV i ( 1 ) + PV n ( 1 ) ) .

• For a receive-fixed, pay equity swap, the fixed rate (r_FIX) for the implied fixed bond that makes the swap’s value (V_EQ) equal to “0” at initiation is:

  r F I X = 1 − P V n ( 1 ) ∑ i = 1 n P V i ( 1 ) .

• The value of an equity swap at Time t (V_EQ,t), after initiation, is:

  V_EQ,t = V_FIX(C₀) – (S_t/S_t–1)NA_E – PV(Par – NA_E)

where V_FIX (C₀) is the Time t value of a fixed-rate bond initiated with coupon C₀ at Time 0, S_t is the current equity price, S_t–1 is the equity price at the last reset date, and PV() is the PV function from the swap maturity date to Time t.