derivatives 5: Pricing and Valuation of Forward Contracts and for an Underlying with Varying Maturities

What is the initial value of a forward contract at inception?

$$V_0(T) = 0$$

the contract is initially structured so neither side has an advantage

its value is zero at trade initiation.

What is the difference between forward pricing and forward valuation?

• Pricing determines the forward price $F_0(T)$ at initiation.

• Valuation determines the contract’s mark-to-market (MTM) value $V_t(T)$ after initiation.

What determines the forward price at initiation?

• Spot price $S_0$

• Risk-free rate $r$

• Time to maturity $T$

• Costs and benefits of ownership (cost of carry)

What is the value of a long forward contract at maturity?

$$V_T(T) =S_T - F_0(T)$$

buyer profits when S_T > F_0(T) (i.e spot price at maturity > forward price you pay)

What is the value of a short forward contract at maturity?

$$V_{T}(T)=F_0\left(T\right)-S_{T}$$

buyer profits when S_{T}<F_0(T) (i.e spot price at maturity < forward price you pay)

outcome for long and short poisitions for

1. S_T > F₀(T)

2. S_T < F₀(T)

3. S_T = F₀(T)

Outcome	VT (T) (long position)	VT (T) (short position)
ST > F0(T)	ST − F0(T) > 0	F0(T) − ST < 0
ST < F0(T)	ST − F0(T) < 0	F0(T) − ST > 0
ST = F0(T)	ST − F0(T) = 0	F0(T) − ST = 0

What is mark-to-market (MTM) value?

MTM value is the current gain or loss on the contract if it were settled immediately.

What causes the value of a forward contract to change over time?

• Changes in spot price

• Passage of time

• Changes in interest rates

• Changes in costs/benefits of carry

What is the present value of the forward price during the life of the contract?

$$PV_{t}(F_0(T))=F_0(T)\left(1+r\right)^{-\left(T-t\right)}$$

What is the MTM value of a long forward contract during the life of the contract?

$$V_{t}(T)=S_{t}-F_0(T)(1+r)^{-\left(T-t\right)}$$

What is the MTM value of a short forward contract during the life of the contract?

$$V_{t}(T)=F_0(T)(1+r)^{-\left(T-t\right)}-S_{t}$$

Short and long position MTM gain or loss for

1. S_t > F₀(T)(1 + r)^{-(T - t)}

2. S_t < F₀(T)(1 + r)^{-(T - t)}

3. S_t = F₀(T)(1 + r)^{-(T - t)}

Outcome	Vt(T) (long position)	Vt(T) (short position)
St > F0(T)(1 + r)-(T - t)	MTM gain	MTM loss
St < F0(T)(1 + r)-(T - t)	MTM loss	MTM gain
St = F0(T)(1 + r)-(T - t)	No MTM gain or loss	No MTM gain or loss

What is the forward pricing formula when ownership benefits and costs exist?

$$F_0\left(T\right)=\left(S_0+PV_{t}\left(C\right)-PV_{t}\left(I\right)\right)\left(1+r\right)^{T}$$

What is the MTM valuation formula for a long forward with costs and benefits?

$$V_{t}(T)=\left(S_{t}+PV_{t}\left(C\right)-PV_{t}\left(I\right))-F_0(T\right)(1+r)^{-\left(T-t\right)}$$

$PV_{t}(C)$ = The present value at time t of remaining ownership costs before maturity.

$PV_{t}(I)$ = The present value at time t of remaining income or benefits before maturity.

What is the no-arbitrage FX forward pricing formula?

$$F_{0,f/d}(T)=S_{0,f/d}e^{(r_f -r_d)t}$$

Which currency trades at a forward premium/discount?

• premium: currency with the lower interest rate

• discount: currency with the higher interest rate

What is the MTM valuation formula for an FX forward?

$$V_{t}\left(T\right)=S_{0,f/d}-F_{0,f/d}(T)e^{-(r_{f}-r_{d})\left(T-t\right)}$$

What is the key difference between interest rate forwards and forwards on equities/commodities?

Interest rates have a term structure (different rates for different maturities)

• pricing depends on multiple spot/zero rates rather than one constant risk-free rate.

What is a zero rate (spot rate)?

The yield-to-maturity on a zero-coupon bond for a specific maturity.

What does a discount factor represent?

The present value (price today) of receiving 1 currency unit in the future.

What is bootstrapping?

The process of deriving zero (spot) rates from coupon bond prices using forward substitution.

Boot strapping method

1. solve for first year zero rate $z_1$

• $$PV_{}=\frac{FV}{1_{}+z_1}$$

2. use $z_1$ to find $z_2$ and so on

• $$PV=\frac{PMT}{1+z_1}+\frac{PMT+FV}{\left(1+z_2\right)^2}$$

this comes from $PV=\sum^{}\frac{CF_{t}}{\left(1+z_{t}\right)^{t}}$

how to solve YTM

use grey buttons on calc

1. N = number of periods

2. PV = -PV given

3. PMT = (interest * FV) → annual so adjust if not annual by dividing

4. FV = face value

5. CPT I/Y

What is the formula for a discount factor?

$$DF_{i}=\frac{1}{\left(1+z_{i}\right)^{i}}$$

z_i is the zero rate for period i.

What is an implied forward rate (IFR)?

The future interest rate implied by current spot (zero) rates that prevents arbitrage.

What does a “2y3y” forward rate mean?

A 3-year interest rate beginning 2 years from today (runs from year 2 to year 5).

What does $IFR_{1,1}$​ represent?

A 1-year forward rate beginning 1 year from today.

What is the intuition behind implied forward rates?

Two investment strategies must produce the same return:

1. Invest short-term and reinvest later

2. Invest long-term immediately

What is the general implied forward rate formula?

$$\left(1+z_{A}\right)\left(1+IFR_{A,B-A}\right)^{B-A}=\left(1+z_{B}\right)^{B}$$

if long-term spot rates are above short-term spot rates, what happens to forward rates?

Forward rates will generally be higher than current short-term rates.

What does the forward curve show?

future interest rates implied by today’s spot curve.

What does the spot (zero) curve show?

shows current zero rates for different maturities.

what is the rate equivalency formula?

$$\left(1+\frac{APR_{m}}{m}\right)^{m}=\left(1+\frac{APR_{n}}{n}\right)^{n}$$

$APR_m =$ annual percentage rate for m periods per year

$APR_{n}=$ annual percentage rate for n periods per year

used to covert rates eg. annual to semi annual

What is a Forward Rate Agreement (FRA)?

An OTC contract where parties agree on an interest rate today for a future borrowing/lending period.

What is the underlying in an FRA?

• Hypothetical deposit

• Based on a market reference rate (MRR)

In an FRA, what does the FRA buyer (long position) do?

• Pays fixed rate

• Receives floating MRR

Used to hedge rising rates on future liabilities.

FRA characteristics

• No exchange of notional principa → used to calc interest payments

• Settled on a net basis

• Similar to a one-period interest rate swap

• Initial value: $V_0(T)=0$ under no-arbitrage conditions.

What is the FRA settlement formula (from fixed-rate payer perspective)?

Net Payment = ($MRR_{B-A} − IFR_{A,B-A})$ × Notional × Period