MODULE 70.1: PRICING AND VALUATION OF FORWARD CONTRACTS AND FOR AN UNDERLYING WITH VARYING MATURITIES

1. Start with the cleanest distinction in the whole reading: price is what you agreed then; value is what the contract is worth now. At trade date, a fair forward is born at zero value. Nobody pays upfront. But once spot prices move, time passes, or rates shift, that zero-value contract starts behaving like an asset for one side and a liability for the other.
2. For a long forward on an asset with no income and no carry costs, the logic is brutally simple:
3. compare the current spot price today, \(S_t\)
4. with the present value of the old agreed forward price, \(F_0(T)\) If spot is higher than the discounted strike, the long is gaining money.

1. At maturity, the contract stops being abstract and turns into raw settlement:
2. Long forward: $\(V_T(T)=S_T-F_0(T)\)$
3. Short forward: $\(V_T(T)=F_0(T)-S_T\)$ If spot explodes above the agreed price, the long wins. If spot crashes below the agreed price, the short wins.

1. During the life of the contract, rates matter too. Why? Because the old forward price is a cash amount you will pay or receive later, not today. And money due later is never treated at full face value today, because you could invest money between now and then. So we discount that future amount back to today. If the risk-free rate rises, future cash gets discounted more aggressively, which means the present value of that old forward price falls.

1. That means:
2. for the long, a higher discount rate can help because the present value of what it owes falls
3. for the short, a higher discount rate can hurt for the exact same reason This is one of those sneaky exam points where spot stays unchanged but value still moves.

1. If the underlying throws off benefits like income, or has costs of carry, then tweak the spot side before comparing it to the discounted forward price: $\(V_t(T)=\big(S_t-\text{PV(benefits)}+\text{PV(costs)}\big)-F_0(T)(1+r)^{-(T-t)}\)$ Benefits reduce forward value to the long. Costs increase it. Same old intuition: what matters is the economic value of carrying the asset from now to maturity. 7

1. FX forwards are kind of same, but now the asset you are "carrying" is one currency and you are funding that position with another currency. That is why two interest rates show up. If you want to lock in buying one currency in the future, you can think of it as:
2. hold the currency you want

3. fund that by borrowing the other currency So one rate helps you and the other rate hurts you. The forward price is driven by that tug-of-war: $\(F_0(T)=S_0e^{(r_f-r_d)T}\)$ where the price currency's rate and the base currency's rate are competing.

4. The intuition is clean:

5. if the foreign/price currency has the higher interest rate, it tends to trade at a forward discount
6. if the domestic/base currency has the higher interest rate, the foreign currency tends to trade at a forward premium It is just no-arbitrage. If one currency earns more in money-market terms, the forward rate has to adjust so people cannot print free money by borrowing one currency, investing in the other, and locking the exchange rate.

1. The value later is still the same forward logic as before: $\(V_t(T)=S_t-F_0(T)e^{-(r_f-r_d)(T-t)}\)$So even if spot has not moved, the contract value can move because the rate gap moved. Change the carry, and you change the value of the old locked FX rate.

1. Now the term-structure part. Interest rates are annoying because there is no single "the rate." A 1-year rate, 2-year rate, and 3-year rate can all be different. So before you can price forwards on rates, you need zero rates and discount factors.

2. A zero rate is the rate for one single cash flow at one future date. A discount factor is just the present value of 1 unit received later: $\(DF_i=\frac{1}{(1+z_i)^i}\)$ The one-period bond is special: its yield-to-maturity equals its zero rate.

1. Once you have zero rates, you can extract the implied forward rate. This is the breakeven future reinvestment rate that makes two strategies give the same final money:
2. invest short, then roll over later

3. invest long right now and hold

4. The master relationship is: $\((1+z_A)^A(1+\text{IFR}_{A,B-A})^{B-A}=(1+z_B)^B\)$ Read it like this: "Go from today to time B either in one jump, or in two linked jumps. No-arbitrage says both paths must end at the same pile of money."

1. Forward-rate notation is pure decoding:
2. 2y3y = a 3-year rate starting 2 years from now
3. F3m,6m = a 6-month rate starting 3 months from now First number tells you when it starts. Second tells you how long it lasts.

1. Sometimes the cash-market rates use different compounding frequencies, so you must align them before solving for the forward rate. If one rate is quoted with semiannual compounding and the other with quarterly compounding, convert them to the same periodicity first. Otherwise the algebra looks clean but the answer is junk.

2. Now the practical instrument: Forward Rate Agreement (FRA). This is a forward on a future borrowing/lending rate for a notional principal. No actual principal changes hands. Only the interest difference is settled.

3. The easiest way to remember FRA sides:

4. Long FRA / fixed-rate payer: pay fixed, receive floating
5. Short FRA / fixed-rate receiver: receive fixed, pay floating So if you fear rates will rise and you want protection, you go long FRA. Why? Because receiving floating becomes valuable when the actual market rate jumps.

1. The FRA fixed rate is the implied forward rate. That is the no-arbitrage rate that makes the contract worth zero at inception. So when you see an FRA question, step one is often: back out the forward rate from the term structure.

2. At the end of the FRA reference period, the undiscounted net payment from the long FRA perspective is: $\(\text{Net payment}=(R-K)\times \text{Notional}\times \tau\)$ where:

3. R = realized market reference rate
4. K = FRA fixed rate

5. \tau = year fraction of the underlying interest period

6. But FRAs usually settle at the beginning of that interest period, not at the end. So discount that net payment back using the realized floating rate: $\(\text{Cash settlement}=\frac{(R-K)\times \text{Notional}\times \tau}{1+R\tau}\)$

1. Quick intuition on signs:
2. R > K means floating beat fixed, so long FRA wins
3. R < K means fixed was too high, so long FRA loses The short FRA is exactly the opposite.

1. Here is the deep link between spot rates, implied forward rates, and FRAs:
2. zero rates tell you how to discount future cash flows
3. implied forward rates tell you the no-arbitrage future rate between two maturities
4. FRAs turn that implied future rate into a tradeable hedge

1. If you want the shortest mental model possible, use this:
2. ordinary forward on stock: lock future price
3. FX forward: lock future exchange rate
4. FRA: lock future interest rate Same skeleton. Only the underlying changes.

MODULE 71.1: FUTURES VALUATION

LOS 71.a: Compare the value and price of forward and futures contracts.
LOS 71.b: Explain why forward and futures prices differ.

1. Unlike a forward that is settled at a future date in totality, the future price difference is settled every day by Exchange. The exchange marks the price of the contract against the current market price. This is known as Mark to market. For example, if I bought a bitcoin future for $100 and the price of bitcoin futures increase to $110. The next day the $10 would be credited to my variation margin account. Daily variation margin cashflows settle the MTM P&L, so the futures contract is reset to ~0 each day.

2. To understand all three kinds of margin, hammer the table below into your head:

1. How much does my futures value change if my MRR moves by 1bp. That is BPV which is notional times time in years times 0.0001. $\(\text{BPV} = \text{Notional} \times \tau \times 0.0001\)$
2. If interest rates are constant or uncorrelated with futures prices, the pricing is effectively the same.
3. When interest rates are high, futures are more attractive than forwards because I receive cash daily, and hence futures are expensive than forwards. This makes interest rate and futures price positively correlated.
4. With negative correlation → futures are less attractive.

MODULE 72.1: SWAP VALUATION

1. Suppose you want to take a mortgage, but you hate interest-rate exposure. Every time rates jump, your payments can jump. So you find a guy willing to take that risk from you. You tell him: "I will pay you a fixed rate every period. You pay me the floating rate every period." Now your floating-rate loan plus the swap starts behaving much more like fixed-rate borrowing. That is the heart of a swap: you are transferring a stream of interest-rate risk to someone else.

2. How will the other guy price this swap? He will look at the floating payments he expects to make or receive in each future period, using today's forward curve as the market's best guess. Then he will ask: "What single fixed rate would make the total present value of my fixed cash flows exactly match the total present value of those expected floating cash flows?" That single break-even fixed rate is the swap rate. If he charges less than that, he gets underpaid. If he charges more, you are the one getting underpaid.

3. Suppose the swap notional is USD 100,000 for one year, and you are pay fixed, receive floating. At inception, the fair fixed swap rate is 10%, so the contract starts at value 0. That means the market originally thought:
4. your fixed payment in one year would be about USD 10,000

5. the floating payment you receive in one year was also expected to be about USD 10,000 So there was no free lunch for either side.

6. Now imagine rates suddenly jump and the new one-year floating rate becomes 20%. What happens to actual cash settlement if that rate is what sets for the period?

7. you still pay fixed = USD 10,000
8. you now receive floating = USD 20,000

9. net swap settlement to you at the end of the year = USD 10,000 That is the cash-flow side of the story: higher floating rates make life better for a pay-fixed, receive-floating position.

10. Now comes the MTM side. Now Mr. Opportunist walks up to you and says, "Hand me this swap." You laugh and say, "Why would I give it to you for free? This contract lets me pay only 10% fixed while receiving floating linked to a market that is now at 20%. That is gold." So Mr. Opportunist would have to pay you money to take that sweet contract away from you. That payment is the positive MTM value of the swap. MTM is just the cash value of how attractive your old contract has become relative to the new market.

11. Now flip the story by keeping the same position but making rates crash instead of jump. Suppose the new floating rate becomes 5%:
12. you still pay fixed = USD 10,000
13. you now receive floating = USD 5,000
14. net settlement = negative USD 5,000 You are stuck in a net negative contract now. So you walk to Mr. Rescuer's shop and say, "Please take this swap." He says, "Why should I? This contract is awful. I would be stuck paying 10% fixed while receiving floating from a market now at only 5%." So you would have to pay him to take that ugly swap away. That is negative MTM or MTM loss. Same position. Different rate move. Completely different outcome.

1. That fixed swap rate is the price of the swap. It is often called the par swap rate. "Par" just means fair. It is the fixed rate that makes: $\(\text{PV(Fixed leg)}=\text{PV(Floating leg)}\)$ so the swap starts with value 0 to both sides.

2. This is the exact same spirit as forward pricing. In a forward, you choose the forward price so nobody can lock in arbitrage on day one. In a swap, you choose the fixed swap rate so the present value of what each side gives and gets is equal on day one.

3. If you want the formula skeleton, it is: $\(\sum \text{PV(floating payments)}=\sum \text{PV(fixed payments)}\)$ The floating payments are built from implied forward rates. The fixed payments are just the same fixed swap rate repeated across periods and discounted back.

4. So the swap rate is really a multi-period breakeven rate. It is the one fixed rate that makes you indifferent between:

5. paying fixed and receiving the whole future floating path
6. receiving fixed and paying the whole future floating path

1. At each reset date, the current period's settlement from the fixed-rate payer's perspective is: $\(\text{Settlement}=(\text{MRR}-s_N)\times \text{Notional}\times \text{Period}\)$ where MRR is the floating market reference rate that set for that period and \(s_N\) is the fixed swap rate locked at inception.

2. That sign tells the whole story:

3. if MRR > fixed swap rate, the fixed-rate payer wins on that period

4. if MRR < fixed swap rate, the fixed-rate payer loses on that period

5. But a swap's value is not just today's settlement. A swap still has a whole life ahead of it. So on any settlement date: $\(\text{Swap value}=\text{current settlement value}+\text{PV of remaining future settlements}\)$ This is the part students often miss. A swap is alive until the final period dies.

1. Now the big valuation intuition. Suppose you are pay fixed, receive floating. Your fixed payments are locked. Your floating receipts depend on expected future rates. If expected forward rates rise after inception, the floating leg becomes richer while your fixed leg stays the same. That gives you an MTM gain.

2. So:

3. higher expected forward rates help the fixed-rate payer

4. lower expected forward rates help the fixed-rate receiver

5. This is exactly why swap positions line up with bond intuition:

6. receive fixed, pay floating behaves like being long a fixed-rate bond
7. pay fixed, receive floating behaves like being short a fixed-rate bond because the floating-rate note side tends to hug par, while the fixed leg is the duration-heavy piece that really moves when rates change.

1. Time alone can also change swap value, even if the whole forward curve stays exactly where it was. Why? Because as early low-rate settlements roll off and the remaining cash flows are farther along an upward-sloping forward curve, the economics of the leftover swap can change.

2. On an upward-sloping forward curve:

3. a pay-fixed swap can drift into a gain as earlier lower floating resets pass

4. a receive-fixed swap can drift into a loss for the mirror-image reason This is not because the fixed rate changed. It did not. It is because the composition of the remaining floating path changed.

5. Another useful lens is bond replication:

6. pay fixed, receive floating = long FRN + short fixed bond
7. receive fixed, pay floating = long fixed bond + short FRN Since FRNs stay near par when rates move, most of the swap MTM comes from the fixed-bond-like side.

1. Practical use follows immediately:
2. a borrower with floating debt who wants certainty will often pay fixed
3. an investor who wants more duration without buying bonds may receive fixed

4. a manager who wants less duration may pay fixed

5. So when you see a swap question, ask these in order:

6. Who is fixed payer and who is fixed receiver?
7. Where is the current floating rate versus the fixed swap rate?
8. Where did expected future forward rates move after inception?
9. Which side now has the more valuable leg?