Pricing Relationships: Swaps, Options and the Forward Connection
This is Part 2 of our retelling of Chapter 2 from “Trading the Fixed Income, Inflation and Credit Markets: A Relative Value Guide” by Neil C. Schofield and Troy Bowler (Wiley, 2011, ISBN: 978-0-470-74229-7).
Previous: The Relative Value Triangle and Spot Pricing
The Spot-Forward Relationship
In Part 1 we covered how spot prices work. Now let’s connect spot to forward. The idea is straightforward: if someone wants a bond delivered in the future, the quoting bank hedges by buying it now and holding it. The forward price reflects that holding cost.
This strategy is called “cash and carry.” The bank buys the bond today, finances the purchase in the repo market, and earns coupon income while holding it. The difference between borrowing cost and coupon income is the net carry.
Forward clean price = Spot clean price + Net carry
When the bond coupon exceeds the repo rate, carry is “positive” and the forward price sits below the spot price. When it’s the other way around, carry is “negative” and the forward price is higher. This has nothing to do with expectations about price direction. It’s pure math.
For credit markets, forward CDS spreads follow similar logic. You can create a synthetic forward position by buying and selling protection at different maturities. A 10-year CDS at 45 bps and a 5-year at 24 bps gives a breakeven forward spread of 66 bps for the second five-year period.
The Spot-Swap Relationship
Here’s a clean way to think about swaps: a swap where you receive fixed and pay floating is economically similar to buying a fixed-rate bond and issuing a floating-rate note.
If both bonds trade at par, the principal amounts cancel out. What’s left is an exchange of fixed cash flows for floating cash flows. Because a swap mirrors a pair of bonds, you can use bond pricing math to value it.
How Swap Pricing Actually Works
The core idea: on the trade date, a swap must be a fair exchange. Neither side should be starting out at a disadvantage. The present value of the fixed leg must equal the present value of the floating leg. The fixed rate that makes this work is the par swap rate.
You derive it by calculating forward LIBOR rates from zero-coupon swap rates, applying discount factors to all cash flows, and solving for the fixed rate that nets everything to zero.
Why would anyone enter a swap that starts at zero value? Because they have a view on where rates are heading. If you think LIBOR will run hotter than the forward curve implies, you receive LIBOR and pay fixed. If you think LIBOR will lag the forwards, you do the opposite.
Forward-Starting Swaps
A forward-starting swap becomes effective at some point in the future. To price one, you just remove the cash flows before the start date and solve for the new fixed rate that brings the NPV back to zero. The rate will be different from a spot-starting swap of the same maturity.
The Forward-Swap Relationship
Here’s another way to see swaps. An FRA is a single-period interest rate swap. String together a bunch of FRAs with the same fixed rate, and you’ve built a full interest rate swap. The swap rate is the weighted average of those forward rates, with discount factors as weights.
Options: Connecting to the Underlying Market
The authors take a practical approach to option pricing. Rather than deep math, they focus on intuition.
The Dice Game
Imagine a gambler pays you $10 for every point above 7 on a dice roll. To figure out the fair price of playing, you calculate each possible outcome, multiply by its probability, and add everything up. That sum is the expected value. It’s also the fair price of an option.
The same logic applies to real options. Take a call option on crude oil struck at $50. List all possible prices at expiry, figure out the payoff for each, weight by probability, and sum. That’s your fair value.
When the underlying price rises, the call gets more valuable. When volatility increases, the range of outcomes widens and the expected payout goes up. Low volatility? Fewer extreme outcomes, lower premium.
The Mean is the Forward Price
For European-style options, the center of the probability distribution is the forward price, not the spot price. This makes sense because you can’t exercise early. Your reference point should be the price at maturity. Volatility is just one standard deviation of that distribution.
Quick shortcut for pricing ATM options: 0.4 x volatility x sqrt(time) x discount factor. It gives you the premium as a percentage of strike. Simple and surprisingly accurate.
Binomial Pricing and Replication
One of the big insights from option theory: you can replicate an option’s payoff by trading the underlying asset in the right proportions. The cost of this replicating portfolio is the fair value of the option.
In the simplest model, an asset either goes up or down. You calculate a hedge ratio (how much of the underlying to hold) that matches the option payoff in both scenarios. The trader borrows the difference, and the math spits out the premium.
This is called “risk-neutral valuation.” Whether you think the price will go up or down, there’s only one fair price. If the option traded anywhere else, someone could exploit the difference.
Monte Carlo Simulation
When there’s no neat formula, Monte Carlo simulation steps in. It generates thousands of random price paths based on how the asset is expected to grow and fluctuate. Each path gives an expiry price and a payoff. Average all the payoffs, discount back to today, and that’s your option value.
Put-Call Parity
This is one of the most useful concepts in options, and the authors think it’s underrated. The basic relationship:
Long call + Short put = Long forward position
As long as the strike, maturity, and amount match, this holds. Rearrange it however you like to create synthetic positions. Sell a call and buy a put? That’s a short forward. This shows up everywhere in structuring trades.
The OIS Discounting Shift
Before the 2008 financial crisis, everyone discounted swap cash flows using LIBOR. Banks assumed their counterparties wouldn’t default, so LIBOR seemed fine.
The crisis blew that assumption apart. The spread between LIBOR and the overnight rate (OIS) spiked. Suddenly the gap mattered.
Now, collateralized swaps are discounted using the OIS curve instead of LIBOR. The overnight rate is the closest thing to a true risk-free rate: shortest tenor, minimal credit risk, targeted by central banks for monetary policy.
For uncollateralized swaps, banks use their own borrowing cost, which means different banks can disagree on the value of the same swap. This creates practical headaches around cross-currency collateral, cheapest-to-deliver options, and collateral substitution rights.
The key takeaway: the discount rate should match how the swap is collateralized or financed. LIBOR for unfunded trades, OIS for collateralized ones.
How OIS Changes the Math
Under OIS discounting, forward LIBOR rates are derived from the OIS curve rather than the LIBOR curve. The resulting forward rates are slightly different, which means the par swap rate changes too. In-the-money swaps gain value under OIS discounting (as long as the LIBOR-OIS basis stays positive), while out-of-the-money swaps lose value.
Monetary Policy and Overnight Rates
The appendix covers how central banks manage overnight rates. The Bank of England uses the Bank Rate, a reserves averaging scheme, and operational standing facilities. The ECB uses a reserve requirement and weekly refinancing operations. These mechanisms keep overnight interbank rates within a corridor, and those rates feed directly into OIS curves used for swap pricing.
That completes Chapter 2. The big picture: every instrument in the fixed income world connects to every other through mathematical relationships. Spots price into forwards, forwards build into swaps, and options link back to the underlying through replication and put-call parity. Understanding these connections is how you find the best way to express any market view.