Term structure models, credit analysis, Black-Scholes-Merton, binomial trees, option Greeks, and MBS analysis for CFA Level II.
Definition first
This guide is designed for first-pass understanding. Start with core terms, then apply the framework in your own account workflow.
Fixed income and derivatives at CFA Level II represent a massive step up from Level I. You move from basic bond pricing to arbitrage-free valuation frameworks, term structure models, credit default swap mechanics, and options pricing using both binomial trees and Black-Scholes-Merton. This guide covers every major model and concept, including the option Greeks, interest rate derivatives, and mortgage-backed securities analysis that dominate the exam.
Term Structure and Interest Rate Models
At Level I, you learned through the fixed income fundamentals that bonds are priced by discounting cash flows at a single yield to maturity. Level II replaces this simplification with the term structure of interest rates — the full set of spot rates for every maturity. The term structure is the foundation for all advanced fixed income analysis because it recognizes a fundamental reality: the market charges different rates for different maturities.
The spot rate (or zero-coupon rate) for maturity T is the yield on a bond that pays no coupons and returns its face value at time T. A 2-year spot rate of 4% means the market demands 4% per year for lending money for exactly two years with no intermediate payments. The set of spot rates across all maturities forms the spot rate curve (or zero curve).
Forward rates are the rates implied by the spot curve for future lending periods. The one-year forward rate one year from now, denoted f(1,1), is the rate you would lock in today for borrowing from year 1 to year 2. Forward rates are derived from spot rates using the no-arbitrage condition: (1 + S₂)² = (1 + S₁)(1 + f(1,1)), where S₁ and S₂ are the 1-year and 2-year spot rates.
Three major theories explain the shape of the term structure:
Pure expectations theory: Forward rates are unbiased predictors of future spot rates. An upward-sloping curve means the market expects rates to rise. This theory has theoretical elegance but poor empirical support.
Liquidity preference theory: Investors demand a premium for holding longer-maturity bonds because they carry more interest rate risk. This premium causes the yield curve to slope upward even when rate expectations are flat. Most practitioners accept some version of this theory.
Segmented markets theory: Different investor groups operate in different maturity segments (banks in short maturities, pension funds in long maturities), and supply/demand within each segment determines rates independently.
Arbitrage-Free Valuation Framework
The arbitrage-free valuation framework is the cornerstone of Level II fixed income. The core principle is that any bond can be decomposed into a portfolio of zero-coupon bonds, and the price of the bond must equal the sum of the present values of those zeros. If it does not, an arbitrage opportunity exists.
To value a bond using this framework, you discount each cash flow at the corresponding spot rate:
Bond Price = C/(1+S₁) + C/(1+S₂)² + ... + (C+FV)/(1+Sn)n
This approach yields a different (and more accurate) price than discounting all cash flows at a single YTM. The difference is most pronounced when the yield curve is steeply sloped.
The binomial interest rate tree extends this framework to value bonds with embedded options (callable bonds, putable bonds). The tree models how interest rates can evolve over time, with rates moving up or down at each node. Key characteristics of the binomial tree used in the CFA curriculum:
It is calibrated to be arbitrage-free: it prices benchmark bonds correctly
Adjacent nodes are related by a factor of e2σ, where σ is interest rate volatility
Risk-neutral probabilities of 0.5 for up and down moves are used at each node
Backward induction is used to determine the bond's value at each node
For a callable bond, at each node you check whether the issuer would exercise the call (i.e., whether the bond value exceeds the call price). If so, the bond value at that node is capped at the call price. For a putable bond, the bondholder exercises if the bond value falls below the put price, so the value is floored at the put price.
The option-adjusted spread (OAS) is the constant spread added to the benchmark spot rates in the binomial tree that makes the model price equal to the market price. OAS strips out the value of embedded options, allowing you to compare bonds with different option features on an apples-to-apples basis. A bond with a higher OAS offers more compensation for credit risk per unit of interest rate risk.
Credit Analysis Models
Level II introduces two major frameworks for modeling credit risk: structural models and reduced-form models. Both aim to estimate the probability and impact of default, but they approach the problem from fundamentally different angles.
Structural Models (Merton Model)
The Merton model treats a company's equity as a call option on its assets. The intuition is powerful: equity holders have a claim on the company's assets after debt holders are paid. If the value of assets exceeds the face value of debt at maturity, equity holders receive the residual. If assets fall below debt, equity holders get nothing (limited liability) and debt holders take a loss.
Formally: Equity = max(VA − D, 0), where VA is the value of assets and D is the face value of debt. This is the payoff of a European call option on the firm's assets with a strike price equal to D. Conversely, risky debt can be viewed as risk-free debt minus a put option on the firm's assets.
Key implications of the Merton model:
Credit spreads increase as leverage increases (more debt relative to assets)
Credit spreads increase as asset volatility increases
Credit spreads decrease as the risk-free rate increases (because higher rates increase the expected growth of assets)
Default occurs only at maturity (a simplification that is addressed by extensions like the Black-Cox model)
Reduced-Form Models
Reduced-form models do not attempt to model the firm's asset value. Instead, they treat default as a random event governed by a hazard rate (default intensity) that can change over time based on observable variables like credit ratings, macroeconomic conditions, and market prices.
The key advantage of reduced-form models is that they can be calibrated to market prices (CDS spreads, bond spreads) and can accommodate default at any time, not just at debt maturity. They are more practical for day-to-day risk management and pricing because they use observable market data rather than requiring estimates of unobservable asset values and volatilities.
Feature
Structural Models
Reduced-Form Models
Default mechanism
Assets fall below debt threshold
Random event driven by hazard rate
Default timing
Only at debt maturity (basic model)
Any time (Poisson process)
Key inputs
Asset value, asset volatility, debt level
Hazard rate, recovery rate, market spreads
Calibration
Difficult (unobservable asset values)
Straightforward (market prices)
Strengths
Economic intuition, link to balance sheet
Practical, flexible, market-consistent
Credit Default Swaps (CDS)
A CDS is essentially an insurance contract on a bond issuer's credit risk. The protection buyer pays a periodic premium (the CDS spread) to the protection seller. If the reference entity defaults, the protection seller compensates the buyer for the loss.
CDS pricing is built on the relationship between the CDS spread, probability of default, and loss given default (LGD):
CDS Spread ≈ Probability of Default × Loss Given Default
More precisely, the CDS spread is set so that the present value of premium payments (premium leg) equals the expected present value of default payouts (protection leg). The hazard rate from the reduced-form model drives the calculation of both legs.
Key CDS concepts tested at Level II:
Single-name CDS: Protection on a single issuer. Used for hedging credit exposure to individual counterparties or for speculating on credit quality changes.
CDS index: Protection on a basket of issuers (e.g., CDX Investment Grade, iTraxx Europe). Used for broad credit market hedging or directional trading.
Basis trade: The difference between the CDS spread and the bond's credit spread (Z-spread). A negative basis means the CDS is cheaper than the bond's credit risk suggests, creating a potential arbitrage opportunity.
Succession and restructuring events: CDS contracts define specific credit events that trigger payouts, including bankruptcy, failure to pay, and restructuring. The exact definition matters, and ISDA protocols govern dispute resolution.
Binomial Option Pricing Model
Building on the Level I derivatives concepts, the binomial model values options by constructing a tree of possible future prices for the underlying asset. At each node, the price can move up by factor u or down by factor d. The model uses risk-neutral probabilities to calculate expected payoffs, then discounts backward through the tree.
The risk-neutral probability of an up move is:
πu = (erΔt − d) / (u − d)
Where r is the risk-free rate and Δt is the length of each time step. The corresponding down probability is πd = 1 − πu.
For a one-period binomial model with a European call option:
Calculate the option payoff at each terminal node: max(ST − K, 0)
Take the risk-neutral expected payoff: πu × Cu + πd × Cd
Discount back one period: C₀ = e−rΔt × Expected Payoff
For American options, at each node you compare the option's continuation value (the discounted expected future value) with the immediate exercise value. The option value at that node is the maximum of the two. This is what makes the binomial model especially valuable — it can handle early exercise, which Black-Scholes-Merton cannot directly accommodate.
Black-Scholes-Merton Model
The BSM model provides closed-form solutions for European option prices. For a European call on a non-dividend-paying stock:
C = S₀N(d₁) − Ke−rTN(d₂)
Where d₁ = [ln(S₀/K) + (r + σ²/2)T] / (σ√T) and d₂ = d₁ − σ√T. N(x) is the cumulative standard normal distribution function.
The BSM model assumptions are critical for exam purposes:
The underlying follows geometric Brownian motion with constant volatility
No dividends during the option's life (the basic model)
European exercise only (no early exercise)
Continuous trading is possible at the risk-free rate
No transaction costs or taxes
The risk-free rate is constant
For dividend-paying stocks, the model adjusts by replacing S₀ with S₀e−qT, where q is the continuous dividend yield. This lowers call values and raises put values because dividends reduce the expected future stock price.
Option Greeks: Measuring Sensitivity
The Greeks measure how an option's price changes in response to changes in its inputs. Level II requires you to understand each Greek conceptually, know its sign for calls and puts, and apply them in risk management scenarios.
Delta (Δ)
Delta measures the change in option price per unit change in the underlying price. For a call option, delta ranges from 0 to 1. For a put, delta ranges from −1 to 0. An at-the-money call has a delta near 0.5, meaning the option price moves about $0.50 for every $1 move in the stock.
Delta is also approximately the probability that the option expires in the money (under risk-neutral pricing). A call with delta 0.7 has roughly a 70% chance of finishing in the money. Delta hedging involves holding Δ shares of the underlying for each short option, creating a locally risk-neutral position.
Gamma (Γ)
Gamma is the rate of change of delta with respect to the underlying price (the second derivative of option price with respect to stock price). Gamma is always positive for long options (both calls and puts) and is highest for at-the-money options near expiration.
High gamma means delta changes rapidly, requiring more frequent rebalancing of a delta hedge. Gamma risk is most acute for short at-the-money options close to expiration, where a small move in the stock can cause a large swing in delta and, As a result, a large profit or loss on the hedged position.
Theta (Θ)
Theta measures time decay — how much the option's value decreases as one day passes, all else equal. Theta is typically negative for long options: options lose value as expiration approaches because there is less time for the underlying to move favorably. Time decay accelerates as expiration nears, especially for at-the-money options.
Vega (v)
Vega measures the sensitivity of the option price to changes in implied volatility. Higher volatility increases the value of both calls and puts because it increases the probability of large moves in either direction. Vega is highest for at-the-money options with longer time to expiration.
Vega is particularly important in practice because implied volatility is the one BSM input that is not directly observable; it must be extracted from market prices. Traders who believe implied volatility will increase buy options (go long vega); those who believe it will decrease sell options (go short vega).
Rho (ρ)
Rho measures the sensitivity of option price to changes in the risk-free interest rate. Higher rates increase call values (because the present value of the strike price decreases) and decrease put values. Rho is typically the least important Greek for short-dated equity options but becomes significant for long-dated options and interest rate derivatives.
Greek
Measures
Long Call
Long Put
Delta
Price sensitivity to underlying
Positive (0 to +1)
Negative (−1 to 0)
Gamma
Rate of change of delta
Positive
Positive
Theta
Time decay
Negative
Negative
Vega
Volatility sensitivity
Positive
Positive
Rho
Interest rate sensitivity
Positive
Negative
Interest Rate Derivatives and Swaptions
Interest rate swaps are the most widely traded derivatives in the world by notional value. A plain vanilla interest rate swap involves exchanging fixed-rate payments for floating-rate payments on a notional principal amount. The fixed rate is set at inception so that the swap has zero initial value.
Swap valuation at Level II uses the concept that a swap is equivalent to a portfolio of forward rate agreements (FRAs). The value of a swap after inception is the present value of the expected net cash flows, discounted using the current term structure. If rates have risen since inception, the fixed-rate payer benefits (and the swap has positive value to them).
A swaption is an option to enter into a swap at a predetermined fixed rate. There are two types:
Payer swaption: The right to enter a swap as the fixed-rate payer. This is valuable when rates rise, as you can lock in paying a below-market fixed rate. Economically similar to a put on a bond.
Receiver swaption: The right to enter a swap as the fixed-rate receiver. This is valuable when rates fall, as you can lock in receiving an above-market fixed rate. Economically similar to a call on a bond.
Swaptions are priced using the Black model (an adaptation of BSM for forward-starting instruments). The key input is the volatility of the forward swap rate. Swaptions are widely used by mortgage servicers to hedge prepayment risk and by pension funds to manage interest rate exposure.
Mortgage-Backed Securities Analysis
Mortgage-backed securities (MBS) are bonds collateralized by pools of residential mortgages. The unique feature of MBS is prepayment risk — homeowners can refinance or sell their homes at any time, returning principal to MBS investors ahead of schedule. This makes MBS analysis significantly more complex than standard bond analysis.
Key MBS concepts for Level II:
Prepayment models: The PSA (Public Securities Association) benchmark assumes prepayments ramp up over the first 30 months and then level off. Actual prepayment speeds are quoted as a percentage of PSA (e.g., 150% PSA means prepayments are 50% faster than the benchmark). Prepayments accelerate when rates fall (refinancing) and slow when rates rise.
Contraction risk: When rates fall, prepayments increase and MBS investors get their principal back sooner than expected. They must reinvest at lower rates. This is the risk that the bond's life contracts.
Extension risk: When rates rise, prepayments slow and MBS investors are stuck with a lower-yielding bond for longer than expected. This is the risk that the bond's life extends.
CMO tranches: Collateralized mortgage obligations (CMOs) redistribute prepayment risk among different tranches. Sequential-pay tranches direct all prepayments to the first tranche until it is retired, then to the second, and so on. PAC (Planned Amortization Class) tranches have a predictable cash flow schedule within a specified range of prepayment speeds, with companion tranches absorbing the variability.
OAS analysis for MBS: Because MBS have embedded prepayment options, OAS is the appropriate spread measure. Monte Carlo simulation is typically used rather than a binomial tree because prepayment behavior is path-dependent (it depends on the history of interest rates, not just the current rate).
Effective Duration and Convexity for Bonds with Options
Standard (modified) duration assumes that cash flows do not change when yields change. This assumption fails for callable bonds and MBS, where cash flows are directly affected by rate changes. Effective duration accounts for this by using the actual price changes observed when the yield curve shifts:
Effective Duration = (P− − P+) / (2 × P₀ × Δy)
Where P− and P+ are the bond prices when the yield curve shifts down and up by Δy, respectively. For callable bonds, effective duration is typically lower than modified duration because the call option caps the upside price appreciation. For MBS, effective duration can actually become negative in extreme scenarios when prepayment speeds respond dramatically to rate changes.
Effective convexity is similarly calculated using the price changes from yield curve shifts. Callable bonds and MBS typically exhibit negative convexity at low yields — their prices rise less than expected when rates fall because the embedded options become more valuable and constrain price appreciation.
Connecting Fixed Income and Derivatives to the Broader Curriculum
Fixed income and derivatives concepts at Level II connect directly to other topic areas. The equity valuation section requires understanding of WACC, which depends on the cost of debt derived from credit analysis. The economics and corporate finance section covers currency forward pricing, which uses the same no-arbitrage framework as interest rate forwards. Financial reporting topics like pension accounting require understanding of discount rates and duration matching.
Derivatives knowledge also feeds into the alternative investments section, where hedge fund strategies often involve complex options positions and credit derivatives. Risk management topics rely heavily on the Greeks for measuring and hedging portfolio exposures.
Common Exam Pitfalls
Confusing spot and forward rates: Make sure you know which rate to use for each discounting step. Forward rates apply to specific future periods; spot rates apply from today to a specific maturity.
Mixing up swaption types: A payer swaption profits when rates rise; a receiver swaption profits when rates fall. Draw the payoff profile if you are unsure.
Ignoring path dependence in MBS: Unlike standard bonds, MBS cash flows depend on the entire path of interest rates, not just the terminal rate. This is why Monte Carlo simulation is needed rather than binomial trees.
Forgetting the BSM assumptions: Many exam questions test whether you know when BSM is inappropriate (early exercise, non-constant volatility, discrete dividends).
Misapplying delta hedging: Delta hedging is a dynamic strategy that requires continuous rebalancing. Gamma tells you how quickly your hedge deteriorates as the underlying moves.
Putting It All Together
Fixed income and derivatives at Level II reward candidates who think in frameworks rather than memorizing formulas. The arbitrage-free framework unifies bond pricing, the binomial model bridges fixed income and options, and the Greeks provide a common language for measuring risk across all derivative instruments.
When working through practice vignettes, always start by identifying the type of instrument and what feature makes it non-standard (embedded option, prepayment feature, credit risk). That identification drives your choice of valuation model and risk measures. Build the habit of checking your answer against economic intuition: if rates rise, do callable bond prices behave as you expect? If volatility increases, do option values move in the right direction?
For a complete picture of the Level II exam and how fixed income fits alongside other topics, see our CFA Level II exam preview. Clarity helps you track fixed income holdings and derivatives positions alongside your equity portfolio, giving you a unified view of your investment exposure and risk profile across all asset classes.