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Arbitrage pricing theory relates expected returns to systematic factor loadings and their risk premiums. With idiosyncratic noise, a diversification or asymptotic argument needs additional assumptions. With no idiosyncratic noise and spanned factors, the formula is an exact consequence of law of one price: portfolios matching factor exposures can differ only by a constant payoff, which absence of arbitrage forces to be zero.

Let $B$ be the $n\times m$ matrix of factor loadings, so $r=a+Bf$. A zero-cost portfolio $v$ with $1^{T}v=0$ and $B^{T}v=0$ again has constant financial payoff $v^{T}a$. Absence of arbitrage impliesTaking expectations givesIf $[1B]$ has full column rank, a unit-cost portfolio with zero exposure to all factors exists and earns the certain rate ${\lambda }_0$. For each $j$, choose a zero-cost portfolio with loading one on factor $j$ and zero on the others; its expected return is ${\lambda }_j$. Hence these coefficients are the rewards per unit systematic exposure, or factor risk premiums, with the financial payoff measured per unit of the chosen factor normalization. A traded risk-free asset fixes ${\lambda }_0$ to its return. Changing the scale or basis of the factors changes the coordinates of the risk premiums but leaves asset prices unchanged. With deficient rank, only the spanned factor directions are identified and the coefficients need not be unique.

One alternative is a stochastic volatility model, such as the Heston model:The parameters satisfy $\kappa ,\theta ,\xi >0$ and $|\rho |\le 1$, and $v_0\ge 0$. The square-root Itô diffusion has a nonnegative solution; $2\kappa \theta \ge {\xi }^2$ is a standard sufficient condition for the positive initial variance not to hit zero. Mean reversion of $v_t$ allows persistent high- and low-volatility episodes, and negative $\rho$ can produce the empirical association between falling prices and rising volatility. Mixtures of conditional return distributions produce richer tails and option implied volatility shapes than constant-volatility geometric Brownian motion.

For data showing volatility clustering and option smiles, this is often a more useful fit. It is not universally superior: more parameters create estimation and calibration uncertainty, the model is more expensive to use, and its continuous price paths still exclude price jumps. With two independent noise directions and only a stock and a bank account, the market is generally incomplete, so a volatility risk premium or an extra traded instrument is needed for unique contingent claim pricing. The alternative improves flexibility at the cost of calibration and hedging complexity.

Form a portfolio with weights $\beta$ in the first factor-exposure asset, $\gamma$ in the second, and $1-\beta -\gamma$ in the risk-free asset. Its total initial investment is one. Its two factor exposures match those of asset $k$, so subtracting its return from $r_k$ leaves a constant. Any nonzero constant difference would allow a zero-cost long-short position with a strictly positive certain payoff after choosing its direction, an arbitrage. Under absence of arbitrage, the returns must therefore have the same expectation. ThusThe coefficients are the two factor risk premiums relative to the risk-free return, not the expected factor levels themselves. This reasoning permits negative weights, as assumed in the unconstrained arbitrage pricing theory model.

A credit derivative separates an exposure to a borrower's credit quality from outright ownership of its loan or bond. In the early-2002 setting, these were a growing, largely dealer-intermediated over-the-counter market. For scale, the contemporaneous BIS comparison reports ISDA member positions of about US$919billionatend-2001,versusUS$632 billion at mid-2001. These are notional amounts, not market values or expected losses, and the survey did not eliminate double-counting; estimates from other surveys cannot simply be combined because their coverage differed. This historical description uses the BIS Quarterly Review, June 2002, derivatives-market comparison, rather than projecting later market totals back into the paper's date.

The basic unfunded contract is a credit default swap. The protection buyer pays a periodic spread until maturity or a defined credit event; the seller pays the contracted default loss, often notional times one minus recovery. Physical settlement instead delivers eligible defaulted debt in exchange for par. A total return swap exchanges the reference asset's coupons and price changes against a funding leg: its receiver earns gains, bears price losses and pays funding, obtaining broader market and credit exposure than default protection alone. A credit spread option pays on widening or narrowing of a quoted credit spread, transferring mark-to-market credit risk even without a default. A credit-linked note is funded: the investor supplies principal and accepts a reduction of coupons or principal after a reference credit event, in return for enhanced yield. The investor is economically selling protection and can also face the note issuer's own credit risk.

Basket products extend these ideas. A first-to-default swap pays when the first member of a reference basket defaults. A synthetic CDO uses credit default swaps to create portfolio credit exposure and allocates aggregate losses among tranches. For monetary portfolio loss $L$ and attachment/exhaustion amounts $A<D$, the credit tranche loss function is $\min ((L-A{)}^{+},D-A)$. These contracts require a model of joint defaults and recoveries, not just a list of individual default probabilities. The separation of credit exposure from financing and the role of synthetic structures were already central to the Federal Reserve's December 2001 discussion of credit derivatives.

For risk management, a bank can buy protection on a concentrated borrower exposure while retaining the loan and customer relationship. It can transfer sectors or names it already holds heavily and diversify the remaining portfolio. Recognition of a hedge for capital purposes depends on the actual transfer and applicable rules, not merely on the instrument's name. Investors can instead sell protection or buy credit-linked notes to obtain chosen credit exposure without buying the cash bond, seek a risk premium, or take relative-value positions between cash bonds and swaps. Such positions still depend on funding and on the credit-spread basis: contractual events, maturity, currency and recovery terms may fail to match the original exposure. Protection also replaces part of the reference risk with counterparty credit risk, particularly if the seller may fail when the protected borrower is distressed.

Pricing equates expected discounted contractual cash flows under a risk-neutral measure, calibrated to traded bonds and derivatives rather than substituting historical default frequencies. To make this concrete, suppose the risk-neutral default intensity $\lambda (u)$ and discount factor $B(0,u)$ are deterministic, fractional recovery $\mathcal{R}$ is fixed, and premiums are continuous. Survival is $G(u)=\exp (-{\int }_0^u\lambda (v)dv)$. A notional-$N$ protection leg pays $N(1-\mathcal{R})$ at default, so its value is $N(1-\mathcal{R}){\int }_0^{T}B(0,u)\lambda (u)G(u)du$. Premium at annual rate $s$ has value $Ns{\int }_0^{T}B(0,u)G(u)du$. A zero-upfront par swap therefore has the continuous-premium credit-default swap spreadFor constant intensity this reduces exactly to $s=(1-\mathcal{R})\lambda$ in this simplified model. Actual discrete premium dates and premium accrued at default change the premium leg; stochastic recovery, discounting and counterparty losses change the expectations. A quoted spread thus cannot universally be read as a physical default probability. Default-risk premiums and other market effects enter the calibrated pricing measure.

A structural alternative is the Merton model. If firm assets at debt maturity are $V_T$ and promised debt is $F$, equity pays $(V_T-F{)}^{+}$ and debt pays $\min (V_T,F)=F-(F-V_T{)}^{+}$. Under the model's diffusion assumptions, debt value is consequently the discounted face value less a put on firm assets. This derives credit prices from asset value, volatility and capital structure, while the simple model allows default only at the debt maturity. Intensity models instead calibrate a default-arrival law directly to market credit prices. Neither approach removes model risk in recovery, dependence or calibration, and incomplete credit markets need not select a unique pricing measure. A useful credit hedge and a fair derivative price both depend on the actual cash-flow and default specifications, not the product label alone.

Compare expected utility of $W(1+\varphi )$, where $\mathbb{E}\varphi =0$, with the certainty equivalent $W(1-\psi )$. A second-order Taylor expansion gives $U(W)+W^2{U}^{\prime \prime }(W)\mathbb{E}{\varphi }^2/2$ for the risky side and $U(W)-W{U}^{\prime }(W)\psi$ for the sure side. Equality gives the displayed proportional risk premium. This requires a genuinely small-shock family with controlled remainders; small variance alone does not control rare large wealth shocks.