Past exam of the mathematics course of the University of Cambridge / 2002 / iii / Paper 32 (Mathematical Models in Financial Management)

Treat each column as the payoff from one unit of initial investment, so the cost of holdings $h=(h_1,h_2,h_3{)}^t$ is $h_1+h_2+h_3$. The first asset weakly dominates the third at the same price, and so does the second. Long the first and short the third gives zero initial cost and terminal payoff $(0,0,2{)}^t$. Long the second and short the third gives payoff $(2,0,0{)}^t$, also at zero cost. These are arbitrages provided the displayed states have positive probability. Combining them gives the zero-cost position $(1,1,-2{)}^t$ with payoff $(2,0,2{)}^t$.

More generally, imposing zero cost means $h_3=-h_1-h_2$, and its payoff is $(2h_2,0,2h_1{)}^t$. Thus every $h_1,h_2\ge 0$ with at least one positive gives a pure-investment arbitrage. The market is not arbitrage-free.

There is nevertheless no arbitrage of the other requested kind, with every future net cash flow identically zero. Indeed the middle-state payoff is $2(h_1+h_2+h_3)$, so a zero terminal payoff already forces zero initial cost. The other two states then force $h_1=h_2=0$, hence $h_3=0$. Equivalently, the return matrix has determinant $8$ and is invertible. ThereforeThe unique vector pricing the three unit-investment payoffs is $(0,\frac{1}{2},0{)}^t$. It is nonnegative but not strictly positive. This illustrates why nonnegative state prices need not exclude arbitrage: strictly positive state prices would assign positive cost to either of the nonzero nonnegative payoffs above.

The current price of one unit of certain future consumption is the sum of the Arrow state prices, namely $0.9$. A unit-price risk-free asset yielding gross return $R$ must satisfy $0.9R=1$. HenceNormalize the Arrow state prices by their sum, or multiply them by $R$, to obtain the risk-neutral probabilities:They are positive and sum to one. Any terminal payoff $Y$ has current price ${\sum }_s{p}_s{Y}_s=R^{-1}{\mathbb{E}}_{Q}Y$. In particular any asset with current price one has ${\mathbb{E}}_Q{r}_i=R$. These pricing probabilities need not equal the physical probabilities governing the investor's utility.

The complete market makes every state-contingent consumption vector attainable. Its budget is ${\sum }_s{p}_s{c}_s=100$, and expected logarithmic utility under the equally likely states is $\frac{1}{4}{\sum }_s\log c_s$, with $c_s>0$. Increasing utility makes the budget bind. For a Lagrange multiplier $\eta$, the first-order conditions areSubstitution in the budget gives $1/\eta =100$, so $\eta =0.01$ andEvery state receives an expenditure $p_s{c}_s=25$, rather than equal consumption. Strict concavity of logarithmic utility makes this feasible stationary point the unique global maximizer. The physical probabilities enter the utility objective; the supporting prices enter the budget, and they must not be interchanged.

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.

Let $M$ denote the displayed matrix of excess-return coefficients. If it were invertible, choose $v$ with $Mv=(1,0,0{)}^t$. Hold $v_i$ units of the three risky investments and short ${\sum }_i{v}_i$ units of the risk-free asset. Initial net cost is zero. The terminal payoff isThis is a guaranteed arbitrage, contradicting the assumption. HenceThe contradiction does not require guessing probabilities for the factors: invertibility would allow simultaneous cancellation of both random exposures while keeping a nonzero constant gain.

When the two factor-exposure rows are independent, every three-column excess-coefficient matrix being singular means that the entire coefficient matrix has rank at most two. The constant excess-return row must therefore be in the span of the two exposure rows. There are common constants ${\theta }_1,{\theta }_2$ such thatTo see that the constants are common, fix two assets with independent exposure columns, solve their two equations for ${\theta }_1,{\theta }_2$, and apply the zero determinant with any third asset; its constant entry is forced to satisfy the same equation. Taking expectations gives the exact two-factor arbitrage pricing theory formulaAn asset with unit loading on one factor and zero loading on the other has excess expected return equal to that factor's premium. This recovers part (i).

The rank qualification matters: singularity by itself would not identify two premiums if both exposure rows were dependent. More generally, write the exposure matrix as $B$. A position $v\in \ker B$ has a constant excess payoff $(a-R1{)}^{t}v$. Absence of arbitrage forces this constant to vanish, so $a-R1$ annihilates $\ker B$ and belongs to the row space of $B$. This proves exact factor pricing without idiosyncratic risk also in a reduced-rank model, though factor premiums need not then be unique. Independence of the random factors is not a substitute for their being spanned by traded exposures.

Assume the covariance matrix $\Sigma$ is positive definite and the excess-mean vector $d=z-R1$ is nonzero. Put $A=d^t{\Sigma }^{-1}d>0$. Substituting $w=\lambda {\Sigma }^{-1}d$ in the mean constraint gives $\lambda A=\mu -R$. ThereforeThese weights are indeed the minimum-variance solution, not just a stationary point. Any other feasible vector is $w+v$ with $d^{t}v=0$. Its variance is $w^t\Sigma w+2\lambda d^{t}v+v^t\Sigma v=w^t\Sigma w+v^t\Sigma v$, which is strictly larger unless $v=0$. If $d=0$, only mean $R$ is feasible, and the minimum-risk portfolio invests entirely in the risk-free asset; the multiplier formula with zero denominator is not applicable.

Using $A$ from part (a), the portfolio variance isThus the minimum-variance correspondence for fixed means has the two branches $\mu =R\pm \sqrt{Av}$. The lower branch is dominated by the upper one with the same variance, and portfolios with mean below $R$ are also dominated by the risk-free position. Consequently the actual mean-variance efficient frontier is the upper capital market line:Its slope in standard-deviation/mean coordinates is the maximal Sharpe ratio $\sqrt{A}$. In variance/mean coordinates it is the upper branch of $v=(\mu -R{)}^2/A$.

The risk-free weight is $1-1^{t}w$, so a zero risk-free weight requires $1^{t}w=1$. Define $H=1^t{\Sigma }^{-1}d$. If $H\ne 0$, this condition gives $\lambda =1/H$ andThese are the requested tangent-portfolio mean and variance under the usual condition $H>0$. Then $\lambda >0$, the normalized portfolio lies on the upper efficient ray, and its Sharpe ratio is $\sqrt{A}$.

There is a real qualification if the data are allowed to be arbitrary. If $H=0$, the efficient risky direction has zero total risky weight, and no finite multiple makes its risky weights sum to one. If $H<0$, the formal normalized portfolio above has negative excess mean and lies on the lower, dominated branch; it is not an efficient tangent portfolio as described in the question. The sign of the normalized tangency portfolio must therefore be checked, rather than inferred from positive definiteness of $\Sigma$ alone.

Use the usual frictionless option model with nonnegative deterministic interest rates and no stock dividends. Let $B(t,T)\le 1$ be the price of a unit zero-coupon bond, $C_E$ the European call price, and $C_A$ the American call price. A call payoff dominates $S_T-X$, so no-arbitrage pricing gives $C_E\ge S_t-XB(t,T)$, and $C_A\ge C_E$. If $S_t>X$, thenThe last term is the immediate exercise payoff. Waiting retains downside optionality and defers payment of the strike. For positive interest and $X>0$ the inequality is strict while exercise has positive payoff. If $S_t\le X$, exercising for a zero payoff provides no advantage over keeping the right. Thus early exercise of a non-dividend-paying call is never strictly beneficial under these assumptions, and $C_A=C_E$. For completeness, discounted European call value is a true martingale in the standard pricing model. For any exercise stopping time $\tau \le T$, its pointwise dominance of immediate payoff gives ${\mathbb{E}}_Q[B(t,\tau )(S_{\tau }-X{)}^{+}\mid {\mathcal{F}}_t]\le {\mathbb{E}}_Q[B(t,\tau )C_E(\tau )\mid {\mathcal{F}}_t]=C_E(t)$. Taking the supremum over exercise rules proves $C_A\le C_E$, and waiting to maturity gives the reverse inequality. This proves no early exercise of a call without dividends.

The contrast for a put is the interest earned on a strike received sooner. The European put has payoff at most $X$, so $P_E\le XB(t,T)$. When interest is positive and $S_t<X(1-B(t,T))$, immediate exercise gives $X-S_t>XB(t,T)\ge P_E$. Hence committing never to exercise before maturity is worse than an available early-exercise strategy in such a deep-in-the-money state. This does not say that every put should be exercised early, only that a general prohibition is not rational.

The interest-rate hypothesis is essential to the printed claim. If $r<0$ and the stock is deterministic, $S_u=S_t{e}^{r(u-t)}$ with no dividends, choose spot sufficiently large that it stays in the money through $T$. The European call price is $S_t-X{e}^{-r(T-t)}<S_t-X$, so immediate exercise is better. At zero interest, early exercise can be indifferent in degenerate cases; it is the usual positive-interest setting that makes the non-exercise comparison strict.

Assume the stated cash dividend is the only dividend before maturity and interest rates are nonnegative. Before its ex-dividend date, compare exercise now with retaining the option until any later deterministic date $u<t_1$. There is no dividend between these dates, and retaining the option has value at least that of a European call expiring at $u$, namely at least $S_t-XB(t,u)$. This is at least the immediate in-the-money payoff, and is strictly larger with positive interest. Thus exercise earlier in the open pre-dividend interval is not optimal; the only possible dividend-induced early-exercise opportunity is the last cum-dividend instant, conventionally denoted $t_1-$.

At that instant write $S_{-}=S_{t_1-}$ and $S_{+}=S_{-}-D$, the ex-dividend price. Immediate in-the-money exercise pays $S_{-}-X$. Continuing across the dividend and retaining the call to maturity gives at least $S_{+}-XB(t_1,T)$, since no further dividend remains. Therefore continuation minus immediate exercise is bounded below byConsequently the dividend-date call-exercise criterion isIf the pre-dividend call is out of the money, exercising gives no benefit in the first place. After the dividend, part (a) excludes later early exercise. The comparison is a sufficient condition: failing it does not prove exercise is optimal, because continuation also includes put/time value not present in this lower bound.

The date convention matters. A dividend can make exercise just before the ex-dividend date optimal. Thus a literal assertion excluding $t_1-$ under all dividends would be false; the pre-dividend non-exercise argument excludes earlier dates, leaving this last cum-dividend opportunity to be tested. Further dividends would require separate tests at their own dates.

For European options with the same strike and maturity, the terminal payoff identity isThus long the call and short the put has the same terminal payoff as one stock less $X$ units of the maturity-$T$ zero-coupon bond. With no dividends, the stock strategy has no intervening payments. The law of one price gives put-call parity:Equivalently the two nonnegative terminal-payoff portfolios, call plus strike bond and put plus stock, agree in every state. The parity proof does not require lognormal stock prices or positive interest.

For American options the exercise times can differ, so equality of their maturity payoffs is not an identity between their full rights. Under the nonnegative-interest, non-dividend assumptions of part (a), $C_A=C_E$ while $P_A=P_E+{\Pi }_P$ with early-exercise premium ${\Pi }_P\ge 0$. Hencewhich generally differs from European parity. For a concrete limiting state, if the stock is zero and remains zero and interest is positive, then $C_A=C_E=0$, $P_A=X$ and $P_E=XB(t,T)<X$. European parity is not a general American-option equality. It holds in a particular case only when the relevant early-exercise premiums cancel, or both vanish.

The two binary terminal payoffs are complementary:in every state, including $S_T=X$. Long one of each is therefore exactly a unit zero-coupon bond. The law of one price gives digital put-call parity:With deterministic interest, the two separate values are $B(t,T)Q(S_T\ge X\mid {\mathcal{F}}_t)$ and $B(t,T)Q(S_T<X\mid {\mathcal{F}}_t)$. There is no need to assume a continuous terminal price distribution: the strict inequality for the put and weak inequality for the call allocate any atom at the strike exactly once.

Take $r>0$, $\sigma >0$ and proportional continuous dividend yield $\delta \ge 0$, the usual perpetual-option setting. Write $L$ for the prescribed downward exercise trigger, with $0<L<X$. Above the trigger there is no dependence on calendar time, so the Black-Scholes equation with continuous stock dividends reduces toSubstitute $V=S^{\beta }$. The characteristic equation and its roots areSince the root product is $-2r/{\sigma }^2<0$, ${\beta }_{-}<0<{\beta }_{+}$. The boundary condition $V(S)\to 0$ as $S\to \infty$ excludes the positive-power solution. Value matching at the trigger requires $V(L)=X-L$, giving the fixed-rule priceThe lower branch means exercise immediately if the starting price is already below the trigger. Above it, the formula also equals ${\mathbb{E}}_Q[e^{-r{\tau }_L}(X-L)1_{\{{\tau }_L<\infty \}}]$, where ${\tau }_L$ is the first passage to $L$. To verify this expectation, apply Itô formula to the bounded stopped discounted solution on $S\ge L$; the generator equation makes it a martingale before the hit, and the residual term on surviving paths vanishes as the time horizon tends to infinity because $r>0$. A fixed trigger is a specified exercise rule, not yet the optimized American price.

For a starting price above the trigger, maximizing $V_L(S)$ amounts to maximizing $g(L)=(X-L)L^{-{\beta }_{-}}$ on $0<L<X$. Its derivative isBecause ${\beta }_{-}<0$, this is positive below ${\beta }_{-}X/({\beta }_{-}-1)$ and negative above it. Both endpoint limits of $g$ are zero. Thus the unique maximizing trigger isEquivalently smooth fit requires ${\beta }_{-}(X-S^{\ast })/S^{\ast }=-1$, matching the derivative of $X-S$ below the boundary. The optimized perpetual put option value is $X-S$ for $S\le S^{\ast }$ and $(X-S^{\ast })(S/S^{\ast }{)}^{{\beta }_{-}}$ above it.

We can verify optimality among all stopping rules, not only constant triggers. The continuation branch is convex, with value and slope matching $X-S$ at $S^{\ast }$, so it dominates $X-S$ above the boundary; it is also positive, and hence dominates $(X-S{)}^{+}$ everywhere. Its discounted generator vanishes in continuation. In the exercise region, the generator of $X-S$ is $\delta S-rX$. The root equation givesso for $\delta \ge 0$ this generator is nonpositive throughout $S\le S^{\ast }$. Smooth fit eliminates a local-time contribution at the boundary. Itô formula and localization therefore make $e^{-rt}V(S_t)$ a nonnegative supermartingale, providing an upper bound for every discounted exercise payoff. The threshold strategy attains that bound by the martingale calculation in part (a). This proves the boundary is globally optimal.

The positive-interest assumption is substantive. If $r=0$ and $\delta \ge 0$, the stock tends to zero almost surely, so the supremum of perpetual put payoffs has value $X$, approached by triggers $L\downarrow 0$. No positive finite trigger attains that supremum for positive spot. The displayed finite-boundary formula belongs to the $r>0$ case. If $r<0$ and $\delta \ge 0$, the stock still tends to zero, while the discount factor grows without bound. Even exercise at deterministic times tending to infinity makes the expected discounted put payoff unbounded, so there is no finite perpetual price in that extension.

Differentiate the negative-root equation $F({\beta }_{-},\delta )=0$, keeping $r$ and $\sigma$ fixed. Its derivatives are $F_{\delta }=-{\beta }_{-}$ andThus $d{\beta }_{-}/d\delta ={\beta }_{-}/F_{\beta }=-{\beta }_{-}/\Delta >0$. Differentiating the trigger ratio then givesThis is the dividend-yield sensitivity of the perpetual put trigger. A larger dividend yield lowers the risk-neutral drift of the ex-dividend stock; waiting for further price reductions becomes more attractive to the put holder, so exercise occurs at a lower price. As checks, at $\delta =0$ the negative root is $-2r/{\sigma }^2$, giving $S^{\ast }=2rX/(2r+{\sigma }^2)$. As $\delta \to \infty$, ${\beta }_{-}\to 0-$ and $S^{\ast }\to 0$.

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.