Past exam of the mathematics course of the University of Cambridge / 2013 / iii / Paper 39 (Advanced Financial Models)

The multidimensional Itô formula includes a mixed second derivative multiplied by the quadratic covariation, with no extra factor $1/2$. HereConsequently the Itô formula for $M_t=U(t,{\sigma }_t,X_t)$ has drift equal to the left side of the stated backward partial differential equation. That drift vanishes, leavingA stochastic integral against Brownian motion with locally square-integrable predictable integrand is a continuous local martingale. The smoothness of $U$ and localization of the diffusion and its coefficients give this integrability on the model's lifetime. Thus $M$ is a local martingale, as required. The partial differential equation cancellation alone does not establish a true martingale or justify replacing $U$ by a terminal-payoff expectation without an additional integrability argument.

For the exponential payoff, substituting $U=e^{\theta X}V$ gives $U_X=\theta U$, $U_{XX}={\theta }^{2}U$, and $U_{\sigma X}=\theta e^{\theta X}{V}_{\sigma }$. Dividing the backward partial differential equation by the nonzero factor $e^{\theta X}$ therefore gives the exponential payoff transform PDEThe terminal value isThe correlation changes the first-derivative coefficient, while the original drift of the log price combines with its variance to give $\theta (\theta -1)/2$ rather than ${\theta }^2/2$.

Set $\tau =T-t$, $\kappa =\lambda -\rho b\theta$ and $k=\theta (\theta -1)/2$. For the Ornstein-Uhlenbeck process volatility, the transformed partial differential equation isUse the Gaussian volatility exponential-quadratic transform ansatz $V=\exp (P+Q\sigma +R{\sigma }^2)$, where the coefficients depend on $\tau$. Its derivatives satisfyMatching the constant, linear and quadratic powers of $\sigma$ givesThe terminal condition requiresThese polynomial ordinary differential equations have a unique local solution by the Picard's theorem for ordinary differential equations. Substitution then proves the desired partial differential equation solution on every horizon for which the coefficient solution remains finite. The Riccati equation for $R$ is solved first; $Q$ subsequently solves a linear equation and $P$ is an integral of known coefficients.

Unrestricted global existence needs a qualification. Take $a=0$, $\lambda =b=1$, $\rho =0$ and $\theta =2$. Then $Q=0$ and the Riccati equation becomesThis solves the initial condition but explodes at $\tau =3\pi /4$. Therefore no finite real exponential-quadratic solution with the required terminal condition exists on an entire horizon $T\ge 3\pi /4$ for these allowed parameters. The correct general claim is local existence, or existence before the Riccati moment-explosion horizon.

A useful sufficient global condition is $0\le \theta \le 1$, so $k\le 0$. If $b\ne 0$, the lower equilibriumtraps the solution in $r_{-}\le R(\tau )\le 0$: the polynomial vector field points inward at the upper endpoint and vanishes at the lower endpoint. If $b=0$, the equation for $R$ is linear. In either case there is no finite-time explosion; $Q$ is then a linear equation with coefficients bounded on compact time intervals, and $P$ is finite on those intervals. This proves the intended ansatz globally under that sufficient parameter restriction, without asserting it for every real $\theta$.

For a deterministic $s>0$, the positive part $(s-K{)}_{+}$ vanishes for $K\ge s$, so direct integration givesAt $s=0$ both sides vanish. Applying this pointwise to the nonnegative random variable proves the power payoff static call representationBy Tonelli theorem, the corresponding moment identity isincluding the possibility that both sides are infinite. No higher-moment assumption is needed to interchange these nonnegative integrals.

Let $A_{\delta }={\sup }_{K\ge 0}{K}^{\delta }C(K)<\infty$. The call-price decay and moment threshold follows by splitting the preceding integral at one. Since $(S-K{)}_{+}\le S$, for $0<\epsilon <\delta$,For $K\ge 1$, the decay bound givesThe power payoff static call representation therefore yieldsThe $\epsilon =0$ case is the given finite first moment. The strict endpoint matters: a Pareto distribution with $\mathbb{P}(S>x)=x^{-(1+\delta )}$ for $x\ge 1$ has $C(K)=K^{-\delta }/\delta$ for $K\ge 1$, but its moment of order $1+\delta$ is infinite. Thus the stated decay condition does not generally imply the endpoint moment.

If $K=0$ or $S\le K$, the proposed sharp power-call inequality is immediate. Otherwise put $x=S/K>1$ and consider the ratioIts logarithmic derivative is $(1+\epsilon )/x-1/(x-1)$, whose sign is that of $\epsilon x-(1+\epsilon )$. The ratio decreases and then increases, with minimum at $x_{\ast }=(1+\epsilon )/\epsilon$. That minimum is $(1+\epsilon {)}^{1+\epsilon }/{\epsilon }^{\epsilon }$. Rescaling proves the sharp power-call inequalityThe constant is sharp because equality holds for $S=(1+\epsilon )K/\epsilon$ when $K>0$. This also explains the hinted minimization: with $a$ equal to that constant times $K^{\epsilon }$, the minimum of $s^{1+\epsilon }-as$ is $-aK$. Its denominator is ${\epsilon }^{\epsilon }$, including when reading the original PDF.

Take expected values in the sharp power-call inequality. If $M(1+\epsilon )$ is finite, then for every $K\ge 0$,The bound is independent of the strike, soCombined with the previous part, this relates finite moments to polynomial decay of expected European call option payoffs, while retaining the distinction at the moment threshold.

In the one-period model, initial holdings are a deterministic vector $H\in {\mathbb{R}}^n$, chosen with trivial initial information, and the terminal payoff of the portfolio is $H^T{P}_1$. A contingent claim ${\xi }_1$ is replicable if some such vector has $H^T{P}_1={\xi }_1$ almost surely; its initial claim replication cost is $H^T{P}_0$. A complete market replicates every claim in the specified terminal-information class; for the arguments below, it suffices that every bounded measurable claim is replicable. The $n$ assets count all assets of the model, including a cash asset if one is traded.

These are one-period definitions. If arbitrary terminal information were already available initially and holdings could depend on it, the finite-atomic conclusion in the next part would not follow; the deterministic-holdings convention is essential.

The span of the $n$ asset payoff functions has dimension at most $n$. If there were $n+1$ disjoint measurable events $A_1,\dots ,A_{n+1}$ of positive probability, their indicator functions would be linearly independent as random variables modulo almost-sure equality. Indeed, restricting a zero linear combination to $A_j$ forces its $j$th coefficient to vanish. Market completeness would put all these independent functions inside a span of dimension at most $n$, a contradiction.

This implies the stronger meaningful partition conclusion: terminal information has at most (n) positive-probability atoms. Start with the whole sample space and split any event which is not a probability atom of a measure into two positive-probability measurable subsets. Each split increases the number of disjoint positive events, so no more than $n-1$ splits are possible. The resulting partition has $m\le n$ components, and each is a probability atom of a measure, since otherwise another split would be possible. Null sets can be included in a component without altering any random variable modulo null sets.

On such an atom every measurable real-valued random variable is constant almost surely: if its distribution on that atom were not concentrated at one value, an appropriate level set would split the atom. Hence every terminal claim is described by its $m$ values. This proves the finite branching bound in a complete market for one period, rather than the vacuous weaker observation that the whole space itself is one event.

For each terminal event $A$, market completeness supplies a portfolio $H_A$ replicating its indicator function. The two pricing identities giveThe constant payoff is also replicable, so these positive pricing variables are integrable under the stated finite pricing expectations. Equivalently, the preceding finite-atom result makes them finite-valued modulo null sets. Taking $A=\{X>Y\}$, the equality says that the nonnegative variable $(X-Y)1_A$ has expectation zero; thus $X\le Y$ almost surely. Reversing the roles givesThis is uniqueness of a one-period pricing density in a complete market. It uses replication of every event, not only matching a few marginal asset expectations.

Let $H$ replicate ${\xi }_1$, so ${\xi }_1=H^T{P}_1$. The positive-definite Gram matrix $Q=\mathbb{E}[P_1{P}_1^T]$ is invertible. Multiplication by $P_1$ and taking expected values giveAll quantities are integrable: the finite-atom representation from completeness gives finite terminal values, and the matrix $Q$ has finite entries. Positive definiteness also guarantees uniqueness of holdings, since $h^T{P}_1=0$ would imply $h^{T}Qh=0$ and hence $h=0$.

Thus the replication cost isConsequently the one-period Gram-matrix replication formula isThe symmetry of $Q^{-1}$ makes the displayed orientations consistent. Positivity of this $Z$ would require an additional no-arbitrage condition; positive definiteness of the payoff Gram matrix alone proves the representation and uniqueness, not positivity of prices across states.

Backward induction makes the Snell envelope integrable and adapted: $|U_t|\le |Y_t|+\mathbb{E}(|U_{t+1}|\mid {\mathcal{F}}_t)$. Its definition gives $U_t\ge Y_t$ and $U_t\ge \mathbb{E}(U_{t+1}\mid {\mathcal{F}}_t)$, so it is a supermartingale dominating the reward.

For a stopping time $\tau$ taking values in $\{0,\dots ,T\}$, expand its stopped value asThe indicators are ${\mathcal{F}}_t$-measurable. Taking conditional expectations in each summand makes its expectation nonpositive by the supermartingale property. Since ${\mathcal{F}}_0$ is trivial, $U_0$ is deterministic andThis proves the finite-horizon optional sampling theorem directly in the instance needed here, without assuming nonnegative rewards.

The first-contact time ${\tau }_{\ast }\le T$ is a stopping time: the event $\{{\tau }_{\ast }\le t\}$ is the finite union of events $\{U_s=Y_s\}$ for $s\le t$. It is finite because $U_T=Y_T$. On $\{{\tau }_{\ast }>t\}$ we have $U_t>Y_t$, so the maximum defining the Snell envelope selects continuation andThe stopped process $U_{t\wedge {\tau }_{\ast }}$ is therefore a martingale: before stopping its conditional increment is zero and after stopping its increment vanishes. Applying the stopped-sum argument with zero conditional increments givesThe upper bound from part (a) now proves that first contact is an optimal stopping time.

Let $v_j$ be the optimal expected value before seeing the next offer when $j$ rounds remain. With one round left the offer must be accepted, so $v_1=1/2$. For $j>1$, observing $x$ gives a choice between $x$ now and the continuation value $v_{j-1}$. Independence of future uniform distributions makes that continuation value independent of past offers. Thus the uniform-offer stopping recursion isIt gives $v_2=5/8$ and $v_3=89/128$. The Snell envelope rule therefore yields the explicit strategyOnly rounds actually reached are played. At a threshold the two actions have identical continuation expected value; either convention is optimal, and exact equality has probability zero. The optimal expected payout before the first offer is

Construct the strictly positive rolling one-period bond account using successive one-period zero-coupon bonds:At time $t-1$, invest the entire account value in the bond maturing at $t$. This gives a positive self-financing portfolio usable as a numéraire.

We use the finite-discrete-time fundamental theorem of asset pricing: in a frictionless market with finitely many adapted assets and trading dates, no arbitrage is equivalent to existence of an equivalent martingale measure for asset prices, including dividends, expressed in a positive traded numéraire. Maturing bond payoffs are reinvested in that numéraire. Let $\mathbb{Q}$ be such a measure and $L_t=\mathbb{E}[d\mathbb{Q}/d\mathbb{P}\mid {\mathcal{F}}_t]$ its positive density process. The martingale pricing relation for a unit zero-coupon bond isDefine $Z_t=L_t/{\beta }_t$. The Bayes formula for conditional expectation then givesThe positive expectations are finite because these are the traded finite bond prices; in particular $\mathbb{E}{Z}_T=Z_0{P}_0(T)$ when initial information is trivial. Normalize $Z_0=1$ without changing any ratio. With nontrivial initial information the same identities are conditional on that information. The state-price density need not be unique when the bond market is incomplete; existence is sufficient.

A unit zero-coupon bond pays one at its maturity, so $P_t(t)=1$. Monotonicity in maturity gives $P_t(t+1)\le 1$. Using the state-price density representation with $T=t+1$,The process is positive and integrable, as noted in part (a), and adapted. Thus it is a supermartingale. Strictly decreasing maturity prices yield a strict one-step conditional inequality; weak decrease is already enough for the conclusion.

The spot interest rate $r_T$ is known at $T-1$. The state-price density price of its floating payment is, by the law of total expectation,Here $P_0(0)=1$ if $T=1$. All these terms are integrable: $Z_T{r}_T$ is bounded in absolute value by $Z_T/P_{T-1}(T)+Z_T$, whose expectation is finite from the one-step pricing relation. Subtracting the fixed payment $f$ givesThis also follows directly from a floating-rate payment bond replication. At time zero buy one unit of the bond maturing at $T-1$ and short $1+f$ units of the bond maturing at $T$. Their initial cost is the displayed ${\xi }_0$. Hold them until $T-1$. The first bond then pays one; spend that one to buy $1/P_{T-1}(T)$ units of the maturity-$T$ bond, leaving the earlier short position in place. This rebalance is self-financing. At maturity the net payment isFor $T=1$, the first unit is time-zero cash, and the same immediate rebalance gives the deterministic payoff. Thus the replication establishes the zero no-arbitrage price without requiring completeness of other claims.

Each floating payment in the interest rate swap has initial value $P_0(t-1)-P_0(t)$ by the previous replication. Their sum telescopes to $1-P_0(T)$. The fixed leg pays $s$ at each of the same dates, so its initial value is $s{\sum }_{t=1}^T{P}_0(t)$. Consequently the par swap rate isThe denominator is positive. This is for unit accrual periods and the printed floating-minus-fixed payments. No extra exchange of principal occurs in the contract; the principal-like terms appear only because the floating-leg replication telescopes.

Assume the usual positive initial asset values. The coefficients determine a unique normalized local martingale deflator. The state-price density and local deflator distinction matters here: calling it a state-price density uses the local convention, while true expectation pricing needs an additional qualification, addressed below.

Let $ZB$ and $ZS$ be local martingales, with $Z_0=1$. The Brownian martingale representation theorem says that every local martingale in the usual natural Brownian filtration is continuous and is a stochastic integral against $W$. Applying it to $ZB$, and dividing by the positive finite-variation account, forcesfor a locally square-integrable predictable $\eta$. The Itô product rule for $ZS$ has driftIt must vanish. Since $S,Z,\sigma$ are positive, this yieldsConversely this drift choice makes both $ZB$ and $ZS$ local martingales. Continuity and strict positivity of $\sigma$ make $\lambda$ bounded along each path on every finite time interval, so its pathwise square integral is finite. The unique linear SDE solution isUniqueness follows from the forced drift and diffusion coefficients and uniqueness of this linear stochastic differential equation.

Continuity alone does not make the density a true martingale. For a true equivalent martingale measure, the stochastic exponential $D_t=Z_t{B}_t/B_0$ must have expectation one, for example under the Novikov condition on each horizon. True martingale pricing of all desired deflated payoffs also requires the relevant integrability. These stronger conclusions do not follow just from pathwise continuity.

An explicit counterexample to the stronger reading uses a three-dimensional Bessel process $R$ with $R_0=1$ and $dR=dW+R^{-1}dt$, which is a positive strong solution in the Brownian filtration. Take $B=1$ and $S=R$. Then $r=0$, $\mu =R^{-2}$ and $\sigma =R^{-1}$ are continuous and $\sigma >0$. The unique local candidate is $Z=R^{-1}$, with $ZS=1$. The reciprocal three-dimensional Bessel strict local martingale is not a true martingale. To verify the loss of expectation, use the standard Bessel transition densityIntegrating $p_t(1,y)/y$ gives $\mathbb{E}[R_t^{-1}]=2\Phi (1/\sqrt{t})-1<1$ for $t>0$, where $\Phi$ is the standard normal cumulative distribution function. A true pricing density for the constant bank account would have expectation one. Hence the printed hypotheses establish the local deflator statement, while a true-density reading needs an additional condition and is false as stated.

Let $h_t$ and $k_t$ be predictable holdings in the stock and account, with the stochastic integrability needed for a self-financing portfolio. Then $V=hS+kB$ and $dV=hdS+kdB$. The Itô product rule and self-financing identity giveSince $V=hS+kB$ and $\sigma \lambda =\mu -r$, the finite-variation term is zero. Thereforeand the deflated wealth is a local martingale.

For zero-capital nonnegative wealth under a local deflator, a nonnegative local martingale is a supermartingale, by localization and the Conditional Fatou lemma. Under the required nonnegative-wealth condition, $ZV\ge 0$ and starts at zero, so $0\le \mathbb{E}(Z_t{V}_t)\le 0$. Thus $Z_t{V}_t=0$ almost surely at every fixed $t$. Strict positivity of $Z$ gives $V_t=0$ almost surely. Taking a countable intersection over rational times and then using continuous wealth paths strengthens this toThis argument only needs the local deflator, so it remains valid without promoting $ZB$ to a true martingale.

Here the bank account is continuous and of finite variation, but for $t>0$ its rate is $r_t=t^{-1/2}$. This is integrable at zero, so the account itself is well defined. The stock satisfies $d{S}_t=S_{t}d{W}_t$. The previous continuity hypothesis on the rate no longer holds at the initial time.

Suppose a positive normalized state-price density existed; even a local martingale deflator would suffice for a contradiction. Then $ZB$ would be a continuous local martingale by the Brownian martingale representation theorem. Dividing by the positive continuous account gives continuous $Z$, with $Z_0=1$, andThe Itô product rule for $ZS$ forces its drift to vanish, so ${\eta }_t=Z_t{t}^{-1/2}$ for almost every positive time. Continuity and $Z_0=1$ imply that each path has a positive interval on which $Z_t\ge 1/2$. But thencontradicting the local square-integrability required for the stochastic integral. ThusThis is the singular initial market-price-of-risk obstruction: the necessary market price of risk is $-t^{-1/2}$, whose squared integral diverges at zero. On an interval starting at a strictly positive time this particular obstruction disappears; the initial-time normalization is essential.