Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 211 (Advanced Financial Models)

$X_t^{x,H}$ is the wealth delivered by the holdings chosen before time $t$, after receiving their time-$t$ dividends. The investor then chooses $H_{t+1}$, whose market value is $H_{t+1}\cdot P_t$, and consumes the remainder $C_t^{x,H}$. A numéraire portfolio is a strategy $\eta$ with zero consumption and strictly positive wealth $N_t=X_t^{\nu ,\eta }$ at every time, where $\nu ={\eta }_1\cdot P_0$.

Write $V_t=X_t^{x,H}-C_t^{x,H}=H_{t+1}\cdot P_t>0$. Set $a_0=1$ and recursivelyThe factors are positive and adapted, so $\eta$ is previsible. For $t\ge 1$,while the same identity at $t=0$ defines $\nu ={\eta }_1\cdot P_0$. Thus consumption is zero and wealth is strictly positive, so $\eta$ is a numéraire portfolio.

Putand defineBecause the numéraire strategy is self-financing,Moreover,This also proves the required wealth formula.

Letwhere the sum is componentwise, and setThenAlso $K_{t+1}\cdot P_t=H_{t+1}\cdot {\tilde{P}}_t$, so subtracting the new holdings value gives $C_t^{x,K}={\tilde{C}}_t^{x,H}$.

The discrete-time fundamental theorem of asset pricing supplies a strictly positive martingale deflator $Y$. Since the maturity-$T$ bond pays one unit at $T$, its deflated price is a martingale:Division by $Y_t>0$ gives the formula.

For the bond maturing one period later,If $r_t\ge 0$, this is at most $Y_{t-1}$, which is exactly the supermartingale property.

Buy one maturity-$(T-1)$ bond. When it pays one at $T-1$, use all proceeds to buy $1/P_{T-1}^T$ maturity-$T$ bonds. Short $1+f$ maturity-$T$ bonds. The terminal payoff isIts initial replication cost iswhich vanishes for $f=P_0^{T-1}/P_0^T-1$.

With $P_0^0=1$, the time-zero value of payment $r_t-s$ isSumming over $t$ telescopes, so the swap value isThe par swap rate is therefore

A claim $\xi$ is replicated if some holdings $h\in {\mathbb{R}}^n$ satisfy $h\cdot P_1=\xi$ almost surely; its replication cost is $h\cdot P_0$. The market is complete when every claim in the specified integrability class can be replicated.

Every indicator random variable must lie in the linear span of the $n$ terminal asset prices. If the sigma-algebra had $n+1$ disjoint events of positive probability, their indicators would be linearly independent, yet completeness would place all of them in an at-most-$n$-dimensional space. Hence, modulo null events, the sample space has at most $n$ positive-probability atoms.

For every replicable claim $\xi =h\cdot P_1$,Completeness makes every bounded claim replicable. Taking increasing bounded truncations of $\mathrm{sgn}(X-Y)$, or using the finite-atom conclusion of part b directly, gives $\mathbb{E}|X-Y|=0$. Thus $X=Y$ almost surely.

The proposed variable satisfiesNo arbitrage and the fundamental theorem of asset pricing provide a strictly positive one-period deflator $Y$ with $\mathbb{E}(Y{P}_1)=P_0$. Part c makes such a deflator unique in a complete market, so $Z=Y>0$ almost surely.

Localize when the predictable integrand first exceeds level $n$, just before that increment is taken. The resulting stopped integrand is bounded, so its martingale transform is a martingale by the stated result. The localization times increase to infinity because each finite collection of $H_t$ is finite almost surely. Hence $X$ is a local martingale.

On $\{H_T\ne 0,X_{T-1}\le 0\}$,Dividing by $\Vert H_T\Vert$ proves $\xi \ge 0$ there; outside that event $\xi =0$.

If $X_{T-1}<0$, then $X_T\ge 0$ forces $H_T\ne 0$ and a strictly positive increment in the $H_T$ direction. If $X_{T-1}=0$, that directional increment is strictly positive exactly when $X_T>0$. These disjoint cases give

If $X_{T-1}=0$ almost surely, the definition of $T$ gives $\mathbb{P}(X_T>0)>0$. Part c then implies $\mathbb{P}(\xi >0)>0$.

If $\mathbb{P}(X_{T-1}<0)>0$, the second term in the identity of part c is positive, so $\mathbb{P}(\xi >0)>0$.

The assumptions of this case imply $X_{T-1}\ge 0$ almost surely and $\mathbb{P}(X_{T-1}>0)>0$. This would satisfy the defining condition for $T$ one period earlier, contradicting the minimality of $T$. Thus the case is impossible.

The one-step predictable integrand $\theta$ is bounded by one. If $M$ were a martingale, its transform $\xi =\theta \cdot (M_T-M_{T-1})$ would be an integrable mean-zero random variable. Parts b–d show instead that it is nonnegative almost surely and strictly positive with positive probability, so its expectation is positive. This contradiction proves that $M$ cannot be a martingale.

A predictable strategy $H$ is self-financing when its wealth $X=H\cdot P$ satisfiesIt is admissible when its wealth obeys the stipulated lower bound, here taken to be nonnegative. A strictly positive Itô process $Y$ is a martingale deflator when every deflated asset price $Y{P}^i$ is a local martingale.

The product rule and self-financing identity show that $XY$ is the stochastic integral of $H$ against the vector of deflated prices $YP$, hence is a local martingale. It is nonnegative by admissibility and positivity of $Y$. Every nonnegative local martingale is a supermartingale, so

The definition of the concave conjugate gives the pointwise Fenchel–Young inequalityApply it to $(X_T,Y_T)$, take expectations, and use part b:If $U^{\prime }(X_T)=Y_T$, the first inequality is equality; if $XY$ is a true martingale, the second is equality.

Applying the Itô product rule to $YB$ makes its drift vanish automatically. For $YS$, the drift isThus both deflated prices are local martingales whenSince self-financing gives $dX=\varphi dB+\pi dS$, another application of the product rule, including $d[X,Y]$, cancels the drift and yields

Normalize $Y_0=1/X_0$. Solving its stochastic differential equation givesFor logarithmic utility, $\hat{U}(y)=-\log y-1$. Part c and $\mathbb{E}{W}_T=0$ therefore giveIf ${\pi }_t=X_t\lambda /(S_t\sigma )$, part d gives $d(XY)=0$. Hence $X_t{Y}_t=X_0{Y}_0=1$, so $U^{\prime }(X_T)=1/X_T=Y_T$ and both inequalities in part c are equalities.

The two-dimensional Itô formula, using $d[W^X,W^Z{]}_t=\rho dt$, gives the drift of $U(t,Z_t,X_t)$ asThe PDE makes this zero, leaving only stochastic-integral terms. Thus $M_t=U(t,Z_t,X_t)$ is a local martingale.

Substitute $U=e^{\theta x}V$. After dividing by $e^{\theta x}$, the PDE becomeswith terminal condition $V(T,z)=1$.

Let $\tau =T-t$ and setThenFor $B(z)=a-bz$ and $C(z)=c$, matching constant, linear, and quadratic coefficients givesandThe terminal condition becomes $P(0)=Q(0)=R(0)=0$. The first equation is a Riccati equation; once it is solved, the second is linear in $Q$, followed by direct integration for $P$.