Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 32 (Advanced Financial Models)

A one-period self-financing strategy is a deterministic holdings vector $h=(h^0,\dots ,h^d)$, with initial cost $h\cdot S_0$ and terminal wealth $h\cdot S_1$. In the zero-initial-cost convention, an arbitrage satisfiesThe convention allowing a nonpositive initial cost is equivalent here: invest any initial proceeds in the strictly positive numéraire to obtain a zero-cost portfolio with nonnegative terminal wealth.

Suppose a strictly positive state-price density $\rho$ satisfies the asset-pricing equations, with the indicated products integrable. By linearity,For an arbitrage, the right-hand side is zero while the left-hand side is strictly positive, since the integrand is nonnegative and is positive with positive probability. This contradiction proves absence of arbitrage.

An attainable European contingent claim $X$ is a terminal random payoff for which some holdings vector $h$ satisfies $h\cdot S_1=X$ almost surely; its replication cost is $h\cdot S_0$. To prove uniqueness in a complete market, suppose ${\rho }^{\prime }$ is another positive state-price density. For every event $A$, attainability gives a portfolio replicating $S_1^{(0)}{1}_A$. Its initial cost has both representations, soThe signed random variable here is integrable, because $\mathbb{E}[\rho S_1^{(0)}]=\mathbb{E}[{\rho }^{\prime }{S}_1^{(0)}]=S_0^{(0)}<\infty$. Take $A$ to be the event where it is positive and then where it is negative. Both its positive and negative parts have expectation zero, and hence vanish almost surely. Since $S_1^{(0)}>0$,This proves that complete markets have unique state-price densities without requiring an extra lower bound on the random terminal numéraire.

For the finite-state example, order the states from top to bottom in the diagram and write the density values as ${\rho }_1,{\rho }_2,{\rho }_3$. The two pricing equations areAdding gives ${\rho }_1+{\rho }_2+{\rho }_3=3$. Put $t={\rho }_3$; substitution then givesThese are all the strictly positive solutions, and each satisfies both equations. The parameter is a state-price-density value, not a probability.

For a concrete unattainable contingent claim, take the payoff vector $(0,1,0)$. A replicating portfolio $(a,b)$ would have to satisfy $3a+6b=0$ and $6a+3b=0$ in the first and third states. These imply $a=b=0$, contradicting the required payoff one in the middle state. Thus the market is an incomplete market, despite having strictly positive state-price densities.

Use the natural filtration ${\mathcal{F}}_t=\sigma (Z_1,\dots ,Z_t)$ and the usual integrable definition of the Snell envelope. Since the factors are positive, its terminal value is $U_T=Y_T$. If $U_{t+1}=Y_{t+1}{c}_{t+1}$, independence of random variables givesThe backward recursion therefore becomesThis proves the assertion by backward induction. All constants are finite and at least one. More explicitly,where the empty product is one. This is the Snell envelope of a multiplicative process. The standard Snell envelope hypothesis includes integrable rewards; independence of random variables and positivity then give finite first moments of the factors. If that hypothesis is dropped, an infinite-mean factor gives an infinite continuation value, so finite constants need not exist.

For the discrete-time binomial market, write $p=(1+r/\epsilon )/2$. The assumptions give $0<p<1$, positive bank account and stock prices, andThusThe finite discrete-time fundamental theorem of asset pricing states that a market with a strictly positive numéraire has no arbitrage if there is an equivalent probability measure under which all numéraire-denominated asset prices are martingales. The original probability measure is already such an equivalent martingale measure, proving no arbitrage.

The complete-market American-option pricing theorem identifies the discounted replication or superhedging value with the Snell envelope of the discounted exercise rewards. HereThe independent factors have common meanThe Snell envelope recursion consequently gives $c_t={\lambda }^{T-t}$. At every time before $T$, the continuation value strictly exceeds immediate exercise, so optimal exercise is at maturity. The undiscounted value process is $V_t=S_t^2{\lambda }^{T-t}$, and the time-zero replication cost isIn fact $V_t/B_t$ is a martingale, not just a supermartingale, because the Snell envelope has no exercise compensator before maturity.

To hedge the optimally exercised payoff, subtract the two next-state portfolio equations. The first-period delta hedge isThe corresponding bank account holding, since $B_0=1$, is $V_0-{\Delta }_0{S}_0=S_0^2{\lambda }^{T-1}(\lambda -2)$. Substitution into either next-state equation verifies both terminal values at time one. Repeating this two-state calculation at every node gives a self-financing strategy replicating the maturity payoff. It also covers earlier exercise: $V_t\ge S_t^2$ at all dates, so the seller can meet any earlier payment with the available wealth.

On a fixed finite horizon, a self-financing strategy consists of predictable stock holdings $h^i$, integrable against the price processes, and a bank account holding $h^0$. Its wealth and gains satisfyAn admissible trading strategy has discounted wealth $X_t/B_t$ bounded below by a deterministic constant throughout the horizon. With the bounded short rate in this part, an undiscounted deterministic lower bound is equivalent after adjusting the constant. An arbitrage is an admissible zero-initial-cost strategy with $X_T\ge 0$ almost surely and $\mathbb{P}(X_T>0)>0$.

Put $1=(1,\dots ,1{)}^{\top }$ and define the market price of riskThe bounded coefficients and bounded inverse make $\lambda$ bounded. Define the stochastic exponentialThe Novikov condition states that $\mathbb{E}\exp (\frac{1}{2}{\int }_0^T\Vert {\lambda }_s{\Vert }^{2}ds)<\infty$ makes this exponential a true martingale with expectation one. Boundedness verifies the condition. Therefore $dQ=L_{T}dP$ defines an equivalent probability measure. The Girsanov theorem states thatis a $d$-dimensional Brownian motion under $Q$. Substituting into the asset equations givesThus $Q$ is an equivalent local martingale measure. The sufficient direction of the fundamental theorem of asset pricing says that such a measure excludes arbitrage among admissible self-financing strategies. Here the reason can be seen directly: $X/B$ is a local martingale under $Q$, and its lower bound makes it a supermartingale. Consequently $X_0=0$ implies ${\mathbb{E}}_Q[X_T/B_T]\le 0$, incompatible with the nonnegative terminal payoff of an arbitrage. Therefore the bounded-inverse market has no arbitrage.

For the requested law of one price counterexample, take a broader continuous-time local-martingale model on $[0,T]$. Let $X$ be a Brownian motion, let $\tau =\inf \{u\ge 0:X_u=-1\}$, and set $q(t)=t/(T-t)$ for $t<T$. DefineThe given hitting-time fact gives $\tau <\infty$ almost surely. The corresponding physical hitting time is $T\tau /(1+\tau )<T$, so $S$ reaches one and then stays one. In particular it is continuous at maturity and $S_t\ge 1$.

For a rigorous closed-horizon localization, stop additionally when $S$ hits $n+2$, or at $T$ if it never does. The stopped process is Brownian motion stopped on first exiting $(-1,n)$ and subjected to the deterministic clock; it is a bounded martingale, including at $T$. The stopping times increase to $T$, since every Brownian path on the finite random interval $[0,\tau ]$ has a finite maximum. Thus $S$ is a positive local martingale on the whole horizon. With the time-changed Brownian filtration, the original probability measure is an equivalent local martingale measure. The admissible-wealth supermartingale argument above proves no admissible arbitrage.

Nevertheless $S_T=B_T=1$ while $S_0=2\ne B_0=1$, so the law of one price fails. Shorting one share and buying two units of cash has zero initial cost and terminal gain one, but its wealth is $2-S_t$. It has no deterministic lower bound: for every $n$, the probability of Brownian motion hitting $n$ before $-1$ is $1/(n+1)>0$, obtained by optional stopping of the bounded stopped Brownian motion. That apparently profitable strategy is therefore inadmissible. The counterexample deliberately uses a strict local martingale; the bounded-coefficient class in the first part has true discounted asset martingales and does satisfy the law of one price.

The boundedness of $a^{\prime }$ makes $a$ globally Lipschitz, so the stock stochastic differential equation has a nonexplosive strong solution. Since $a$ is bounded, $S_t-S_0={\int }_0^{t}a(S_u)d{W}_u$ is a square-integrable martingale on every finite horizon. Apply Itô formula to the option price:Thus both the stock and the option are local martingales under the original probability measure, while cash is constant. The relevant no-arbitrage theorem is the sufficient direction of the fundamental theorem of asset pricing: a common equivalent measure under which all numéraire-denominated traded prices are local martingales excludes arbitrage by admissible trading strategies. In this case the original measure itself is an equivalent local martingale measure. Hence the augmented market has no admissible arbitrage. The nonnegative option-value assumption also makes the replicating wealth below admissible.

SetDifferentiating the pricing equation in the stock-price variable givesThe limiting terminal derivative is $1_{\{S>K\}}$ away from the strike. Its value at the strike may be assigned as the stated $1_{\{S\ge K\}}$: the smooth positive diffusion coefficient gives a continuous terminal distribution, so that point has probability zero. This is the call delta equation for a driftless local volatility diffusion.

Hold ${\pi }_t$ shares and ${\eta }_t=V(t,S_t)-{\pi }_t{S}_t$ units of cash. The portfolio value is $V(t,S_t)$, and the preceding Itô calculation gives $dV={\pi }_{t}d{S}_t+{\eta }_{t}d{B}_t$, with $d{B}_t=0$. It is therefore a self-financing portfolio, with initial wealth $V(0,S_0)$ and terminal wealth $(S_T-K{)}^{+}$. This proves replication by the stated delta hedge.

A further application of Itô formula, now to $U$, yieldsThe density process is the strictly positive stochastic exponentialIts integrand is bounded, so the Novikov condition proves that it is a true martingale. The product formula for $M=Z\pi$ givesThe drift in $Zd\pi$ cancels this quadratic covariation term, leavingThus $M$ is a local martingale; the expression also identifies its stochastic integral integrand explicitly.

Under the stated assumption that $M$ is a true martingale on $[0,T]$, its terminal value givesSince $0\le Z_T{1}_{\{S_T\ge K\}}\le Z_T$ and $\mathbb{E}[Z_T\mid {\mathcal{F}}_t]=Z_t>0$, division givesThis is the derivative-weighted call delta martingale argument. It uses true martingality through maturity, not just local martingality before the terminal date.

For $v>0$ and $m>0$, let $\Phi$ and $\varphi$ be the standard normal distribution function and density, and putThe exercise event is $Z>(\log m+v/2)/\sqrt{v}=-d_2$. Complete the square in the first truncated expectation:The probability of exercise is $\Phi (d_2)$. Hence the normalized Black-Scholes call function isAt $v=0$ it is $(1-m{)}^{+}$. If $m\le 0$, the positive underlying always exceeds $m$, so $F(v,m)=1-m$; the reciprocal identity below is understood on its natural domain $m>0$.

For the reciprocal strike, $d_1(v,1/m)=-d_2(v,m)$ and $d_2(v,1/m)=-d_1(v,m)$. ThusThereforeThe $v=0$ case follows either by continuity or by checking the two positive parts directly.

For the stock model, the original probability measure is already risk-neutral: $e^{-rt}{S}_t=S_0{e}^{\sigma W_t-{\sigma }^{2}t/2}$ is a true martingale. Write $\tau =T-t$. Conditional on ${\mathcal{F}}_t$, the stock at maturity iswith $Z$ independent of current information and standard normal distribution. ThereforeEquivalently,which is a true martingale, since the payoff is integrable. The sufficient direction of the fundamental theorem of asset pricing says that an equivalent measure making all discounted traded prices local martingales excludes arbitrage by admissible trading strategies. The original measure works simultaneously for the bank account, stock and call. Thus the call-augmented market has no arbitrage.

For the put, hold one call, short one share, and hold $K{e}^{-rT}$ units of the bank account. This self-financing portfolio has current value $C_t-S_t+K{e}^{-r\tau }$ and terminal payoffIts discounted value is also the conditional expectation of that payoff. Thus the put-call parity priceadds another discounted martingale to the same market, proving no arbitrage with both call and put included.

Now put $m=K{e}^{-r\tau }/S_t$. The reciprocal identity givesThe resulting Put-call symmetry in the Black-Scholes model isThe final formula printed in the PDF has two errors: its discount-rate symbol is undefined, and the reciprocal argument has the wrong sign in the interest-rate exponential. Both are resolved by the displayed calculation. Even replacing the undefined symbol by $r$ leaves an incorrect expression with $e^{-r\tau }$ inside $F$. For $\tau >0$, $r>0$ and $\sigma >0$, $F(v,m)$ is strictly decreasing in $m$, because ${\partial }_{m}F=-\mathbb{P}(e^{-v/2+\sqrt{v}Z}>m)<0$. The printed argument is smaller than the reciprocal argument and hence gives a strictly larger put price. Thus the literal printed symmetry cannot be established; the boxed expression is the corrected identity.

A unit-face-value zero-coupon bond pays one unit of currency at its maturity $T$ and makes no earlier payments. Write its time-$t$ price as $P(t,T)$, with $P(T,T)=1$. The instantaneous forward rate is related to the bond price byThe short rate is $r_t=f_t(t)$, and the money-market account is $B_t=\exp ({\int }_0^t{r}_{s}ds)$.

Apply Itô formula to $f_t(T)=F(T-t,r_t)$, holding maturity fixed:The supplied equation identifies its volatility and drift asThis is exactly the one-factor Heath-Jarrow-Morton model drift restriction with zero market price of risk. The initial condition also verifies $f_t(t)=F(0,r_t)=r_t$.

To verify absence of arbitrage directly, without merely naming the drift restriction, putIntegrate the equation in its first variable. Since $F(0,r)=r$ andwe obtainThe bond price from the forward curve is $P(t,T)=e^{-G(T-t,r_t)}$. Its logarithm therefore satisfiesA second application of Itô formula givesThus every discounted bond price is a positive local martingale under the original measure. The sufficient no-arbitrage theorem is: if a common equivalent probability measure makes all discounted traded prices local martingales, then there is no arbitrage using self-financing strategies whose discounted wealth has a deterministic lower bound. Its proof is the supermartingale bound on such wealth. Apply it to any finite collection of bond maturities, using the original measure itself. This establishes no admissible arbitrage in the bond market, on horizons where the stated short rate model and smooth forward curve are defined. It is the Forward-rate equation for a Markov short-rate diffusion construction.

For the constant-coefficient case, substitute $F(\theta ,r)=A(\theta )r+B(\theta )$. The initial condition requires $A(0)=1$ and $B(0)=0$. Since $F_r=A$ and $F_{rr}=0$, the equation becomesThe right-hand side does not depend on $r$, so $A^{\prime }=0$ and $A=1$. It follows that $B^{\prime }=a_0-b_0^2\theta$, givingThe corresponding zero-coupon bond price isIt has discounted volatility $-b_0(T-t)$, which is deterministic and square-integrable on finite horizons. Its discounted price is consequently a true martingale, so the no-arbitrage conclusion follows again. This is the Gaussian short-rate model with constant coefficients.

As a separate check of the signs, conditional on ${\mathcal{F}}_t$,The final integral is centered Gaussian with variance $(T-t{)}^3/3$, obtained by integrating $\mathrm{Cov}(W_s-W_t,W_u-W_t)=\min (s-t,u-t)$. Its negative exponential expectation gives exactly the positive $b_0^2(T-t{)}^3/6$ term in the bond-price exponent above.