Past exam of the mathematics course of the University of Cambridge / 2001 / iii / Paper 23 (Mathematical Models in Financial Management)

The no-borrowing portfolio constraint is $0\le x\le T$. Net asset returns must be added to the original principal. Thus the two final portfolio wealths areThe expected utility isThe probabilities are those of the investor's beliefs; no risk-neutral measure is being used in this preference calculation.

Risk neutrality means indifference between a random portfolio wealth and its certain expected value. For increasing twice differentiable expected utility preferences, this corresponds to an affine function $u(w)=Aw+C$ with $A>0$ on the relevant wealth interval. Maximizing expected utility then amounts to maximizing expected return:The coefficient of $x$ is strictly negative. Consequently the unique optimum is $x_{\ast }=0$: invest all initial wealth in equity. Risk neutrality here is a preference property, distinct from the pricing use of a risk-neutral measure.

Risk aversion means that a sure mean is preferred to the corresponding risky portfolio wealth. An increasing concave utility function, with $u^{\prime }>0$ and $u^{\prime \prime }\le 0$, has this property by the Jensen inequality; strictly concave utility gives strict preference for nondegenerate risks. Differentiating the expected utility from part (a) givesAn interior optimum therefore satisfiesThis balances the expected loss of marginal utility in the good state against its expected gain in the bad state. The borrowing constraint also requires checking the endpoints: $x_{\ast }=0$ if $U^{\prime }(0)\le 0$, while $x_{\ast }=T$ would require $U^{\prime }(T)\ge 0$. At $x=T$ both state wealths equal $T(1+r)$, soHence an optimum never puts all wealth into deposits. If $U^{\prime }(0)>0$, continuity and the displayed endpoint sign give an interior root, and concavity makes every such root globally optimal. Under strict concavity it is unique.

Assume first that $T>0$ and $-1<b<r<g$, so the logarithmic utility is defined throughout the allowed portfolio interval. Put $A=g-r>0$ and $B=r-b>0$. ThenEquating this to zero and clearing the positive denominators givesThe second derivative is strictly negative, and $U^{\prime }(T)<0$ by part (c), so $x_0<T$. The two-state logarithmic portfolio with a borrowing constraint is thereforeIn particular, the threshold for all equity isFor $p<p_c$ the optimum is interior. The probability bound printed in the PDF has $1+r$ where the all-equity threshold has $1+b$. Since $1+r>1+b$, that printed lower bound is strictly below $p_c$ and does not determine which of these two cases occurs. For example, with $(b,r,g)=(0,0.1,0.3)$ its lower bound is $13/35$, while $p_c=13/33$. At $p=0.38$ all the given inequalities hold and $x_{\ast }=0.23T$; at $p=0.4$ they also hold and $x_{\ast }=0$. Thus the boxed constrained formula applies to the printed data without replacing its probability assumption.

If bad-state gross wealth can be nonpositive, the logarithmic utility domain must instead be imposed explicitly. With $1+r>0$, $b\le -1$, and $0<p<1$, admissibility requires $x>-T(1+b)/(r-b)$; the same stationary root lies inside this positive-wealth interval and is optimal. If even the fully safe gross return is nonpositive, there is no admissible positive-wealth portfolio. These domain issues are implicit in using $\log w$.

Use unrestricted long and short portfolios, as required by the usual arbitrage pricing theory argument. A vector $v\in {\mathbb{R}}^n$ of initial monetary positions with $1^{T}v=0$ costs zero. Its terminal financial payoff, in the exact one-factor model, isIf also $v^{T}b=0$, the financial payoff is the constant $v^{T}a$. A nonzero value would be an arbitrage after choosing the sign of $v$. Absence of arbitrage thus says that $a$ annihilates $\ker [1b{]}^T$. By finite-dimensional linear algebra,Write $a={\lambda }_{0}1+\kappa b$. Taking expectations, assuming the factor has finite mean, provesThis is exact factor pricing without a traded risk-free asset. The PDF prints an asset-indexed ${\lambda }_i$; the stronger valid result has the same ${\lambda }_1$ for every asset, so it also satisfies the printed relation by taking all its ${\lambda }_i$ equal. If every loading is equal, the spanning vectors are dependent and the coefficients need not be unique. For a nondegenerate factor and a zero-exposure unit-cost portfolio, ${\lambda }_0$ is that portfolio's certain return; if a risk-free asset is explicitly traded, law of one price makes it the risk-free asset's return. No equilibrium preferences are needed for this argument.

The capital asset pricing model givesThus it has the same affine relation between expected returns and a single systematic exposure. When the factor is the market portfolio's excess return, the loadings are the beta of an asset, ${\lambda }_0=r_f$, and ${\lambda }_1=\mathbb{E}{r}_M-r_f$. More generally, for a nondegenerate exact one-factor model and a market portfolio with loading $b_M\ne 0$, ${\beta }_i=b_i/b_M$, and the pricing relation can be rewritten using that exposure.

The interpretations and assumptions differ. The CAPM is an equilibrium relation obtained with a mean-variance optimization framework, common beliefs, and suitable borrowing/lending and market-clearing assumptions. Arbitrage pricing theory uses absence of arbitrage, factor structure, and in realistic models diversification; it does not identify a generic factor with the market portfolio. Moreover the CAPM permits asset-specific risk, whereas the exact model in part (a) has no residual risk. The two formulas coincide for a suitable market factor, but no-arbitrage factor pricing by itself does not establish the CAPM's equilibrium assumptions.

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.

For weights summing to one, collect the exact portfolio return asThe desired approximation means that $e_n\to 0$, for example in mean-square convergence. Centering the residuals gives $\mathbb{E}{e}_n=0$, but its variance isThis identifies an omission in the printed hypotheses: bounded individual variances do not control the cross terms. Indeed, take ${ϵ}_i=Z$ for every $i$, where $Z$ is a nondegenerate centered random variable with $\mathrm{Var}Z<s^2$. Equal weights $1/n<W/n$ for any $W>1$ retain $e_n=Z$ at every $n$. The claimed diversification conclusion does not follow from the printed assumptions alone.

Under the usual extra assumption that the residuals are pairwise uncorrelated and the weights are nonnegative, the advertised calculation isThe Chebyshev inequality then gives $\mathrm{Pr}(|e_n|>\delta )\le s^{2}W/(n{\delta }^2)\to 0$. Alternatively, signed weights satisfying $|w_i|\le W/n$ give the bound $s^2{W}^2/n$. These are instances of the covariance criterion for diversification of factor residuals. More generally, with that absolute weight bound, ${\sum }_{i,k}|\mathrm{Cov}({ϵ}_i,{ϵ}_k)|=o(n^2)$ suffices.

If short selling are allowed, the printed one-sided weight bound is another insufficiency. Choose $0<c<W-1$, put $w_1=-c$ and $w_i=(1+c)/(n-1)$ for $i\ge 2$, and let only ${ϵ}_1$ be a nondegenerate centered residual. For all sufficiently large $n$ the one-sided bounds hold, but $e_n=-c{ϵ}_1$ never vanishes. Thus nonnegative weights or an absolute bound is needed.

With the repaired assumptions, well-diversified portfolios become approximately exposed only to the common factors, and their residual standard deviation is $O(n^{-1/2})$ in the uncorrelated case. The averaged coefficients may depend on $n$; convergence to fixed coefficients needs additional assumptions. In the associated asymptotic arbitrage pricing theory, diversifiable risk cannot command a persistent premium in well-diversified portfolios under absence of asymptotic arbitrage, while common-factor exposure can. This does not prove an exact pricing equation for every individual noisy asset from the finite-market hypotheses alone.

On a filtered probability space carrying a standard Brownian motion $W$ with independent increments relative to the filtration, a strictly positive security price follows geometric Brownian motion with constant drift $\mu$ and volatility $\sigma$ if it solves the stochastic differential equationHere $W_t-W_s$ is independent of ${\mathcal{F}}_s$ and has the normal distribution $N(0,t-s)$ for $s<t$. The solution isIndeed applying the Itô formula to this exponential gives the specified drift and volatility. Consequently the price is positive and continuous, andThus disjoint log-return increments are independent and stationary. The parameter $\mu$ is the instantaneous expected relative return under the physical probability measure, not the mean log-return. Under the pricing risk-neutral measure it becomes the risk-free rate for a non-dividend-paying stock.

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.

For an Itô process $d{X}_t=a_{t}dt+b_{t}d{W}_t$ and $F\in C^{1,2}$, the differential Itô formula iswhere the derivatives are evaluated at $(t,X_t)$. A multidimensional version replaces $b_t^2{F}_{xx}$ by the contraction of the Itô diffusion covariance matrix with the spatial Hessian.

For a heuristic derivation, partition time and use the Euler increment $\Delta X=a\Delta t+b\sqrt{\Delta t}Z$, where $Z$ is standard normal. A second-order Taylor expansion givesThe term $b^2(\Delta W{)}^2$ has order $\Delta t$, so it survives summation; ordinary first-order calculus would incorrectly discard it. For independent Brownian motion increments,This gives the quadratic variation $[W{]}_t=t$. The summed mixed $dtdW$ terms and $d{t}^2$ terms vanish, while the linear random increments converge to an Itô integral. After localizing bounded coefficients and derivatives, Taylor remainders are negligible and the surviving terms give the formula. Thus $(dW{)}^2=dt$ is shorthand for a statement about summed quadratic variation, not an equality of individual random increments.

Assume a frictionless market with no taxes or transaction costs, unrestricted short selling and borrowing/lending at the same constant rate $r$, continuous trading, and absence of arbitrage. The non-dividend-paying stock has geometric Brownian motion dynamics $dS=\mu Sdt+\sigma SdW$ with constant $\sigma >0$, and the bank account satisfies $dB=rBdt$. A contingent claim with no interim cash flows has sufficiently regular value $f(t,S)$, meaning $f\in C^{1,2}$ before maturity; terminal financial payoff kinks can be handled by the smooth value for earlier times. Trading strategies are adapted, self-financing portfolios, and admissible trading strategies so that doubling strategies are excluded.

The Itô formula givesChoose $\Delta =f_S$ shares and put the remaining value $f-S{f}_S$ in the bank account. Its self-financing gain isMatching the contingent claim's gain to this replicating gain cancels the random increment and equates the drifts:This is the Black-Scholes equation. The argument uses the self-financing gain $\Delta dS$, not a product differential that incorrectly treats a changing $\Delta$ as free cash. Rebalancing purchases are financed from the bank account. Conversely, a smooth solution with suitable growth and boundary conditions defines a delta hedge whose value replicates the contingent claim; law of one price identifies this value with the contingent claim price. Here $S$ is the current stock price, $t$ is time, $\sigma$ is proportional volatility, $r$ is the risk-free rate, and $f$ is the contingent claim price. The physical expected return $\mu$ cancels because exposure to price risk has been hedged.

For strike $K>0$ and maturity $T$, a European call option and a European put option have boundary conditionsThe Black-Scholes equation is solved backward from these conditions. At zero stock price, the zero state is absorbing in geometric Brownian motion, soAt the other boundary, with $\tau =T-t>0$,These conditions and a suitable growth bound select the financial solution and provide boundary data for numerical truncations. In the constant-coefficient model, under the risk-neutral measure,For $d_1=[\log (S/K)+(r+{\sigma }^2/2)\tau ]/(\sigma \sqrt{\tau })$ and $d_2=d_1-\sigma \sqrt{\tau }$, the exercise event is $Z>-d_2$. Completing the square in $e^{\sigma \sqrt{\tau }Z}$ times the normal density gives the truncated first moment and hencewhere $\Phi$ is the standard normal cumulative distribution function. These are the Black-Scholes formula values and satisfy the specified terminal and boundary conditions.

Put-call parity is the relationA call minus a put has terminal financial payoff $S_T-K$. A portfolio of one share minus a zero-coupon bond paying $K$ has the same financial payoff, so absence of arbitrage and law of one price equate their current values. The formula here assumes the same strike and maturity and no dividends; known dividends modify the stock-side value.

An American option allows exercise at a stopping time $\tau \in [t,T]$. For its exercise financial payoff $H$, the pricing problem isThe discounted value is the Snell envelope of discounted exercise payoffs. Define the pricing operatorIn the continuation region the Black-Scholes equation holds, $\mathcal{L}V=0$. In the exercise region $V=H$, while the ability to wait and the supermartingale property give $\mathcal{L}V\le 0$. Together these conditions are the obstacle problemThus an unknown exercise boundary must be found alongside the value. At a regular boundary in the nondegenerate Itô diffusion, value matching is $V=H$, and smooth pasting is $V_S=H^{\prime }$ where the financial payoff is differentiable. A finite difference method or a binomial options pricing model steps backward taking the larger of continuation and immediate exercise values at each node.

For a non-dividend-paying American call option with $r\ge 0$, no early exercise of a call without dividends follows directly from the European lower boundAt every candidate exercise time, keeping the European call has value at least the immediate exercise financial payoff, so early exercise cannot improve the value; the American and European prices coincide. The first inequality follows from Jensen inequality or put-call parity and nonnegative put value. With $r>0$ an American put option can benefit from early receipt of its strike at sufficiently low stock prices, so its exercise boundary is generally nontrivial and its value is at least the European put value. Dividends can make call exercise worthwhile, and negative interest invalidates the stated no-early-exercise call argument.

For a continuous strictly increasing cumulative distribution function $F$ and a uniform distribution $U$ on $(0,1)$, monotonicity givesThus each transformed draw has distribution function $F$. This is inverse transform sampling. The conclusion extends to discontinuous or non-strictly increasing $F$ by its quantile function $Q(u)=\inf \{x:F(x)\ge u\}$: right continuity gives $Q(u)\le y$ exactly when $u\le F(y)$. Independent uniforms produce independent transformed draws, but marginal uniformity alone does not establish independence.

If “uniformly distributed sequence” instead means deterministic equidistribution, the empirical fraction of ${\eta }_i\le y$ is the empirical fraction of ${\xi }_i\le F(y)$ and tends to $F(y)$. That is an empirical distribution statement, not a claim that the deterministic sequence consists of independent random variables.

One method applies inverse transform sampling to the standard normal cumulative distribution function: $Z_i={\Phi }^{-1}(U_i)$, followed by $X_i=\mu +\sigma Z_i$ for mean $\mu$ and standard deviation $\sigma >0$. Numerical approximations to ${\Phi }^{-1}$ are needed, and the endpoints $U=0,1$ must be avoided.

A convenient alternative is the Box-Muller transform. From independent uniforms $U_1,U_2\in (0,1)$ setThe radial probability density function is $r{e}^{-r^2/2}$ for $r>0$, and the angle is independently uniform on $[0,2\pi )$. Dividing their joint density by the polar-coordinate Jacobian determinant $r$ givesThus the outputs are independent standard normal distribution draws. Apply $\mu +\sigma Z_j$ to obtain other normal means and variances. With finite pseudorandom number generators, this is an approximation to the ideal independent-uniform model; poor dependence in the input stream is not cured by the transform.

Let $Y=h(U)$ have finite variance $v$ and mean $I$. An independent-sample Monte Carlo estimator has variance $v/N$, so its root-mean-square random error is $\sqrt{v/N}$. Three techniques reduce the variance coefficient rather than changing this ordinary $N^{-1/2}$ rate.

For antithetic variates, couple $Y=h(U)$ with $Y^{\prime }=h(1-U)$ and average the pair. With $M$ independent pairs, the variance is $(v+\mathrm{Cov}(Y,Y^{\prime }))/(2M)$, compared with $v/(2M)$ for $2M$ independent function evaluations. If $h$ is monotone in a scalar uniform, the covariance is nonpositive: for independent uniforms $U,V$,Opposite fluctuations therefore cancel. The reduction factor at equal evaluation count is $1+\rho$, where $\rho$ is the paired correlation coefficient; it can approach zero, but positive covariance would make the method worse. This is appropriate for monotone or otherwise negatively coupled payoffs, with cheap complementary paths.

For control variates, use a jointly simulated $C$ with known mean $m_C$ and average $Y-\beta (C-m_C)$. It remains an unbiased estimator for fixed $\beta$, andCompleting the square gives ${\beta }_{\ast }=\mathrm{Cov}(Y,C)/\mathrm{Var}C$ and minimal variance $v(1-{\rho }^2)$. For contingent claim simulation, an analytically priced related financial payoff, such as a simpler option or discounted terminal stock, can be a useful control. It works best when correlation has large absolute value and simulation of the control adds little cost. A coefficient fixed from an independent pilot keeps the elementary unbiasedness argument valid; estimating it from the same sample can introduce finite-sample bias.

For stratified sampling, partition the sampling space into strata $A_h$ of probabilities $p_h$, and independently simulate $N_h$ samples conditional on each stratum. The estimator ${\sum }_h{p}_h{\bar{Y}}_h$ is unbiased and has varianceWith proportional allocation $N_h=N{p}_h$, ignoring integer rounding, this equals ${\sum }_h{p}_h{v}_h/N$. The law of total variance shows the reduction from ordinary sampling is $\mathrm{Var}(\mathbb{E}[Y\mid A_h])/N$: fixed representation removes randomness in the stratum counts. At equal per-draw cost, minimizing the displayed variance gives $N_h\propto p_h\sqrt{v_h}$ and minimum $({\sum }_h{p}_h\sqrt{v_h}{)}^2/N$ for positive within-stratum variances. This follows by Cauchy-Schwarz inequality and is useful when a low-dimensional coordinate strongly predicts the financial payoff and conditional simulation is easy.

There is no universal ranking. At comparable cost the variance factors depend respectively on paired covariance, squared control correlation, and within-stratum variation. A nearly perfect control can outperform a weak antithetic pair; an appropriate stratification can be almost exact for a nearly stratum-constant financial payoff. Extra construction and sampling costs must be included in a fair accuracy comparison, and the methods can be combined.

The instantaneous forward rate is defined by $f(t,T)=-{\partial }_T\log P(t,T)$, with $P(t,t)=1$. Integrating in maturity gives the zero-coupon bond price and its diagonal short rate:The solution of the continuous-time bank account equation $d{B}_t=r_t{B}_{t}dt$, $B_0=1$, is $B_t=\exp ({\int }_0^t{r}_{s}ds)$; the integration variable is time. Its discounted bond value isAll these are adapted processes when the Heath-Jarrow-Morton model coefficients are adapted and the integrals exist.

For market completeness, stochastic integration over maturity givesThe Itô formula for the exponential then yieldsConsequently under a risk-neutral measure the no-arbitrage drift restriction isThe last equality needs sufficient maturity regularity and has a positive sign despite the negative definition of $\Sigma$.

Under the specified risk-neutral measure, the Itô product rule and $d(B_t^{-1})=-r_t{B}_t^{-1}dt$ giveThus discounted traded zero-coupon bonds have zero drift. Assume the usual integrability and admissibility conditions that make discounted replicating values true martingales, not merely local martingales. If $X$ is an attainable financial payoff with ${\mathbb{E}}_Q|B_T^{-1}X|<\infty$, its self-financing portfolio value $V_t$ has terminal value $V_T=X$ and satisfiesMultiplying by $B_t$ yieldsThis proves the requested conditional pricing formula. In a Brownian filtration, the Martingale representation theorem realizes the discounted conditional expectation as a stochastic integral. If an available bond has nonzero Itô diffusion exposure, its holdings can match that integral, with the remaining value in the bank account, giving replication under the usual square-integrability conditions.

These attainability or market completeness hypotheses matter for a completely arbitrary contingent claim: existence of a risk-neutral measure alone need not imply a unique price for every unspanned financial payoff, and a zero-drift local martingale alone need not satisfy the displayed terminal expectation identity. In the general path-dependent Heath-Jarrow-Morton model, write the value as $V_t$; notation $V(t,r_t)$ is justified only when the current short rate is a sufficient state for the financial payoff and dynamics.

Suppose the forward field is differentiable in maturity with the stochastic integrability needed for differentiation. As time increases, both arguments of $r_t=f(t,t)$ increase. The diagonal short-rate dynamics in the Heath-Jarrow-Morton model are thereforeUnder the HJM drift restriction from part (a), ${\alpha }^Q(t,t)=0$, givingMore explicitly the slope isIt depends on the forward curve and potentially the whole history, not necessarily on $r_t$ alone. Without the stated maturity regularity, the original forward equations do not by themselves guarantee this classical Itô process formula for the diagonal.

The short rate need not be Markov, even with one Brownian factor and deterministic volatility. An explicit example of One Brownian factor does not imply a Markov short rate takes $f_0(T)=0$ and $\sigma (t,T)=T-t$. Its no-arbitrage forward drift is ${\alpha }^Q(t,T)=(T-t{)}^3/2$, soThe continuous rate history determines $W_t$ from the left derivative of $I_t$. Hence for $h>0$,But $W_t$ is not determined by $I_t$: joint normality, $\mathrm{Var}{I}_t=t^3/3$, and $\mathrm{Cov}(W_t,I_t)=t^2/2$ giveThus the conditional mean of the future given the whole rate history cannot be a function of the current rate alone, contradicting the Markov property. Enlarging the state to $(I_t,W_t)$ gives a two-dimensional Markov process; special HJM specifications can instead close on a single short rate.