Past exam of the mathematics course of the University of Cambridge / 2010 / iii / Paper 39 / 6

Write $R=1+r$, $m=\mathbb{E}{\xi }_1$ and $c=\mathrm{Cov}(S_1,{\xi }_1)=\mathbb{E}[(S_1-\mu )({\xi }_1-m)]$. The riskless gross return must be nonzero for the requested formulas; in the usual positive-bank-account model $R>0$. If $R=0$, the bank holding cannot affect the terminal payoff, and the later moment equation for pricing $B_0=1$ is impossible. We use the intended nondegenerate case $R\ne 0$.

The claim and asset prices are square integrable. Centering separates the bias from the stochastic error:For fixed $\pi$, the unique minimizing bank holding removes the bias. The invertible covariance matrix $V$ is positive definite, and completing the square givesConsequently the unique one-period quadratic hedge isIts minimum expected squared error is $\mathrm{Var}({\xi }_1)-c^T{V}^{-1}c$. This is a least-squares hedge, not necessarily exact claim replication.

Put $h=S_0-\mu /R$. The initial capital of the one-period quadratic hedge isDefine the minimum-norm one-period pricing weightIt is square integrable, and $\mathbb{E}[(S_1-\mu ){\xi }_1]=c$, soThe weight depends only on the asset prices and their first two moments, not on the claim. It can be negative, so this identity alone does not supply an equivalent martingale measure or a positive state-price density.

The centered term in ${\rho }^{\ast }$ has zero mean, giving $\mathbb{E}{\rho }^{\ast }=1/R$ and hence $\mathbb{E}[{\rho }^{\ast }{B}_1]=1=B_0$. For the risky assets, symmetry of the covariance matrix givesTherefore ${\rho }^{\ast }$ reproduces every asset's initial price.

For another square-integrable $\rho$ satisfying the price moment constraints, set $\delta =\rho -{\rho }^{\ast }$. The bank equation and $R\ne 0$ give $\mathbb{E}\delta =0$, while the risky equations give $\mathbb{E}[\delta S_1]=0$. Since ${\rho }^{\ast }$ is an affine combination of $1$ and the coordinates of $S_1$,The Pythagorean theorem in an inner-product space now givesEquality holds exactly when $\rho ={\rho }^{\ast }$ almost surely. If a competing weight has infinite second moment, the inequality is immediate in the extended sense. The minimum itself isThus ${\rho }^{\ast }$ is the orthogonal projection of any feasible square-integrable pricing weight onto the linear span of the traded payoffs.

Let $A=V^{1/2}$ be the symmetric positive square root of the covariance matrix, and write $S_1=\mu +AZ$ with $Z$ having the multivariate normal distribution $N_d(0,I)$. Bounded gradient makes $g$ globally Lipschitz, so $g(S_1)$ has at most linear growth and is square integrable. This also justifies the permitted Gaussian integration by parts formula for each coordinate.

By the chain rule, ${\nabla }_{z}g(\mu +Az)=A^T\nabla g(\mu +Az)$. Applying Gaussian integration by parts givesThus the Gaussian quadratic hedge holds ${\pi }^{\ast }=\mathbb{E}[\nabla g(S_1)]$. Substitution into its initial capital formula yieldsOnly the hedge calculation is being used here; the Gaussian model need not possess a positive pricing weight for this least-squares identity.