Past exam of the mathematics course of the University of Cambridge / 2012 / iii / Paper 43 / 3

For centered square-integrable variables with positive variances, the correlation coefficient isPut $v=\sqrt{\mathbb{E}[X^2]}$ and $v^{\prime }=\sqrt{\mathbb{E}[(X^{\prime }{)}^2]}$. ThenThus $\boxed{\rho =1⟺X=(v/v^{\prime })X^{\prime }\text{ almost surely}}$, with positive proportionality factor. Conversely positive proportionality immediately gives correlation one. For variables with nonzero means, this criterion applies to their centered versions and yields an affine, not necessarily proportional, relationship. That distinction is essential for the positive stock prices below.

Assume positive deterministic initial stock prices and finite, positive terminal variances, as required to define the correlation coefficient. The explicit stochastic-exponential solutions with constant volatilities giveThe common, possibly random, accumulated interest rate cancels. If $\sigma ={\sigma }^{\prime }$, this ratio is a positive constant, so the prices are perfectly correlated.

Conversely, perfect correlation would give $S_t=c{S}_t^{\prime }+d$ almost surely for constants $c>0,d$. If $\sigma \ne {\sigma }^{\prime }$, the displayed ratio is a nonconstant variable with a lognormal distribution with full support $(0,\infty )$. If $d>0$, the ratio would always exceed $c$, and if $d<0$ it would always be below $c$, both impossible. If $d=0$, the ratio would be constant, also impossible. Therefore $\boxed{\rho (S_t,S_t^{\prime })=1⟺\sigma ={\sigma }^{\prime }}$, whenever the stated terminal variances exist and are positive. No deterministic-interest assumption is needed. Degenerate terminal prices, for example zero volatilities and deterministic rates, have undefined correlation and must be excluded.

The principle terminal correlation does not identify adapted stock volatility exposes a false printed assertion for the correlation coefficient defined in the introduction. Take $r=0$, $S_0=S_0^{\prime }=1$, andThen $d{S}_t=S_{t}d{B}_t$ andHence ${\sigma }_t=1$ and ${\sigma }_t^{\prime }=S_t/(1+S_t)$ are bounded. Both terminal prices have positive finite variance and are perfectly correlated, because one is a positive affine function of the other. Nevertheless $\boxed{{\int }_0^1({\sigma }_t-{\sigma }_t^{\prime }{)}^{2}dt={\int }_0^1\frac{dt}{(1+S_t{)}^2}>0}$ on every continuous finite stock path. Equal initial prices fix the means, not the affine slope.

Here is the exact general conclusion and the intended qualified proof. Bounded volatilities make these zero-rate stochastic exponentials true square-integrable martingales. Writing $s_0=S_0=S_0^{\prime }$, perfect terminal correlation gives $S_1-s_0=c(S_1^{\prime }-s_0)$ for some $c>0$. Conditional expectation therefore gives $S_u-s_0=c(S_u^{\prime }-s_0)$ for every $0\le u\le 1$. Comparing stochastic integrals, the Itô isometry yieldsThis relation need not imply equality of the volatilities.

The terminal proportionality identifies bounded stock volatility criterion repairs the statement. If one additionally assumes equal terminal variances, then $c=1$ and the whole stock paths coincide, so positivity gives $\sigma ={\sigma }^{\prime }$ almost everywhere and the requested integral is zero. The same repair follows if terminal proportionality $S_1=c{S}_1^{\prime }$ is assumed: equal martingale means force $c=1$, after which the conditional-expectation and Itô isometry argument applies. This proves the intended conclusion under a sufficient additional hypothesis without treating centered correlation as uncentered proportionality.

Yes: different adapted volatilities can give perfectly correlated terminal prices. The counterexample in part ii already establishes this with one nonconstant volatility. Both can be nonconstant as well. Put $h(x)=1+\frac{1}{2}\tanh x$ and takeThe Doléans-Dade exponential has volatility $h(B_t)$, bounded between $1/2$ and $3/2$. The second stock has volatility $h(B_t)S_t/(1+S_t)$, also bounded and nonconstant. Their squared volatility difference integrates toThe Novikov condition and boundedness give true square-integrable martingales, and the Itô isometry gives positive variance for each $t>0$. Yet $S_t^{\prime }=(1+S_t)/2$ gives correlation one. The mechanism is positive affine dependence, not equality of the multiplicative volatility coefficients; it also explains the failure of the literal part-ii premise.