Past exam of the mathematics course of the University of Cambridge / 2001 / iii / Paper 23 / 3

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.