Past exam of the mathematics course of the University of Cambridge / 2007 / iii / Paper 41 / 2

Consider a binomial market with dates $0,\dots ,n$, bank account $B_k=b^k$ for $b>0$, and stock price $S_{k+1}=S_{k}u$ or $S_{k}d$, where $0<d<u$. Both moves have positive physical probability, and the information at date $k$ is the history of the first $k$ moves. Absence of arbitrage requires and is implied by $d<b<u$. If $b\le d$, borrowing cash to buy stock has nonnegative payoff with a strict gain in one state; if $b\ge u$, the reverse trade does. In the interior range, the risk-neutral probability of an up move isIt makes ${\mathbb{E}}_Q[S_{k+1}\mid {\mathcal{F}}_k]=b{S}_k$, so $S_k/B_k$ is a martingale. This proves absence of arbitrage by pricing discounted gains under the equivalent measure.

At a node with stock price $s$, let the desired next-period portfolio values be $V_u,V_d$. Solve the two replication equations $\Delta su+\beta B_{k+1}=V_u$ and $\Delta sd+\beta B_{k+1}=V_d$. The replicating portfolio in a binomial market hasIts current value isRebalance at the next node using exactly the value supplied by the previous holdings, so the strategy is self-financing. Backward induction from any terminal contingent claim $H$ gives its hedge and the unique priceThus the finite binomial market is a complete market. The construction works for path-dependent claims on the full history tree, even when their values do not recombine merely as functions of the current stock price. For $H=f(S_n)$, independence under $Q$ gives the explicit time-zero priceThe stock hedge is a difference quotient of successor values, the discrete analogue of the delta hedge.

Suppose under the physical measure $P$ the up indicators are independent with probability $p\in (0,1)$. If $N_k$ is the number of up moves through date $k$, a particular history has probability $p^{N_k}(1-p{)}^{k-N_k}$ under $P$ and $q^{N_k}(1-q{)}^{k-N_k}$ under $Q$. Thus the binomial-market probability density, the Radon-Nikodym derivative on ${\mathcal{F}}_k$, isIt is strictly positive. Its next multiplier has conditional expectation $p(q/p)+(1-p)((1-q)/(1-p))=1$, so $L_k$ is a mean-one martingale and $L_k={\mathbb{E}}_P[L_n\mid {\mathcal{F}}_k]$. These facts verify normalization and equivalence of the two probability measures.

The state-price density for terminal cash is ${\zeta }_n=L_n/B_n=b^{-n}{L}_n$. In particular $V_0={\mathbb{E}}_P[{\zeta }_{n}H]$. At an intermediate date the conditional change-of-measure identity givesIf physical up probabilities depend on the past, replace the constant-probability expression by the product over periods of $q/p_k$ on an up move and $(1-q)/(1-p_k)$ on a down move. Each multiplier still has conditional mean one, provided $0<p_k<1$, and the pricing measure retains conditional up probability $q$.

In a complete market, terminal-wealth optimization can first be solved over terminal payoffs and then implemented by replication. Let initial wealth be $x_0>0$, and take an increasing differentiable strictly concave utility function $U$ on $(0,\infty )$ satisfying the Inada conditions. The state-price budget constraint for a terminal wealth $X$ is ${\mathbb{E}}_P[{\zeta }_{n}X]\le x_0$. Monotonicity spends the whole budget at the optimum.

Let $I=(U^{\prime }{)}^{-1}$ be the inverse marginal utility. Choose $y>0$ to satisfy ${\mathbb{E}}_P[{\zeta }_{n}I(y{\zeta }_n)]=x_0$. In a finite tree with positive state probabilities, this left side is continuous and strictly decreasing from infinity to zero, so $y$ exists uniquely. The complete-market terminal utility optimizer isTo prove optimality, concavity gives $U(X)\le U(X^{\ast })+U^{\prime }(X^{\ast })(X-X^{\ast })=U(X^{\ast })+y{\zeta }_n(X-X^{\ast })$. Taking expectations and using the budget proves $\mathbb{E}U(X)\le \mathbb{E}U(X^{\ast })$. Strict concavity gives uniqueness of the optimal terminal wealth. The finite-market setting avoids additional infinite-state existence and integrability issues.

Replicate $X^{\ast }$ using the backward recursion of part (i). Its wealth process is $V_k=B_k{\mathbb{E}}_Q[X^{\ast }/B_n\mid {\mathcal{F}}_k]$, and the two-successor difference quotient gives its stock holdings. This self-financing strategy starts with $x_0$ by the budget equality and remains positive at every node, since it is the discounted conditional expectation of a strictly positive terminal wealth.

For logarithmic utility, $U(x)=\log x$, $I(z)=1/z$, the multiplier is $y=1/x_0$ and $X^{\ast }=x_0/{\zeta }_n=x_0{B}_n/L_n$. For CRRA utility with risk aversion $\gamma >0$, $\gamma \ne 1$, the normalized payoff isConstraints such as prohibiting short sales change the attainable terminal-payoff set and require a constrained optimization; the unconstrained complete-market formula uses the stated assumptions.

For the Cox-Ross-Rubinstein diffusion approximation, fix calendar maturity $T$, take $h=T/n$, and choose $u=e^{\sigma \sqrt{h}}$, $d=e^{-\sigma \sqrt{h}}$, and the one-step bank factor $b=e^{\rho h}$. For sufficiently fine mesh, $d<b<u$. The risk-neutral probability expands asEach stock log-increment is $\pm \sigma \sqrt{h}$. Its mean under $Q$ is $(\rho -{\sigma }^2/2)h+O(h^2)$ and its variance is ${\sigma }^{2}h+O(h^2)$. Independent increments, their vanishing maximum size, and the Donsker invariance principle for this triangular array give the path limitExponentiation gives the geometric Brownian motion of the Black-Scholes model, and the bank account tends to $e^{\rho t}$.

The physical model is matched by taking $p_h=(e^{\mu h}-d)/(u-d)$. Its corresponding limit has log drift $\mu -{\sigma }^2/2$, so $d{S}_t=\mu S_{t}dt+\sigma S_{t}d{W}_t^P$. The likelihood ratios from part (ii) have the matching continuous-time limit. With $\lambda =(\mu -\rho )/\sigma$, a Taylor expansion of their two possible log-multipliers giveswhere ${\epsilon }_k$ is $+1$ or $-1$. Thus the limiting Radon-Nikodym derivative is the Girsanov theorem density $\exp (-\lambda W_T^P-{\lambda }^{2}T/2)$, turning the physical stock drift into $\rho$.

The binomial risk-neutral pricing formula converges to $e^{-\rho T}{\mathbb{E}}_Q[f(S_T)]$ for bounded continuous terminal payoffs by convergence in distribution. For ordinary call and put payoffs, uniform stock second-moment bounds give the required uniform integrability and hence convergence despite unboundedness. Bounded payoffs with discontinuities only at events of zero limiting probability are handled by the same weak-convergence criterion. Corresponding continuous path functionals can be treated at the process level. Arbitrary discontinuous path claims do not follow from weak convergence without further conditions. In the smooth limiting model, the discrete hedging difference quotient becomes the delta hedge $V_x$, and its value satisfies the Black-Scholes equation.