Shouting at zero locks in $S_0-K=-1$, producing terminal payoff $\max (S_2,100)-101$. Its pathwise values are 20 on $uu$ and $-1$ on each other path. Waiting and applying part (c) instead gives values 20 on $uu$, 9 on $ud$, and zero on $du,dd$. Therefore waiting dominates shouting now, strictly on events of positive probability.

Under the optimal binomial shout option with signed lock-in, discount the latter terminal payoffs using independent risk-neutral transitions:The time-one option value is $115/7$ after an up move and zero after a down move. These values also reproduce the price by one-step backward induction.

The recombining stock price 99 has two different claim payoffs, because $ud$ has an earlier lock-in while $du$ does not. Keeping the path history is essential.

If the first move is up, $S_1=110$ and $S_2$ can be 121 or 99. A shout now gives $\max (S_2,110)-101$, hence payoffs 20 and 9. Waiting until maturity or never shouting gives 20 and 0. The first policy is no worse on either branch and strictly better on the down branch, so shout after the first up move.

If the first move is down, $S_1=90$ and the terminal prices are 99 or 81. A shout would give $-2$ or $-11$, while waiting gives zero. Thus do not shout after the first down move. These comparisons use terminal payoffs directly, so no intermediate discounting can change their ordering.

At maturity a shout has payoff $S_2-K$, while no shout gives $(S_2-K{)}^{+}$. If $S_2<K$, shouting creates a negative payoff and is strictly worse. If $S_2\ge K$, the two choices give exactly the same payoff, so shouting is optional and gives no improvement.

Thus do not shout when out of the money; be indifferent when at or in the money. The terminal stock values are 121, 99, and 81, so only the up-up node permits an indifferent shout with the given strike.

The risk-neutral one-step mean multiplier must equal the bank growth factor:Thus the risk-neutral probability in a binomial market isIt lies strictly between zero and one, consistent with the no-arbitrage inequality $d<e^r<u$.

Linearity separates the terminal call payoff from the running payment stream. The first contributes the supplied European call price. Under the risk-neutral measure, $\mathbb{E}(S_v\mid S_t=s)=s{e}^{r(v-t)}$, so the stream contributesConsequentlyThis is the claim dividends proportional to the underlying stock identity. The underlying itself is still non-dividend-paying; the cash flow belongs to the derivative.

Use the auxiliary diffusion $d{X}_v=\mu X_{v}dv+\sigma X_{v}d{W}_v$, started at $X_t=x$. Under sufficient regularity and integrability for the Feynman-Kac formula,To verify the signs, apply Itô's formula to $e^{-r(v-t)}F(v,X_v)$. Its drift is $e^{-r(v-t)}(F_t+\mu X_v{F}_x+{\sigma }^2{X}_v^2{F}_{xx}/2-rF)=-e^{-r(v-t)}d(v,X_v)$. Adding the accumulated source integral makes a martingale; taking expectations through $T$ proves the formula.

Here the auxiliary drift is the printed $\mu$, rather than automatically $r$: this part is a general terminal-value PDE, whose diffusion must match its generator.

Invest each cash payment $q_v{S}_{v}dv$ received at time $v$ in the bank account until $T$. Its accumulated terminal amount is $q_v{S}_v{e}^{r(T-v)}dv$, so the entire future stream is equivalent, path by path, to the terminal payment ${\int }_t^T{q}_v{S}_v{e}^{r(T-v)}dv$.

The risk-neutral pricing formula, including this reinvested payoff, iswhere $d{\tilde{S}}_v=r{\tilde{S}}_{v}dv+\sigma {\tilde{S}}_{v}d{W}_v^{\mathbb{Q}}$. Equivalently, discount each cash flow directly to time $t$. This is the discounted-gains martingale principle underlying the PDE in part (a).

A self-financing replicating strategy must fund both the claim's price change and its outgoing cash flow:Apply Itô's formula to $V(t,S_t)$. Matching the diffusion gives $\Delta =V_s$. Matching drift and eliminating $\beta B=V-s{V}_s$ gives the Black-Scholes equation with claim dividendsThe physical drift cancels because stock risk is hedged. The cash-flow term has a plus sign because it is a payment to the holder.

The call spread in the hint has payoffIt is between zero and one and converges to $1_{\{s>K\}}$ as $ϵ\downarrow 0$. In the Black-Scholes model with positive volatility and $t<T$, the conditional terminal stock distribution has no atom at $K$. The strict and non-strict indicator therefore have the same discounted expectation. The dominated convergence theorem in the risk-neutral pricing formula givesEquivalently this is $e^{-r(T-t)}\Phi (d_2)$. The no-atom condition explains the endpoint subtlety: the displayed spread has value zero at $s=K$, whereas the printed digital pays one there. At maturity the strike derivative can be one-sided or undefined, so the smooth pricing identity is understood before expiry.