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A barrier claim paying a fixed cash amount at the first time the stock reaches a higher level, provided this happens before expiry. In the Black-Scholes model, take $a=\log (c/S_0)/\sigma$ and $b=({\sigma }^2/2-\rho )/\sigma$. Its unit-cash price is ${\mathbb{E}}_Q[e^{-\rho T_{a,b}}{1}_{\{T_{a,b}\le T\}}]$, given by truncated discounted Brownian first passage. The effective square root $\sqrt{b^2+2\rho }=|\rho +{\sigma }^2/2|/\sigma$ is real for every real interest rate.

A barrier contract paying a specified fixed amount if the underlying touches a specified barrier before expiry. Payment at touch and payment at expiry are different conventions. For payment at touch, valuation uses the truncated discounted Brownian first passage after converting a geometric stock barrier to a log-price level; discounting solely at expiry would price a different contract.

Let $M_t={\sup }_{s\le t}{W}_s$ and write ${\varphi }_t(y)=(2\pi t{)}^{-1/2}{e}^{-y^2/(2t)}$ for the Brownian terminal density. By the Brownian reflection principle, for $y<a$ the reflected paths that hit $a$ and end at $y$ have terminal density ${\varphi }_t(2a-y)$. HenceThe Girsanov theorem changes zero drift into drift $\nu$ using the finite-horizon density $e^{\nu W_t-{\nu }^{2}t/2}$. The finite-horizon maximum of Brownian motion with drift therefore has distribution functionIn the first integral term, complete the square to get a normal density with mean $\nu t$. In the second, substitute $z=2a-y$ and complete the square with mean $-\nu t$; the prefactor is $e^{2a\nu }$. Here $\Phi$ is the standard normal distribution function, $a>0$ and $t>0$.

Let $T_{a,b}=\inf \{s\ge 0:W_s=a+bs\}$, with $b$ real. Survival means that $W_s-bs$ stays below $a$. Tilt by $e^{\theta W_t-{\theta }^{2}t/2}$; under the tilted probability, $W_s$ has drift $\theta$. Applying the maximum formula with drift $\theta -b$ gives the exponential terminal moment below a linear Brownian boundaryThis holds for every real $\theta$. The maximum distribution has no atom at the positive barrier, so strict or weak survival inequalities give the same probability here.

For the hitting-time moment, first use drift $-b$ in the maximum formula and take the complementary probability:If $z_1=(-a-bt)/\sqrt{t}$ and $z_2=(-a+bt)/\sqrt{t}$, their standard normal densities satisfy $\varphi (z_1)=e^{-2ab}\varphi (z_2)$. Differentiating the two terms, with $z_1^{\prime }=(a-bt)/(2t^{3/2})$ and $z_2^{\prime }=(a+bt)/(2t^{3/2})$, yields the drifted Brownian first-passage densityIts total mass need not be one, because the upward-sloping boundary may never be reached.

For $2\theta \le b^2$, put $\kappa =\sqrt{b^2-2\theta }\ge 0$. Multiplication by $e^{\theta s}$ gives $e^{\theta s}{f}_b(s)=e^{a(\kappa -b)}{f}_{\kappa }(s)$. Integrating to $t$ produces truncated discounted Brownian first passage, with negative discount rate allowed when the square root remains real:This includes the equality case $\kappa =0$, where it becomes $2e^{-ab}\Phi (-a/\sqrt{t})$. The expression integrates only finite hitting times up to $t$; no value is assigned to an exponential at an infinite hitting time.

For the cash-at-hit digital call, use the risk-neutral measure in the Black-Scholes model and denote expiry by $T=t_0$. ThenHitting $c>S_0$ corresponds to $W_s^Q=a+bs$, where $a=\log (c/S_0)/\sigma >0$ and $b=({\sigma }^2/2-\rho )/\sigma$. The unit-cash payment is made at the hitting time, so risk-neutral valuation discounts at that time, not at expiry. Set $\theta =-\rho$ andThe identity on the right verifies the square-root condition even for negative interest rates. The time-zero price isThis is in pounds for the unit-pound contract. It differs from a maturity-paid digital call option, whose discounting and exercise event are different.

Under the Black-Scholes model pricing measure, write $\log S_t/\sigma =B_t+ct$, where $c=(r-{\sigma }^2/2)/\sigma$. Brownian continuity and immediate crossing on reaching a level make the strict up-crossing time equal almost surely to the hitting time $H_a$. Its finite-time law has no atom at $T$, so the convention before rather than at expiry does not affect the price.

For truncated discounted Brownian first passage, since payment occurs at the hitting time, discount there, not at expiry. The one-touch option price isSet $d=\sqrt{c^2+2r}=|r+{\sigma }^2/2|/\sigma$. The expression under the square root is nonnegative for every real $r$, by this Black-Scholes identity. The discounted density satisfies $e^{-rt}{h}_c(t)=e^{a(c-d)}{h}_d(t)$. Applying part i yields the closed priceThis formula also covers $d=0$ and negative rates. When $r\ge 0$, one may use $d=(r+{\sigma }^2/2)/\sigma$, so the two prefactors simplify to $e^{-\sigma a}$ and $e^{2ra/\sigma }$. At $r=0$ the expression is the undiscounted hitting probability, as a consistency check.