Past exam of the mathematics course of the University of Cambridge / 2012 / iii / Paper 43 / 6

For zero drift, the reflection principle gives the known Brownian first-passage time densityTo obtain a general drift, on a fixed finite horizon use the exponential martingale $M_t=e^{c{B}_t-c^{2}t/2}$ as change-of-measure density. The Novikov condition holds for constant $c$, and the Girsanov theorem says that the coordinate process $B$ has drift $c$ under the new measure. Thus the hitting law of $B$ under that measure is the law of $B+ct$ under the original measure. Conditional expectation of $M_T$ at a hitting time bounded by $T$ equals its stopped value, by the martingale optional sampling theorem. At $H_a=t$ this value is $e^{ca-c^{2}t/2}$. The drifted Brownian first-passage density is obtained by multiplying $h_0$, giving $\boxed{h_c(t)=a{e}^{-(a-ct{)}^2/(2t)}/\sqrt{2\pi t^3}}$.

To verify the integrated formula, put $u=(a-cT)/\sqrt{T}$ and $v=(-a-cT)/\sqrt{T}$. The identity $e^{2ac}\varphi (v)=\varphi (u)$ for the standard normal density givesBoth terms vanish as $T\downarrow 0$, so $\boxed{\mathbb{P}(H_a\le T)=\overline{\Phi }((a-cT)/\sqrt{T})+e^{2ac}\Phi ((-a-cT)/\sqrt{T})}$. The distribution has total mass one for $c\ge 0$ and mass $e^{2ac}$ for $c<0$; in the latter case the remaining probability is an atom at infinity.

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.