1.1 Girsanov’s Theorem

Let \(\left\{W_t\right\}\) be a \(\mathbb{P}-\)Brownian motion on \((\Omega,\mathcal{F})\) that generates the filtration \((\mathcal{F}_t)\) and \(\left\{\theta_t\right\}\) is an adapted process with \[\mathbb{E}^{\mathbb{P}}\left(\exp\left(\frac{1}{2}\int_0^T\theta_t^2dt\right)\right)<\infty.\] Then \[W_t^{\mathbb{Q}}=W_t+\int_0^t\theta_sds\] is a \(\mathbb{Q}-\)Brownian motion, where \[\frac{d\mathbb{Q}}{d\mathbb{P}}|_{\mathcal{F}_t}=Z_t=\exp\left(-\int_0^t\theta_sdW_s-\frac{1}{2}\int_0^t\theta_s^2ds\right).\]

1.2 Geometric Brownian Motion

Assume the stock price process \(\left\{S_t\right\}\) is governed by the SDE \[dS_t=\mu S_tdt+\sigma S_tdW_t.\] Let \(f(t,x)=e^{-rt}x,\) then \(f_t(t,x)=-e^{-rt}rx,f_x(t,x)=e^{-rt},f_{xx}(t,x)=0\) so \[\begin{align*} d(e^{-rt}S_t) &= f_t(t,S_t)dt+f_x(t,S_t)dS_t+\frac{1}{2}(\sigma S_t)^2f_{xx}(t, S_t)dt\\ &= -e^{-rt}rS_tdt+e^{-rt}(\mu S_tdt+\sigma S_tdW_t)\\ &= e^{-rt}(\mu-r)S_tdt+e^{-rt}\sigma S_tdW_t. \end{align*}\] Replace \(dW_t=dW_t^{\mathbb{Q}}-\theta dt\) gives \[d(e^{-rt}S_t)=e^{-rt}(\mu-r-\sigma\theta)S_tdt+e^{-rt}\sigma S_tdW_t^{\mathbb{Q}}\] so \(\left\{e^{-rt}S_t\right\}\) is a \(\mathbb{Q}-\)martingale if and only if \(\theta=(\mu-r)/\sigma.\) Then \[dS_t=(\mu-\sigma\theta)S_tdt+\sigma S_tdW_t^{\mathbb{Q}}=rS_tdt+\sigma S_tdW_t^{\mathbb{Q}}\] so \[S_t=S_0\exp\left(\left(r-\frac{\sigma^2}{2}\right)t+\sigma W_t^{\mathbb{Q}}\right).\]

1.3 Arithmetic Brownian Motion

Assume the stock price process \(\left\{S_t\right\}\) is governed by the SDE \[dS_t=\mu dt+\sigma dW_t.\] \[\begin{align*} d(e^{-rt}S_t) &= f_t(t,S_t)dt+f_x(t,S_t)dS_t+\frac{1}{2}(\sigma S_t)^2f_{xx}(t, S_t)dt\\ &= -e^{-rt}rS_tdt+e^{-rt}(\mu dt+\sigma dW_t)\\ &= e^{-rt}(\mu-rS_t)dt+e^{-rt}\sigma dW_t. \end{align*}\] Replace \(dW_t=dW_t^{\mathbb{Q}}-\theta_tdt\) gives \[d(e^{-rt}S_t)=e^{-rt}(\mu-rS_t-\sigma\theta_t)dt+e^{-rt}\sigma dW_t^{\mathbb{Q}}\] so \(\left\{e^{-rt}S_t\right\}\) is a \(\mathbb{Q}-\)martingale if and only if \(\theta_t=(\mu-rS_t)/\sigma.\) Then \[dS_t=(\mu-\sigma\theta_t)dt+\sigma dW_t^{\mathbb{Q}}=rS_tdt+\sigma dW_t^{\mathbb{Q}}\] so \[S_t=e^{rt}S_0+\sigma\int_0^te^{r(t-s)}dW_s^{\mathbb{Q}}.\]

1.4 First Passage Time

Let \(\alpha,\beta\in\mathbb{R}\) be constants with \(\alpha\neq0,\) then under the measure \(\mathbb{Q}\) defined as \[\left.\frac{d\mathbb{Q}}{d\mathbb{P}}\right|_{\mathcal{F}_t}=Z_t=\exp\left(-\frac{1}{2}\int_0^t\beta^2ds+\int_0^t\beta dW_s^{\mathbb{P}}\right)=\exp\left(-\frac{\beta^2t}{2}+\beta W_t^{\mathbb{P}}\right)\] the process \[W_t^{\mathbb{Q}}=W_t^{\mathbb{P}}-\beta t\] is a \(\mathbb{Q}-\)Brownian motion. Then the first passage time of \(W_t^{\mathbb{P}}\) hitting a slope line can be expressed as \[\tau_{\alpha,\beta}^{W_t^{\mathbb{P}}}=\inf\left\{t\geq0:W_t^{\mathbb{P}}=\alpha+\beta t\right\}=\inf\left\{t\geq0:W_t^{\mathbb{Q}}=\alpha\right\}=\tau_{\alpha}^{W_t^{\mathbb{Q}}}\]

so the cdf of \(\tau_{\alpha,\beta}^{W_t^{\mathbb{P}}}\) is

\[\begin{align*} F_{\tau_{\alpha,\beta}^{W_t^{\mathbb{P}}}}(t) &= \mathbb{P}\left(\tau_{\alpha,\beta}^{W_t^{\mathbb{P}}}\leq t\right)=\int_{\Omega}\chi_{\tau_{\alpha,\beta}^{W_t^{\mathbb{P}}}\leq t}d\mathbb{P}\\ &= \int_{\Omega}Z^{-1}_{\tau_{\alpha,\beta}^{W_t^{\mathbb{P}}}}\cdot\chi_{\tau_{\alpha,\beta}^{W_t^{\mathbb{P}}}\leq t}d\mathbb{Q}=\int_{\Omega}Z^{-1}_{\tau_{\alpha}^{W_t^{\mathbb{Q}}}}\cdot\chi_{\tau_{\alpha}^{W_t^{\mathbb{Q}}}\leq t}d\mathbb{Q}\\ &= \mathbb{E}^{\mathbb{Q}}\left(\exp\left(-\frac{\beta^2\cdot\tau_{\alpha}^{W_t^{\mathbb{Q}}}}{2}-\beta\cdot W_{\tau_{\alpha}^{W_t^{\mathbb{Q}}}}^{\mathbb{Q}}\right)\cdot\chi_{\tau_{\alpha}^{W_t^{\mathbb{Q}}}\leq t}\right)\\ &= \int_0^t\exp\left(-\alpha\beta-\frac{\beta^2u}{2}\right)\cdot f_{\tau_{\alpha}^{W_t^{\mathbb{Q}}}}(u)du\\ &= \int_0^t\exp\left(-\alpha\beta-\frac{\beta^2u}{2}\right)\cdot\frac{|\alpha|}{u\sqrt{2\pi u}}\cdot e^{-\frac{\alpha^2}{2u}}du\\ &= \int_0^t\frac{|\alpha|}{u\sqrt{2\pi u}}\cdot e^{-\frac{(\alpha+\beta u)^2}{2u}}du \end{align*}\]

and hence

\[f_{\tau_{\alpha,\beta}^{W_t^{\mathbb{P}}}}(t)=\frac{|\alpha|}{t\sqrt{2\pi t}}\cdot e^{-\frac{(\alpha+\beta t)^2}{2t}},t\geq0.\]

2.1 Black-Scholes-Merton Equation

Assume the stock price process \(\left\{S_t\right\}\) is governed by the SDE \[dS_t=\mu(t,S_t)dt+\sigma(t,S_t)dW_t.\] Denote \(V_t=u(t,S_t)\) the value of the European call on \(S_t\) at time \(t,\) then \[dV_t=\left[u_t+\frac{1}{2}\sigma^2u_{xx}+\mu u_x\right]dt+\sigma u_xdW_t.\] We establish a hedging portfolio \(\Pi\) by buying \(1\) call and short sell \(\Delta\) stock shares. Then \[d\Pi_t=dV_t-\Delta dS_t=\left[u_t+\frac{1}{2}\sigma^2u_{xx}+\mu(u_x-\Delta)\right]dt+\sigma(u_x-\Delta)dW_t.\] Eliminating randomness by choosing \(\Delta=u_x,\) we obtain \[\left(u_t+\frac{1}{2}\sigma^2u_{xx}\right)dt=d\Pi_t=r\Pi_t=r(V_t-\Delta S_t)=r(V_t-u_xS_t)\] where \(r\) is the risk-free interest rate compounded continuously. It implies \[u_t(t,S_t)+\frac{1}{2}\sigma(t,S_t)^2u_{xx}(t,S_t)+rS_tu_x(t,S_t)-ru(t,S_t)=0,\] known as the Black-Scholes-Merton equation. In case the interest rate \(r(t)\) varies non-randomly over time, \[u_t(t,S_t)+\frac{1}{2}\sigma(t,S_t)^2u_{xx}(t,S_t)+r(t)S_tu_x(t,S_t)-r(t)u(t,S_t)=0.\]

2.2 Feynman-Kac Formula

Consider the Black-Scholes-Merton equation in the risk-neutral world, i.e. \(\mu(t,S_t)=r(t)S_t.\) The boundary condition \(u(T,S_T)=\Phi(S_T)\) and notations \[g(v)=\exp\left(-\int_t^vr(s)ds\right)\textrm{ and }Z_v=g(v)u(v,S_v)=f(v,S_v)\] imply \(g'(v)=-r(v)g(v)\) and \[\begin{align*} dZ_v &= f_vdv+f_xdS_v+\frac{1}{2}\sigma^2f_{xx}dv\\ &= \left(g'u+gu_v+\frac{1}{2}\sigma^2gu_{xx}\right)dv+gu_x(\mu dv+\sigma dW_v)\\ &= (-ru+u_v+\frac{1}{2}\sigma^2u_{xx}+rS_tu_x)gdv+gu_x\sigma dW_v=gu_x\sigma dW_v \end{align*}\] so \(\left\{Z_t\right\}\) is a martingale, deriving the Feynman-Kac formula \[u(t,S_t)=Z_t=\mathbb{E}^{\mathbb{Q}}(Z_T|\mathcal{F}_t)=\mathbb{E}^{\mathbb{Q}}\left(\Phi(S_T)\exp\left.\left(-\int_t^Tr(s)ds\right)\right|\mathcal{F}_t\right).\]

2.3 Black-Scholes-Merton Formula

Under the geometric Brownian motion, the conditional cdf of \(S_T\) given \(\mathcal{F}_t\) is \[\begin{align*} F_{S_T|\mathcal{F}_t}(x) &= \mathbb{Q}(S_T\leq x|\mathcal{F}_t)\\ &= \mathbb{Q}\left(S_t\cdot\exp\left.\left(\left(r-\frac{\sigma^2}{2}\right)(T-t)+\sigma\left(W_T^{\mathbb{Q}}-W_t^{\mathbb{Q}}\right)\right)\leq x\right|\mathcal{F}_t\right)\\ &= \mathbb{Q}\left(\left.\left(r-\frac{\sigma^2}{2}\right)(T-t)+\sigma W_{T-t}^{\mathbb{Q}}\leq\ln\frac{x}{S_t}\right|\mathcal{F}_t\right)\\ &= \mathbb{Q}\left(\left.\mathcal{Z}\leq\frac{\ln x-\ln S_t-(r-\sigma^2/2)(T-t)}{\sigma\sqrt{T-t}}\right|\mathcal{F}_t\right)\\ &= \Phi(-d_-(x)) \end{align*}\] where \(\mathcal{Z}\sim\mathcal{N}(0,1)\) and \(\Phi=F_{\mathcal{Z}},\phi=f_{\mathcal{Z}}=\Phi'\) and \[d_{\pm}(x)=\frac{\ln S_t-\ln x+(r\pm\sigma^2/2)(T-t)}{\sigma\sqrt{T-t}}.\] It implies that \[f_{S_T|\mathcal{F}_t}(x)=F_{S_T|\mathcal{F}_t}'(x)=\frac{\phi(-d_-(x))}{x\sigma\sqrt{T-t}}\] and \[\begin{align*} A &= \int_K^{\infty}\frac{e^{-r(T-t)}x\phi(-d_-(x))}{x\sigma\sqrt{T-t}}dx\\ &= \int_K^{\infty}\frac{xdx}{x\sigma\sqrt{T-t}\sqrt{2\pi}}\exp\left(-\frac{d_-(x)^2}{2}-r(T-t)\right)\\ &= \frac{S_t}{\sqrt{2\pi}}\int_K^{\infty}\frac{dx}{x\sigma\sqrt{T-t}}\exp\left(-\frac{d_-(x)^2}{2}-r(T-t)+\ln\frac{x}{S_t}\right)\\ &= \frac{S_t}{\sqrt{2\pi}}\int_K^{\infty}\frac{dx}{x\sigma\sqrt{T-t}}\exp\left(-\frac{d_+(x)^2}{2}\right)\\ &= S_t\int_K^{\infty}\frac{\phi(-d_+(x))}{x\sigma\sqrt{T-t}}dx=S_t\int_K^{\infty}\frac{d}{dx}\Phi(-d_+(x))dx \end{align*}\] yielding the Black-Scholes-Merton pricing formula \[\begin{align*} C(t,S_t) &= \mathbb{E}^{\mathbb{Q}}\left(e^{-r(T-t)}\max(S_T-K,0)|\mathcal{F}_t\right)\\ &= e^{-r(T-t)}\int_{\mathbb{R}}\max(x-K,0)f_{S_T|\mathcal{F}_t}(x)dx\\ &= e^{-r(T-t)}\int_K^{\infty}(x-K)f_{S_T|\mathcal{F}_t}(x)dx\\ &= A-e^{-r(T-t)}K\int_K^{\infty}f_{S_T|\mathcal{F}_t}(x)dx\\ &= S_t\int_K^{\infty}\frac{d}{dx}\Phi(-d_+(x))dx-e^{-r(T-t)}K\int_K^{\infty}\frac{d}{dx}\Phi(-d_-(x))dx\\ &= S_t\Phi(d_+)-e^{-r(T-t)}K\Phi(d_-) \end{align*}\] where \(d_{\pm}=d_{\pm}(K).\) The put-call parity implies the European put value \[P(t,S_t)=C(t,S_t)+Ke^{-r(T-t)}-S_t=Ke^{-r(T-t)}\Phi(-d_-)-S_t\Phi(-d_+).\]

2.4 Arithmetic Brownian Motion

Under the risk-neutral arithmetic Brownian motion

\[S_T=S_t+r(T-t)+\sigma\left(W_T^{\mathbb{Q}}-W_t^{\mathbb{Q}}\right)\sim\mathcal{N}(\alpha,\beta^2)\]

where

\[\alpha=S_t+r(T-t)\textrm{ and }\beta^2=\sigma^2(T-t),\]

the conditional cdf of \(S_T\) given \(\mathcal{F}_t\) is

\[F_{S_T|\mathcal{F}_t}(x)=\mathbb{Q}\left(\mathcal{N}(\alpha,\beta^2)\leq x|\mathcal{F}_t\right)=\Phi\left(\frac{x-\alpha}{\beta}\right)\] so \[f_{S_T|\mathcal{F}_t}(x)=\frac{1}{\beta}\phi\left(\frac{x-\alpha}{\beta}\right)\] and denoting \(u=(x-\alpha)/\beta\) gives \[\begin{align*} \frac{C(t,S_t)}{e^{-r(T-t)}} &= \int_{\mathbb{R}}\max(x-K,0)f_{S_T|\mathcal{F}_t}(x)dx\\ &= \int_K^{\infty}(x-K)f_{S_T|\mathcal{F}_t}(x)dx\\ &= \int_K^{\infty}\frac{xe^{-u^2/2}}{\beta\sqrt{2\pi}}dx-K\int_K^{\infty}f_{S_T|\mathcal{F}_t}(x)dx\\ &= \int_{\frac{K-\alpha}{\beta}}^{\infty}\frac{(\alpha+\beta u)e^{-u^2/2}}{\sqrt{2\pi}}du-K\Phi\left(\frac{\alpha-K}{\beta}\right)\\ &= \alpha\int_{\frac{K-\alpha}{\beta}}^{\infty}\phi(u)du+\beta\int_{\frac{K-\alpha}{\beta}}^{\infty}\frac{ue^{-u^2/2}}{\sqrt{2\pi}}du-K\Phi\left(\frac{\alpha-K}{\beta}\right)\\ &= (\alpha-K)\Phi\left(\frac{\alpha-K}{\beta}\right)+\beta\phi\left(\frac{\alpha-K}{\beta}\right) \end{align*}\] or \[C(t,S_t)=\hat{\sigma}d\Phi(d)+\hat{\sigma}\phi(d)\textrm{ and }P(t,S_t)=\hat{\sigma}\phi(-d)-\hat{\sigma}d\Phi(-d)\] where \[\hat{\sigma}=e^{-r(T-t)}\beta=e^{-r(T-t)}\sigma\sqrt{T-t}\textrm{ and }d=\frac{\alpha-K}{\beta}=\frac{S_t+r(T-t)-K}{\sigma\sqrt{T-t}}.\]

Under the real-world arithmetic Brownian motion\[S_T=e^{r(T-t)}S_t+\sigma\int_t^Te^{r(T-s)}dW_s^{\mathbb{Q}}\sim\mathcal{N}(\alpha,\beta^2)\] where \[\alpha=e^{r(T-t)}S_t\textrm{ and }\beta^2=\sigma^2\int_t^Te^{2r(T-s)}ds=\frac{\sigma^2\left(e^{2r(T-t)}-1\right)}{2r},\] we have \[C(t,S_t)=\hat{\sigma}d\Phi(d)+\hat{\sigma}\phi(d)\textrm{ and }P(t,S_t)=\hat{\sigma}\phi(-d)-\hat{\sigma}d\Phi(-d)\] where \[\hat{\sigma}=e^{-r(T-t)}\beta=\sigma\sqrt{\frac{1-e^{-2r(T-t)}}{2r}}\textrm{ and }d=\frac{\alpha-K}{\beta}=\frac{S_t-Ke^{-r(T-t)}}{\hat{\sigma}}.\]

3.1 Digital Options

The put-call parity for European digitals is \[DC(t,S_t)+DP(t,S_t)=e^{-r(T-t)}.\]

Under the geometric Brownian motion, the prices of European digitals are \[\begin{align*} DC(t,S_t) &= \mathbb{E}^{\mathbb{Q}}\left(e^{-r(T-t)}\cdot\chi_{S_T>K}|\mathcal{F}_t\right)= e^{-r(T-t)}\cdot\mathbb{E}^{\mathbb{Q}}\left(\chi_{S_T>K}|\mathcal{F}_t\right)\\ &= e^{-r(T-t)}\cdot\mathbb{Q}(S_T>K|\mathcal{F}_t) = e^{-r(T-t)}(1-\Phi(-d_-))\\ &= e^{-r(T-t)}\Phi(d_-) \end{align*}\] and \[DP(t,S_t)=e^{-r(T-t)}\Phi(-d_-).\]

Under the risk-neutral arithmetic Brownian motion, \[\begin{align*} DC(t,S_t) &= e^{-r(T-t)}\cdot\mathbb{Q}(S_T>K|\mathcal{F}_t) = e^{-r(T-t)}(1-\Phi(-d))\\ &= e^{-r(T-t)}\Phi(d) \end{align*}\] and \[DP(t,S_t)=e^{-r(T-t)}\Phi(-d).\]

Under the real-world arithmetic Brownian motion, \[\begin{align*} DC(t,S_t) &= e^{-r(T-t)}\cdot\mathbb{Q}(S_T>K|\mathcal{F}_t) = e^{-r(T-t)}(1-\Phi(-d))\\ &= e^{-r(T-t)}\Phi(d) \end{align*}\] and \[DP(t,S_t)=e^{-r(T-t)}\Phi(-d).\]

3.2 Asset-or-Nothing Options

The prices of asset-or-nothing call options are \[C_{AN}(t,S_t)=C(t,S_t)+K\cdot DC(t,S_t)\] and the prices of asset-or-nothing put options are \[P_{AN}(t,S_t)=K\cdot DP(t,S_t)-P(t,S_t).\]

3.3 American Digital Options

For an American digital call, let \(\tau\) be the waiting time from current time \(t\) to the first time the stock price hits \(K>S_t.\) Then under the risk-neutral arithmetic Brownian motion, \(\tau\) can be expressed as

\[\begin{align*} \tau &= \inf\left\{u\geq t:S_u=K\right\}\\ &= \inf\left\{u\geq t:S_t+r(u-t)+\sigma\left(W_u^{\mathbb{Q}}-W_t^{\mathbb{Q}}\right)=K\right\}\\ &= \inf\left\{u-t\geq0:W_{u-t}^{\mathbb{Q}}=\frac{K-S_t-r(u-t)}{\sigma}\right\}\\ &= \inf\left\{x\geq0:W_{x}^{\mathbb{Q}}=\frac{K-S_t}{\sigma}-\frac{r}{\sigma}x\right\}=\tau_{\alpha,\beta}^{W_t^{\mathbb{Q}}} \end{align*}\]

where

\[\alpha=\frac{K-S_t}{\sigma}>0\textrm{ and }\beta=-\frac{r}{\sigma}.\]

Letting \(u_{\pm}=\sqrt{\beta^2s+2sr}\pm\alpha/\sqrt{s},\) it implies that

\[\begin{align*} I_{\pm} &= \int_0^{T-t}\frac{s\sqrt{\beta^2+2r}\mp\alpha}{2s\sqrt{2\pi s}}\cdot\exp\left(-\frac{\alpha^2+(\beta^2+2r)s^2}{2s}\right)ds\\ &= \mp e^{\pm\alpha\sqrt{\beta^2+2r}}\int_{\frac{(T-t)\sqrt{\beta^2+2r}\pm\alpha}{\sqrt{T-t}}}^{\infty}\phi(u_{\pm})du_{\pm}\\ &= \mp e^{\pm\alpha\sqrt{\beta^2+2r}}\cdot\Phi\left(\frac{-\alpha\mp(T-t)\sqrt{\beta^2+2r}}{\sqrt{T-t}}\right) \end{align*}\]

so the price of an American digital call is

\[\begin{align*} DC_A(t,S_t) &= \mathbb{E}^{\mathbb{Q}}(e^{-r\tau}\cdot\chi_{\tau\leq T-t}|\mathcal{F}_t)=\int_0^{T-t}e^{-rs}f_{\tau_{\alpha,\beta}^{W_t^{\mathbb{Q}}}}(s)ds\\ &= \int_0^{T-t}\frac{\alpha}{s\sqrt{2\pi s}}\cdot \exp\left(-rs-\frac{(\alpha+\beta s)^2}{2s}\right)ds\\ &= \frac{1}{e^{\alpha\beta}}\int_0^{T-t}\frac{\alpha}{s\sqrt{2\pi s}}\cdot \exp\left(-\frac{\alpha^2+(\beta^2+2r)s^2}{2s}\right)ds\\ &= \frac{1}{e^{\alpha\beta}}\cdot\left(I_--I_+\right)=e^{\lambda_+(S_t-K)}\cdot\Phi(d_+)+e^{\lambda_-(S_t-K)}\cdot\Phi(d_-)\\ \end{align*}\]

where

\[d_{\pm}=\frac{S_t-K\pm(T-t)\sqrt{r^2+2\sigma^2r}}{\sigma\sqrt{T-t}},\]

\[\lambda_{\pm}=\frac{-r\pm\sqrt{r^2+2\sigma^2r}}{\sigma^2}.\]

Under the geometric Brownian motion, \(\tau\) can be expressed as

\[\begin{align*} \tau &= \inf\left\{u\geq t:S_u=K\right\}\\ &= \inf\left\{u\geq t:S_t\cdot\exp\left(\left(r-\frac{\sigma^2}{2}\right)(u-t)+\sigma\left(W_u^{\mathbb{Q}}-W_t^{\mathbb{Q}}\right)\right)=K\right\}\\ &= \inf\left\{u-t\geq0:W_{u-t}^{\mathbb{Q}}=\frac{\ln K-\ln S_t-(r-\sigma^2/2)(u-t)}{\sigma}\right\}\\ &= \inf\left\{x\geq0:W_x^{\mathbb{Q}}=\frac{\ln K-\ln S_t}{\sigma}+\left(\frac{\sigma}{2}-\frac{r}{\sigma}\right)x\right\}=\tau_{\alpha,\beta}^{W_t^{\mathbb{Q}}} \end{align*}\]

where

\[\alpha=\frac{1}{\sigma}\cdot\ln\frac{K}{S_t}>0\textrm{ and }\beta=\frac{\sigma}{2}-\frac{r}{\sigma}.\]

Then

\[DC_A(t,S_t)=\frac{1}{e^{\alpha\beta}}\cdot\left(I_--I_+\right)=\left(\frac{S_t}{K}\right)^{\lambda_+}\cdot\Phi(d_+)+\left(\frac{S_t}{K}\right)^{\lambda_-}\cdot\Phi(d_-)\]

where

\[d_{\pm}=\frac{\ln S_t-\ln K\pm(T-t)\sqrt{(r-\sigma^2/2)^2+2\sigma^2r}}{\sigma\sqrt{T-t}},\]

\[\lambda_{\pm}=\frac{-(r-\sigma^2/2)\pm\sqrt{(r-\sigma^2/2)^2+2\sigma^2r}}{\sigma^2}.\]

For an American digital put, let \(\tau\) be the waiting time from current time \(t\) to the first time the stock price hits \(K<S_t.\) Then under the risk-neutral arithmetic Brownian motion, \(\tau\) can be expressed as

\[\tau=\inf\left\{x\geq0:W_{x}^{\mathbb{Q}}=\frac{K-S_t}{\sigma}-\frac{r}{\sigma}x\right\}=\tau_{\alpha,\beta}^{W_t^{\mathbb{Q}}}\]

where

\[\alpha=\frac{S_t-K}{\sigma}>0\textrm{ and }\beta=\frac{r}{\sigma}.\]

Then

\[DP_A(t,S_t)=\frac{1}{e^{\alpha\beta}}\cdot\left(I_--I_+\right)=e^{\lambda_+(K-S_t)}\cdot\Phi(d_+)+e^{\lambda_-(K-S_t)}\cdot\Phi(d_-)\]

where

\[d_{\pm}=\frac{K-S_t\pm(T-t)\sqrt{r^2+2\sigma^2r}}{\sigma\sqrt{T-t}},\]

\[\lambda_{\pm}=\frac{r\pm\sqrt{r^2+2\sigma^2r}}{\sigma^2}.\]

Under the geometric Brownian motion, \(\tau\) can be expressed as

\[\tau=\inf\left\{x\geq0:W_x^{\mathbb{Q}}=\frac{\ln K-\ln S_t}{\sigma}+\left(\frac{\sigma}{2}-\frac{r}{\sigma}\right)x\right\}=\tau_{\alpha,\beta}^{W_t^{\mathbb{Q}}}\]

where

\[\alpha=\frac{1}{\sigma}\cdot\ln\frac{S_t}{K}>0\textrm{ and }\beta=\frac{r}{\sigma}-\frac{\sigma}{2}.\]

Then

\[DP_A(t,S_t)=\frac{1}{e^{\alpha\beta}}\cdot\left(I_--I_+\right)=\left(\frac{K}{S_t}\right)^{\lambda_+}\cdot\Phi(d_+)+\left(\frac{K}{S_t}\right)^{\lambda_-}\cdot\Phi(d_-)\]

where

\[d_{\pm}=\frac{\ln K-\ln S_t\pm(T-t)\sqrt{(r-\sigma^2/2)^2+2\sigma^2r}}{\sigma\sqrt{T-t}},\]

\[\lambda_{\pm}=\frac{(r-\sigma^2/2)\pm\sqrt{(r-\sigma^2/2)^2+2\sigma^2r}}{\sigma^2}.\]