Formula sheets

Itô's lemma $\;df=\left(f_t+\mu f_x+\tfrac12\sigma^2 f_{xx}\right)dt+\sigma f_x\,dW$

Black–Scholes $\;C=S\Phi(d_1)-Ke^{-rT}\Phi(d_2),\quad d_{1,2}=\dfrac{\ln(S/K)+(r\pm\sigma^2/2)T}{\sigma\sqrt T}$

Vasicek $\;dr=a(b-r)\,dt+\sigma\,dW;\quad P(t,T)=A(t,T)e^{-B(t,T)r_t},\;B=\dfrac{1-e^{-a(T-t)}}{a}$

CIR $\;dr=a(b-r)\,dt+\sigma\sqrt r\,dW$ (Feller: $2ab>\sigma^2$)

CDS spread (flat hazard $\lambda$, recovery $R$) $\;s\approx\lambda(1-R)$

Merton distance to default $\;DD=\dfrac{\ln(V/D)+(\mu-\sigma_V^2/2)T}{\sigma_V\sqrt T}$