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Questions tagged [differential-equations]

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27
votes
2answers
20k views

Transformation from the Black-Scholes differential equation to the diffusion equation - and back

I know the derivation of the Black-Scholes differential equation and I understand (most of) the solution of the diffusion equation. What I am missing is the transformation from the Black-Scholes ...
2
votes
1answer
251 views

Riccati Equation in spot rate model

Given that $dr=(\eta-\gamma r)dt+\sqrt{\alpha r+\beta}dW$ Let $Z(r,t)=e^{A(t;T)-rB(t;T)}$, \begin{matrix} \frac{dA}{dt}=\eta B-\frac{1}{2}\beta {{B}^{2}} \\ \frac{dB}{dt}=\frac{1}{2}\alpha {{...
2
votes
1answer
154 views

Prove $E_{\mathbb Q}[X_t | \mathscr F_u] = X_u$ given $Y_t$ is a martingale

We are given a filtered probability space $(\Omega, \mathscr{F}, \{\mathscr{F}_t\}_{t \in [0,T]}, \mathbb{P})$, where $\{\mathscr{F}_t\}_{t \in [0,T]}$ is the filtration generated by standard $\mathbb ...
1
vote
1answer
327 views

PDE for Pricing Interest Rate Derivatives

Suppose that interest rate $r(t)$ follows some short-rate models, say Vasicek, so that$dr = a(b-r) dt + \sigma dZ$, with constants $a,b,\sigma$. It is well known that the price of zero-coupon bond $...
1
vote
1answer
139 views

Prove uniqueness, and prove $Y_t$ is a martingale by considering $dZ_t$ and $dL_t$

Suppose we are given a filtered probability space $(\Omega, \mathscr{F}, \{\mathscr{F}_t\}_{t \in [0,T]}, \mathbb{P})$, where $\{\mathscr{F}_t\}_{t \in [0,T]}$ is the filtration generated by standard $...