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3

I'm not sure I agree with that being a very difficult task... The black formula for a caplet (using notation from Hull's book) is given by: $caplet = L \delta_k P(0, t_{k+1}) [F_k N(d_1) - R_kN(d_2)]$ where: $d_1 = \frac{ln(F_k/R_k) + \sigma_k^2t_k/2}{\sigma_k\sqrt{t_k}}$ and $d_2 = d_1 - \sigma_K \sqrt{t_k}$ The delta will just be the first derivative of ...

0

Short answer: I believe the quantification of the impact of an increase in the shorter-dated vol on the longer-dated vol depends on what type of vol function we select to model the Volatility term-structure of the options. Long answer (focusing on piece-wise volatility term-structure function): Variances of log-returns in the GBM model are proportional to ...

4

Let $V$ denote the variance and $v$ the volatility, i.e. $V=v^2$. The natural argument for the option price under a stochastic volatility model is typically the variance, i.e. $C_\text{SV}=C_\text{SV}(S_0,V_0,...)$. However, using the chain rule, we can compute vega in terms of the volatility: \nu=\frac{\partial C_\text{SV}}{\partial v}=\frac{\partial C_\...

2

The relationship between theta and gamma is the Black-Scholes PDE. Let's take normal B-S dynamics with $r=0$: $dS_t = \sigma S_t dW_t$ The pricing PDE for a derivative $g(S_T)$ is (with terminal condition $g$): $\frac{dp}{dt} = \frac{1}{2}\sigma^2S^2 \frac{d^2p}{dS^2}$ Or $\Theta = \frac{1}{2}\sigma^2S^2 \Gamma$ This PDE has a solution (Feynman-Kac Theorem): ...

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