New answers tagged vega
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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