Characteristic Function Integrability and the COS Option Pricing Method

# COS Method and existence of density


# COS Method and existence of density












Hey in the COS method we use characteristic function of $\ln{S_T}$ to price european options (by recovering density from characteristic function). But how do we know that density exists? For example I would like to use this method to price options in Kou, NIG or CGMY model, do these Levy processes has density and we can use this method? I know the theorem which states that if characteristic function is integrable, i.e $\int_{\mathbb{R}}|\phi (t)|dt<\infty $ then there exists continuous density with respect to Lebesque measure. Is characteristic function integrable for processes which I mentioned and in general for all Levy processes?

## Answer by Kevin (score 4, accepted)

https://quant.stackexchange.com/a/61503

## Example: Kou Model

Well, let's have a look at the Kou model $$\phi_{\ln(S_t)}(u)=\exp\left(\underbrace{\ln\left(S_0e^{(r-q+\omega)t}\right)iu}_{=(\star)}\;\;\underbrace{-\frac{1}{2}\sigma^2tu^2}_{=(\star\star)}+\underbrace{\lambda t\left(\frac{p\eta_+}{\eta_+-iu}+\frac{p'\eta_-}{\eta_-+iu}-1\right)}_{=(\star\star\star)}\right),$$ where

- $r$ and $q$ are interest rate and dividend yield

- $\sigma$ is the volatility

- $p$ and $p'$ are the probabilities of up and down jumps ($p+p'=1$)

- $\eta_+>1$ and $\eta_->0$ relate to the (inverse) mean jump size

- $\omega$ is the jump compensation ensuring that $S_te^{-(r-q)t}$ is a martingale.

I claim we can easily prove that \begin{align*} \int_\mathbb{R}|\phi(u)|\text{d}u<\infty. \end{align*}

- Clearly, $|e^{(\star)}|\leq 1$ by Euler's formula. All these points lie on the unit circle.

- Next, $|e^{(\star\star)}|$ decays super quickly, it's like a Gaussian bell curve. So $\int_\mathbb{R} |e^{(\star\star)}|<\infty$.

- The $(\star\star\star)$ case is not much harder. Firstly, the $-\lambda t$ can be taken out of the integral and doesn't bother us. Next, rationalising the denominator, we get \begin{align*} \left|\exp\left(\frac{\lambda tp\eta_+}{\eta_+-iu}\right)\right| &= \left|\exp\left(\frac{\lambda tp\eta_+(\eta_++iu)}{\eta_+^2+u^2}\right)\right| \\ &= \left|\exp\left(\frac{\lambda tp\eta_+\eta_+}{\eta_+^2+u^2}\right)\exp\left(\frac{\lambda tp\eta_+ui}{\eta_+^2+u^2}\right)\right| \\ &\leq \exp\left(\frac{\lambda tp\eta_+\eta_+}{\eta_+^2+u^2}\right). \end{align*} This generates further exponential decay as $u\to\pm\infty$ and remains finite at $u=0$. In fact, we can estimate the term by a constant. The same applies to the $\eta_-$ fraction.

Thus, all in all, $(\star)$ is bounded by one, $(\star\star)$ looks like a bell curve and $(\star\star\star)$ is also bounded. Taken together, $\phi$ decays exponentially fast and is in $L^1(\mathbb{R})$.

A few notes

- The Riemann-Lebesgue Lemma tells us that characteristic functions (Fourier transforms) converge to zero for $u\to\pm\infty$. All we need to show is that they converge quickly enough to zero.

- You can do similar bounds for the NIG model and other characteristic functions

- For these exponential Lévy processes, the probability density function is rarely known in closed-form. The characteristic function is very simple though. You can always obtain the PDF by (numerically) Fourier inverting the characteristic function.

## Sidenote: Uncertainty Principle

One thing to keep in mind is the uncertainty principle: If $f$ spreads out, then its Fourier transform $\hat{f}$ is very `compact' and vice versa. The density of the log stock price is very spread out if the time horizon $t$ is large: there is a wide range of potential values for a stock price in 5 years time. However, the density for tomorrow's stock price is very peaked around today's value. Accordingly, the characteristic function is very spread out for short maturities but decays super quickly for long maturities. Example for the imaginary part of the characteristic function of the normal distribution (real part looks similar):