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I'm looking for a resource, a book or an article, that describes the framework of change of numéraire in a broader context than just Brownian motions or Normal distributions. I'm only really interested in 1 time step, i.e. not a full stochastic process but just 1 random variable.

I'm asking this because in the context of a computation I'm making for the price of a European call, with the underlying having a terminal value that is distributed according to a Log Laplace distribution, I noticed that the formula for the price takes the same form as the formula in case of the classical Black Scholes formula, but with the normal cdfs replaced by skew Laplace distributions. That is:

$$EC(r,K,T,X,b) = X \; SL(d,b \sqrt{T}) - K e^{-rT} \; SL(d,0)$$

with

$$SL(x,s) = \begin{cases}\frac{1}{2}(1+s)e^{(1-s)x} &, \; \text{ for } \; x \leq 0 , \\ 1-\frac{1}{2}(1-s)e^{-(1+s)x} &, \; \text{ for } \; x > 0 \end{cases}$$

and

$$d = \frac{\ln(X/K)+rT+\ln(1-b^2 T)}{b\sqrt{T}} \; .$$

Here, $X$ is the initial stock price, $K$ is the strike, $T$ is the time to maturity, $r$ is the interest rate and $b=\sigma/\sqrt{2}$ with $\sigma$ the volatility.

I derived the formula quite easily, but then I wanted to derive this formula with the change of numéraire technique, but I made a naïve assumption that under the new measure, the log returns would still be Laplace distributed but it turns out they are skew Laplace distributed. Note that $SL(d,0) = L(d)$ though. But I don't need a change of numéraire for that term. It's the first term that in principle could be derived with a change of numéraire.

EDIT: $SL$ is the cumulative distribution, the density is given by the derivative w.r.t. $x$:

$$SL'(x,s) = \frac{1}{2}(1-s^2)e^{-|x|-sx} \; .$$

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I can't provide any references that explicitly cover this but came across a similar question before and worked out some results. Below is a quick overview of my findings.

In summary, you can show that the two distributions under the bank account and spot asset numeraire are the same if you model logarithmic returns to follow a mixture of natural exponential families.

Mixtures Over Natural Exponential Families

The density function of the skew Laplace distribution in your question can be re-written as

\begin{eqnarray} f(x; s) & = & \frac{1 - s^2}{2 (1 - s)} (1 - s) e^{(1 - s) x} \mathrm{1} \{ x \leq 0 \} \\ & & + \frac{1 - s^2}{2 (1 + s)} (1 + s) e^{-(1 + s) x} \mathrm{1} \{ x > 0 \}. \end{eqnarray}

We note that this is a weighted mixture distribution over two natural exponential families $f_\pm \left( x; \theta_\pm \right)$ with parameters $\theta_\pm$ and weights $w_\pm$, i.e.

\begin{equation} f(x; s) = w_- f_- \left( x; \theta_- \right) + w_+ f_+ \left( x; \theta_+ \right). \end{equation}

The lower tail can for example be expressed as

\begin{equation} f_- \left( x; \theta_- \right) = a_-(x) \exp \left\{ \theta_- x - b_-\left( \theta_- \right) \right\} \end{equation}

where we defined $\theta_- = 1 - s$,

\begin{equation} a_-(x) = \mathrm{1} \{ x \leq 0 \}, \qquad b_- \left( \theta_- \right) = -\ln \left( \theta_- \right) \end{equation}

and weight

\begin{equation} w_- = \frac{1 - s^2}{2(1 - s)} \end{equation}

We can find a similar expression for the upper tail and its weight. The corresponding characteristic functions are given by

\begin{equation} \phi_\pm \left( \omega; \theta_- \right) = \exp \left\{ b_\pm \left( \theta_\pm + \mathrm{i} \omega \right) - b_\pm \left( \theta_\pm \right) \right\} \end{equation}

To keep the answer brief, we just focus on the lower tail going forward and drop the $\pm$ subscripts.

Esscher Transform Probability Measure

Consider some random variable $X$ whose law under $\mathbb{P}$ is given by $F_X(x)$ with corresponding characteristic function $\phi_X(\omega)$. Following Esscher (1932), the Esscher transform measure $\hat{\mathbb{P}}(\beta)$ equivalent to $\mathbb{P}$ is defined through the Radon-Nikodym derivative

\begin{equation} \frac{\mathrm{d}\hat{\mathbb{P}}}{\mathrm{d}\mathbb{P}} = \frac{e^{\beta X}}{\phi_X(-\mathrm{i} \beta)} \end{equation}

for some transform parameter $\beta \in \mathbb{R}_+$ and conditional on the $\beta$-th exponential moment in the denominator being finite. See also Gerber and Shiu (1994). The corresponding characteristic function is

\begin{equation} \hat{\phi}_X(\omega) = \mathbb{E}_\mathbb{P} \left[ e^{\mathrm{i} \omega X} \right] = \mathbb{E} \left[ \frac{\mathrm{d}\hat{\mathbb{P}}}{\mathrm{d}\mathbb{P}} e^{\mathrm{i} \omega X} \right] = \frac{\phi_X(\omega - \mathrm{i} \beta)}{\phi_X(-\mathrm{i} \beta)} \end{equation}

Escher Transform of Exponential Families

It is now easy to show that if some random variable $X$ follows a natural exponential family under $\mathbb{P}$ with parameter $\theta$, then

\begin{equation} \hat{\phi}_X(\omega) = \exp \left\{ b(\theta + \beta + \mathrm{i} \omega) - b(\theta + \beta) \right\}. \end{equation}

Thus, $X$ follows the same natural exponential family under $\hat{\mathbb{P}}$ but with parameter $\hat{\theta} = \theta + \beta$.

Link to Change of Numeraire

Assume that the asset price is modelled as

\begin{equation} S_t = S_0 e^{\gamma T + b \sqrt{T} X}, \end{equation}

where $\gamma$ is such that

\begin{equation} \mathbb{E}_\mathbb{P} \left[ S_t \right] = S_0 e^{r t}. \end{equation}

Then the change of numeraire from the bank account to the spot asset corresponds to an Esscher transform with parameter $\beta = b \sqrt{T}$.

In your example, we thus have $\hat{\theta}_- = \theta_- + b \sqrt{T}$ and $\hat{s} = s - b \sqrt{T}$.

Remarks

  1. Note that the weights $\hat{w}_i$ under $\hat{\mathbb{P}}$ also change. I omitted the details here for brevity.

  2. A similar result can be obtained when considering jump-diffusion models where the jump-size distribution follows a natural exponential mixture. A prominent example is the Kou (2002) double exponential jump-diffusion model. The result can be further generalized to additive jump-diffusion processes with potentially time-dependent jump-intensities and jump-size distributions.

References

Esscher, Frederik (1932) "On the Probability Function in the Collective Theory of Risk," Scandinavian Actuarial Journal, Vol. 15, No. 3, pp. 175-195

Gerber, Hans U. and Elias W. Shiu (1994) "Option Pricing by Esscher Transforms," Transactions of Society of Actuaries, Vol. 46, pp. 99-191

Kou, Steven G. (2002) "A Jump-Diffusion Model for Option Pricing," Management Science, Vol. 48, No. 8, pp. 1086-101

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  • $\begingroup$ Note that either me or you have a typo. I get $\hat{s} = s - b \sqrt{T}$ while you write $\hat{s} = s + b \sqrt{T}$ with $s = 0$. I will double-check my answer later. $\endgroup$ – LocalVolatility May 1 '18 at 9:25
  • $\begingroup$ Wow, I didn't expect such a fantastic answer so soon! I'll need some time to take it in completely though. Thanks! $\endgroup$ – Raskolnikov May 1 '18 at 9:43
  • $\begingroup$ Actually, the density is $f(x,s)=(1-s^2)\exp(-|x|-bx)/2=0.5(1-s^2)\exp((1-b)x)\;1\{x\leq 0\}+0.5(1-s^2)\exp(-(1+b)x)\;1\{x> 0\}$. $\endgroup$ – Raskolnikov May 1 '18 at 9:46
  • $\begingroup$ Could you correct this in your question? $\endgroup$ – LocalVolatility May 1 '18 at 9:56
  • $\begingroup$ But it is correct. In my question, I didn't write the density but the cumulative distribution. $\endgroup$ – Raskolnikov May 1 '18 at 10:02

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