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My starting point is https://bintanvictor.wordpress.com/2016/06/29/probability-density-clarified-intuitively/. Look at the cross section at X=7.02. This is a 2D area, so volume (i.e. probability mass) is zero, not close to zero. Hard to work with. In order to work with a proper probability mass, I prefer a very thin but 3D “sheet” , by cutting again at X=7.02001 i.e 7.02 + deltaX. The prob mass in this sheet divided by deltaX is a number. I think it’s the marginal density value at X=7.02.
The standard formula for marginal density function is on https://www.statlect.com/glossary/marginal-probability-density-function:
How is this formula reconciled with our “sheet”? I prefer to start from our sheet, since I don’t like to deal with zero probability mass. Sheet mass divided by the thickness i.e. deltaX:
Since f(x,y) is assumed not to change with x, this expression simplifies to
Now it is same as formula . The advantage of my “sheet” way is the numerator always being a sensible probability mass. The integral in the standard formula  doesn’t look like a probably mass to me, since the sheet has zero width.
The simplest and most visual bivariate illustration of marginal density — throwing a dart on a map of Singapore drawn on a x:y grid. Joint density is a constant (you can easily work out its value). You could immediate tell that marginal density at X=7.02 is proportional to the island’s width at X=7.02. Formula  would tell us that marginal density is
Ito’s formula in a nutshell — Given dynamics of a process X, we can derive the dynamics of a function f() of x .
 The original “dynamics” is usually in a stoch-integral form like
dX = m(X,t) dt + s(X,t) dB
In some problems, X is given in exact form not integral form. For an important special case, X could be the BM process itself:
 the “function” or the dependent random variable “f” is often presented in exact form, to let us find partials. However, in general, f() may not have a simple math form. Example: in my rice-cooker, the pressure is some unspecified but “tangible” function of the temperature. Ito’s formula is usable if this function is twice differentiable.
The new dynamics we find is usually in stoch-integral form, but the right-hand-side usually involves X, dX, f or df.
Ideally, RHS should involve none of them and only dB, dt and constants. GBM is such an ideal case.
Advice: When possible, I would work with CoN rather than CoM. I believe once we identify another numeraire (say asset B) is useful, we just know there exists an equivalent measure associated with B (say measure J), so we could proceed. How to derive that measure I don’t remember. Maybe there’s a relatively simple formula, but very abstract.
In one case, we only have CoM, no CoN — when changing from physical measure to risk neutral measure. There is no obvious, intuitive numeraire associated with the physical measure!
CoN is more intuitive than CoM. Numeraire has a more tangible meaning than “measure”.
I think even my grandma could understand 2 different numeraires and how to switch between them. Feels like simple math.
CoM has rigorous math behind it. CoM is not just for finance. I guess CoM is the foundation and basis of CoN.
I feel we don’t have to have a detailed, in-depth grasp of CoM to use it in CoN.
The drift rate “j” is defined for BM not GBM
dAt = j dt + dW term
Now, for GBM,
dXt = r Xt dt + dW term
So the drift rate by definition is r Xt, Therefore, it’s confusing to say “same drift as the riskfree rate”. Safer to say “same growth rate” or “same expected return”
Volatility, in the context of pure math (not necessarily finance), refers to the coefficient of dW term. Therefore,
* it implies a measure,
* it implies a process, a stoch process
Therefore, if a vol number is 5%, it is, conceptually and physically, different from a stdev of 0.05.
* Stdev measures the dispersion of a static population, or a snapshot as I like to say. Again, think of the histogram.
* variance parameter (vol^2) of BM shows diffusion speed.
* if we quadruple the variance param (doubling the vol) value, then the terminal snapshot’s stdev will double.
At any time, there’s an instantaneous vol value, like 5%. This could last a brief interval before it increase or decreases. Vol value changes, as specified in most financial models, but it changes slowly — quasi-constant… (see other blog posts)
There is also a Black-Scholes vol. See other posts.
Lida pointed out CoM (change of measure) means that given a pdf bell curve, we change its mean while preserving its “shape”! I guess the shape is the LN shape?
I guess CoM doesn’t always preserve the shape.
Lida explained how to change one Expectation integral into another… Radon Nikodym.
The concept of operating under a measure (call it f) is fundamental and frequently mentioned but abstract…
Aha – Integrating the expectation against pdf f() is same as getting the expectation under measure-f. This is one simple, if not rigorous, interpretation of operating under a given measure. I believe there’s no BM or GBM, or any stochastic process at this stage — she was describing how to transform one static pdf curve to another by changing measure. I think Girsanov is different. It’s about a (stochastic) process, not a static distribution.
See also the backgrounder – blog post on discount asset prices and martingale
Q: Among all numeraires, which one has a known future time-T value?
A: The bond maturing at time T is the only numeraire I know. Therefore the measure behind this numeraire is uniquely useful.