marginal probability density: clarified #with equations

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My starting point is 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

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 [1]. The advantage of my “sheet” way is the numerator always being a sensible probability mass. The integral in the standard formula [1] 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 [1] would tell us that marginal density is


Applying Ito’s formula on math problems — learning notes

Ito’s formula in a nutshell — Given dynamics of a process X, we can derive the dynamics[1] of a function[2] f() of x .

[1] 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:


[2] 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.

change of .. numeraire^measure

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.

physical measure is impractical

Update: Now I think physical probability is not observable nor quantifiable and utterly unusable in the math including the numerical methods.  In contrast, RN probabilities can be derived from observed prices.

Therefore, now I feel physical measure is completely irrelevant to option math.

RN measure is the “first” practical measure for derivative pricing. Most theories/models are formulated in RN measure. T-Forward measure and stock numeraire are convenient when using these models…

Physical measure is an impractical measure for pricing. Physical measure is personal feeling, not related to any market prices. Physical measure is mentioned only for teaching purpose. There’s no “market data” on physical measure.

Market prices reflect RN (not physical) probabilities.

Consider cash-or-nothing bet that pays $100 iff team A wins a playoff. The bet is selling for $30, so the RN Pr(win) = 30%. I am an insider and I rig the game so physical Pr() = 80% and Meimei (my daughter) may feel it’s 50-50 but these personal opinions are irrelevant for pricing any derivative.

Instead, we use the option price $30 to back out the RN probabilities. Namely, Calibrate the pricing curves using liquid options, then use the RN probabilities to price less liquid derivatives.

Professor Yuri is the first to point out (during my oral exam!) that option prices are the input, not the output to such pricing systems.

drift ^ growth rate – are imprecise

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”

vol, unlike stdev, always implies a (stoch) Process

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.

Radon-Nikodym derivative #Lida video

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.

discounted asset price is MG but "discount" means…@@

The Fundamental Theorem

A financial market with time horizon T and price processes of the risky asset and riskless bond (I would say a money-market-account) given by S1, …, ST and B0, …, BT, respectively, is arbitrage-free under the real world probability P if and only if there exists an equivalent probability measure Q (i.e. risk neutral measure) such that
The discounted price process, X0 := S0/B0, …, XT := ST/BT is a martingale under Q.

#1 Key concept – divide the current stock price by the current MMA value. This is the essence of “discounting“, different from the usual “discount future cashflow to present value
#2  key concept – the alternative interpretation is “using MMA as currency, then any asset price S(t) is a martingale”
I like the discrete time-series notation, from time_0, time_1, time_2… to time_T.
I like the simplified (not simplistic:) 2-asset world.
This theorem is generalized with stochastic interest rate on the riskless bond:)
There’s an implicit filtration. The S(T) or B(T) are prices in the future i.e. yet to be revealed [1]. The expectation of future prices is taken against the filtration.
[1] though in the case of T-forward measure, B(T) = 1.0 known in advance.
–[[Hull]] P 636 has a concise one-pager (I would skip the math part) that explains the numeraire can be just “a tradable”, not only the MMA. A few key points:

) both S and B must be 2 tradables, not something like “fwd rate” or “volatility”
) the measure is the measure related to the numeraire asset
) what market forces ensure this ratio is a MG? Arbitragers!