__The Theory of Speculation__

The focus of Bachelier’s work (The Theory of Speculation) was to establish the probability law for the price fluctuations that the market admits at this instant. Indeed, while the market does not foresee fluctuations, it considers which of them are more or less probable, and this probability can be evaluated mathematically.

There are innumerable influences on the movements of stock exchange.

Stock Exchange acts upon itself, implies: its current movement is a function not only of earlier fluctuations, but also of the present market position.

The determination of these fluctuations is subject to an infinite number of factors: it is therefore impossible to expect a mathematically exact forecast.

*(An excellent translation of the original paper: http://www.radio.goldseek.com/bachelier-thesis-theory-of-speculation-en.pdf)*

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__A few important terms defined__

**Spot Price:** The current price at which a particular security can be bought or sold at a specified time and place. (A securities futures price is the expected value of the security.)

**Futures Contract:** A contractual agreement to buy or sell a particular commodity or financial instrument at a pre-determined price in the future.

**Option:** offers the buyer the right, but not the obligation, to buy (call) or sell (put) a security or other financial asset at an agreed-upon price (the strike price) on a specific date (exercise date).

**Mathematical advantage**: Mathematical expectation does not provide a coefficient representing, in some sense, the intrinsic value of the game, it only logically tells whether a game will be profitable or not, Hence the concept of Mathematical advantage.

We can understand mathematical advantage from the example of a gambler’s odds as explained in the paper by Louis Bachelier: Define the mathematical advantage of a gambler as the ratio of his positive expectation and the arithmetic sum of his positive and negative expectations. Mathematical advantage varies like probability from zero to one, it is equal to 1/2 when the game is fair.

Mathematical expectation of gain / (mathematical expectation of gain + absolution value of mathematical expectation of loss).

The mathematical expectation of the speculator is zero

__The General Form of probability curve__

The probability that a price y be quoted at a given epoch is a function of y.

The price considered by the market as the most likely is the current true price.

Prices can vary between −x_{o} and +∞: x_{o} being the current absolute price.

It will be assumed that it can vary between −∞ and +∞. The probability of a spread greater than x_{o} being considered a priori entirely negligible.

The probability of a deviation from the true price is independent of the absolute level of this price, and that the probability curve is symmetrical with respect to the true price.

Only relative prices will matter because the origin of the coordinates will always correspond to the current true price.

__Probability law of Financial Product__

**Method 1:**

- p
_{x,t}dx: probability for price to be in the interval [x,x+dx) at time t

p_{x,t}dx denotes the probability that, at epoch t, the price is to be found in the elementary interval x, x + dx.

The desired probability is therefore

\[

p_{x,t_1}p_{z-x,t_2} dx dz

\]

- Markov property: p
_{z,t1+t2}dz =_{−∞}∫^{∞}p_{x,t1}p_{z−x,t2}dxdz

Because at epoch t_{1}, the price could be located in any the intervals dx between −∞ and +∞, so this is the probability of the price z being quoted at epoch t_{1}+ t_{2}. - Homogeneous Gaussian process:

\[

p_{x,t} = {1\over 2\pi k\sqrt t}\exp\left(-{1\over 4\pi k^2 t}\right)

\]

Where \[k=\int_0^\infty p_{x,1} dx\] is the gain at time 1.

**Method 2: Alternate derivation of the Probability Law**

- conduct m Bernoulli trials in [0,t), with success rate p

Suppose that two complementary events A and B have the respective probabilities p and q = 1 – p. The probability that, on m occasions, it would produce α equal to A and m − α equal to B, is given by the expression.

\[{P}(\alpha; m,p) = {m\choose \alpha}p^\alpha q^{m-\alpha}\]

- mathematical expectation of gain

\[\sum_{h\ge0}\mbox{P}(mp+h;m,p)\]

\[ = pq\cdot\left(\partial_{p}\sum_{h\ge0}\mbox{P}(mp+h;m,p)-\partial_{q}\sum_{h\ge0}\mbox{P}(mp+h;m,p)\right) \]

\[= \frac{m!}{(mp)!(mq)!}p^{mp}q^{mq}mpq\]

The quantity h is called the spread.

Let us seek for the mathematical expectation for a gambler who would receive a sum equal to the spread whenever this spread be positive.

We have just seen that the probability of a spread h is the term from the expansion of (p + q)^{m} in which the exponent of p is mp + h, and that of q is mqh.

To obtain the mathematical expectation corresponding to this term, it is necessary to multiply this probability by h. Now,

h = q(mp + h) − p(mqh)

mp + h and mqh being the exponents of p and q in a term from (p + q)^{m}.

- Apply Stirling’s formula

\[n!=e^{-n}n^n\sqrt{2\pi n}\]

\[{E}={dh\over \sqrt{2\pi m p q}}\exp\left(-{h^2\over 2 m p q}\right)\]