Animated Dadim operators

2001-Apr-20

Convection Diffusion Process in 1 dimension

The image below shows how the theoretically computed option price changes when moving backward in time. As described by the Black-Scholes Equation, this process is governed by three different operators: Scale, Shift and Blur. The Blur operator (Diffusion) smoothens the function by averaging the values of different possible underlying prices according to their probabilities. The shift operator (Convection) models the underlyings drift by copying a value to the position determined by the drift's speed.


click to proceed to the next image

Blur operator
The Blur operator models a diffusion process, which in turn is determined by a probability distribution of parameter changes. The animation shows how an images is affected by a diffusion process with two correlated dimensions. The correlation yields a much faster smothening in the (1,1) direction, where both parameters change by the same amount. Only little diffusion occurs where parameters move in opposite directions (1, -1).

Definition:

B f(x) = Integral f(x)P(x-s)ds

Applications:

  • Stochastic time step
  • Correlation

Shift operator

The animation shows a shift process where the drift's speed is defined by the x-coordinate and the drift direction is along the y-axis. This makes it possible to implement discount processes with variable factors. Another important application is a summation step in Asian derivatives. Here the value of the new function is determined by the value of the old function where the y-coordinate is increased by x.

Definition:

S f (x,y) = f (y + x, y)
or, on a logarithmic scale
S f (x, y) = f (y*(1+x), y)

Applications:

  • Summation for Asian options
  • Parameter depentent discounting

Maximum operator

The maximum operator takes two functions and returns a single function which is greater than each individual function. Consider the two original functions to be the expected payoff of two investment strategies. The maximized function represents the present value of having the option between those two. An important application is an American option where the choice of immediate excercising is exists in every time step.

M A B (x, y) = max { A(x,y), B(x,y) }

Exotic derivatives

Example: Pricing an Asian binary call

  • X-Axis: Stock price
  • Y-Axis: Sum of all past stock prices
  • Initial Payoff funcition: 1 where the sum exceeds the strike, 0 otherwise.
  • Even steps: The shift operator adds the current price to the Y-axis
  • Odd steps: A diffusion process along the X-axis
  • Final result: The resulting option price can be optained at the point (0, S). Where 0 is the initial value for the sum and S is the lastest stock price.

Modell fitting

Example: Volatility estimation

  • X-Axis: Volatilty
  • Y-Axis: Stock price
  • Initial Function: A distribution with equal probability for all volatilities and Probability 1 for Y=S0
  • Even steps: Diffusion with variable volatility
  • Odd steps: A new market price arrives and all points where Y != Sn are set to 0
  • Final result: A distribution of volatility and the next stock price SN+1

Efficient data structures

Sparse Grids

Sprase grids store data points only at certain positions which where proved to allow optimal function reconstruction. Further improved adaptive data structures can exploit a posteriori results for eliminating further points.

Grid Data points Error
full 16641 3.05e-03
sparse 2817 4.03e-03
adaptive 1066 5.39e-03

Dadim-XML: Storing operator trees

Dadim-XML

The final goal is to find approximations for several kinds of functional operators by only using a small set of strongly optimized implementations. Dadim-XML identifies the types of operators and their possible combinations which maximize computational capability while keeping an intuitive and consistent mathematical framework. 
    <Dadim>
      <Binary underlying="S">
      <Shift underlying="S" length="strike"/>
      <Loop count="10">
	 <Shift underlying="S" target="sum"/>
         <GaussianBlur underlying="S" deviation="volatility"/>
      </Loop>
      <Project underlying="sum" value="0"/>
      <Project underlying="strike" value="100"/>
    </Dadim>

Stefan Dirnstorfer