II. Mathematical Tools and Principles
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17. September 2023
The reflection principle in layman’s terms…
The reflection principle in layman’s terms…
Imagine a serene lake with a mirror-like surface. Standing at a height of "a" units above the water, you drop a stone into the lake.
The stone traces a path from your hand, descending towards the water.
Upon making contact with the water's surface, the stone's reflection moves in the opposite direction beneath the water. This reflective path is the exact opposite of the stone's initial descent.
Consider a situation where the stone is 1 unit above the water. In this case, its reflection is 1 unit below. If the stone were to be 2 units above, its reflection would be 2 units below, and this pattern continues.
To determine the "height" of the reflected stone from the water's surface, you would use the formula: (2a - height of the actual stone).
This analogy closely mirrors the principle seen in the Wiener process. When W_t surpasses a particular threshold "a" (much like the stone touching the water), its subsequent movements around "a" are reflected, forming a mirrored trajectory. For every time t > T, the height of this mirrored process from the "surface" (level "a") is calculated by 2a - W_t.
Where W_t represents the position of the particle (or the value of the process) at time (t).
Now, regarding the reflection principle and the notion of a barrier at (a): If the Wiener process hits a level (a) at some time (t), it’s just as likely that, if there had been no barrier and the process had been free to move, it would’ve reached a level (-a) by that same time (t). This symmetry is the essence of the reflection principle.
Envision a particle experiencing Brownian motion, starting from W_0 = 0. This particle moves erratically. The main question is the likelihood of this particle reaching a level a > 0 before time t.
One approach is picturing a "barrier" at level a. As the particle first collides with this barrier, think of it as spawning a mirrored counterpart that "reflects" back downwards.
The beauty of this reflection concept is its grounding in the symmetry of Brownian motion: The chances of W_t being over level a equate to it being under level -a.
With the reflection principle in play, a complex question about the particle encountering a barrier morphs into a simpler query: When will two independent Brownian motions touch the level 0? This switch greatly simplifies the math and associated calculations in numerous stochastic process problems.
This principle becomes especially handy when studying American options, where the time of exercise matches a stopping time. Paired with the stopping time, the reflection principle offers a way to determine the joint distribution of the peak Ws value between 0 and t and Wt. This insight proves valuable when pricing unique options like barrier and lookback options.
#ReflectionPrinciple #WienerProcess #BrownianMotion#StochasticProcesses #BarrierReflection #AmericanOptions#StoppingTime #OptionPricing #MathInNature #BarrierOptions#LookbackOptions
Imagine a serene lake with a mirror-like surface. Standing at a height of "a" units above the water, you drop a stone into the lake.
The stone traces a path from your hand, descending towards the water.
Upon making contact with the water's surface, the stone's reflection moves in the opposite direction beneath the water. This reflective path is the exact opposite of the stone's initial descent.
Consider a situation where the stone is 1 unit above the water. In this case, its reflection is 1 unit below. If the stone were to be 2 units above, its reflection would be 2 units below, and this pattern continues.
To determine the "height" of the reflected stone from the water's surface, you would use the formula: (2a - height of the actual stone).
This analogy closely mirrors the principle seen in the Wiener process. When W_t surpasses a particular threshold "a" (much like the stone touching the water), its subsequent movements around "a" are reflected, forming a mirrored trajectory. For every time t > T, the height of this mirrored process from the "surface" (level "a") is calculated by 2a - W_t.
Where W_t represents the position of the particle (or the value of the process) at time (t).
Now, regarding the reflection principle and the notion of a barrier at (a): If the Wiener process hits a level (a) at some time (t), it’s just as likely that, if there had been no barrier and the process had been free to move, it would’ve reached a level (-a) by that same time (t). This symmetry is the essence of the reflection principle.
Envision a particle experiencing Brownian motion, starting from W_0 = 0. This particle moves erratically. The main question is the likelihood of this particle reaching a level a > 0 before time t.
One approach is picturing a "barrier" at level a. As the particle first collides with this barrier, think of it as spawning a mirrored counterpart that "reflects" back downwards.
The beauty of this reflection concept is its grounding in the symmetry of Brownian motion: The chances of W_t being over level a equate to it being under level -a.
With the reflection principle in play, a complex question about the particle encountering a barrier morphs into a simpler query: When will two independent Brownian motions touch the level 0? This switch greatly simplifies the math and associated calculations in numerous stochastic process problems.
This principle becomes especially handy when studying American options, where the time of exercise matches a stopping time. Paired with the stopping time, the reflection principle offers a way to determine the joint distribution of the peak Ws value between 0 and t and Wt. This insight proves valuable when pricing unique options like barrier and lookback options.
#ReflectionPrinciple #WienerProcess #BrownianMotion#StochasticProcesses #BarrierReflection #AmericanOptions#StoppingTime #OptionPricing #MathInNature #BarrierOptions#LookbackOptions
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