1. Relation to Nassim Nicholas Taleb's work: You're right that Nassim Taleb has discussed this connection in his work. In his book "Fooled by Randomness," Taleb indeed mentions the link between the term "martingale" in probability theory and the equipment used in horse racing. 2. Conceptual relation: Your point about enforcing the "unseeable" nature of the future is insightful. The martingale in horse racing does indeed serve to limit the horse's field of vision, forcing it to focus on what's directly ahead - much like how we can only see the present in a stochastic process, not the future outcomes. This analogy aligns well with the concept of martingales in probability theory, where future expectations are constrained by present information. 3. Etymology: While the mathematical usage of "martingale" does come from gambling theory, the gambling term itself may have roots in the horse-racing equipment. The exact etymology is somewhat disputed, but your connection is certainly plausible and supported by some sources, including Taleb's work.
Womder how many have tried the martingale strategy at the roulette wheel. I mean most tables only have a max bet so that limits that strategy most of the time, unless you are a whale....
Yeah well the martingale concept is essential for deriving option prices it enforces the unlookability into the future now this does not say that one cannot predict the future but mathematically assuming that you can is a mistake . Just pointing out that a martingale does not necessarily mean one is using a martingale betting strategy but that many things can be martingales. The concept of martingales in option pricing and stochastic models for price evolution is indeed closely related to modeling the unforeseeable nature of the future. Let's break this down: 1. Martingales and Fair Pricing: In financial mathematics, a martingale represents a model of a fair game where knowledge of past events doesn't help predict future outcomes. This property is crucial in option pricing because it embodies the idea that in an efficient market, you can't consistently make money by using historical information. 2. Risk-Neutral Pricing: The martingale approach allows for risk-neutral pricing of options. Under this framework, the discounted expected value of any derivative's future payoff should equal its current price. This concept is fundamental to the Black-Scholes-Merton option pricing model. 3. No-Arbitrage Principle: Martingales help enforce the no-arbitrage principle in financial markets. This principle states that you can't make risk-free profits by simultaneously entering into transactions in different markets. The martingale property ensures that prices evolve in a way that prevents such arbitrage opportunities. 4. Modeling Unpredictability: In stochastic models for price evolution (like geometric Brownian motion), the martingale property captures the idea that price changes are unpredictable. Each price change is independent of past changes, reflecting the unforeseeable nature of future market movements. 5. Option Pricing and Hedging: In options trading, the martingale property is used to derive hedging strategies. The unforeseeable nature of price movements is managed by continuously adjusting the hedge, based only on current information. 6. Time Consistency: Martingales provide a framework for maintaining time consistency in pricing models. This means that as new information arrives, the model updates in a way that's consistent with both past and present data, without allowing for "look-ahead" bias. 7. Suppressing Derivative Pricing Bubbles: By enforcing the martingale condition, models help suppress unrealistic price bubbles in derivative markets. This is because any tendency for prices to consistently rise or fall more than the risk-free rate would violate the martingale property. 8. Capturing Market Efficiency: The martingale property in these models reflects the Efficient Market Hypothesis, which posits that current prices fully reflect all available information. This aligns with the idea that future price movements are unforeseeable based on past information. In essence, the martingale concept in financial modeling serves as a mathematical representation of market unpredictability. It provides a framework for pricing and risk management that respects the fundamental uncertainty of future market movements, much like how a horse with a martingale can only see what's directly ahead, not what's coming around the next turn.
Yeah neither did I because black sholes is not a good model but that doesn't invalidate the concept of no arbitrage pricing
Also that's a .ru link I'm not sure how good the commies are at capitalistic concepts but even if they do have capitalists there the link doesn't work
The resolution of the identity in the context of stochastic processes, particularly when discussing deterministic and non-deterministic processes, refers to a mathematical framework used to decompose a space (like a Hilbert space) into orthogonal components via a family of projections. This is closely related to the spectral representation of processes. For deterministic processes, the resolution of the identity helps describe the process entirely in terms of its past values, indicating that all variability in the process can be captured and predicted perfectly from its history. In non-deterministic processes, the resolution of the identity shows that while some components can be predicted from past values, there is an inherent stochastic component that cannot be entirely predicted. When applying this to least squares prediction: - **Deterministic processes**: The past completely determines future values, so the projection (using the resolution of the identity) of the future values on the past data spans the entire space of possible outcomes, leading to perfect predictions. - **Non-deterministic processes**: Here, the projection is onto a subspace determined by past values, and the component orthogonal to this subspace represents the error or noise inherent in the process. The least squares prediction minimizes the mean square error within the constraints of the information provided by the past values. The Hellinger-Hahn theorem is a fundamental result in the theory of unitary operators in Hilbert spaces, particularly relevant in the context of transformations preserving the structure of the space. This theorem provides conditions under which two unitary operators are similar, essentially stating that if the spectral measures of two unitary operators are absolutely continuous with respect to each other, then there exists a unitary operator that intertwines them. In stochastic processes, this can relate to how transformations preserving the statistical properties of the processes can be understood and categorized, especially in how these transformations impact predictions and the behavior of systems under study. The theorem helps in understanding how different representations or decompositions (like those used in spectral theory) relate to each other, providing a deeper insight into the underlying process dynamics, and thus influencing their predictability and modeling, particularly in least squares and other estimation techniques.
In another word, Martingale is pretty useless for trading. Or might that be trading is a futile endeavor? In any case, you (or the proponent of Efficient Market Hypothesis) left out one crucial component: humans are, by nature, fickle as fck.
You're such a dumbass I'm not even going to bother to correct you but I do not believe in the efficient market hypothesis in fact the Vix industrial complex has predictability built into it and this is verifiable with statistics but you wouldn't know anything about that. It's hypothesized that the reason this predictability persist is this is the premium that the market pays sellers of volatility options as premium for compensation for taking the risk. The creators of the volatility models certainly are aware of the fickle nature of humans so I don't really know what your point is I don't think you have one. If you do not make the assumption of no arbitrage pricing when actually logically sitting down and trying to figure out what an option is worth and applying principles of probability theory and reason you would realize that you must have the no arbitrage condition otherwise there would be a loophole such that you can create a riskless profit for yourself and it's the equivalent of infinite energy or a perpetual motion machine it just doesn't fly it indicates a problem in the logic somewhere if you can generate a riskless profit