#booksummaries #binomialmodelinfinance #llms > Last night I decided to try out [Claude 3 Opus 200k](https://poe.com/Claude-3-Opus-200k) and see how well it could do at assisting me in one of my long standing goals of expanding my general literacy in Mathematics and especially in relation to new and different areas that I haven't studied. I dug through my directory of Math books I'll probably never get around to reading... until now? > I just recently discovered a treasure trove of FinTech / FinMath pdfs on [Github](https://github.com/PlamenStilyianov/FinMathematics/) and so Binomial Model in Finance seems like a great choice. Let's see how far it takes us. This is just the first chapter and I did ask a few follow ups and and added bolded headings, I would like to continue this and make it through the whole book. > TLDR: For those interested in the ideas more than the Mathematics, try skipping to the last section on [Motivation and Discussion](#Motivation%20and%20Discussion) and discussion, it's very interesting and accessible without a deep Mathematical background. # Overview Here is a summary of the key points from Chapter 1 of [Binomial Models in Finance](https://github.com/PlamenStilyianov/FinMathematics/blob/master/Binomial%20Models%20in%20Finance.pdf) by John van der Hoek and Robert J. Elliott: # Introduction The book focuses on modeling prices of underlying assets (stocks, currencies, bonds, etc.) and derivative assets (whose prices depend on underlying assets) using binomial models. The fundamental axiom is that there are no arbitrage opportunities. No Arbitrage and Its Consequences - An arbitrage opportunity is a portfolio whose value is zero today but has non-negative value in all future states and strictly positive value in at least one future state. - The Law of One Price theorem states that if two assets have equal payoffs in all future states, they must have the same price today. This is proved by showing an arbitrage would exist otherwise. - A forward contract is an agreement to buy/sell an asset at a future time T for an agreed price F. No payments are made initially. The rational forward price is F = S(0)R where S(0) is the spot price and R is the risk-free return over [0,T]. - Call and put options give the right but not obligation to buy or sell at a strike price K on or before expiry T. Their payoffs at T are C(T) = max(S(T)-K, 0) for calls and P(T) = max(K-S(T),0) for puts. - Various non-arbitrage inequalities are derived, e.g. C(t) ≥ max(S(t)-PV(K), 0), it is never optimal to exercise an American call early on a non-dividend paying stock, put-call parity C-P = S-PV(K) for European options, etc. The key takeaways are: 1) Absence of arbitrage is the fundamental axiom 2) It implies the Law of One Price 3) Allows rational pricing of forwards 4) Leads to various restrictions on option prices 5) Enables relative pricing of derivatives from underlying So in summary, Chapter 1 lays out the basic framework, axioms and initial results that form the foundation for the binomial option pricing approach developed in the rest of the book. The no-arbitrage principle is highlighted as the cornerstone. # Mathematical Details Let's dive deeper into the mathematical details of the Law of One Price Theorem and the derived inequalities. Law of One Price Theorem: Suppose there are two assets A and B with prices at time 0 P0(A) ≥ 0, P0(B) ≥ 0. If at some future time T the prices of A and B are equal in all states of the world, i.e., PT(A) = PT(B), then P0(A) = P0(B). Proof: We prove by contradiction. Suppose P0(A) > P0(B). Consider the following strategy at time 0: 1. Short sell asset A. This realizes P0(A). 2. Buy asset B. This costs -P0(B). 3. Pocket the positive difference P0(A) - P0(B). At time T, the value of this portfolio is PT(B) - PT(A) = 0 in all states of the world. So we have made a riskless profit of P0(A) - P0(B) > 0 at time 0, and have no liabilities at time T. This is an arbitrage opportunity, contradicting our fundamental axiom. Therefore, P0(A) ≤ P0(B). Similarly, assuming P0(B) > P0(A) leads to an arbitrage opportunity. Therefore, we must have P0(A) = P0(B). Derived Inequalities: ### 1. Lower bound for European calls: For a European call option C with strike K and expiry T, C(t) ≥ max(S(t) - PV(K), 0) = (S(t) - PV(K))+ where S(t) is the underlying asset price at time t and PV(K) is the present value of K at time t. Proof: Assume C(t) < S(t) - PV(K). Then at time t: 1. Short sell one unit of asset S. This realizes S(t). 2. Invest PV(K) in a risk-free asset. 3. Buy one call option C. This costs -C(t). 4. Pocket the positive difference S(t) - PV(K) - C(t). At expiry T: - If S(T) > K, exercise the call, sell the asset received to cancel the short position, and collect K from the risk-free investment. The net payoff is 0. - If S(T) ≤ K, let the call expire worthless, use K from the risk-free asset to buy one unit of S to close the short position. Again, the net payoff is 0. In both cases, there are no further liabilities, but we pocketed a profit at time t. This is an arbitrage, hence a contradiction. Therefore, C(t) ≥ S(t) - PV(K). Since call prices are non-negative, we have C(t) ≥ max(S(t) - PV(K), 0). ### 2. Put-Call Parity for European Options: For European put (P) and call (C) options with the same strike K and expiry T, C(t) - P(t) = S(t) - PV(K) Proof: Suppose C(t) - P(t) - S(t) + PV(K) > 0. Consider the following strategy at time t: 1. Short the call C, realizing C(t). 2. Buy the put P, costing -P(t). 3. Buy one unit of asset S, costing -S(t). 4. Borrow PV(K), realizing PV(K). 5. Pocket the positive difference C(t) - P(t) - S(t) + PV(K). At expiry T, the payoff of this portfolio is: - C(T) - P(T) - S(T) + K = - max(S(T)-K, 0) + max(K-S(T), 0) - S(T) + K = 0 So there are no liabilities at expiry, but we made a profit at time t. This is an arbitrage. Similarly, C(t) - P(t) - S(t) + PV(K) < 0 leads to an arbitrage by reversing the above trades. Therefore, the put-call parity C(t) - P(t) = S(t) - PV(K) must hold. These are some of the key mathematical derivations in Chapter 1 that demonstrate the power of the no-arbitrage principle in determining rational prices and relations between different assets. The proofs rely on constructing portfolios that would lead to riskless profits if the stated equalities or inequalities were violated. # Motivation and Discussion The Arbitrage-Free Axiom is more than just a theoretical curiosity; it's a foundational principle in modern financial theory and practice. While it's true that arbitrage opportunities can and do exist in real markets from time to time, they tend to be short-lived and quickly exploited by vigilant traders. The assumption of no arbitrage is a useful simplification that allows for the development of tractable models and provides a basis for rational pricing in financial markets. Motivations for the Arbitrage-Free Axiom: 1. Market Efficiency: The axiom is closely tied to the concept of market efficiency. In an efficient market, prices should reflect all available information, and it should not be possible to consistently make riskless profits. If arbitrage opportunities persisted, traders would quickly exploit them, driving prices to an equilibrium where the opportunities disappear. 2. Rational Pricing: The absence of arbitrage is a necessary condition for rational pricing. If two portfolios have the same payoffs in all future states but different prices, rational investors would buy the cheaper one and sell the expensive one, pushing prices to parity. The Law of One Price, which follows from the no-arbitrage axiom, ensures that assets with identical cash flows are priced consistently. 3. Risk-Neutral Pricing: The Arbitrage-Free Axiom is the foundation of risk-neutral pricing, a key concept in derivatives valuation. In a risk-neutral world, the expected return on all assets is the risk-free rate, and the actual probabilities of future states are replaced by risk-neutral probabilities. This simplifies the pricing of complex derivatives. 4. Tractability: Assuming no arbitrage allows for the development of tractable mathematical models. Without this assumption, models would need to account for the complexities of real-world market inefficiencies, making them much harder to solve and interpret. Relevance in Practice: While the Arbitrage-Free Axiom is a simplification of reality, it has proven to be a highly useful one in practice. 1. Derivatives Pricing: Models based on the no-arbitrage principle, such as the Black-Scholes model and the Binomial Option Pricing Model, are widely used in the financial industry to price and hedge derivatives. 2. Risk Management: Banks and financial institutions use these models to manage their risk exposures. The models help in computing hedge ratios, assessing the sensitivity of portfolios to various risk factors, and setting risk limits. 3. Market Making: Market makers, who provide liquidity by constantly quoting bid and ask prices, rely on no-arbitrage relationships to ensure their quotes are consistent across different assets and maturities. 4. Anomaly Detection: Deviations from no-arbitrage conditions can signal mispricing in the market. Traders and quantitative analysts often monitor these deviations to identify trading opportunities. In conclusion, while the Arbitrage-Free Axiom is a simplification of reality, it provides a powerful framework for understanding and modeling financial markets. Models based on this axiom have become indispensable tools in derivatives pricing, risk management, and quantitative trading strategies. The axiom's relevance lies in its ability to provide a tractable and consistent basis for rational pricing and decision-making in finance.