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Finance & Options

No-arbitrage pricing, Black-Scholes, and the Greeks.

Prerequisites: Stochastic Calculus + Probability

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  1. 1Options: Calls, Puts and PayoffsOptions work like insurance on a stock: what calls and puts pay at expiry, how to draw and combine their payoffs, and what in, at and out of the money mean.
  2. 2No-Arbitrage and Put-Call ParityTwo portfolios with the same payoff must cost the same, and that one rule gives put-call parity, C − P = S − K e^(−rT), with no model at all.
  3. 3Risk-Neutral PricingWhy an option’s price doesn’t depend on the probability the stock goes up.
  4. 4A Coin-Flip Tree That Becomes Black-ScholesPaths that end at the same price fold into one node, so a 252-step tree prices in well under a millisecond, and as the steps shrink its price settles on the Black-Scholes number.
  5. 5Fair Games and the Markov PropertyUnder the risk-neutral odds every discounted price is a fair game, and carrying the running maximum in the state turns a path-dependent lookback into a small backward recursion.
  6. 6The Black-Scholes PDEHedge away all the risk, and the option’s value must satisfy one equation.
  7. 7Black-Scholes Is the Heat EquationA few substitutions turn the Black-Scholes PDE into the equation for heat spreading along a rod, with the payoff as the starting temperature.
  8. 8The Black-Scholes Formula: N(d₂) as a ProbabilityTake the risk-neutral expectation of the call payoff and the formula falls out, with N(d₂) as the risk-neutral chance of exercise and N(d₁) as the same chance with paths weighted by the stock.
  9. 9Delta and Gamma HedgingDelta says how much an option moves when the stock moves one dollar, and gamma says how fast delta itself changes, which is why a hedge has to be redone as the stock moves.
  10. 10Monte Carlo Option PricingPrice an option by averaging simulated payoffs, then shrink the slow 1/√n error with antithetic pairs and control variates.
  11. 11American Options and Early ExerciseExercising early is an optimal stopping problem: a deep in-the-money put is sometimes worth exercising before expiry, and a call on a non-dividend stock never is.
  12. 12Interest-Rate Trees, Forwards and FuturesLet the interest rate move on a coin-toss tree, price zero-coupon bonds from it, and see why forward and futures prices come apart once rates are random.

Practice

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