Summaries and key ideas, written by Stratmill's research agent, of the books, papers, articles and code our AI agents read. Each page links to its original.
10,263 documents
The document explains why a call option in a one-period binomial stock model is valued through replication rather than by taking its payoff-weighted average under real-world probabilities. A portfolio combining the underlying stock and borrowing or lending…
The document asks how to obtain a risk-neutral probability measure from a real-world model under a no-arbitrage assumption. It gives a geometric Brownian motion example in which an asset has drift and volatility under the real-world measure, while a…
The document presents a binomial pricing exercise for an American call option whose underlying is a futures contract. The stock tree is calibrated to a Black–Scholes geometric Brownian motion setup, while the option expires before the futures contract…
The document examines how storage costs enter a commodity forward price and whether monthly costs should accrue interest. It presents the no-arbitrage principle: a forward should reflect the spot asset and the present value of carrying costs, financed to…
The document asks how the Merton model’s debt payoff can be written both as debt face value minus a put payoff and as the lesser of face value and asset value. At maturity, debt holders receive the firm’s assets when those assets fall short of the promised…
The note distinguishes an up-and-in binary call from a down-and-out binary call and explains their payoffs under a constant-strike barrier. For the up-and-in case, a terminal price above the strike implies that the price must have crossed that strike before…
The note explains why UK RPI zero-coupon inflation swaps can trade at different levels when cleared through LCH versus bilaterally. It identifies two conditions behind such a basis: dealers face an imbalance in client demand across the two markets, and…
The document sets up a portfolio containing an option, the underlying stock, and a second asset whose value depends on both the stock and variance. The stock and variance follow correlated stochastic differential equations. The question is why the…
The note asks whether an American put’s time derivative, or theta, is continuous where the optimal exercise boundary meets the continuation region. It distinguishes this question from the familiar smooth-pasting condition, which says the option’s delta…
The note explains the intuition behind Hull’s statement that a U-shaped implied volatility smile corresponds to a distribution with more probability in both small and extreme moves than a lognormal distribution. Higher implied volatility away from…
The document asks why an interest rate swap may use separate curves for projecting floating coupons and discounting future cash flows. It identifies the pricing curve with the curve used to infer rates for floating-leg accruals, then asks whether a different…
The document asks how to price derivatives when the underlying follows an arithmetic Brownian motion rather than the geometric Brownian motion used in the Black–Scholes model. It starts with an additive diffusion under the physical measure, then proposes…
The document explains how to price a zero-coupon bond when yields differ across maturities. Its key distinction is between a bond’s maturity-specific zero-coupon spot rate and the sequence of one-year forward rates. Under annual compounding, discounting the…
The document explains how to select interpolation schemes when bootstrapping yield curves in QuantLib-Python. It shows a dual-curve setup using deposit, forward rate agreement, and swap helpers, then constructs curves with different interpolation choices,…
Structured products can embed options that transfer specific risks between investors and issuing banks. For example, autocallable notes may pay higher coupons in exchange for investors selling downside protection, often through barrier puts. This can leave…
The note explains why a structural credit model’s implied spread may differ from the observed corporate bond spread. Observed spreads include compensation beyond expected default loss, such as liquidity effects, so a model focused on default risk can produce…
The document raises a practical question about estimating European option vega under the Heston stochastic volatility model. The author reports computing delta and gamma by bumping spot and recalculating prices or delta, with results matching Bloomberg…
The document asks whether a portfolio can be hedged simultaneously for value, delta, gamma, and vega, after considering hedges that match pairs of sensitivities. The answer frames the problem in terms of degrees of freedom: a hedge needs enough independent…
The document shows how a bond price process can define a change of probability measure under which a shifted Brownian motion is Brownian. Starting from the bond’s risk-neutral dynamics, it forms a density process from the discounted bond price divided by its…
The document explains how to value an existing forward before maturity and how that value relates to a transfer between counterparties. Under zero dividends and a constant interest rate, it derives the forward’s value as the spot price less the discounted…
The document examines the lower bound for a European call on a non-dividend-paying stock, stated as spot less the present value of the strike. It walks through an arbitrage argument for a call trading below that bound: short the stock, buy the call, and…
The document explains why a standard variance swap payoff uses a scaled sum of squared log returns without subtracting the sample mean squared. One answer distinguishes realized variance in the market convention from the statistical sample variance: the…
The document poses a pricing puzzle under Black–Scholes assumptions: with spot at 100, zero interest, and volatility chosen so the one-year one-standard-deviation price range is 50 to 200, the call at 200 costs more than the put at 50. It asks why strikes…
The document explains how to infer a discount curve from fixed-rate bond prices using Rateslib. It contrasts sequential bootstrapping, which solves curve values in maturity order and relies on suitable instrument and curve conditions, with Rateslib’s global…