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 considers how to define quantitative finance and whether finance belongs primarily to mathematics or economics. One proposed definition treats quantitative finance as applied mathematics drawing on statistics, economics, econometrics, game…
A European put with strike K pays no more than K at maturity, since its payoff is the greater of K minus the underlying price and zero. The document applies the no-arbitrage principle: the put cannot be worth more before maturity than a zero-coupon bond…
The document presents a Python implementation of Longstaff–Schwartz least-squares Monte Carlo for an American-style Asian call with exercise available at discrete dates. The payoff depends on the average underlying price observed up to exercise. The reported…
The document explains why option volatility surfaces may use forward moneyness instead of raw strike. Moneyness measures a strike’s distance from a reference price; forward moneyness compares the strike with the forward price for that option’s maturity. This…
The document defines rolling for credit default swap contracts, distinguishing the creation of a new series from the maturity of an existing contract. For CDS indices, a new series is introduced on semiannual dates, and its basket of reference entities may…
The document explains two ways to create forward scenarios for a bond yield series. One approach resamples historical yield changes: repeatedly draw changes from the observed series, accumulate them from the latest yield, and use the resulting paths to…
The document asks how a risk surface can turn sensitivities from a model’s implied volatility function into sensitivities to market quoted volatilities. The described method adds an interpolated surface of risk bumps to the model volatility. At the base…
The document raises a question about delta for American-style exotic options whose holders can make choices at specified times during the option’s life. The holder is assumed to follow an optimal policy that maximizes the option’s value, described in terms…
The document distinguishes an analytic swap PV01 from a market risk measure calculated by shifting rates. Analytic PV01 measures the value change from moving the fixed coupon by one basis point while holding discount factors unchanged; it is useful for…
The document discusses estimating the volatility and mean-reversion parameters of a Hull–White short-rate model when the market has daily overnight rates and a swap-based zero-coupon curve but no actively traded options. It distinguishes estimation from…
The document derives probabilities for the first observation date on which a geometric Brownian motion underlier exceeds an autocall barrier. It writes the asset as an exponential of drift and Brownian motion, then converts the barrier condition at each…
The document examines whether combining a long butterfly, which benefits from a price near its center at expiration, with a long straddle, which benefits from a large move, can produce a profit across all terminal prices. The accepted answer explains the…
The document asks whether negative short-term risk-free rates require changes to the Black-Scholes option pricing formula. Its answer says the standard model does not require the rate to be positive, so a negative rate can be used directly. Under the stated…
The document addresses why interest rate swaps are often valued using forward rates derived from today’s yield curve instead of simulating each future floating payment. It explains that spot rates are bootstrapped from traded instruments and imply forward…
The document introduces the risk-neutral density as the probability distribution of an asset price under an equivalent martingale measure. It asks whether that distribution can be described by a density with respect to ordinary length on the price axis, and…
The note addresses whether a rates Bermudan callable or swaption remains convex as its strike changes, even though possible exercise dates correspond to swaps with different remaining maturities. It represents the exercise payoff at a given date as the…
The document discusses how to express the Heston stochastic volatility pricing equation using a forward price, log-moneyness, time to expiry, and an undiscounted option value. It gives the spot and variance processes, their correlated Brownian shocks, and a…
The document explains why the Black–Scholes delta of an in-the-money call can first decline and later increase as volatility rises. Interpreting delta as the risk-neutral probability of exercise, it separates two effects: greater volatility makes the strike…
The document examines how a Black–Scholes call price approaches its terminal payoff as time to maturity shrinks to zero. It rewrites the price into a term involving the discounted intrinsic-value difference and a second term expressed as an integral of the…
The document introduces Fischer Black’s idea that the nominal short rate can be viewed as an option: it is the greater of zero and the sum of a shadow real interest rate and inflation. The shadow rate is described as tied to the investment opportunity set,…
The note explains how delta-hedging error arises when an option is valued using an assumed volatility that differs from the underlying asset's true volatility. The option's Black–Scholes pricing equation links its theta to its price, delta, gamma, interest…
The note asks why options on leveraged exchange-traded funds can appear cheaper than options on an unleveraged fund tracking the same index and expiry, despite an expectation that leverage should multiply implied volatility. It explains that a daily…
The note addresses why a call spread built from two outer strikes does not create arbitrage simply because its price exceeds the price of an intermediate-strike call. The intermediate call is cheaper than the weighted combination, so buying it and selling…
The document compares two numerical approaches for pricing European calls under the Heston–Nandi GARCH model using an inversion integral. One implementation evaluates the integrand through adaptive quadrature, while another precomputes…