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 a curve trade described as buying five-year bonds and hedging with two- and ten-year bonds. The basic idea is to offset the purchased bond’s interest-rate exposure by selling bonds at shorter and longer maturities. It illustrates the…
The document explains why a cash flow in one currency may be discounted using the overnight rate associated with the collateral currency. Under a USD collateral agreement remunerated at USD OIS, the foreign-currency cash flow is converted through the…
The document shows how to transform the Black–Scholes partial differential equation into a simpler equation for the discounted option value as a function of the discounted underlying. It begins with the original equation and defines both discounted…
The document asks about an arbitrage proof for the value of a long forward contract and whether borrowing the contract’s value creates a repayment obligation that changes the proof’s final balance. The answer recasts the problem as replication under a…
The document explains how CDS premiums and protection payments relate to survival and default probabilities. The premium leg is paid while the reference entity survives, with discounting for the time value of cash flows. Accrued premium accounts for the…
The answer explains why a curve steepener can carry negative convexity when it sells a longer-maturity rate product and buys a shorter-maturity product in matched delta. Longer-maturity bonds and swaps generally have greater convexity, and convexity tends to…
The discussion compares buying a zero-dividend stock with buying a hypothetical American call on one share that has a zero strike and no expiry. It explains that, without dividends and under no-arbitrage assumptions, the discounted stock price is a…
The document considers whether the required or expected return on equity can be inferred when equity is modeled as a call option on the firm’s assets, even though no corresponding option trades in the market. It presents a calculation that treats equity…
The document examines whether a credit default swap spread approximates the spread between a risky par bond’s coupon or yield and a risk-free bond’s rate. It presents a general expression in terms of discount factors, survival probabilities, and coupon…
The document explains two ways to value rebates in barrier options under Black–Scholes assumptions. For a down-and-out call with a rebate paid at maturity, the payoff can be separated into the ordinary surviving call payoff and a rebate paid if the barrier…
The document discusses how to express the price of a European call as a discounted expectation under a risk-neutral measure, starting from a binomial stock-price model and considering its continuous-time limit. The key pricing correction is to discount the…
Local volatility models can be calibrated to European vanilla option prices across strikes and maturities, reproducing the current implied-volatility surface. Dupire’s relationship derives a risk-neutral diffusion consistent with a sufficiently smooth…
The question describes estimating a risk-neutral density by choosing distribution parameters that minimize squared differences between model and observed call and put prices. The answer says that when the assumed terminal distribution is lognormal, its…
The answer explains the square-root-of-time rule of thumb for the vega of at-the-money options. Under Black–Scholes, vega is proportional to the underlying price, the normal density evaluated at d1, and the square root of time to expiry. Near the money, for…
The document asks which LIBOR observations are used for the stub rate when calculating a floater’s discount margin. It refers to a formula reproduced from a credit-spread paper and proposes that a Bloomberg yield-analysis page’s index-to-next-coupon field…
The document poses a derivatives-pricing question: how to derive the stochastic differential equation for a futures price when the underlying stock follows geometric Brownian motion but its interest rate and volatility vary randomly over time. It notes the…
The discussion asks why a stock's expected return does not directly appear in the Black-Scholes option-pricing formula. Several replies explain the result through replication and hedging: a dynamically adjusted stock position can offset exposure to the…
The document asks about simulating an integral driven by a Black–Scholes asset process, with the integrand switching according to whether the asset price is below a strike. The author describes this as a naive hedge for a European put and notes that such a…
The document poses an asymptotic option-pricing question. With the underlying price, interest rate, and time to expiry normalized, it defines a put price as the larger of a linear function of strike above a threshold and the put’s intrinsic value. Below or…
The document shows how to choose a drift adjustment for an exponential Lévy asset price so that its discounted, dividend-reinvested value is a martingale under the risk-neutral measure. The key step conditions on the natural filtration: the current process…
The document asks how to hedge foreign exchange exposure on a long-dated EUR bond while retaining interest rate exposure. Its answer recommends hedging the bond’s future cash flows with a fixed-for-fixed currency swap, which can be viewed as a series of…
The discussion outlines a cash-flow-based framework for valuing dual-currency bonds. Discount cash flows using the swap curve for the payment currency, adjusted by the relevant cross-currency basis when valuing in another currency. The example values the…
The discussion explains how an FX option table organized by strike relates to a volatility chart organized by delta. Delta links a strike to the option’s moneyness: at a given strike and maturity, the call and put have opposite-side deltas and one is in the…
The discussion explains why volatility measured on log returns cannot generally be converted into a unique standard deviation of price changes using an exponential transformation. Log-return volatility is scale invariant: multiplying every observed price by…