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Market liquidity plays a vital role in the field of market micro-structure, because it is the vigor of the financial market. This paper uses a variable called convexity to measure the potential liquidity provided by order-book. Based on the high-frequency data of each stock included in the SSE (Shanghai Stock Exchange) 50 Index for the year 2011, we report several statistical properties of convexity and analyze the a…
There is an observed basis between repo discounting, implied from market repo rates, and bond discounting, stripped from the market prices of the underlying bonds. Here, this basis is explained as a convexity effect arising from the decorrelation between the discount rates for derivatives and bonds. Using a Hull-White model for the discount basis, expressions are derived that can be used to interpolate the repo rates…
The performance of trend following strategies can be ascribed to the difference between long-term and short-term realized variance. We revisit this general result and show that it holds for various definitions of trend strategies. This explains the positive convexity of the aggregate performance of Commodity Trading Advisors (CTAs) which -- when adequately measured -- turns out to be much stronger than anticipated. W…
In this work we derive an approximated no-arbitrage market valuation formula for Constant Maturity Credit Default Swaps (CMCDS). We move from the CDS options market model in Brigo (2004), and derive a formula for CMCDS that is the analogous of the formula for constant maturity swaps in the default free swap market under the LIBOR market model. A "convexity adjustment"-like correction is present in the related formula…
Automated market makers (AMMs) are a new prototype of decentralised exchanges which are revolutionising market interactions. The majority of AMMs are constant product markets (CPMs) where exchange rates are set by a trading function. This work studies optimal trading and statistical arbitrage in CPMs where balancing exchange rate risk and execution costs is key. Empirical evidence shows that execution costs are accur…
We study the problem of the optimal execution of a large trade in the presence of nonlinear transient impact. We propose an approach based on homotopy analysis, whereby a well behaved initial strategy is continuously deformed to lower the expected execution cost. We find that the optimal solution is front loaded for concave impact and that its expected cost is significantly lower than that of conventional strategies.…
This paper discusses the valuation of credit default swaps, where default is announced when the reference asset price has gone below certain level from the last record maximum, also known as the high-water mark or drawdown. We assume that the protection buyer pays premium at fixed rate when the asset price is above a pre-specified level and continuously pays whenever the price increases. This payment scheme is in fav…
In this three-part series of papers, we argue that the conventional spread measures are not well defined for credit-risky bonds and introduce a set of credit term structures which correct for the biases associated with the strippable cash flow valuation assumption. We demonstrate that the resulting estimates are significantly more robust and remain meaningful even when applied to deeply distressed bonds. We also sugg…
We present a simple framework for dynamic portfolio management that uses nothing but daily prices, trading volumes, and market capitalizations. Its state is three fixed-size matrices built from the price history: the distance matrix of the return correlations and the transition matrices of two Markov chains that rank the S\&P 500 names monthly by trailing return and by trailing volatility. These three matrices rest o…
In financial markets, accurately measuring the risk of future fluctuations in asset prices is of paramount importance. Studies such as Carr and Madan have shown that the expected value of the quadratic variation of log prices can be expressed as an integral of European option prices over a continuum of strikes. This has led to the widespread estimation of model-free volatility (implied variance). However, this theore…
We develop a comprehensive framework to measure the impact of the climate transition on investment portfolios. Our analysis is enriched by including geographical, sectoral, company and ISIN-level data to assess transition risk. We find that investment funds suffer a moderate 5.7% loss upon materialization of a high transition risk scenario. However, the risk distribution is significantly left-skewed, with the worst 1…
Executing even moderately large derivatives orders can be expensive and risky; it's hard to balance the uncertainty of working an order over time versus paying a liquidity premium for immediate execution. Here, we introduce the Time Is Money model, which calculates the Equilibrium Trading Horizon over which to execute an order within the adversarial forces of variance risk and liquidity premium. We construct a hypoth…
We revisit the problem of pricing options with historical volatility estimators. We do this in the context of a generalized GARCH model with multiple time scales and asymmetry. It is argued that the reason for the observed volatility risk premium is tail risk aversion. We parametrize such risk aversion in terms of three coefficients: convexity, skew and kurtosis risk premium. We propose that option prices under the r…
We derive an arbitrage free relationship between recovery swap rates, digital default swap spreads and conventional CDS spreads, and argue that the fair forward recovery rate used in recovery swaps must contain a convexity premium over the expected recovery value.
This paper studies a type of periodic utility maximization problem for portfolio management in incomplete stochastic factor models with convex trading constraints. The portfolio performance is periodically evaluated on the relative ratio of two adjacent wealth levels over an infinite horizon, featuring the dynamic adjustments in portfolio decision according to past achievements. Under power utility, we transform the …
This paper compares mathematical models for automated market makers including logarithmic market scoring rule (LMSR), liquidity sensitive LMSR (LS-LMSR), constant product/mean/sum, and others. It is shown that though LMSR may not be a good model for Decentralized Finance (DeFi) applications, LS-LMSR has several advantages over constant product/mean based automated market makers. However, LS-LMSR requires complicated …
Leverage does not create manipulation or informed trading in event markets, but it changes their economics. We separate four conduct channels: market-price manipulation, real-world outcome manipulation, resolution-process manipulation, and informed trading that exploits non-public information without changing the event or resolution rule. A capital-constrained amplification model shows that gross directional gains sc…
In this article, we look at the effect of volatility clustering on the risk indifference price of options described by Sircar and Sturm in their paper (Sircar, R., & Sturm, S. (2012). From smile asymptotics to market risk measures. Mathematical Finance. Advance online publication. doi:10.1111/mafi.12015). The indifference price in their article is obtained by using dynamic convex risk measures given by backward stoch…
A convertible is a bond plus an embedded call on the issuer’s stock — credit with equity convexity, or equity with a coupon, depending on the delta.
Crisis alpha is return earned from persistent trends that form after a market crisis starts — not a prediction of the crash day, and not a put that pays on a two-day dip.
Whipsaw is the range-bound regime where trend signals flip, scratch, and bleed — the ordinary cost of owning tail convexity.
Slow trend: lookbacks of roughly 6–12 months, low turnover, fewer whipsaws, later entries, and the bulk of classic CTA crisis convexity.
A managed-futures book that is structurally long options or long VIX-curve convexity — pays carry, aims to print in jumps and persistent stress.
A butterfly is long one wing, short two bodies, long the other wing — a bet on a pin or on the curvature of the smile.
A call option is the right, not the obligation, to buy the underlying at a strike by expiry — convex upside for a premium.
Gamma is the sensitivity of delta to the underlying — how fast the hedge ratio moves, and who is chasing whom.
Skew measures the relative richness of downside versus upside implied volatility, helping track hedging demand and asymmetry in market risk pricing.
Theta is the sensitivity of option value to the passing of time — the daily rent of holding convexity.
Convexity Risk — Non-linear price response to yield changes, especially relevant in MBS and long bonds.
The 2s10s Treasury curve measures the spread between 10-year and 2-year Treasury yields and is a key indicator of growth expectations, policy path, and term structure dynamics.
A convertible is a bond plus an embedded call on the issuer’s stock — credit with equity convexity, or equity with a coupon, depending on the delta.
Convexity Risk — Non-linear price response to yield changes, especially relevant in MBS and long bonds.
Whipsaw is the range-bound regime where trend signals flip, scratch, and bleed — the ordinary cost of owning tail convexity.
Slow trend: lookbacks of roughly 6–12 months, low turnover, fewer whipsaws, later entries, and the bulk of classic CTA crisis convexity.
A managed-futures book that is structurally long options or long VIX-curve convexity — pays carry, aims to print in jumps and persistent stress.
Theta is the sensitivity of option value to the passing of time — the daily rent of holding convexity.