Abstract
In this communication, based upon the stochastic Gompertz law of population growth, we have reformulated the Leaky Competing Accumulator (LCA) model with multiple alternatives such that the ...positive-definiteness of evidence accumulation is automatically satisfied. By exploiting the Lie symmetry of the backward Kolmogorov equation (or Fokker–Planck equation) assoicated with the modified model and applying the Wei–Norman theorem, we have succeeded in deriving the
N
-dimensional joint probability density function (p.d.f.) and marginal p.d.f. for each alternative in closed form. With this joint p.d.f., a likelihood function can be constructed and thus model-fitting procedures become feasible. We have also demonstrated that the calibration of model parameters based upon the Monte Carlo simulated time series is indeed both efficient and accurate. Moreover, it should be noted that the proposed Lie-algebraic approach can also be applied to tackle the modified LCA model with time-varying parameters.
This paper proposes a simple bounded stochastic motion to model equity price dynamics under shocks. The stochastic process has a quasi-bounded boundary which can be breached if the probability ...leakage condition is met. The quasi-boundedness of the process at the boundary can thus provide an indicator of the possible risk of equities under price shocks or in distress. Empirical calibration of the model parameters of the proposed process for equities can be performed easily due to the availability of an analytically tractable probability density function which generates fat-tailed distributions consistent with empirical observations. The volatility and mean-reversion of the S&P500 dynamics calibrated by the process are positively and negatively co-integrated, respectively, with the VIX index representing the level of market distress. The process captures the high likelihood of Hertz’s default about two months earlier, using only information until that point, and before the firm filed for Chapter 11 bankruptcy in May 2020 as a result of the COVID-19 pandemic. Empirical calibration of the process for GameStop’s stock price shows that the short squeeze in the stock occurred when the condition for breaching the upper boundary was met on 14 January 2021, i.e., about two weeks before major short-sellers closed out their positions with significant losses. The trading volume of the stock was positively co-integrated with the probability leakage ratio.
In this paper, based upon the Gompertz model of population growth, we have reformulated the stochastic Lotka-Volterra Competition model with N interacting species. By means of Ito's lemma and some ...simple changes of variables, we have succeeded in deriving the N-dimensional joint probability density function of the stochastic modified multi-species Lotka-Volterra model in closed form. With this joint probability density function, an analytical likelihood function can be constructed readily, and thus model-fitting procedures become feasible and efficient.
We derive the Bitcoin exchange rate dynamics by solving the exchange rate equation of the standard flexible-price monetary model to investigate any characteristics of Bitcoin like a currency. The ...dynamics is driven by an asymmetric mean-reverting fundamental shock which can be attributed to a money demand shock. A crash occurs when the exchange rate with a weakened mean-reverting force breaches a lower boundary where a smooth-pasting condition is imposed. The empirical results show the exchange rate dynamics can be calibrated according to the model, in which the mean reversion of the dynamics is positively co-integrated with the Bitcoin transaction volume indicating demand for Bitcoin; and with the risk reversals of the commodity currencies (Australian dollar and Canadian dollar) in currency option markets. The analysis shows that the Bitcoin exchange rate shares some characteristics of commodity currencies with crash risk. This suggests that Bitcoin behaves as a currency between fiat money and a crypto-commodity used for trading and investment purposes.
•The analysis shows that Bitcoin behaves like a currency with crash risk.•A fundamental shock as a money demand shock drives the exchange rate dynamics.•Bitcoin crash is allowed under the smooth-pasting boundary condition.•The model can empirically describe the Bitcoin exchange rate dynamics.•Co-integration analyses support inclusion of money demand shocks and crash risk.
Asymmetric behaviour has been documented in unemployment rates which increase quickly in recessions but decline relatively slowly during expansions. To model such asymmetric dynamics, this paper ...provides a rigorous derivation of the asymmetric mean-reverting fundamental dynamics governing the unemployment rate based on a model of a simple labour supply and demand (fundamental) relationship, and shows that the fundamental dynamics is a unique choice following the Rayleigh process. By analogy, such a fundamental can be considered as a one-dimensional overdamped Brownian particle moving in a logarithmic-harmonic potential well, and a simple prototype of stochastic heat engines. The solution of the model equation illustrates that the unemployment rate rises faster with more flattened potential well of the fundamental, more ample labour supply, and less anchored expectation of the unemployment rate, suggesting asymmetric unemployment rate dynamics in recessions and expansions. We perform explicit calibration of both the unemployment rate and fundamental dynamics, confirming the validity of our model for the fundamental dynamics.
Our paper presents a crude oil price model in which the price is confined in a wide moving band. A price crash occurs when the price breaches the lower boundary where a smooth-pasting condition is ...imposed. Using an asymmetric mean-reverting fundamental (supply/demand) shock, the solution derived from the oil price equation for the model shows the oil price follows a mean-reverting square-root process, which is quasi-bounded at the boundary. The oil price dynamics generates left-skewed price distributions consistent with empirical observations. A weakened mean-reverting force for the price increases the probability leakage for the price across the boundary and the risk of a price crash. The empirical results show the oil price dynamics can be calibrated according to the model, where the mean reversion of the price dynamics is positively co-integrated with the oil production reaction to negative demand shocks, and with the risk reversals of the commodity currencies, the Canadian dollar and the Australian dollar in currency option markets. The results are consistent with an increased price crash risk with negative demand shocks and negative risk reversals. The forecasting performance of the oil price model is better than the futures-spread models and random walk models during the crash periods. While the price of oil was above the lower boundary for most of the time, the conditions for breaching the boundary were met in 2008 and 2014 when the price fell sharply.
On 6 September 2011, a ceiling on the value of the Swiss franc was imposed, at CHF 1.2 per euro. With the continuous weakness of the euro area economy, this exchange rate limit was abandoned on 15 ...January 2015. This paper proposes a quasi-bounded process for the Swiss franc exchange rate dynamics under a one-sided target zone during this period, in which the exchange rate can breach the strong-side limit under a restricted condition of the relationship between the parameters of the drift term and stochastic part of the process. The empirical results using market data during 6 September 2011–14 January 2015 with a rolling one-year window suggest that this model can describe the dynamics of the Swiss franc under a one-sided target zone, where the drifting force is an increasing function of foreign reserves. While the exchange rate was bounded below the strong-side limit during most of the time, as indicated by its dynamics, the condition for breaching the limit was met in November 2014 using only information until that point, i.e., about two months before abandoning the limit.
•We model the Swiss franc exchange rate dynamics.•The dynamics is found to be quasi-bounded under the one-sided target zone.•The mean-reverting drift is an increasing function of foreign reserves.•The model signals for the Swiss franc breaching the strong-side limit.
•Option-implied correlation in CDS indexes is estimated by a basket-option model.•Correlation between iTraxx Financials and Non-Financials Indexes is estimated.•Option-implied correlation measures ...the spillover in the European debt crisis.•Information flow was from option-implied correlation to realized correlation.•Sovereign and funding liquidity risks are determinants of the correlation.
This paper proposes an analytic method to estimate the option-implied correlation embedded in options on the iTraxx Europe CDS indexes. The option-implied correlation is suggested as a measure of the spillover effect of default risk between the financial and corporate sectors in Europe. In particular, the correlation between the iTraxx Financials and Non-Financials sub-indexes is estimated from options on the iTraxx Main Index, which is considered as a basket option with the two sub-indexes being its underlyings. The abrupt changes of the realized correlation anticipated information of the corresponding option prices. The sovereign default risk, funding liquidity risk, level of risk aversion, and equity market performance are identified to be significant determinants of the option-implied correlation, implying inter-dependence amongst various markets during the European debt crisis.
Using data on Brazil, Colombia, Mexico, the Philippines, Russia and Turkey, our empirical results show that the exchange rates of their currencies have adequate explanatory power in explaining their ...US dollar-denominated sovereign bonds, particularly in the post-global financial crisis period. We develop a two-factor pricing model with closed-form solutions for the sovereign bonds in which the correlated factors are foreign exchange rates and US risk-free interest rates that follow a double square-root process relevant in the low interest rate environment. The numerical results and associated error analysis show that the model credit spreads can broadly track the market credit spreads.
Since the pioneering paper of Black and Scholes was published in 1973, enormous research effort has been spent on finding a multi-asset variant of their closed-form option pricing formula. In this ...paper, we generalize the Kirk Managing Energy Price Risk, 1995 approximate formula for pricing a two-asset spread option to the case of a multi-asset basket-spread option. All the advantageous properties of being simple, accurate and efficient are preserved. As the final formula retains the same functional form as the Black-Scholes formula, all the basket-spread option Greeks are also derived in closed form. Numerical examples demonstrate that the pricing and hedging errors are in general less than 1% relative to the benchmark results obtained by numerical integration or Monte Carlo simulation with 10 million paths. An implicit correction method is further applied to reduce the pricing errors by factors of up to 100. The correction is governed by an unknown parameter, whose optimal value is found by solving a non-linear equation. Owing to its simplicity, the computing time for simultaneous pricing and hedging of basket-spread option with 10 underlying assets or less is kept below 1 ms. When compared against the existing approximation methods, the proposed basket-spread option formula coupled with the implicit correction turns out to be one of the most robust and accurate methods.
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