Dear Colleagues,

The origin of the application of rigorous mathematical and stochastic methods for asset pricing can be traced back to Louis Bachelier’s 1900 doctoral thesis Théorie de la spéculation. From the early 1950s, economists, including Paul Samuelson, started to model asset prices using geometric Brownian motion. The Black and Scholes paper in 1973 is the first to show that the European option price is the solution of a partial differential equation that is derived from a self-financing hedging argument, which is also followed by some extensions by Merton in 1973. Merton, in 1976, further extended the Samuelson–Black–Scholes geometric Brownian motion asset pricing model to a geometric jump-diffusion model and derived a pricing formula for a European call option under a jump-diffusion model. Margrabe, in 1978, extended the Black–Scholes call/put option pricing formula under geometric Brownian motion dynamics to price a European-style exchange option. This was an early example of pricing an option written on multiple assets, which was extended by Carmona and Durrleman, in 2003, to more general spread options. The work by Harrison and Pliska, in 1981, also formalised the relationship between risk-neutral valuation and equivalent martingale measures, which also led to the Fundamental Theorems of Asset Pricing by Delbaen and Schachermayer in 1994. The original Black–Scholes 1973 model was also extended to include stochastic volatility (e.g., Hull and White in 1987 and Heston in 1993), both stochastic volatility and jumps (e.g., Bates in 1996). Other option pricing models that have been introduced include the Variance–Gamma model by Madan and Seneta in 1990 and Hidden Markov models by Elliott et al. in 1995. Apart from option pricing models, there is also a wealth of literature on interest rate term structure models and option pricing with stochastic interest rates.

In addition to stochastic analysis in asset pricing, option and derivatives pricing, and interest rate modelling, techniques around stochastic optimal control, forward–backward stochastic differential equations (FBSDEs), and stochastic filtering, among others, have gained greater traction in addressing financial problems, such as portfolio management and optimization, risk management and measurement, algorithmic trading and trading strategies, among many other areas of application.

In this Special Issue, we call for original papers that further extend the frontiers of the existing rich literature, as well as shorter insightful review and survey papers on the applications of stochastic analysis in financial mathematics.

Dr. Gerald H. L. Cheang
Dr. Len Patrick Garces
Guest Editors

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• financial mathematics
• stochastic analysis
• stochastic optimal control
• asset pricing