Levy Process

Levy Process is a specific stochasitic process, named by Paul Levy who is the French mathematician. It is defined as a continuous stochastic process which has properties of independent increments and stationary increments. The typical exapmle is as we know, Brownian motion. In practice, Black Sholes formulation is often used by plain vanilla option, such as FX Option, Stock option, Swaption, and so on. Black Sholes is based on the assumption that the underlying is followed by Brownian Motion. I want to pay attention to what this process is continuous path. On the other hand, Levy process is not supposing that it is continuous path. Therefore, a Levy Process is including Brownian Motion at all. Let us think about realistic from now. Market behavior can not always be continuous. For example, when it was Black Monday or Lehman Brothers bankruptcy, the market changed in a extream range. Hence, Levy process model is more and more worth of studying in Mathematical Finance. Of course, many financial reserchars has already studied it...

ISDA

ISDA is an abbreviation for International Swaps and Derivatives Association. It is the largest global financial trade assosiation, which develop the ISDA Master agreetment. It contains a lot of leagal rule for trading derivatives. When we trade derivative producst, we have to issue some formal documents to leave a evidence to prevent each others different claim. That is to say, derivative trading is safely contracted each other by refering ISDA rule. For example, when we trade some swap which menas exchage some fixed rate for floating rate, we must calculate total present value of this trading. In the way of this calculation, we have to build exact cash flows and this accrued interest rate needs to be caluculated, which means "timeinterval" * couponrate. Of course, it is easily described as mathematical word like r*t , but in practice, it has delicate modification for timeinterval because interval has a lot of meaning. 1 year is 365day?? 366day?? This detail problem is resolved by refering ISDA document.

Reconcile

Reconcie is a professional slang among a trading business. It means what traders verify profit and loss of their own position, which should be done every day. For doing this "reconcile" task, they need some instrument, that is, market data and risk value are needed. Every day, traders can check risk value such as Delta, Gamma, Vega, Theta and so on, every day from thier model, and they can pick up market data such as IRS Swap rate, FXSpot, FXOption Volatility and so on.
they evaluate their profit and loss by means of easy caluculation with usuful computing machine in their hand from these data. If thier result in their hand is different from official profit and loss which is caluculated by a big computating machine set by company regulation, they including qunats must check our model and we must explain what the cause is. Especially about exotic trading, "reconcile" is more difficult than others because exotic position is consist of many parametars and many risk factors.

Propagation Property

Propagation property is a technical word in the study of partial differential equations.
It is closely related with a phenomena of conduction of heat. Particulary explaining from the view of probability theory, by giving a time-homogeneous Markov process, we can make a function which has propargation property. It can lead us to several application in our dairy life. More detail, As a qunatitive engineering in finance, we can generate a lot of interest rate model by using propagation property. Of course, this approach is, I think, new challenging in the field of mathematical finance, but it has alot to offer. That is why many researchers would be interested in it for the time being.