Delta Hedge including FXVega

Delta Hedge is the most important risk hedge operation of exotic trading. Whenever exotic traders has done a exotic FX derivative, they hedge at first by Spot FX. This operation is of course, for a purpose of erasing a risk for delta. But, in this operation, it also enables us to hedge a FXVega. This explanation is as follows.
FX Volatility is quoted in the market as Black Sholes volatility. Black Sholes is supposing that underylying is followed by geometric Brownian Motion. That is, this quoted volatility shows a relative variation for this underlying, not a absolutely variation. Hence, this quoted volatility is propotional to underlying price. From this definition of quoted volatility, quoted volatility mostly increases when spot FX is going down. This correlation is often true at markets.
Therefore, exotic trader take into consideration FX Vega hedging when they hedge delta.

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.

EOD

EOD is an abbrevation of End of Day which is qute often used financial company. It means the time when we estimate the profit or loss about products we have. It must be done every day, and it must be of cousre correct. Especially it is very important for financial company, particulary swap houses which deal exotic derivatives. There are some reason for this.
At first, derivatives are not real products. They are not fruits, fishes, sweets, nor oils. They are only the kind of contracts which promise specialized cash flow. So we have to estimate how much the price is every day.
Secondly, Exotic derivatives are difficult to price. Exotic, as you can imagine easily by this word, is not a simple product. So we have to simulate them by using complexity logic. it might be impossible only by using simple formulation.
From this point, EOD are quite related with quants developers. It is not too much to say that quats developers' task are almost to manage that EOD has been done every day correctly.

Plain Vanilla

Plain Vanilla is a technical term about financial products. It means the most basic or standard version of a financial instrument, options, bonds, futures, swaps. In particular, a plain vanilla option is the standard type of option, whose pricing model was published by Fischer Black and Myron Scholes in 1973. This theory is called "Black-Scholes formula", and anyone who engaged with financial markets knows about it. Moreover, Black-Scholes formula is quite related with partial differential equation, more exactly speaking, by supposing some conditions, a solution of some partial differential equation is expressed with the expectation of some random variables.

The most symple case is a Black-Sholes equation. From Black-Sholes equation, we can get the risk value of option products easily. From this equation, we are able to manage the option as much as our risk reducing. If option traders have done the call option, the first they do is almost "Delta" hedging, that is reducing the risk for underlying changing. Even they can hedge the delta perfectully, they remain the risk for time-values, which is called "Theta". But this risk can be cancelled if they do "Gamma" dealing. Black-Sholes equation is certainly explaining these way.

Quantitive Meeting

In Tokyo, there are at least over 30 swap houses, which are included U.S., U.K. and E.U. companys. I had dinner party with quants who work in Otemachi, Marunouchi, Nihonbashi, and Roppongi and so on.
Around many quants, topics are always about the latest model, the latest product.
"What is your new model in your company??" "Has your company already developed XX product ??" "How is YY model calibrated in the market??"
Quants are defined as people who love finance, who love a lot of rewards for ourselves, and who love mathmatics.
Quants are living with an ambitious for developing our original model and our original theory.
The most exciting time for me is of course when I have done them.

Brownian Bridge

Brownian Bridge is defined as the conditional expected distribution of a standard Brownian Motion given the condition that we know values of Winner process at a starting point and some points. We of course know that this expectations and covariances.
As a mathematical instrument, Brownian Bridge is used when we simulate a extotic product pricing. Concretely, interest rate models are simulated at some discrete-fixed times which are often called "simulation grids". Though we sometimes have to price products whose payoffs do not lie in the simulation grids, we only know information of simulation grids from this calculation. But If we use Bronwian Bridge, we can get a good view of the process at not even simulation grids. Thus by using it, there may happen that we can get a better result.

Quasi-Monte Calro

Quasi-Monte Calro is a method for the computation fot the integral.
This method is based on low-discrepancy sequences.
On the other hand, Monte Calro method has a long history, and it derives from the Law of Large numbers.
The most advantage of Quasi-Monte Carlo compared with Monte Carlo is the faster convergence. We need 100 times samples to make the convergence accurate 10 times if we use Monte-Calro method, while we need only 10times samples to get the same result with Monte Calro if we use Quasi-Monte Calro.
But, Quasi-Monte Calro method works well only how integral function is smooth, and how its dimension is small. These are problems for Quasi-Monte Calro.
In finance, we often use the above ways to price some financial products. To price accurately, or to price quickly, Financial developer must know these ways.

Paris suburbs

I have visited one of university in Paris, excalty in Paris suburbs to attend a financial seminar.
One of their topic was about some estimation of correlations. A speaker who is a famous researcher talked in English very fulently, but it was difficult to understand it. The claim was that we can estimate the correlation among company's stocks by using asymptotic expansions even in condition that we only 'observe' them during some discrete-time. He called them 'observation time', and he formulated them by poisson-time.
I think it is very interesting because a correlation is important for finance. Of course we have many ways of its estimation, but implying or hedging the correlation may be impossible by using financial instruments. If this research is more advanced, it will be tradable.

First diary

I am a business man in Tokyo, majoring in Mathematics, using Probability Theorem.
This blog is for mostly purpose that I just want to keep making diary in English,
Of course I am very glad if many peaple will read this.