## November 02, 2004

### The state price

The state price is today's price for $1 in the according future state. In another word, to secure $1 in some future state the investor has to pay the state price.

Assume that the world has n possible scenarios tomorrow, denoted by {S1, S2, …, Sn} with RISK-NEUTRAL probabilty{p1, p2, …, pn} and in each scenario the payoff of the risky asset/contingent claim he hold will be {C1, C2, …, Cn} respectively. Assume that there is a risk-free bond with return Rf. So the price for the risky thing will be the discounted expectation under risk-neutral measure, i. e.,

Price today = P = 1/Rf * SUM{pi*Ci, i=1:n}

= 1/Rf * (p1*C1+p2*C2+p3*C3+...+pn*Cn)

The investor is not sure which scenario the world is going into tomorrow, but he wants to secure the payoff Ci if scenario i comes into being tomorrow. He wants to do this for every future scenario. So he has to pay for each and every future payoffs at the appropriate state price. Recall that the state price is the price for one future dollar in a given scenario, and we find in the above equation that each payoff is multiplied by the probability divided by risk-free interest rate. That is the state price. For example, to secure payoff C1 in scenario 1, the investor pays p1/Rf for every future dollar in scenario/state 1, and so on. The total amount the investor has to pay is the sum of the prices for each future payoff, as in the above equation.

Now we reach the idea that, in discrete time, the state price will be the RISK-NEUTRAL probability divided by risk-free interest rate, p/Rf. What if in continuous time?

## October 31, 2004

### Jack–o'–lantern and more

Create a Jack-o'-lantern by carving on a pumpkin is one of the best things that makes he feels the flow of time and concentration. Using a Swiss army kinfe needs speciall concentration; otherwise either it goes too far on the target or does it to somewhere else, my hand, for example. I never dreamed of pumpkins in high demand but they are just at the Halloween night. Sold out. We had nearly given up our search at Tesco's before Ellla cleverly found Butternut squash as substitutes. They are very much like pumpkins except for the long neck. What's more we wer not sure if they were hollow inside, as a real pumpkin was.

It was the carving homework back home, which I had neve done before, neither had Ella, for sure…It was the plan on a paper, the paint on the squash, and finally the carve. We started from his mouth as there were more space to make mistakes and the only point is the teeth. 30 minutes later the squash opened his mouth and smiled, with two teeth, safe, for both him and me. My plan was far from making a horror (neither a horrible, of course) Jack. So the eyes should be 8:20 and the nose should be round at the lower end. It was so great since everything went as my plan, and the knife seemed have no interest in trying on my fingers. Good luck.

Carving time. My thought suddenly jumps here. Seldom had I done something whole-heartedly for a long time. Time flows without being noticed and everything I did turned out to be gone without any trait left in my heart. It is that I ran with time and I even wanted to run faster than time! I then suddely remember when I was a child and standing by a brook, I put my hand in it and feel the flow. It was carving. As time passes by, shall we carve it as well? Feel the water and you know it. When I feel the water flow I feel concentration, consciousness, and peace of mind. What if feel the flow of time? Carving a pumpkin and he gets a latern. Carving water and he feels peace and consciousness. Carving time, and there will be traits of life left in his heart, probably.

## October 28, 2004

### CRR

Cox, Ross and Rubinstein discretised Black-Scholes description to the financial world by means of a binomial tree. I was wrong on the tree when I thought that the expectation of return for each timestep should be equal to the instantaneous return of the stock. In fact it is the continuous limit of the tree that we should care about. The limit of the discretisation has to be the values in the continuous time. That is, the upward/downward move (u and d), and the porbability of both, have to be chosen in such a way that in the limit the return and volatility goes to the value in a continuous time world.

Another thing is the Central Limit Theorem. In a world generated by Brownian motion the stock price process may be regarded as the sum of many many tiny movements, each of which happens in a small time interval and follows the same distribution, either normal (continuous time) or binomial (binomial tree, discrete case). Therefore the stock price, by Central Limit Theorem, is normal. What's more the limit of discrete model becomes Black-Scholes as we go to the limit.

Risk neutrality. Risk-neutral measure comes into being when CRR found that the value of a call option at each tree node can be expressed in the form of a expectation discounted by risk-free interest rate. Risk-neutral measure is also found in Martingale pricing theory, of course. In Martingale pricing theory we have to either 1)numeraire 2)change measure or 1)change to risk-neutral measure under which stock drift is equal to r, the risk-free interest rate and 2) numeraire. By either way we reach a stock price process with zero drift and therefore a martingale. In the first case, after 1) we have a drift mu-r; after 2) we kill the drift. In the latter one 1) gives a drift of r and 2)kil the drift again.

Want to do something about pricing theory? a long way to go.

## October 06, 2004

### Scottie Pippen and more

I had never imagined a quiet retirement announcement will belong to Scottie Pippen, the one who can not be the all-time greatest Basketball player as such a versatile small forward only because the god himself plays in the same era. However, after 17 years of journey in NBA, travelling coast to coast to beat each and every team consisting of best talented player in the world, Scottie leaves so quiet.

One of my best friend, Neo (No, not the one who wears sun glasses all the time), plays in the way that Scottie does, clever, precise, and fast. Neo's been so energetic all the time. He stands after hours of game of basketball in the killing Beijing Summer sunlight. I can never do that. I have to take each and every minutes to have a rest. I cant run and play so consistently as he does, never can I. And his field goal is always to be described as precise and, sometimes, deadly. I took several moves before I shoot, which is the very imitation of Michael Jordan and shows that I am superficial. For Neo, it is as simple as, shoot and nothing but net. I have been always proud of my turning fadeaway but I admit he plays basketball while what I have been doing is to achive self-satisfaction. It is this difference that pushes me further from the essence of the game of basketball and makes him the one I should learn from.

Well, Scottie's gone, and I am leaving Beijing again in such a short peroid of time that there is little opportunity that I play again this year with Neo. Maybe next year but I am getting old and I am lazy. I have lost almost all the stuff I had 4 years ago. Next year? Who knows…MJ is gone, so is Scottie, so is my dream. What's left? Nothing but nonsense like this and endless throwback. One day I thought I should take some photos and maybe even videoclips of my plays on the court. It was Neo who reminded me of those stuff that I have lost and could not be kept again. Why bother? Love for Basketball will go on. What's more, there's much more things to do, to enjoy, to achieve. I think I should go back to work and finish my PhD studies asap, say, in two years. One day I will play Basketball with my children, and argue with them that MJ and Scottie is the best.

### Start of wind

It was just 20 minutes ago that I opened a blog owned by Nic, my friend and a talented basketball player. 5 minutes after that I decided to start my own one. The name of this blog, Wind, had suddenly flashed in my mind even before I actually started to think about naming it. Well, let's follow the instinction.

In fact I don't know the exact reason why I sign up for a blog. I even had never known what it was. However, it was the instinction that drove me here. I guess it is partly because I just appreciated my friend Edwin's personal webpage. It was excellently bulit to accommodate his photos during his exciting while sophisticated trip to America, where he attended a great conference. I wish I could be there presenting my paper some day in the future, although I have done essentially nothing for my PhD thesis.

Anyway, I hope this entry would not be the last one here. The first thing I want to do is to upload photos, yes, a gallery of myself. But the thing before the first is to take photos. I am eager to do that with my new DC, and nice weather in the UK, if possible.