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Application: pricing and hedging of Barrier options代写

Barrier option is a contract that pays to its holder a function of the terminal value of the underlying asset and of its maximum or minimum value.

Brownian motion accumulates quadratic variation at rate one per unit time.

Brownian Motion （BM）

BM is a martingale
—Applications: ruin probabilities, first passage time
BM is a strong Markov process
Reflection principle
—Applications
Quadratic variation of BM

Martingales

Why martingales?

Usually expect (discounted) stock price process to be a submartingale.
Reason: higher risk calls for higher returns (on average).
Still, will think of (discounted) stock price as a martingale!

Stochastic Process and Markov Chains代写

For example, the price of a particular stock at the close of each day’s trading would be a stochastic process indexed by time. Of course, the index does not have to be time, it may be a spatial indicator. For example, the number of defects in specied regions of a computer chip. In fact, the indexing may be almost anything. Indeed, if we consider the index to be individuals, we can consider a random sample。

As with individual random variables, we shall be interested in the set S of values which the random variables may take on, but we shall generally refer to this set as the state space in this context. Again, as with single random variables, the state space may be either discrete or continuous. In addition, however, we must now also consider whether the index set T is discrete or continuous.

Markov Chain代写. The simplest sort of stochastic processes are of course those for which the random variables Xt are independent. However, the next simplest type of process, and the starting point for our journey through the theory of stochastic processes, is called a Markov chain.