Financial Markets

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Summery of arbitrage pricing

• Under risk neutral probability measure, the
  discounted stock
• price and portfolio processes are martingales

• The binomial lattice model is complete, i.e., it
  is hedgeable.

• The discounted value process of a European
  derivative security
• is a martingale based on the APT.

2
Summery of Markov property

• A stochastic process Xk is said to be Markov
  if the distribution
• of Xk1 conditioned on X0 ,., Xk is the same
  as the distribution
• of Xk1 conditioned on Xk only.

• A process with independent increments is a
  Markov process

• Conditional expectation of h(Xk1) can be
  computed as a function
• of Xk if the underlying stochastic process is
  Markov

• The APT value of a European derivative security
  is computed
• on lattice backward based on the Markov property

3
- Stopping times and American options
- Properties of value processes of American
options

• The Radon-Nikodym Theorem and the state price
  density
• process

4
American options

• European option (with an expiration N and a
  strike price K)

You can exercise the option at the expiration only

• American option

You can exercise the option at any time until the
expiration
- You have an additional choice
When is the optimal to exercise?
5
Pricing and hedging American options
- g(Sk) the payoff of an American option
- Vk the value of an American option

• We solve an backward recursion algorithm on the
  binomial
• lattice with the above constraint

6
Binomial lattice pricing model
2. Keep it until time N (next step)
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Binomial lattice pricing model
K - uSN-1
VN-1?
K - dSN-1
Step 1 Find the value of European put at time N
- 1
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9
Stopping time

• t is a time when you can decide to stop playing
  our game.
• Whether or not you stop immediately after the
  kth game
• depends only on the history up to (and
  including) k

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Examples
- Convince yourself t2 can not be a stopping
time.
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Examples

• It is reasonable to think that t1 is a stopping
  time

- I will quite this game first time when I lose
10 box
- I will retire as soon as I finish paying the
mortgage

• I will quit finding a job as soon as I find a
  company
• which pays more than 5 digit

• Be careful!

• I wont stop gambling at Las Vegas until I
  become a
• millionaire

It might work, but t can be infinite.
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Information up to a Stopping time
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Stop at time k j what ever happens
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Value of stochastic process at a stopping time
The result follows that,
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H go up
T go up
- The value of Xk on each circle is Xt
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Stopped martingales
Proof
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Martingales on the information up to a stopping
time
Proof
We first note that
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• An estimate of XN given this information is Xt ,
  a value
• to stop, if Xk is a martingale

19

• Assume that we have
• another stopping time,

stop at time N

• We want to estimate XN
• based on the information up
• to the other stopping time t

• This estimate is given by Xt
• if Xk is a martingale

- In particular,
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Optimal Sampling Theorem (OST)

• More generally,

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• We want to estimate Xr
• based on the information up
• to the other stopping time t

• This estimate is given by Xt
• if Xk is a martingale

- In particular,
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OST for super-, sub-, martingale cases