连续时间金融ctf-chp7.ppt

plete Model of Warrant pricing that Maximizes Utility
Chen Rong
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The stock price motion
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The stock price motion
mon stock without dividends; Current price: Xt
The price of the stock n periods later: Xt+n,
a finite-variance multiplicative probability distribution
Where the price ratios Xt+n/X=Z=Z1Z2…Zn are assumed to be products of uniformly and independently distributed distributions of the form prob{Z1≤Z}=P(Z;1), and where for all integral n and m, the chapman-Kolmogorov relation
Is satisfied.
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The stock price motion(2)
All these mean that the geometric Brownian motion, which at lease asymptotically approach the familiar log-normal.
Another hypothesis: risk averter with concave utility and diminishing marginal utility. So,
Where αis the mean expected rate of return on the stock per unit time. (a constant independent of n.)
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A special case
A special case: n=1,λ>1
This simple geometric binomial random walk leads asymptotically to the log-normal distribution.
()es
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The 1965 Model of Samuelson
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The 1965 Model of Samuelson
Some assumptions
American warrants
A arbitrarily postulated gain as exogenously given for stocks (α) and warrants (β)
No-arbitrage pricing
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No-arbitrage pricing
Rational pricing (arbitrage value )
like binomial trees pricing for American options
One-step binomial tree
Then, for any length of life,
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The perpetual warrant
The perpetual warrant
Where C is the critical level that the warrant will be exercised at once or not.
This critical level will be definite if β>α.
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β=α
When β=α, the case is particularly simple.
When β=α, the conversion will never be profitable.
The value of the warrants of any duration can be evaluated by mere quadrature:
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