Math3075

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26 Terms

1
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Single Period Model Theorems


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Arbitrage Definition


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Q is an EMM if:


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1st FTAP


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Claim X is replicable if and only if…


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Market is complete if and only if…

EMM is unique

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Tower Property for Conditional Expectation

Where we assume G1 is a sigma field containing LESS information than G2

<p>Where we assume G<sub>1 </sub>is a sigma field containing LESS information than G<sub>2</sub></p>
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Martingale Requirements for adapted/stochastic process


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Super Martingale Requirements


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Sub-Martingale Requirements



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Self financing portfolio if:


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Arbitrage Pricing Principle

Assuming NA holds for ℳ (B, S1 , …. , Sn),

πt(X) is consistent with NA if:
πT(X) = X and there is NA for ℳ ~ (B, S1 , …. , Sn, Sn+1 = πt(X) )

<p>Assuming NA holds for ℳ (B, S<sup>1</sup> , …. , S<sup>n</sup>), <br><br>π<sub>t</sub>(X) is consistent with NA if:<br> π<sub>T</sub>(X) = X and there is NA for ℳ <sup>~ </sup>(B, S<sup>1</sup> , …. , S<sup>n</sup>, S<sup>n+1</sup> = π<sub>t</sub>(X) )</p>
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Martingale Measure


What makes it equivalent is when Q is equivalent prob measure.

<p></p><p>What makes it equivalent is when Q is equivalent prob measure.</p>
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Risk Neutral Pricing Formula


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2nd FTAP

Market Complete ⇔ EMM is Unique

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American Option Price



The solution here being rational stopping time

<p></p><p></p><p>The solution here being rational stopping time</p>
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Rational stopping time


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Properties of Brownian Motion

  1. Wt - Ws ~ N(0, t - s)

  2. Wt4 - Wt3 is independent of Wt2 - Wt1

  3. Cov(Wt , Ws) = min( t, s)

  4. W is a martingale


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Bank Account Dynamics


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Stock price dynamics


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Given X = f (ST), what is πt(X) under BM?


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Delta/Hedge Ratio


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BS Formula

πt(X) = St N(d+) - Ke-r( T-t) N(d-)

<p>π<sub>t</sub>(X) = S<sub>t</sub> N(d<sub>+</sub>) - Ke<sup>-r( T-t) </sup>N(d<sub>-</sub>)</p>
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d± =


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= S0