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Quantitative Finance · Glossário

O que é State-price density?

Definition 1.8 Derivatives and Volatility · Capítulo 1 — No Arbitrage and the Fundamental Theorems

Given the real-world probabilities P(s)>0\P(s)>0 of the states, the state-price density is ms=qs/P(s)m_s=q_s/\P(s); every price is an expectation under P\P: pj=E[mDj]p_j=\E[mD_j]. It is the ratio of the risk-neutral to the real-world probability times the discount factor, and it is high in the states in which investors most value a payoff.

Exemplos

Example 1.9 (A binomial market)

A bond costs 0.98 and pays 1; a share costs 100 and pays 120 or 90. The equations qu+qd=0.98q_u+q_d=0.98 and 120qu+90qd=100120q_u+90q_d=100 give qu=0.3933q_u=0.3933, qd=0.5867q_d=0.5867, both positive: no arbitrage. The risk-neutral probability of the up state is 0.3933/0.98=0.40140.3933/0.98=0.4014, whatever its real-world probability. A call struck at 100 pays (20,0)(20,0) and is worth 20qu=7.866720q_u=7.8667.

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