定價競爭策略英文-閱讀頁
2025-02-03 02:40本頁面
【正文】 This is the fundamental partial differential equation for derivatives. The solution for an specific derivative is determined by boundary conditions. For example, the European call option is determined by boundary condition: cT = max(0,STK). ? Risk neutral pricing The drift term ? does not appear in the fundamental equation. Rather, the reiskfree rate r is there. Under risk neutral measure, the stock price dynamics is dS = rSdt + ?Sdz. If interest rate is constant as in BS, the European option can be priced as c = exp[r(Tt)] E*[max(0,STK)] where E* denotes the expectation under risk neutral probability. ? The BlackScholes Formula for European Options (with dividend yield q) c = exp[r(Tt)] ?[0,?] max(0,STK)g(ST)dST where g(ST) is the probability density function of the terminal asset price. By using Ito’s lemma, we can show ln(ST) ~ N(lnS + (r 189。s change with respect to an increase in sock price ?c=?p=N39。(d1)?eqT/(2T1/2) + qSN(d1)eqTrXerTN(d2) ?p=SN39。(d1)eqT Rho: with respect to an increase in interest rate ?c=XTerTN(d2) ?p=XTerTN(d2) Example1: X=$70, T= S= X= T= r= . = q = European Option Prices d1= N(d1)= d2= N(d2)= Call= Put= Delta= Gamma= Theta = Vega= Rho= ? Synthetic option Set aside cash in the amount equal to the model value. Maintain the stock position equal to the delta of the target option. Cash balance is invested in riskfree assets to earn interests. Close the position at the desired matuirity. If the model is good, the terminal payoff of this dynamic strategy should be close to the payoff of the target option at the maturity. Example: Synthetic put option Day Closing price Daily Return Maturity Delta Stock Position Overall Cash 07/04 08/04 09/04 10/04 11/04 14/04 15/04 16/04 17/04 18/04 21/04 22/04 23/04 24/04 25/04 28/04 29/04 30/04 01/05 02/05 05/05 0 0 0 0 0 0 0 0 0 Mean . a. m. 250 d a. . 250 d % % X= T= r= P= ? Duration of an option An optio
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