定價策略black-scholesoptionpricingformula(1)-免費(fèi)閱讀
【正文】 2023年 2月 12日星期日 上午 12時 58分 14秒 00:58: 1最具挑戰(zhàn)性的挑戰(zhàn)莫過于提升自我。 2023年 2月 12日星期日 12時 58分 14秒 00:58:1412 February 2023 1空山新雨后，天氣晚來秋。 上午 12時 58分 14秒 上午 12時 58分 00:58: 沒有失敗，只有暫時停止成功！。 , February 12, 2023 雨中黃葉樹，燈下白頭人。 ?2S2?2C/?S2 – rC = 0 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。*(1+??h /?), ?=??h ? The continuous time limit It cab be shown that the process P(t) has the following three properties: 1. For any t1 and t2 such that 0 ? t1 t2 ? T: P(t1)P(t2) ??(?(t2t1), ?2(t2t1)) 2. For any t1, t 2 , t3, and t4 such that 0 ? t1 t2 t1 t2 ? t3 t4? T, the increment P(t2) P(t1) is statistically independent of the increment P(t4) P(t3). 3. The sample paths of P(t) are continuous. P(t) is called arithmetic Brownian motion or Winner process. If we set ?=0, ?=1, we obtain standard Brownian Motion which is denoted as B(t). Accordingly, P(t) = ?t + ?B(t) Consider the following moments: E[P(t) | P(t0)] = P(t0) +?(tt0) Var[P(t) | P(t0)] = ?2(tt0) Cov(P(t1),P(t2) = ?2 min(t1,t2) Since Var[ (B(t+h)B(t))/h ] = ?2/h, therefore, the derivative of Brownian motion, B’(t) does not exist in the ordinary sense, they are nowhere differentiable. ? Stochastic differential equations Despite the fact, the infinitesimal increment of Brownian motion, the limit of B(t+h) = B(t) as h approaches to an infinitesimal of time (dt) has earned the notation dB(t) and it has bee a fundamental building block for constructing other continuous time process. It is called white noise. For P(t) define earlier we have dP(t) = ?dt + ?dB(t). This is called stochastic differential equation. The natural transformation dP(t)/dt = ?+ ?dB(t)/dt doesn’t male sense because dB(t)/dt is a not well defined (althrough dB(t) is). The moments of dB(t): E[dB(t)] =0 Var[dB(t)] = dt E[ dB dB ] = dt Var[dB dB ] = o(dt) E[dB dt] = 0 Var[dB dt ] = o(dt) If we treat terms of order of o(dt) as essentially zero, the (dB)2 and dBdt are both nonstochastic variables. ? | dB dt dB | dt 0 dt | 0 0 Using th
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