經濟,財務,統計學,數理科學與政治評論

Hello there:

  這裡要先複習一下傳統的put-call parity與其相關內容;爾後再談一下Black-Schole-Merton option pricing與其相關的equity options風險管理!!

 I. 我們知道歐式與美式的買權其payoff各自為c(T) = max(0, S(T) - X)與C(T) = max(0, S(T) - X),美式的買權可以再到期前任一天去執行,而歐式則一定是到期日才能執行!而歐式與美式的賣權其payoff則為p(T) = max(0, X - S(T))與P(T) = max(0, X - S(T)).從這裡出發來談各自的Boundary condition,會談Boundary condition的原因當然是因為一開始做這些計算用的是偏微分方程式,相對上面其boundary conditions對於最終解集合有非常重大的影響!!(小寫的c,p是歐式,大寫的C與P是美式選擇權)

(如果用利率的角度來看，利率選擇權的買權自然是收浮動利率支付固定利率，利率選擇權的賣權自然是收固定利率支付浮動利率。)

  (a) c(0) >= 0, C(0) >= 0, p(0) >= 0, P(0) >= 0,

  (b) c(0) <= So, C(0) <= So,

  (c) c(0) <= C(0), p(0) <= P(0),

  (d) c(0) >= max(0, So - X*exp(-Rf*T))

      C(0) >= max(0, So - X*exp(-Rf*T))

      p(0) >= max(0,X*exp(-Rf*T) - So)

      P(0) >= max(0,X - So)

  (e) X1 <= X2 ,所以 c(X1) >= c(X2)

                     C(X1) >= C(X2)

                     p(X2) >= p(X1)

                     P(X2) >= P(X1)

  (f) T2 >= T1, 因此 p(T2)可能比p(T1)大,也可能比p(T1)小(因為比較晚被支付金錢有時間成本)

                     P(T2) >= P(T1)

 II. The put-call parity for European Options:

   So + p(0) = c(0) + X*exp(-Rf*T) 

這裡任何的證明都是從Cash flow出發,只要能夠證明兩者所產生出來的cash flows相同即可!!

 III.The Black-Scholes-Merton option model for European options:

   c = So*N(d1) - X*exp(-Rf*T)*N(d2)

   p = X*exp(-Rf*T)[1 - N(d2)] - So*[1 - N(d1)]

  where

   d1 = [ln(So/X) + (rf + 0.5* sigma*sigma)*T]/[sigma*(T**0.5)]

   d2 = d1 - sigma*(T**0.5)

      =  [ln(So/X) + (rf - 0.5* sigma*sigma)*T]/[sigma*(T**0.5)]

 IV. The Greeks of BSM of European equity options for Risk Management

  (a) The Delta defines the sensitivity of the option price to a change in the price of the underlying

  (b) The Gamma will tend to be larger when the option expire in or out-of-the money and close to expiration

   i.e., the Delta will be a poor approximation for the option's price sensitivity when it is at-the-money amd close to the expiration date

  (c) the price of European option on an asset is not very sensitive to the risk-free rate

  (d) Option prices are higher the longer the time to expiration. But there are some exceptions for Europeam put option.

  (e) Option price is higher as the volatility is higher.

 V. The put-call parity of options on forwards

    c(0) +[X - F(0,T)]*exp(-Rf*T) = p(0)

  as F(0,T) = So*exp(Rf*T)

所以其BSM的修正式為

  c = exp(-Rf*T)[F(0,T)*N(d1)-X*N(d2)]

  p = exp(-Rf*T)[X*(1 - N(d2))-F(0,T)*(1 - N(d1))]

  其中

   d1 = [ln(F(0,T)/X) + 0.5*sigma*sigma*T]/[sigma*Square-Root(T)] 

   d2 = d1 - sigma*square(T) = [ln(F(0,T)/X) - 0.5*sigma*sigma*T]/[sigma*Square-Root(T)]