# Black Scholes模型：支付功能的條件

For a claim with payout $\xi_T$, $T>0$, if $\xi_T>0$, $\mathcal{F}_T$-measurable and such that $\xi_TY_T$ is integrable, then there exists an admissible strategy $(\pi_t)_{t \in [0,T]}$ that replicates the European claim with payout $\xi_T$.

## 最佳答案

$$V_t = \ alpha_tS_t + \ beta_t B_t$$

$$dV_t = \ alpha_tdS_t + \ beta_tdB_t$$

$$DC = \ partial_tCdt + \ partial_sCdS + \壓裂{1} {2} \西格瑪^ 2S_t ^ 2 \ partial_ {SS} CDT = \ partial_SCdS_t +（\ partial_tC + \壓裂{1} {2} \西格瑪^ 2S_t ^ 2 \ partial_ {SS} C）dt的$$

$$\ partial_tC + \ frac {1} {2} \ sigma ^ 2S_t ^ 2 \ partial_ {SS} C = rC-rS_t \ partial_S C$$

$$DC = \ partial_SCdS_t +（C-S_T \ partial_SC）RDT$$

$$dC = \ partial_SC \ cdot dS_t + \ left（\ frac {C_t} {B_t} - \ frac {S_t} {B_t} \ partial_SC \ right）\ cdot dB_t$$

$$\左（\ alpha_t = \ partial_SC，\，\ beta_t = \ dfrac {C_T} {B_T} - \ dfrac {S_T} {B_T} \ partial_SC \右）$$