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Black Scholes模型:支付功能的條件

鑒於:

考慮一個雙資產,連續時間模型(B,S),其中 $$ dB_t = B_t r dt,\ quad dS_t = S_t(\ mu dt + \ sigma dW_t)$$ 顯然,鞅平減指數是: $$ Y_t = e ^ {( - r - \ frac {\ lambda ^ 2} {2})t - \ lambda W_t} $$

有一個定理說明如下:

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 $\xi_T$.

問題:

假設$ \ xi_T = g(X_T)$,其中$ g $是一個有界平滑的Borel函數。 (然後滿足上述定理中$ \ xi_t $的假設。)

進一步假設$ g $具有有界導數並且是遞增函數。

表明存在一個可接受的復制策略$(\ pi_t)_ {t \ in [0,T]} $,這樣$ \ pi_t \ geq 0 $ a.s.對於所有$ t \ geq 0 $。

最佳答案

投資組合$ V_t(\ alpha_t,\ beta_t)$(對於股票$ S_t $和zerobond $ B_t $)是自籌資金iff:

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

它進一步暗示

$$ dV_t = \ alpha_tdS_t + \ beta_tdB_t $$

要通過自籌資金的股票和債券投資組合復制衍生品$ C(S_t,t)$,請設置:$$ dV_t = dC_t $$

可以使用Ito的引理在$ C(S_t,t)$指定$ dC $的動態:

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

接下來假設$ C $滿足BS-PDE:

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

將其插入$ dC $:

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

現在我們進一步得到債券動態$ dB_t = B_trdt $,所以:

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

最後,$ dS_t $和$ dB_t $之前的系數正是自籌資金投資組合的權重:

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

所以權重是$ \ pi_t = \ partial_SC \ geq 0 $,因為$ C = g $是OP的增加函數。

轉載註明原文: Black Scholes模型:支付功能的條件