バタフライスプレッドモデル価格

Consider a butterfly spread with strikes $K_1, K_2, K_3$. My professor wrote the model price, $V$, was equal to the following: $$V = exp(-rT) * P(K_1

where $\Delta K = K_2-K_1 = K_3 - K_2$. I asked after class why this was true. He said it was obvious and that its just the probability times the area of the spread or something like that. I understand that he is discounting the expected payoff of the option. The option only has value when $S_T$ is between $K_1$ and $K_3$, but why multiply by $0.5\Delta k$. What am I not seeing? Can someone provide a rigorous proof with more steps?

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1 答え

リスク中立措​​置の下では、バタフライ・ペイオフの期待現在価値は以下のとおりです。 $ _V_0 = e ^ { - rT} * \ int_ {S_T = K_1} ^ {K_3} P(T、S_T)f_ {S_T} dS_T $$

そして、$ f_ {S_T} $が$ K_1 $から$ K_3 $まで一定であると仮定すると、次のようになります。

【数1】【数2】【数3】【数4】【数5】【数6】【数7】【数8】【数8】 \ delta ^ 2} {\ Delta K} $$

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