不用put-call-parity推导Black-Scholes看跌期权定价公式

Derive the Black-Scholes Formula for a European put option without making use of the put-call-parity. Proceed along the lines of the corresponding calculations for the European call option.

求解呀\\\

Call=exp(-rt).E(max(0,S-K))=exp(-rt)E(S-K+max(0,K-S))=exp(-rt)E(S-K)+put
Put=Call-exp(-rt)E(S-K)
E(S-K)=E(S)-K,在风险中性测度下，E(S)=S0*exp(r-δ)t。
相减可以得到Put的公式了。

P(S,K,T)={K|S<=K}-{S|S<=k}
the first term involves K, the dollar amount, and given that N(-d2)= P(S<=K), the first term is K*e^(-rt)*N(-d2)
the second term involves S, the stock, so you use partial expectation, second term is S*e^(-δt)*N(-d1)
I never use Put-call parity to find the value of put since the black-scholes formula is so simple.

dmwt2007 发表于 2011-4-18 22:26
P(S,K,T)={K|S

The above method consider a put as a combination of K dollar-or nothing option and a stock-or-nothing option
to consider the value of put directly, treat K as underlying asset and S as the payment
then use the generalized Black-scholes formula for exchange option

budong

Call=exp(-rt).E(max(0,S-K))=exp(-rt)E(S-K+max(0,K-S))=exp(-rt)E(S-K)+put
Put=Call-exp(-rt)E(S-K)
E(S-K)=E(S)-K,在风险中性测度下，E(S)=S0*exp(r-δ)t。
相减可以得到Put的公式了。

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围观

可以使用风险中性定价，风险中性定价三部曲：1、假设收益率是无风险利率，2、求出预期收益，3、按无风险利率贴现。
那么根据c=max(St-K,0)
可得，c=exp{-rT}E[max(St-K,0)]
由于St服从对数正太分布，ln(St/S0)~N(ut-sig2/2*t,sig2*t)
可以得到E[max(St-k,0)]=E(St)N(d1)-KN(d2)
就可以了。