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In addition to StackG's answer, here is a good introductory overview of several (approximate and semi-analytical) methods to price baskets in a Black-Scholes framework: Krekel et al - An analysis of pricing methods for basket options

8

I'm not completely certain from your question, but I'm going to assume you have a basket of $n$ stocks with prices $S_0(t)$ to $S_n(t)$, and you want to price an option with payoff at $C(\tau)$ at time $\tau$ equal to \begin{align} C(\tau) = \max\Bigl({\frac 1 n}\sum^n_{i=1} S_i - K, 0\Bigr) \end{align} where $K$ is the strike of the option I'm also going to ...

1

Following the notation in Hull, let $H$ be the barrier level. I list the prices of European-style down-and-out barrier options with continuously observed barrier. If $H\leq K$, then $$c_{di}=S_0e^{-qT}(H/S_0)^{2\lambda}N(y)-Ke^{-rT}(H/S_0)^{2\lambda-2}N(y-\sigma\sqrt{T})$$ and $$c_{do}=c-c_{di}.$$ If $H>K$, then $$c_{do}=S_0N(x_1)e^{-qT}-Ke^{-rT}N(x_1-\... 4 SPY pays dividends ~1.8%, and the expiry is ~3y (as of date was 2018, 2021 expiry), so the it looks like there is a discount Assuming 0 time value$$OptionValue=Intrinsic Value+Time Value OptionValue= (S-K)-DividendOptionValue=267x(1-0.02)-267x1.8\%x3=\\\$247$\$

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I suspect that the reference value is, well, only for reference and not the real price. Maybe the price of SPY when the call contract is bought is lower

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