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If $EC(S_0, K, \sigma, r, T)$ represents the price of a European call option with strike $K$, expiry $T$, initial price $S_0$, volatility $\sigma$ and where the constant interest rate is $r$, then I want to express the price of a down-and-out call option in terms of $EC$.

Specifically, I wish to show that the price of a down-and-out call with strike $K$ and a barrier at $S_0 e^b < min\{S_0, K\}$ can be expressed as:

$$EC(S_0, K, \sigma, r,T) - e^{2\mu b / \sigma^2} EC(S_0, K, \sigma, r, T),$$

where $\mu = r - \frac{1}{2} \sigma^2$.

Any help would be really appreciated. Thanks!

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No, you cannot decompose a barrier option as a linear combination of European options. You can find the derivation of the formula in Musiela & Rutkowsi pg.235, for example.

But I can tell you that your formula is wrong because if $S_t<S_0e^b$ the price should be zero but in your equation this does not happen. Also note that your equation is nothing else than $$ (1 - e^{2\mu\sigma^2})C(T,K) $$ which implies that a barrier option is equivalent to a leveraged call option, and this is not true.

Edit: The knock-out call formula for completeness. Strike $K$, barrier level $H>K$ $$ \Pi_{\text{KO-Call}}(S_0;T,K,H) = C(S_0;T,K) - \left(\frac{S}{H}\right)^{\frac{2(r-q)}{\sigma^2}-1}C\left(\frac{H^2}{S_0};T,K\right) \ . $$ This formula is valid for $S_0>H$. Note that the price is zero when $S_0=H$ .

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It looks like $\left(\frac{S}{H}\right)$ should be $\left(\frac{H}{S}\right)$. – bcf Jan 7 at 13:33

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