Correct Pricing Formula for a Down-and-Out Call
Summary
The document challenges a proposed expression that would price a down-and-out call as a scaled European call with the same initial asset price, strike, volatility, rate, and expiry. The response explains that a barrier option cannot generally be represented by that simple scaling: the knock-out feature depends on the asset path and requires a formula that reflects the barrier level.
For a continuously monitored knock-out call in the stated setting, the response gives a formula involving the European call price at the original spot less a transformed call price evaluated at a reflected spot, with a multiplicative power term involving the barrier and model parameters. It notes the formula’s stated barrier configuration and that the value reaches zero when the initial spot is at the barrier. The exchange discussion offers a correction and reference, but does not provide a full derivation or cover variations such as discrete monitoring or different model assumptions.
Key ideas
- A down-and-out call generally cannot be priced as a simple multiple of a European call at the same spot.
- The barrier feature requires a term that accounts for the asset path and barrier level.
- The provided formula uses a reflected spot in a second European call term.
- The stated formula applies under specific assumptions and reaches zero when the initial spot equals the barrier.
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Full text
# Price of a down-and-out call in terms of European call
# Price of a down-and-out call in terms of European call
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!
## Answer by FKaria (score 5)
https://quant.stackexchange.com/a/10036
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 $S_0=H$ .Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.