Pricing Down-and-Out Calls with Barrier Option Formulas
Summary
The document considers a leveraged turbo certificate whose value depends on the underlying price and a financing level. The financing level grows over time with daily interest, while a barrier level triggers a total loss if reached. The question asks whether the instrument can be priced with Black–Scholes or approximated by simulating price paths, discarding those that hit the barrier, and averaging discounted terminal payoffs.
The answer gives closed-form formulas for European-style down-and-out calls with continuously observed barriers, distinguishing cases according to whether the barrier lies below or above the strike. It defines the relevant variables and relates the barrier price to vanilla call values. This shows that a barrier feature does not by itself rule out a closed-form valuation. However, the formulas rely on standard option-model assumptions and do not directly establish how to value the described leveraged product, including its changing financing level and barrier. The proposed simulation also needs careful payoff, survival, and discounting specifications.
Key ideas
- Continuously monitored European down-and-out calls have closed-form pricing formulas under the stated model setup.
- The formula changes depending on whether the barrier is below or above the strike.
- A barrier feature alone does not imply that Black–Scholes style analytical methods are unavailable.
- A leveraged product with a financing level that changes over time may not match the quoted vanilla barrier formulas directly.
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Full text
# Can a down-and-out barrier call option be priced using the Black & Scholes formula or should it be approximated?
# Can a down-and-out barrier call option be priced using the Black & Scholes formula or should it be approximated?
I am trying to price of a Down-and-Out Barrier call option with leverage. When the price of the underlying asset hits a certain barrier (B), the option becomes worthless. The issuer of these options indicates that their price is calculated as follows: $$P = \frac{S - F}{ratio} $$
(Note: the ratio is used in case the price of the underlying asset is high, like in the case of Google stock which is around $1,500. The ratio is often 10 or 100.)
The issuer of the Turbo charges an interest of around 2% on the financing level $F$, which is paid daily by increasing the level of $F$ and consequently $B$ everyday. So that:
$$ F_t = F_0 (1+r)^t$$
The diagram below explains the structure.
Closed form formula Assume I want to estimate the value of this option after 1 year. As far as I understand, the Black & Scholes Model cannot be used. This is because this Barrier resembles an American option, as the Barrier option can be exercised at any time (i.e. when the price of the stock hits the Barrier). As the Black & Scholes Model applies only to European options which can be exercised only at expiration, a closed form model seems difficult to me.
Price approximation My question is if the price of this Barrier option could be approximated as follows.
- Simulate the price of the underlying stock using Brownian motion with say `n` simulations.
- Take all price paths that never touch the Barrier and list their end prices.
- Estimate the value of the option by calculating the option value for each of these end prices $S_i$ using $(S_i - F_t)$, sum them and discount to the present day, and then divide by the total number of price paths $n$:
$$ P = \sum_i\frac{(S_i - F_t)e^{-rt}}{n} $$
Does this make sense? Any suggestions on improving this, to make the approximation more realistic?
## Answer by Alex (score 1, accepted)
https://quant.stackexchange.com/a/57039
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 $$c_{do}=S_0N(x_1)e^{-qT}-Ke^{-rT}N(x_1-\sigma\sqrt{T})-S_0e^{-qT}(H/S_0)^{2\lambda}N(y_1)+Ke^{-rT}(H/S_0)^{2\lambda-2}N(y_1-\sigma\sqrt{T})$$ and $$c_{di}=c-c_{do}.$$
Here, \begin{align} d_{1,2} &= \frac{\ln(S_0/K)+(r-q\pm0.5\sigma^2)T}{\sigma\sqrt{T}}, \\ \lambda &= \frac{r-q+0.5\sigma^2}{\sigma^2}, \\ y&=\frac{\ln(H^2/(S_0K))}{\sigma\sqrt{T}}+\lambda\sigma\sqrt{T},\\ x_1 &= \frac{\ln(S_0/H)}{\sigma\sqrt{T}}+\lambda\sigma\sqrt{T}, \\ y_1 &= -\frac{\ln(S_0/H)}{\sigma\sqrt{T}}+\lambda\sigma\sqrt{T}. \end{align}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.