Can a down-and-out barrier call option be priced using the Black & Scholes formula or should it be approximated?

Question

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.

1. Simulate the price of the underlying stock using Brownian motion with say n simulations.
2. Take all price paths that never touch the Barrier and list their end prices.
3. 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 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 $$c_{do}=S_0N(x_1)e^{-qT}-Ke^{-rT}N(x_1-\sigma\sqrt{T})-S_0e^{-qT}(H/S_0)^{2\lambda}N(y_1)+Ke^{-rT}(H/S_0)^{2\lambda-2}N(y_1-\sigma\sqrt{T})$$ 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}