Quantitative Finance Asked by twhale on December 12, 2020

I am wondering how to calculate option Greeks for Down-and-out barrier Call options with leverage.

The option characteristics are as follows. The buyer of the option pays a fraction of the spot price `S`

of the underlying asset, with the rest of the spot price financed by the issuer of the call. This is the financing level `F`

, and the buyer pays an interest on F. If the price of the underlying asset is high (for example the Amazon stock is around $3000), a `ratio`

is applied.

Once the price of the underlying asset falls to hit a Barrier `B`

, the option expires and becomes worthless. The Financing level is always below the Barrier: `F < B`

The Barrier `B`

gradually increases along with the `F`

.

So the current price `P`

of the option is as follows:

```
P = (S - F) / ratio
```

To calculate the Greeks for such options, **can I simply use the normal formulas based on the Black and Scholes formula?**

If not, could someone help me with the maths or point me to the right formula’s for calculating the Greeks?

Below are the formulas I am currently using.

The buyer of the call pays a daily interest. The interest is charged by raising the financing level F:

```
F(t) = F(0) * (1 + r)^t
```

This means future price of the option:

```
P = ( (S - F) / (1 + r)^t ) / (ratio * exchange)
S = spot price of the underlying
r = the risk-free rate of return
t = time period under consideration
The future spot price S, I estimate using Monte Carlo analysis.
```

I would be interested to calculate the delta (dP/dS), rho (dP/dr), theta (dP/dt). Ideally also the vega. But how to do this is not clear to me.

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