How to calculate Greeks for leveraged Barrier options?

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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