# Pricing and Correlation Risk

Pricing and Correlation Risk

To see how the quanto option should be priced, 2 new variables have to be introduced:

1. Correlation between BP's price and FX rate

2. FX volatility

1. Correlation between BP's price and FX rate

The US investor who is buying an ATM USD quanto call long or short this correlation (FX = Foreign/Investor)?

Assume the correlation is positive --> if GBP gets more valuable vs USD, BP's price increases.

Investor is therefore short this correlation.

If the correlation increases, the \$ increase of a plain vanilla call option (qto option with zero correlation) is more than the \$ increase of the USD quanto call option with a fixed rate.

2. FX Volatility

It is less obvious whether the US investor is long or short this FX volatility.

Let us introduce a model that describes the stock price in USD for a small time interval.

It has 2 additional variables vs the BS model:

- correlation between log S and log FX

- FX volatility

The following equation defines a new share, Ft, quoted in the currency the option is quanto'd into rather than the share's own currency.

Ft is defined in such a way that the regular \$ option on Ft is in fact a quanto option on St This quanto model is quite intuitive as the volatility used to price the quanto option should be the same as the IV of the underlying stock since the FX rate is fixed and therefore the option payout depends solely on the actual movement of the stock.

The drift part is slightly different as the delta hedge is affected by the FX rate movements!

From the equation, it is obvious that:

- long quanto call --> short correlation equity/FX

- long quanto put --> long correlation equity/FX

Depending on the sign of the correlation, this formula makes clear whether the holder of a quanto option is long or short the FX volatility.

If correlation is negative:

- long quanto call --> long FX volatility

- long quanto put --> short FX volatility

If correlation is positive:

- long quanto call --> short FX volatility

- long quanto put --> long FX volatility