For a given valuation date , expiry date , and time remaining to expiry (i.e. ), the interest rate and carry are defined as

Asset prices are assumed to be log-normally distributed with interest rates and asset yields assumed deterministic. Under these assumptions, forwards and european options are given by closed-form formulas.

## Forwards

A forward deal pays at the maturity date . where is the forward price. The value of the forward contract at time is

## European Options

A European option pays at the expiry date , where either for a call option or for a put option. The value of the European option for annualized implied volatility is given by Black's formula:

where is given by:

and (for ):

- is the standard normal cumulative function