Compare this rule of thumb to the value of a 1-year at-the-money-forward call on an underlying stock with spot price 100 and volatility 40%, assuming no dividends and a 5% interest rate.

Rule of thumb for the value of an ‘at-the-money-forward’ call

Consider an ‘at-the-money-forward’ European call, i.e. a plain vanilla call whose strike K is equal to the underlying forward price F₀.

(a) What does the closed-form formula in the lognormal model become in this case?

(b) Using a first-order Taylor expansion of the standard normal cumulative distribution N(·), show that the value of the call is approximately:

(c) Compare this rule of thumb to the value of a 1-year at-the-money-forward call on an underlying stock with spot price 100 and volatility 40%, assuming no dividends and a 5% interest rate.

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