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a binary valued function of two variables, the price and the time. The exercise strategyπ(S,t) speciﬁes whether to exercise or not at the state (S,t), The American call option and the European call option are equivalent. To prove this, we will simply show that it is never optimal to exercise. Consider time t where

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Conceptually calculating what a 110 OTM call option should be worth if the present price of the stock is 100 9.00 1.20 (1) We know how to calculate the 0.40 0.60 0.80 1.00 3.00 4.00 5.00 6.00 7.00 The Black-Scholes Options Pricing Model Author: Gary R. Evans Subject: The primary options pricing model

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A binary call option is, at long expirations, similar to a tight call spread using two vanilla options. One can model the value of a binary cash-or-nothing option, C, at strike K, as an infinitessimally tight spread, where is a vanilla European call:

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Foundations of Finance: Options: Valuation and (No) Arbitrage 3 • Notation S, or S0 the value of the stock at time 0. C, or C0 the value of a call option with exercise price X and expiration date T P or P0 the value of a put option with exercise price X and expiration date T