Sample Question #147 (finance – option pricing)

[This is a difficult question… Enjoy!]

For a European call option on a non-dividend-paying stock, what’s the lower bound on its price? How can you verify this lower bound using a portfolio approach?

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ANSWER

The lower bound is P(t) – K*exp{-r(T-t)}, where K is expiration day and K is the strike price. This is a fundamental result and you should know it (and the one for puts) well.

To verify this with a portfolio approach, consider two portfolios:

1. Portfolio 1 has one European call plus cash in the amount of K*exp{-r(T-t)} invested at the risk-free rate r

2. Portfolio 2 has one share of the underlying stock

You can then consider what happens to the values of the two portfolios at time T. The result gives you the lower bound formula.

Bonus question: how do you mathematically derive the lower bound formula? (Note this is different from the portfolio verification question above.)