Finance

Finance derivative& security

Assignment 2

Question:

Let C(K) denote a European vanilla call option with strike price K. assume the all options are identical except for strike price, and strike prices satisfy K1<K2<K3 and 2K2=K1+K3.

Q1：what are the no-arbitrage lower bound, and the no-arbitrage upper bound, of the vertical spread C (K1)-C (K2)?

Option buyers have the right, not the obligation. European option is an option that can only be exercised at its maturity. European options tend to sometimes trade at a discount to its comparable American option. This is because European options allow investors less opportunities to exercise the contract. European options normally trade over the counter, while American options usually trade on standardized exchanges. A buyer of a European option that does not want to wait for maturity to exercise it can sell the option to close the position.

Vertical spreads: in this spread, a trader use an options trading strategy to make a simultaneous purchase and sale of two options of the same type that have the same maturity but different strike prices.

The question required the vertical spreads CK1-C(K2), so the situation will be K2-K1<CK1-C(K2), the table below is the payoff of vertical spread under this situation.

Portfolio | Current Price | ST>K2 | ST<K1 | K1≤ST≤K2 |

Buy C1 | K1 | ST-K1 | 0 | ST-K1 |

Sell C2 | -K2 | 0 | 0 | 0 |

Subtotal | K1-K2 | ST-K1 | 0 | ST-K1 |

Lend | K2-K1 | >=( K2-K1) | >=( K2-K1) | >=( K2-K1) |

Total | 0 | >0 | >0 | >0 |

As the result, total is >0, there is arbitrage opportunities such as the table above, pay zero today to get a guaranteed positive payoff in the future, so investor can gain profits by doing this many times. In fact, this cannot for long, the prices will become equivalent very quickly.

Expiry Payoff

Expiry Payoff

ST

ST

K1

K2

Expiry Payoff

Expiry Payoff

ST

ST

K1

K2

(From the lecture note of Curtin)

The...