The previous discussion tells us that the price is somewhere between zero and maximum, which is either the underlying price, the exercise price, or the present value of the exercise price - a fairly wide range of possibilities. The range will be tightened up a little on the low side by establishing a

For American options, which are exercisable immediately:

C

P

If the option is in-the-money and is selling for less than its intrinsic value, it can be bought and exercised to net an immediate risk-free profit.

However, European options cannot be exercised early; thus, there is no way for market participants to exercise an option selling for too little with respect to its intrinsic value. Investors have to determine the lower bound of a European call by constructing a portfolio consisting of a long call and risk-free bond and a short position in the underlying asset.

First the investor needs the ability to buy and sell a risk-free bond with a face value equal to the exercise price and current value equal to the present value of the exercise price. The investor buys the European call and the risk-free bond and sells short (borrows the asset and sells it) the underlying asset. At expiration the investor shall buy back the asset.

This combination produces a non-negative value at expiration, so its current value must be non-negative. For this situation to occur, the call price has to be worth at least the underlying price minus the present value of the exercise price:

The lower bound of a European put is established by constructing a portfolio consisting of a long put, a long position in the underlying, and the issuance of a zero-coupon bond. This combination produces a non-negative value at expiration so its current value must be non-negative. For this situation to occur, the put price has to be at least as much as the present value of the exercise price minus the underlying price.

For both calls and puts, if this lower bound is negative, we invoke the rule that an option price can be no lower than zero.

- All options expire in 60 days, have the same exercise price (X) of $60 and the same underlying asset.
- The current price of the underlying (S
_{0}) is $50. - The risk-free rate (r) is 5%.
- Find the lower bounds of American and European calls and puts.

- Time to expiration (T) = 60/365 = 0.1644
- European Call (c
_{0}): MAX[0, 50 - 60/(1 + 5%)^{0.1644}] = MAX[0, -9.52] = 0 - American Call (C
_{0}): MAX[0, 50 - 60/(1 + 5%)^{0.1644}] = MAX[0, -9.52] = 0 - European Put (p
_{0}): MAX[0, 60/(1 + 5%)^{0.1644}- 50] = MAX[0, 9.52) = 9.52 - American Put (P
_{0}): MAX[0, 60 - 50) = 10 - Note that the lower bound of the American put is above the lower bound of the European put.

6162: the reason behind this is that the buyer can use his part of money for some other investment purposes when the interest rates r high in the market but paying only a margin amount of the option whereas the writer or the seller cannnot sell the goods until the buyer wishes so he is stuck with the stock and cannot convert it into cash and benefit the high rate of interest in the market. |

mtcfa: Am I the only one that thinks this section is somewhat off base? If I have an equity call that goes in my favor (ie deep in the money), you can bet I'm going to excercise that option and take my profits. Why would I sit around and wait for the market to turn againat me and lose what ever profits I already made? |

cwrolfe: I'd just short the call option and lock-in my gains |

antarctica: Here it's assumed that price rises continuously. So if you exercise now you'll be worse off than keeping it until expiration, assuming that stock price either goes up or stay the same. |

Winner: Euro Call Max I get 9.92 rather than 9.52 as detailed above. Does anyone else get this answer? |

Winner: Sorry, Now I see how they get the 9.52, itsclearer if you see it like this 50 - [(60/(1.05^.1644)] = -9.52 |

uberstyle: kind of a strong assumption to assume prices continue to rise - I am not disagreeing but wonder why it is not noted? |

Tom81: Solution* Time to expiration (T) = 60/365 = 0.1644. * European Call (c0): MAX[0, 50 - 60/(1 + 5%)0.1644] = MAX[0, -9.52] = 0. * American Call (C0): MAX[0, 50 - 60/(1 + 5%)0.1644] = MAX[0, -9.52] = 0. * European Put (p0): MAX[0, 60/(1 + 5%)0.1644 - 50] = MAX[0, 9.52) = 9.52. * American Put (P0): MAX[0, 60 - 50) = 10. * Note that the lower bound of the American put is above the lower bound of the European put. Should that not just be for an American Max(0,50-60), i.e. not discounted. |

olagbami: Why is the American call option discounted? |

tschorsch: You would just sell the option to lock in profits.It is still worth more than the intrinsic value because the underlying could still go more in your favor. Also, exercising an option would give exposure to the underlying, and you would also usually incure more transaction costs by exercising. |

tschorsch: Also, suppose you are long a call. When the underlying price hits your target, you want to exit the position. When you exercise, you are still net long the position and must sell the underlying to remove your exposure. Exercising is not instantaneous and you will have some time exposure when you cannot exit (unless you sell the underlying short).The moral of the story is, unless you want to hold the underlying for the longer term (or for puts you already have the underlying and want to unload it) it is really not worth exercising just to take profits. |

Shammel: @ olagbami: I think the American call option is discounted as we are finding the lower bound. |

cardinal08: The call is using leverage, "borrowing," to buy the underlying later. The put is holding the premium. In a high interest rate environment, This is beneficial to the call holder because he is achieving leverage without needing to finance through high rates, and this disadvantages the put holder because he is holding a premium that would otherwise be paying a high rate. |

Juhee: American call can be exercised before expire date not like European call. so that is why we need to discount to get the present value |

anova: The lower bounds of American and European Calls are the same else the American Call lower bound would be lower the European Call lower bound.co>= Max[0,So-X/(1+r)^T] Co>= Max[0,So-X/(1+r)^T] |

papajeff: They are referring only to exercising the option, not selling the calls. if you sell the call you lock in the time value. |

moneyguy: At least this stuff is not at all confusing. |

Emily1119: Why a long term european put can be worth more or less than a short term european？ |

Oksanata: in this case they should have discounted American put as well, shouldn't they? |

Oksanata: Oh, now I see..All the explanations are in No.9 of this subject.. |

Oksanata: If Volatility, Time to Expiration is UP - Call and Put values are UPAll other factors,if UP, affecting option prices in opposite directions as follows: Dividents, Strike Price UP: Call Down, Put Up Intr.Rate, Underlying UP: Call UP, Put Down |

gill15: Why would it be confusing to you? You're the Moneyguy. |

SKIA: This section should have a couple of questions to hammer on the relationships. |