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# Minyan Mailbag: Option Valuation

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## In general, the longer term an option is and the more in the money it is, the value of being able to exercise it at any time increases.

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Prof. Succo,

Hope all is well. I had a quick option valuation question. Are European put options valued differently than American ones? The reason I ask is that a fund I follow owns European put options expiring in '07 (see below). The issue I have is that if these options expired today, I get a value of \$18.8 million versus the presented value of \$12.1 million (the S&P was at approx. 1294 at the time of valuation). Do you think I'm missing something in my calculations, i.e. does this make sense to you?

 Number of Contracts Put Options Purchased Value 54,651 S&P European, expiring October 2007 at US\$1,639 \$12,114,047

Minyan Brian

MB,

This is a good example to explain the value of an option attributed to the right of exercise. The value of the right to exercise changes over time and the price of an option: in general, the longer term an option is and the more in the money it is, the value of being able to exercise it at any time increases.

By definition, any option (no matter how long term) must trade above parity if it is exercisable at any time during its life. If it traded under parity, an arbitrager could simply buy the option, hedge it one up against the cash, exercise it and immediately earn a riskless profit. For example, if I can buy a stock for \$48 and buy a \$50 put for \$1.5 (trading \$0.50 under parity), I could buy the stock and put and exercise the put (thus selling the stock at \$50) and earn \$0.50 of instant and riskless profit. An option that is exercisable at any time is called an "American" option.

But some options cannot be exercised right away. A "European" option can only be exercised at expiration. This makes its value different (less) than an American option, for the arbitrage explained above cannot occur. The value as an option is not much different, but the value as a "futures" contract is much different: the difference is the value of the right to exercise.

When an option is out, at the money, or even somewhat in the money it acts mostly like an option: it has gamma, which means there is likely to be a change in its delta. When an option is so deep in the money (the delta is near 1) and it is unlikely that the delta will change again (it has almost no gamma) it has lost most of its "optionality" and acts basically like a futures contract.

So when a European option is at or out of the money, the value is very similar to an American option because the right to exercise is not very valuable (no one would exercise an at the money option or out of the money option guaranteeing an instant loss). When it goes in the money, and the more it does so, its value becomes very different than an American option because the value of exercise becomes much greater. Let me explain.

The difference is attributable to the carry (the cost of the hedge). If I buy a European put (like your SPX index put) that is deep in the money, I must buy the SPX cash or futures contract to hedge the delta. The deeper in the money it is, the more I have to buy. The more I buy, the more carrying cost I have (I assume a borrowing cost against the cash hedge; this borrowing cost is embedded in the futures contract through premium over spot paid). The more cost I have, the less I am willing to pay for the put.

For an American option, when it gets to the point where these carrying costs equal the option value (premium over intrinsic), the delta will be 1 and the holder will likely exercise the put. Since there is no gamma left and the option is now basically a futures contract, the carrying costs of the hedge warrant the holder of the option to exercise. If you notice, options like these (very deep in the money puts) have no open interest because they have all been exercised.

Since a European option cannot be exercised, they will begin to trade at a discount to American options as they go in the money (the early exercise feature is gaining value) to reflect these increased carrying costs. Hedgers who were long these puts sell them to arbitragers at the appropriate price to earn close to the risk free rate.

The put you own is trading at a discount to parity because it does not have any option value left (the delta is close to 1) and the carrying costs exceed parity. Think of it this way. You are essentially short a futures contract (not quite, but close) out to October 30 2007. A futures contract out that far would trade at about a \$58 premium to cash. If you wanted to cover that futures contract you would have to pay a \$58 premium over cash. If you hold that future until October 2007 the future premium will amortize away to the cash price. Right now, you instead own a put that is commensurately worth a \$73 discount to parity. If you hold that put until October 2007, that discount will amortize away to the cash price. So you are indifferent in holding a short futures contract or long a deep in the money put that cannot be exercised. As an arbitrager I would be indifferent if I bought the futures contract at a \$58 premium and bought the put from you at a \$73 discount. In the end both wash and I earn a risk free rate (\$73-\$58 less my carry costs on both).

By the way, I have your puts worth about \$2.5 million more than where you have them marked (although still below parity). I assume these are over the counter puts, therefore your counter-party who sold them to you is "helping" you mark them. This is alarming but typical. This is what I refer to when I say that I believe many derivatives are marked incorrectly (aggressively) by dealers. By marking a position they are short below fair value they 1) get to book profits early since the puts will eventually go to parity and 2) hopefully get you to sell them out cheaply to them. They begin by marking them fairly and little by little (a few pennies a day), begin to mark them down. Imagine multiplying your \$2.5 million by trillions of derivative positions and you can see why I say there is a significant mis-marking problem with dealers. I do believe that certain large dealers may actually be bankrupt if they re-marked their derivative positions to fair value.

When and if you go to sell them because they are over the counter, you normally go back to the counter party. They are depending on this. If you hit their bid you will forego \$2.5 million in value. It would behoove you to get several bids in line either to force your counter-party to pay the right closing price, or if they don't, go to another counter-party who will pay the right price. You will be forced to carry a long/short position with two counter-parties, but it definitely is in your best interest to do so.
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