73323306-15-Greeks - T he Greek: Letters A financial...

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The Greek: Letters A financial institution that sells an option to a client in the over-the-counter markets is faced with the problem of managing its risk. If the option happens to be the same as one that is traded on an exchange, the financial institution can neutralize its exposure by buying on the exchange the same option as it has sold. But when the option has been tailored to the needs of a client and does not correspond to the st'andardized products traded by exchanges, hedging the exposure is far more difficult. In this chapter we discuss some of the alternative approaches to this problem. We cover what are commonly referred to as the "Greek letters", or simply the "Greeks". Each Greek letter measures a different dimension to the risk in an option position and the aim of a trader is to manage the Greeks so that all risks are acceptable. The analysis presented in this chapter is applicable to market makers in options on an exchange as well as to traders working in the over-the-counter market for financial institutions. Toward the end of the chapter, we will consider the creation of options synthetically. This turns out to be very closely related to the hedging of options. Creating an option position synthetically is essentially the same task as hedging the opposite option position. For example, creating a long call option synthetically is the same as hedging a short position in the call option. 15.1 IllUSTRATION In the next few sections we use as an example the position of a financial institution that has sold for $300,000 a European call option on 100,000 shares of a non-dividend- paying stock. We assume that the stock price is $49, the strike price is $50, the risk-free interest rate is 5% per annum, the stock price volatility is 20% per annum, the time to maturity is 20 weeks (0.3846 years), and the expected return from the stock is 13% per annum. I With our usual notation, this means that So = 49, K = 50, r = 0.05, (J = 0.20, T = 0.3846, JL = 0.13 The Black-Scholes price of the option is about $240,000. The financial institution has 1 As shown in Chapters 11 and 13, the expected return is irrelevant to the pricing ofan option. It is given here because it can have some bearing on the effectiveness of a hedging scheme. 341
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342 CHAPTER 15 therefore sold the option for $60,000 more than its theoretical value, but it is faced with the problem of hedging the risks. 2 15.2 NAKED AND COVERED POSITIONS One strategy open to the financial institution is to do nothing. This is sometimes referred to as adopting a nakedposition. It is a strategy that works well if the stock price is below $50 at the end of the 20 weeks. The option then costs the financial institution nothing and it makes a profit of$300,000. A naked position works less well if the call is exercised because the financial institution then has to buy 100,000 shares at the market price prevailing in 20 weeks to cover the call. The cost to the financial institution is 100,000 times the amount by which the stock price exceeds the strike price. For example, if after 20 weeks the stock price is $60, the option costs the financial institution $1,000,000. This
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This note was uploaded on 01/30/2012 for the course MATH 174 taught by Professor Donblasius during the Spring '11 term at UCLA.

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73323306-15-Greeks - T he Greek: Letters A financial...

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