The Greek Letters Based on Options, Futures, and Other Derivatives, 8th Edition, Copyright John C. Hull 2012
Introduction Each of the Greek letters measures a different dimension to the risk in an option position. The aim of a trader is to manage the Greeks so that all risks are acceptable. A bank has sold for $300,000 a European call option on 100,000 shares of a non-dividend paying stock. The points that will be made apply to other types of options and derivatives. S 0 = 49, K = 50, r = 5%, s = 20%, T = 20 weeks (0.3846 years), m = 13% The Black-Scholes-Merton value of the option is $240,000(i.e., $2.40 for an option to buy one share). A good hedge would ensure that the cost is always equal or at least close to $240,000. 2
Naked and Covered Positions Strategy A: Naked position Suppose that the bank takes a naked position against risk: it takes no specific action to hedge risk. If the stock price at the end of the 20 weeks remains below K=$50, then this strategy works well. A naked position does not work well if the call is exercised because the bank 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. 3
Naked and Covered Positions Strategy B: Covered position As an alternative to a naked position, the financial institution can adopt a covered position. This involves buying 100,000 shares as soon as the option has been sold. If the option is exercised, this strategy works well, but in other circumstances it could lead to a significant loss. For example, if the stock price drops to $40, the financial institution loses $900,000 on its stock position. 4
Stop-loss Strategy Neither a naked nor a covered position provides a good hedge on its own. An alternative hedging procedure is the stop-loss strategy, which is a combination of both naked and covered strategies. This involves: Buying 100,000 shares as soon as the stock price rises above $50. Selling 100,000 shares as soon as price falls below $50. The objective is to hold a naked position whenever the stock price is less than K and a covered position whenever the stock price is greater than K. 5
Stop-loss Strategy The stop-loss strategy involves buying the stock at time t1, selling it at time t2, buying it at time t3, selling it at time t4, buying it at time t5, and delivering it at time T. 6
Stop-loss Strategy The procedure is designed to ensure that at time T the institution owns the stock if the option closes in the money and does not own it if the option closes out of the money. Options are referred to as in the money, at the money, or out of the money. A call option is in the money when S > K, at the money when S = K, and out of the money when S < K. A put option is in the money when S < K, at the money when S = K, and out of the money when S > K. 7
Stop-loss Strategy The cost of setting up the hedge initially is S 0 if S 0 > Kand zero otherwise. It seems as though the total cost, Q, of writing and hedging the option is the option s initial intrinsic value: Q = max(s 0 K, 0) The intrinsic value of an option is defined as the maximum of zero and the value the option would have if it were exercised immediately. For a call option, the intrinsic value is max S 0 K, 0 For a put option, it is max (K S 0, 0) As we can see, all purchases and sales subsequent to time 0 are made at price K. 8
Stop-loss Strategy There are two key reasons why the cost equation is incorrect. The first is that the cash flows to the hedger occur at different times and must be discounted. The second is that purchases and sales cannot be made at exactly the same price K. If we assume a risk-neutral world with zero interest rates, we can justify Q ignoring the time value of money. But we cannot legitimately assume that both purchases and sales are made at the same price. The hedger cannot know whether the stock price equals K, it will continue above or below K. 9
Stop-loss Strategy In practice, purchases are made at a price K + e and sales are made at a price K + e, for some small e>0. Thus, every purchase and subsequent sale involves a cost (apart from transaction costs) of 2e. If the path of the stock price crosses the strike price level many times, the procedure is quite expensive. Assuming that stock prices change continuously, e can be made arbitrarily small by monitoring the stock prices closely. But as e is made smaller, trades tend to occur more frequently. Thus, the lower cost per trade is offset by the increased frequency of trading. As e 0, the expected number of trades. 10
Delta hedging The delta (Δ) of an option is defined as the rate of change of the option price with respect to the price of the underlying asset. It is the slope of the curve that relates the option price to the underlying asset price. Suppose that the delta of a call option on a stock is 0.6. This means that when the stock price changes by a small amount, the option price changes by about 60% of that amount. In general: Δ = c S where c is the price of the call option and S is the stock price. 11
Delta hedging The figure shows the relationship between a call price and the underlying stock price. When the stock price corresponds to point A, the option price corresponds to point B and is the slope of the line indicated. Call option price B Slope = D = 0.6 A Stock price 12
Delta hedging Suppose that the stock price is $100 and the price of a call option is $10. An investor has sold 20 call option contracts -that is, options on 2,000 shares. The investor s position could be hedged by buying: 0.6 x 2,000 = 1,200 shares The delta of the trader s short position in 2,000 options is: 0.6 x (-2,000) = -1,200 shares 13
Delta hedging If the stock price goes up by $1 (producing a gain of $1,200 on the shares purchased), the option price will tend to go up by 0.6 x $1 = $0.60 (producing a loss of $1,200). If the stock price goes down by $1 (producing a loss of $1,200 on the shares purchased), the option price will tend to go down by $0.60 (producing a gain of $1,200 on the options written). The gain (loss) on the stock position would then tend to offset the loss (gain) on the option position. The delta of the stock position offsets the delta of the option position. A position with a delta of zero is referred to as delta neutral. 14
Delta hedging The delta of an option does not remain constant. Therefore, the trader s position remains delta hedged (or delta neutral) for only a relatively short period of time. The hedge has to be adjusted periodically. This is known as rebalancing. Suppose that delta rises from 0.60 to 0.65. An extra 0.05 x 2,000 = 100 shares would then have to be purchased to maintain the hedge. A procedure such as this, where the hedge is adjusted on a regular basis, is referred to as dynamic hedging. It can be contrasted with static hedging, where a hedge is set up initially and never adjusted. Static hedging is sometimes also referred to as hedge-andforget. 15
Delta hedging For a European call option on a non-dividend-paying stock, it can be shown that: Δ(call) = N(d1) where: N(x) is the cumulative distribution function for a s.n. distribution. ln( S K r 2 T d 0 / ) ( / 2) 1 = T The above formula gives the delta of a long position in one call option. The delta of a European futures call option is defined as the rate of change of the option price with respect to the futures price (not the spot price). Δ(call) = e rt N(d1) The delta of a short position in one call option is given by: Δ(call) = - N(d1) 16
Delta hedging For a European put option on a non-dividend-paying stock, it can be shown that: Δ(put) = N(d1) - 1 Delta is negative, which means that a long position in a put option should be hedged with a long position in the underlying stock. A short position in a put option should be hedged with a short position in the underlying stock. 17
Delta hedging Variation of delta with stock price for (a) a call option and (b) a put option on a non-dividend-paying stock. 18
Delta hedging: Exercise What is the delta of a short position in 1,000 European call options on silver futures? The options mature in eight months, and the futures contract underlying the option matures in nine months. The current nine-month futures price is $8 per ounce. The exercise price of the options is $8. The risk-free interest rate is 12% per annum. The volatility of silver is 18% per annum. 19
Theta The theta (Θ) of a portfolio of options is the rate of change of the value of the portfolio with respect to the passage of time with all else remaining the same. Theta is sometimes referred to as the time decay of the portfolio. The theta of a call or put is usually negative. A negative theta means that, if time passes with the price of the underlying asset and its volatility remaining the same, the value of a long call or put option declines. 20
Gamma The Gamma (G) is the rate of change of delta (D) with respect to the price of the underlying asset. Gamma is the second partial derivative of the portfolio with respect to asset price: If gamma is small, delta changes slowly, and adjustments to keep a portfolio delta neutral need to be made only relatively infrequently. However, if gamma is highly negative or highly positive, delta is very sensitive to the price of the underlying asset. It is then quite risky to leave a delta-neutral portfolio unchanged for any length of time. 21
Relationship Between Delta, Gamma, and Theta For a portfolio of derivatives on a stock paying a continuous dividend yield at rate q it follows from the Black-Scholes-Merton differential equation that: 1 2 2 rsd S = r 2 For a delta-neutral portfolio (Δ = 0), we obtain: 1 2 2 S 2 = r This shows that, when Θ is large and positive, gamma of a portfolio tends to be large and negative, and vice versa. 22