1 Option Trading and Positioning Pooya Tavana Option Trading and Positioning Professor Bodurtha 5/7/2011 Pooya Tavana
2 Option Trading and Positioning Pooya Tavana I. Executive Summary Financial options have become increasingly important in finance. Many types of options are traded by financial institutions, hedge fund managers and corporate treasurers for various reasons such as hedging, speculation and arbitrage. In this paper, I will focus on hedging aspect of the options by demonstrating a real life example, and I will analyze the result of this hedging exercise by using Black-Scholes and Binomial option pricing methods. In particular, I will demonstrate how we can decompose different risk elements by using option Greeks and Taylor series expansion. I assume I have 100 call options on Cisco Systems Inc. maturing in May 2011 at strike price of $17. The hedging exercise will be broken into three periods: the first period is devoted to delta hedging strategy; the second period deals with the construction of a delta-gamma neutral hedge; and finally in the last period, I attempt to implement a delta-gamma-vega neutral hedge. Here is summary of my findings: Delta hedging could be easily achieved by taking a position in the underlying asset. However, to maintain the neutrality, position in the underlying assets should be regularly changed. Moreover, delta hedge could be regarded as a volatility trade too. Gamma is an important hedging element that could reduce our exposure to changes in the asset price. Particularly, if it is hedged along with delta. Theoretically, a deltagamma neutral portfolio is free of exposure to changes in the underlying asset price. In practice however, this not true. For hedging vega and gamma in the same time, we need to add two options to the portfolio. This method is easy to construct, but it does not capture the fact that implied volatility for options with different strike prices are different and they could even move in opposite directions. This phenomenon undermines the effectiveness of our approach. After going through this exercise, it is obvious that risk management is a complicated practice. A wise trader should completely understand the mathematics of option pricing in order to successfully implement hedging strategies.
3 Option Trading and Positioning Pooya Tavana II. Introduction Financial options have become increasingly important in finance. Many types of options are traded by financial institutions, hedge fund managers and corporate treasurers for various reasons such as hedging, speculation and arbitrage. In this paper, I will focus on hedging aspect of the options by demonstrating a real life example, and I will analyze the result of this hedging exercise by using Black-Scholes and Binomial option pricing methods. In particular, I will demonstrate how we can decompose and hedge different risk elements by using option Greeks. I assume that we own a specific at-the-money call option and try to hedge the exposure to changes in share price and volatility as well as time decay of the option. The hedging exercise would be performed by taking either short or long positions in the stock and also other options on the same stock. III. Option Pricing Models Binomial and Black-Scholes models are among the most frequently used models for pricing options. I used a Binomial model for calculating option premium and also most of the Greeks and compared my results from the Binomial with the analytical values derived from Black-Scholes. For the Binomial method, I developed VBA codes (Appendix 1) to perform all the necessary calculations. In my model, number of steps in the Binomial tree is user-defined. For my analysis, I used 200 steps to get close to Black-Scholes analytic results. The formulas and methods used in this report are adopted from Option, Future, and Other Derivatives (Hull & Basu, 2010). Some of the most important formulas for constructing the Binomial tree and performing other necessary calculations are shown below. Binomial tree formulas is number of steps in the Binomial tree
4 Option Trading and Positioning Pooya Tavana As previously mentioned, I used the Black-Scholes formula for validating the value derived from the Binomial method. The Balck-Scholes formulas for valuing call option (c) and put option (p) are reproduced here for easy reference. In these formulas, S 0, E, T, r, σ and D refer to initial stock price, exercise price, time to maturity, risk-free rate, stock volatility and dividend yield, respectively.
5 Option Trading and Positioning Pooya Tavana IV. Option Greeks Owning an option exposes you to different types of risk that could drastically change the value of your portfolio if not hedged. For example, understanding how the value of an option changes with respect to change in the stock price is the first step to understand the risk you are facing by owning a stock option. Change in the stock price could change the value of your option substantially. Think about a call option with the exercise value of the $17. If the stock price drops from $18 to $16, your option goes from being in-the-money (ITM) to being out-of-themoney (OTM), and loses all of value. Hull & Basu have a chapter about the Greek letters (Chapter 17) which has been used for the purpose of this report. It is assumed that readers are familiar with the basic concepts of the Greeks and the mathematics behind them. Therefore, I am going to discuss Taylor series expansion and its application in hedging. Based on the Taylor series expansion, a change in the value of portfolio Π (t,s, σ) can be decomposed to the following elements: (i) All of above partial derivatives have names and could be easily calculated and hedged. The first term could be eliminated by making your portfolio delta neutral. The second term in the equation is non-stochastic and it shows the time decay of the portfolio. The third term could also be eliminated if the portfolio is vega (exposure to volatility) neutral. Forth term in the equation measures change in the value of portfolio in connection with the second partial derivative of the portfolio in respect to stock price. This exposure could be hedged away with making the portfolio gamma neutral. The fifth term and other terms which are not shown here have negligible effect on the value of the portfolio. For this study, we are going to focus on the first four terms of the above equation, and try to hedge our exposure by making the portfolio delta, gamma, and vega neutral. We can rewrite the formula by using the Greek letters: (ii) The table below summarizes the analytical formula of the Greeks for a call and a put option based on the Black-Scholes formula. I also approximated delta, gamma, and theta by using the Binomial tree. A Numerical method for calculating the Greeks is show in Appendix 2.
6 Option Trading and Positioning Pooya Tavana Greek Letters Call Option Put Option Delta Gamma Theta Vega Rho V. Hedging Exercise I use Cisco Systems Inc. (CSCO) options in order to apply the aforementioned models and implement a hedging strategy using the Greeks. I assume a portfolio consisting of 100 call options on CSCO with an exercise price of $17 maturing on May 20 th, 2011. I manage this portfolio from April 4 th, 2011 to May 6 th, 2011 and try to hedge my exposures based on the Greeks. We are currently in an extremely low interest rate environment; for this exercise risk free rate is hovering around 0.020%. Since CSCO is not paying any dividend during my holding period; I treat the stock as a non-dividend paying stock. It is obvious this assumption will not hold if the option maturity was after an ex-dividend date. Moreover, implied volatility is calculated using my Binomial model. I uses the goal seek function in Excel to back out the implied volatility based on the market data. This exercise has four parts; the first part deals with Taylor series expansion and decomposition of daily change in the option premium based on equation (2); in the second part, I tried to put on a delta hedge; in the third part, I extended my coverage to make my portfolio gamma neutral as well; and in the last step, I implemented a delta-gamma-vega neutral strategy to minimize my exposure to uncertainties arising from changes in the stock price and volatility. Decomposing the Option Price Movement Taylor series expansion helps us to explain daily changes in the value of my position (CSCO C17). Table 1 shows this exercise for the first week. Stock prices and option premiums
7 Option Trading and Positioning Pooya Tavana are all retrieved from Bloomberg. Based this market data, I calculate implied volatility and Greeks for CSCO C17. Using all these data, I calculate the daily changes in option premium and contributing factors. For example, column (14) demonstrates the first term of Taylor series, which is change in the option premium explained by delta. Table 1- Decomposing daily change in option premium (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) (15) (16) (17) (18) (19) Stock Price Implied Vol Option Premium Greek Letters Market Movements Daily Change in Option Premium Date Open Close Open Close Open Close Delta Gamma Theta Vega d S dσ dt Delta Gamma Vega Theta Other Total 4/4/2011 17.01 17.06 30.45% 29.21% 0.740 0.730 0.525 0.217 (0.008) 2.404 0.05-1.24% 1 2.62 0.03 (2.98) (0.80) 0.13 (1.00) 4/5/2011 17.16 17.22 28.17% 31.33% 0.760 0.860 0.559 0.233 (0.008) 2.378 0.06 3.16% 1 3.35 0.04 7.51 (0.75) (0.15) 10.00 4/6/2011 17.44 18.07 30.86% 31.98% 0.980 1.420 0.616 0.205 (0.008) 2.313 0.63 1.12% 1 38.81 4.06 2.59 (0.82) (0.64) 44.00 4/7/2011 18.16 17.91 31.06% 32.97% 1.470 1.320 0.750 0.164 (0.007) 1.980 (0.25) 1.91% 1 (18.75) 0.51 3.78 (0.72) 0.19 (15.00) 4/8/2011 17.97 17.65 29.98% 32.61% 1.300 1.110 0.726 0.183 (0.007) 2.032 (0.32) 2.63% 1 (23.22) 0.94 5.35 (0.73) (1.33) (19.00) On 4/4/2011, stock price changed 5 cents, therefore, ds is equal to $0.05. Option premium, on the other hand, slid 1 cents resulting in change of 1$ ($0.01 * 100 options) for the portfolio. Now, let s use Taylor series to decompose changes in portfolio value (. As you can see, the result from the left hand side and the right hand side of Taylor series does not equal. This problem is due to all sorts of errors such as modeling errors, rounding errors, violation of Black-Scholes assumptions, etc. Column (18) shows this error on the daily basis and it seems that is fairly low when ds is low. However, for days like 4/7/2011 when ds is relatively high, error term is larger in magnitude. In order to decrease the error term, one should spend more time and effort for modeling and try to calculate portfolio changes on shorter intervals (eg. hourly or even every minute). Also please note that in the formula, I used dt equal to 1, which is also an approximation since from open to close only 8 hours elapsed. Delta Hedging Strategy After decomposing the daily change in the option premium, we can see that the most significant factor is delta term. The easiest way to hedge delta is delta hedging strategy where one goes long (short) stock based on the delta of the option portfolio. This strategy makes the portfolio delta neutral by eliminating the first term in Taylor series. Please note that ds in Taylor
8 Option Trading and Positioning Pooya Tavana series is assumed to be a change in the stock price within a small timeframe, therefore, this strategy will work better if changes in stock price are measured more frequently. We will talk about hedging risks due to bigger stock price movements later on when we talk about gamma neutral portfolios. There are some assumptions that make trading easier compared to a real life experience: (1) Trading Cost assumed to be zero. This might not be the case in the real world; however, for big financial institutions with sizable trading volume this cost is negligible. Additionally, I assumed I can borrow (lend) with zero interest. (2) In the real world, I cannot buy a fraction of a stock; however, I assume I can go long (short) any fraction of stock. This is a valid assumption since my portfolio could be scaled up. (3) I can buy (sell) any option or stock at any day based on the open price. This assumption is appropriate for the purpose of this assignment. However, a trader might not be able to buy (sell) at the open price. Continuing with our position of 100 on CSCO C17, I am going to short CSCO stocks based on delta of the portfolio on the daily basis. For example on 04/04/2011, delta of C17 option is 0.5249, therefore, delta of the portfolio is 52.49 (0.5249 * 100 options). In order to hedge this exposure away, I short 52.49 CSCO stock. Our short position on the stock will offset the movement in option portfolio due to delta and makes the portfolio delta neutral (Table 2). To better understand how this strategy works, we should look at the profit & loss section (columns 14-16) in Table 2. Column 14 shows the open profit and loss calculated by marking to market the position we had the day before to the opening price, while Column 15 shows the close profit and loss calculated by marking to market the position we had during the day and we held till end of the day (as part of my assumptions, I only buy (sell) positions in the mornings using the open price). For example on 4/6/2011, our option position gained $44 during the day. $38.81 out of this $44 is due to the delta element from Taylor expansion (see column 6). On the same day we were short 61.60 shares at open price of $17.44 per share for implementing the delta hedge. As you can see our hedge lost $38.81 (see column 17) during the day. This loss on the hedge completely offsets the profit on the options.
9 Option Trading and Positioning Pooya Tavana 1) C17 CSCO Table 2- Profit and Loss for Delta Hedging Strategy (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) (15) (16) Option Premium Market Movements Daily Changes Position Profit & Loss Option Date Open Close ds dσ dt Delta Gamma Vega Theta Other Total (C17) (Open) (Close) Total 4/4/2011 0.74 0.73 0.05-1.24% 1 2.62 0.03 (2.98) (0.80) 0.13 (1.00) 100.00 - (1.00) (1.00) 4/5/2011 0.76 0.86 0.06 3.16% 1 3.35 0.04 7.51 (0.75) (0.15) 10.00 100.00 3.00 10.00 13.00 4/6/2011 0.98 1.42 0.63 1.12% 1 38.81 4.06 2.59 (0.82) (0.64) 44.00 100.00 12.00 44.00 56.00 4/7/2011 1.47 1.32 (0.25) 1.91% 1 (18.75) 0.51 3.78 (0.72) 0.19 (15.00) 100.00 5.00 (15.00) (10.00) 4/8/2011 1.30 1.11 (0.32) 1.82% 1 (23.22) 0.94 3.70 (0.73) 0.32 (19.00) 100.00 (2.00) (19.00) (21.00) 4/11/2011 1.10 0.97 (0.24) 1.65% 1 (16.30) 0.59 3.42 (0.81) 0.11 (13.00) 100.00 (1.00) (13.00) (14.00) (13.50) 6.17 18.02 (4.65) (0.05) 6.00 17.00 6.00 23.00 2) CSCO Stock Price Market Movements Daily Changes Position Profit & Loss Date Open Close ds Delta Gamma Vega Theta Other Total Stock (Open) (Close) Total 4/4/2011 17.01 17.06 0.05 (2.62) - - - - (2.62) (52.49) - (2.62) (2.62) 4/5/2011 17.16 17.22 0.06 (3.35) - - - - (3.35) (55.86) (5.25) (3.35) (8.60) 4/6/2011 17.44 18.07 0.63 (38.81) - - - - (38.81) (61.60) (12.29) (38.81) (51.09) 4/7/2011 18.16 17.91 (0.25) 18.75 - - - - 18.75 (75.02) (5.54) 18.75 13.21 4/8/2011 17.97 17.65 (0.32) 23.22 - - - - 23.22 (72.57) (4.50) 23.22 18.72 4/11/2011 17.71 17.47 (0.24) 16.30 - - - - 16.30 (67.94) (4.35) 16.30 11.95 13.50 - - - - 13.50 (31.94) 13.50 (18.44) 3) Portfolio Daily Changes Position Profit & Loss Date Delta Gamma Vega Theta Other Total Stock (Open) (Close) Total 4/4/2011-0.03 (2.98) (0.80) 0.13 (3.62) 100 (52.49) - (3.62) (3.62) 4/5/2011-0.04 7.51 (0.75) (0.15) 6.65 100 (55.86) (2.25) 6.65 4.40 4/6/2011-4.06 2.59 (0.82) (0.64) 5.19 100 (61.60) (0.29) 5.19 4.91 4/7/2011-0.51 3.78 (0.72) 0.19 3.75 100 (75.02) (0.54) 3.75 3.21 4/8/2011-0.94 3.70 (0.73) 0.32 4.22 100 (72.57) (6.50) 4.22 (2.28) 4/11/2011-0.59 3.42 (0.81) 0.11 3.30 100 (67.94) (5.35) 3.30 (2.05) - 6.17 18.02 (4.65) (0.05) 19.50 (14.94) 19.50 4.56 Delta is the most important risk element for my portfolio, and delta hedging as explained could significantly reduce the risk of holding an option portfolio. Delta hedge is particularly effective when trader wants to create a volatility trade with minimum directional exposure. Delta hedge is not as effective as a straddle for purpose of volatility trade; however, it neutralizes most of the risk due to direction of stock price. Delta hedging compared to a straddle is cheaper to implement in the real world especially for dynamic hedging purposes where the trader needs to rebalance her portfolio on daily or hourly basis.
10 Option Trading and Positioning Pooya Tavana Even though delta hedging is a good method for risk management, traders usually look at other measure such as gamma and vega. In the next section, I am going to further expand my hedging exercise by introducing ways to manage gamma and vega risks. Delta and Gamma Neutral Hedge As we discussed, delta hedging could easily be achieved by assuming a position in the underlying stock. However, we cannot use the same technique for hedging gamma due to the fact that stock does not have gamma and no matter how many stocks you own, a portfolio consisting of one type of option and stocks will not be gamma neutral. To hedge away the gamma of the option, you need an instrument which has a gamma that can offset your initial exposure. One instrument is another call or put option with the same maturity but with a different strike price. I used an OTM put option for the purpose of gamma hedging. Since I initially had 100 C17 CSCO options, I decided to short P15 CSCO to achieve a gamma neutral hedge. Formula (iii) shows the number of put options required to construct the gamma hedge. One more thing before I implement the strategy is the fact that by assuming position in the put options, I am breaking my delta hedge unless I alter my stock position based on the delta of the portfolio. Formula (iv) shows that the new delta of the portfolio is the sum of deltas of options multiplied by the number of options. (iii) (iv) It could be argued that gamma contribution to change in the option price is not significant compared to the other elements such as delta and vega for C17 CSCO, so any attempt to hedge gamma exposure will not be effective. Additionally, trading cost for shorting put options might be even higher than the value that gamma hedge will add to our portfolio. This argument could not be generalized to the gamma hedging in real world since gamma for other options might be significant. Moreover, if the initial position is big as it could be for some of the financial institutions then the benefit of gamma hedging will justify the cost of this strategy. Another important attribute of a delta-gamma neutral hedge is explained by Taylor series expansion. Looking back at the Taylor series we can see that the only two terms which depend
11 Option Trading and Positioning Pooya Tavana on stock price movement (ds) are delta and gamma terms. Therefore, by hedging both delta and gamma away, we are essentially making our portfolio free of any exposure to changes in stock price. In practice however, you cannot continuously hedge delta and gamma and as a result you will always be exposed to some degree of risk due to changes in stock price especially if movements are very big. Table 3 on the next page demonstrates the effect of delta-gamma neutral hedging strategy. As you can see in the second panel, position in CSCO P15 changes every morning based on the formula (iii) to ensure gamma neutrality. Additionally, compared to the delta hedge strategy we need to short more stock in a delta-gamma neutral strategy. This is due to the fact that P15 also has a delta and therefore, delta of portfolio will be sum of deltas. Comparing to table 2, you can see the number of shorted stocks in table 3 is higher. Finally, comparing the portfolio panel to C17 panel in Table 3, one could argue that the delta-gamma neutral strategy has a drawback. If you look carefully, it is obvious that vega of the portfolio increases as a result of introducing the new option. In the next section, I explain how to hedge our exposure to vega. Delta, Gamma, and Vega Neutral Hedge Vega of an option is the rate of change of the value of the option with respect to the volatility of the underlying asset. A position in underlying asset has zero vega, therefore, for hedging vega, we should add a position in another traded option. In other words, vega could be hedged in a similar way we hedged gamma in previous section. However, in this section we are interested in constructing a delta-gamma-vega neutral portfolio. Delta hedging is not a problem and could be achieved by buying (shorting) required amount of underlying stock. How about hedging gamma and vega?
12 Option Trading and Positioning Pooya Tavana 1) C17 CSCO Table 3- Profit and Loss for Delta and Gamma Neutral Hedge (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) (15) (16) (17) Option Premium Market Movements Daily Changes Position Profit & Loss Option Date Open Close ds dσ dt Delta Gamma Vega Theta Other Total (C17) (Open) (Close) Total 4/12/2011 0.950 0.97 (0.06) 3.10% 1 (3.82) 0.04 6.58 (0.85) 0.05 2.00 100.00 (2.00) 2.00-4/13/2011 0.940 0.81 (0.25) 1.50% 1 (15.95) 0.70 3.13 (0.86) (0.02) (13.00) 100.00 (3.00) (13.00) (16.00) 4/14/2011 0.770 0.75 0.02-1.01% 1 1.11 0.00 (2.15) (0.96) (0.00) (2.00) 100.00 (4.00) (2.00) (6.00) 4/15/2011 0.750 0.63 (0.16) -1.11% 1 (9.06) 0.31 (2.33) (0.93) 0.01 (12.00) 100.00 - (12.00) (12.00) 4/18/2011 0.580 0.48 (0.15) -0.99% 1 (7.34) 0.28 (1.98) (1.00) 0.03 (10.00) 100.00 (5.00) (10.00) (15.00) 4/19/2011 0.470 0.41 (0.04) -1.73% 1 (1.72) 0.02 (3.30) (0.99) 0.00 (6.00) 100.00 (1.00) (6.00) (7.00) 4/20/2011 0.520 0.54 0.03 0.79% 1 1.47 0.01 1.52 (0.95) (0.05) 2.00 100.00 11.00 2.00 13.00 4/21/2011 0.530 0.52 - -0.01% 1 - - (0.02) (0.97) (0.01) (1.00) 100.00 (1.00) (1.00) (2.00) (35.32) 1.37 1.44 (7.49) 0.01 (40.00) (5.00) (40.00) (45.00) 2) P15 CSCO Option Premium Market Movements Daily Changes Position Profit & Loss Option Date Open Close ds dσ dt Delta Gamma Vega Theta Other Total (P15) (Open) (Close) Total 4/12/2011 0.090 0.08 (0.06) -1.29% 1 (1.53) (0.04) 3.37 1.28 (0.19) 2.89 (289.02) - 2.89 2.89 4/13/2011 0.090 0.09 (0.25) -2.06% 1 (6.48) (0.70) 5.40 1.33 0.45 - (293.17) (2.89) - (2.89) 4/14/2011 0.090 0.09 0.02 0.71% 1 0.48 (0.00) (1.61) 1.09 0.04 - (244.50) - - - 4/15/2011 0.090 0.09 (0.16) -1.24% 1 (4.10) (0.31) 3.00 1.22 0.19 - (265.08) - - - 4/18/2011 0.080 0.11 (0.15) 1.79% 1 (3.49) (0.28) (3.66) 1.03 (0.54) (6.94) (231.20) 2.65 (6.94) (4.29) 4/19/2011 0.110 0.13 (0.04) 1.98% 1 (1.00) (0.02) (3.91) 1.06 0.02 (3.85) (192.69) - (3.85) (3.85) 4/20/2011 0.100 0.09 0.03-0.14% 1 0.87 (0.01) 0.34 1.42 (0.05) 2.56 (255.99) 5.78 2.56 8.34 4/21/2011 0.080 0.08-0.64% 1 - - (1.46) 1.35 0.11 - (274.61) 2.56-2.56 (15.26) (1.37) 1.47 9.79 0.04 (5.34) 8.10 (5.34) 2.76 3) CSCO Stock Price Market Movements Daily Changes Position Profit & Loss Date Open Close ds Delta Gamma Vega Theta Other Total Stock (Open) (Close) Total 4/12/2011 17.50 17.44 (0.06) 5.35 - - - - 5.35 (89.20) (2.04) 5.35 3.31 4/13/2011 17.50 17.25 (0.25) 22.43 - - - - 22.43 (89.72) (5.35) 22.43 17.08 4/14/2011 17.15 17.17 0.02 (1.59) - - - - (1.59) (79.49) 8.97 (1.59) 7.38 4/15/2011 17.19 17.03 (0.16) 13.16 - - - - 13.16 (82.24) (1.59) 13.16 11.57 4/18/2011 16.88 16.73 (0.15) 10.84 - - - - 10.84 (72.23) 12.34 10.84 23.17 4/19/2011 16.65 16.61 (0.04) 2.73 - - - - 2.73 (68.15) 5.78 2.73 8.50 4/20/2011 16.90 16.93 0.03 (2.33) - - - - (2.33) (77.79) (19.76) (2.33) (22.10) 4/21/2011 16.94 16.94 - - - - - - - (76.85) (0.78) - (0.78) 50.58 - - - - 50.58 (2.44) 50.58 48.14 4) Portfolio Daily Changes Position Profit & Loss Option Option Date Delta Gamma Vega Theta Other Total (C17) (P15) Stock (Open) (Close) Total 4/12/2011 - - 9.95 0.43 (0.14) 10.24 100.00 (289.02) (89.20) (4.04) 10.24 6.20 4/13/2011 - - 8.52 0.47 0.43 9.43 100.00 (293.17) (89.72) (11.24) 9.43 (1.81) 4/14/2011 - - (3.76) 0.13 0.04 (3.59) 100.00 (244.50) (79.49) 4.97 (3.59) 1.38 4/15/2011 - - 0.67 0.29 0.19 1.16 100.00 (265.08) (82.24) (1.59) 1.16 (0.43) 4/18/2011 - - (5.64) 0.04 (0.50) (6.10) 100.00 (231.20) (72.23) 9.99 (6.10) 3.89 4/19/2011 - - (7.22) 0.07 0.02 (7.13) 100.00 (192.69) (68.15) 4.78 (7.13) (2.35) 4/20/2011 - - 1.85 0.47 (0.10) 2.23 100.00 (255.99) (77.79) (2.98) 2.23 (0.76) 4/21/2011 - - (1.48) 0.39 0.09 (1.00) 100.00 (274.61) (76.85) 0.78 (1.00) (0.22) - - 2.90 2.29 0.04 5.24 0.67 5.24 5.90
13 Option Trading and Positioning Pooya Tavana For hedging gamma and vega, we need to add two more options to our initial position. I decided to use P15 and C19 CSCO for the same maturity. On 4/25/2011, we have the following information: Option No. of Options Gamma Vega C17 CSCO 100 0.284 1.768 C19 CSCO N C19 0.116 0.737 P15 CSCO N P15 0.103 0.724 To make the portfolio gamma and vega neutral, we need to solve for number of C19 and P15 options required. Gamma Neutrality: (0.284) (100) + (0.116) N C19 + (0.103) N P15 = 0 Vega Neutrality: (1.768) (100) + (0.737) N C19 + (0.724) N P15 = 0 N C19 = -208.45, N P15 = 41.42 If we buy 41.42 of P15 CSCO and short 208.45 of C19 CSCO, our portfolio will be gamma and vega neutral. Now that we know the weights for new options in the portfolio, we can hedge the delta of the portfolio by adding position in the underlying stock and make our portfolio delta, gamma, and vega neutral. I repeated this exercise on a daily basis based on the open prices from 4/25/2011 to 5/6/2011. The results are summarized in table 4. Table 4 has a panel for each position and at the end a panel for the portfolio. Looking at the last panel under daily changes columns, it is obvious that our portfolio is delta and gamma neutral; however, it does not seem like vega neutral. Why? The hedging method we used for hedging vega exposure implicitly assume that volatility is going to be the same for different options during the day. However, this is not the case in the real life. Even options on the same underlying and same maturity have different implied volatilities depending on their intrinsic value (e.g ATM, OTM, or ITM). There are other limitations that my model could not capture. For example my model back out the implied volatility based on the market data assuming that market data are always in sync with all the data, in other words, it assumes that efficient market hypothesis holds. This however, might not happen in the market. Specially in the case of a single company like CSCO, C19 and P15 options which are out of the money could have different volatility compared to C17 ( our
14 Option Trading and Positioning Pooya Tavana 1) C17 CSCO Table 4- Profit and Loss for Delta-Gamma-Vega Neutral Hedge Using P15 & C19 (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (15) (16) (17) (18) Option Premium Market Movements Daily Changes Position Profit & Loss Option Date Open Close ds dσ dt Delta Gamma Vega Theta Other Total (C17) (Open) (Close) Total 4/25/2011 0.53 0.63 0.17 1.31% 1 8.46 0.41 2.31 (1.13) (0.05) 10.00 100.00 1.00 10.00 11.00 4/26/2011 0.65 0.87 0.31 2.31% 1 17.92 1.39 3.99 (1.11) (0.19) 22.00 100.00 2.00 22.00 24.00 4/27/2011 0.95 0.69 (0.41) 0.52% 1 (27.52) 1.98 0.83 (1.21) (0.08) (26.00) 100.00 8.00 (26.00) (18.00) 4/28/2011 0.69 0.77 0.09 2.44% 1 5.15 0.11 4.04 (1.32) 0.03 8.00 100.00-8.00 8.00 4/29/2011 0.75 0.87 0.20 0.46% 1 12.09 0.53 0.74 (1.34) (0.03) 12.00 100.00 (2.00) 12.00 10.00 5/2/2011 0.89 0.90 0.07-1.35% 1 4.55 0.06 (1.95) (1.58) (0.08) 1.00 100.00 2.00 1.00 3.00 5/3/2011 0.86 0.79 (0.12) 1.72% 1 (7.96) 0.19 2.38 (1.52) (0.07) (7.00) 100.00 (4.00) (7.00) (11.00) 5/4/2011 0.77 0.81 0.10-0.52% 1 6.19 0.13 (0.71) (1.71) 0.10 4.00 100.00 (2.00) 4.00 2.00 5/5/2011 0.81 0.82-2.12% 1 - - 2.78 (1.68) (0.10) 1.00 100.00-1.00 1.00 5/6/2011 0.90 0.86 (0.07) 2.21% 1 (4.88) 0.06 2.66 (1.68) (0.16) (4.00) 100.00 8.00 (4.00) 4.00 13.99 4.87 17.06 (14.29) (0.63) 21.00 13.00 21.00 34.00 2) P15 CSCO Option Premium Market Movements Daily Changes Position Profit & Loss Option Date Open Close ds dσ dt Delta Gamma Vega Theta Other Total (P15) (Open) (Close) Total 4/25/2011 0.07 0.07 0.17 2.92% 1 (0.64) 0.06 0.87 (0.21) (0.08) - 41.42 2.75-2.75 4/26/2011 0.05 0.03 0.31 0.19% 1 (3.93) 0.72 0.21 (0.83) 0.04 (3.79) 189.45 (0.83) (3.79) (4.62) 4/27/2011 0.04 0.05 (0.41) -1.77% 1 (2.32) (0.55) 0.91 0.45 0.40 (1.10) (110.45) 1.89 (1.10) 0.79 4/28/2011 0.05 0.05 0.09 1.97% 1 (1.84) 0.10 3.31 (1.47) (0.10) - 304.68 - - - 4/29/2011 0.05 0.05 0.20 3.36% 1 (0.59) 0.07 0.81 (0.24) (0.05) - 45.54 - - - 5/2/2011 0.04 0.04 0.07 2.32% 1 (0.03) 0.00 0.07 (0.04) (0.01) - 7.43 (0.46) - (0.46) 5/3/2011 0.03 0.04 (0.12) 2.62% 1 (0.06) (0.00) (0.10) 0.05 0.00 (0.11) (11.03) (0.07) (0.11) (0.18) 5/4/2011 0.04 0.04 0.10 2.73% 1 (0.22) 0.01 0.45 (0.23) (0.01) - 40.63 - - - 5/5/2011 0.04 0.03 - -1.44% 1 - - (0.21) (0.23) 0.07 (0.37) 37.16 - (0.37) (0.37) 5/6/2011 0.03 0.03 (0.07) 0.55% 1 (0.14) (0.01) (0.08) 0.25 (0.03) - (45.18) - - - (9.76) 0.41 6.25 (2.51) 0.23 (5.38) 3.28 (5.38) (2.09) 3) C19 CSCO Option Premium Market Movements Daily Changes Position Profit & Loss Option Date Open Close ds dσ dt Delta Gamma Vega Theta Other Total (C19) (Open) (Close) Total 4/25/2011 0.06 0.09 0.17 2.07% 1 (4.43) (0.47) (4.28) 1.34 (0.57) (8.41) (280.45) - (8.41) (8.41) 4/26/2011 0.09 0.14 0.31 0.91% 1 (12.10) (2.11) (2.56) 1.93 (0.30) (15.15) (303.05) - (15.15) (15.15) 4/27/2011 0.14 0.11 (0.41) 4.28% 1 6.96 (1.44) (4.64) 0.77 1.10 2.75 (91.81) - 2.75 2.75 4/28/2011 0.11 0.11 0.09-0.54% 1 (4.51) (0.21) 1.81 2.78 0.13 - (347.71) - - - 4/29/2011 0.11 0.14 0.20 0.55% 1 (5.69) (0.60) (1.01) 1.57 0.02 (5.71) (190.50) - (5.71) (5.71) 5/2/2011 0.15 0.14 0.07-1.13% 1 (1.84) (0.06) 1.66 1.61 0.04 1.41 (141.15) (1.90) 1.41 (0.49) 5/3/2011 0.12 0.11 (0.12) 2.07% 1 2.84 (0.18) (2.77) 1.47 0.06 1.41 (141.43) 2.82 1.41 4.24 5/4/2011 0.11 0.12 0.10 0.63% 1 (2.75) (0.15) (0.98) 1.94 0.12 (1.82) (181.95) - (1.82) (1.82) 5/5/2011 0.11 0.12-2.61% 1 - - (3.78) 1.90 0.18 (1.71) (170.69) 1.82 (1.71) 0.11 5/6/2011 0.11 0.11 (0.07) 2.79% 1 1.43 (0.06) (2.96) 1.43 0.15 - (124.04) 1.71-1.71 (20.09) (5.28) (19.52) 16.73 0.93 (27.23) 4.44 (27.23) (22.78) 4) CSCO Stock Price Market Movements Daily Changes Position Profit & Loss Date Open Close ds Delta Gamma Vega Theta Other Total Stock (Open) (Close) Total 4/25/2011 16.9 17.1 0.17 (3.39) - - - - (3.39) (19.96) 0.77 (3.39) (2.62) 4/26/2011 17.2 17.5 0.31 (1.89) - - - - (1.89) (6.11) (2.20) (1.89) (4.09) 4/27/2011 17.6 17.2 (0.41) 22.87 - - - - 22.87 (55.79) (0.49) 22.87 22.39 4/28/2011 17.2 17.3 0.09 1.20 1.20 13.39 (0.56) 1.20 0.65 4/29/2011 17.3 17.5 0.20 (5.81) (5.81) (29.04) 0.40 (5.81) (5.41) 5/2/2011 17.5 17.6 0.07 (2.68) - - - - (2.68) (38.26) 0.29 (2.68) (2.39) 5/3/2011 17.5 17.4 (0.12) 5.18 - - - - 5.18 (43.19) 1.91 5.18 7.10 5/4/2011 17.4 17.5 0.10 (3.22) - - - - (3.22) (32.19) 1.73 (3.22) (1.49) 5/5/2011 17.5 17.5 - - - - - - - (36.77) (0.32) - (0.32) 5/6/2011 17.6 17.6 (0.07) 3.59 - - - - 3.59 (51.27) (5.52) 3.59 (1.93) 50.58 - - - - 15.86 (2.44) 50.58 11.88 5) Portfolio Daily Changes Position Profit & Loss Option Option Option Date Delta Gamma Vega Theta Other Total (C17) (P15) (C19) Stock (Open) (Close) Total 4/25/2011 - - (1.10) (0.01) (0.70) (1.81) 100.00 41.42 (280.45) (19.96) 4.51 (1.81) 2.71 4/26/2011 - - 1.63 (0.02) (0.45) 1.17 100.00 189.45 (303.05) (6.11) (1.02) 1.17 0.14 4/27/2011 - - (2.90) 0.00 1.42 (1.48) 100.00 (110.45) (91.81) (55.79) 9.41 (1.48) 7.93 4/28/2011-0.00 9.16 (0.01) 0.05 9.20 100.00 304.68 (347.71) 13.39 (0.56) 9.20 8.65 4/29/2011 - - 0.54 (0.01) (0.05) 0.48 100.00 45.54 (190.50) (29.04) (1.60) 0.48 (1.12) 5/2/2011-0.00 (0.21) (0.00) (0.05) (0.27) 100.00 7.43 (141.15) (38.26) (0.07) (0.27) (0.34) 5/3/2011 - - (0.50) (0.01) (0.01) (0.51) 100.00 (11.03) (141.43) (43.19) 0.66 (0.51) 0.15 5/4/2011-0.00 (1.24) (0.01) 0.21 (1.04) 100.00 40.63 (181.95) (32.19) (0.27) (1.04) (1.31) 5/5/2011 - - (1.22) (0.01) 0.15 (1.08) 100.00 37.16 (170.69) (36.77) 1.50 (1.08) 0.42 5/6/2011 - - (0.38) 0.00 (0.04) (0.41) 100.00 (45.18) (124.04) (51.27) 4.19 (0.41) 3.78-0.00 3.79 (0.07) 0.54 4.26 16.75 4.26 21.01 initial position), therefore, it is not easy to hedge vega of C17 using the other two options.
15 Option Trading and Positioning Pooya Tavana Another complication here is regarding the trading volumes of the OTM options for CSCO, I have not looked at these data, but it is safe to assume that CSCO OTM options maturing in May 2011 are not liquid enough to capture all the market information. This could result in wrong implied volatilities from my model. To address some of the issues I mentioned regarding hedging vega exposure, I will redo the exercise on the same timeframe by using C18 and P16 instead of C19 and P15, respectively. I am hoping these options which are closer to ATM option (C17) will have a closer implied volatility to that of C17. Moreover, these options are more liquid compared to previous options, which means that the market prices are closer to the efficient market equilibrium prices. If this is true my overall portfolio s vega should be lower than the vega of portfolio with C19 and P15. Table 5 demonstrates the new hedging strategy. Looking at the last panel, we see that vega of the portfolio has changed compared to the earlier portfolio (Table4). These data are obviously not enough to test any hypothesis regarding the effectiveness of the new strategy compared to the earlier strategy. However, you can see that daily vegas are less dispersed when we used C18 and P16 instead of C19 and P15. In order to improve volatility hedge, one should develop a model to include the implied volatility surface and update the model to account for stochastic volatilities. Such models are superior to the Black-Scholes and Binomial models which assume the volatility is constant. VI. Conclusion This report demonstrated the simple strategies that could be used for hedging exposure to delta, gamma and vega. I used the Binomial model and the Black-Scholes formula in my model to drive option premiums and the Greeks. Here is the list of main findings: Delta hedging could be easily achieved by taking a position in the underlying asset. This method is useful; however, to maintain the neutrality, position in the underlying asset should be regularly changed to reflect the current delta of the option. Additionally, this method is very accurate if changes in stock price are small. For larger changes however, delta hedging loses its accuracy. Based on this characteristic, delta hedge could be regarded as a volatility trade too.
16 Option Trading and Positioning Pooya Tavana Table 5- Profit and Loss for Delta-Gamma-Vega Neutral Hedge Using P16 & C18 1) C17 CSCO (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (15) (16) (17) (18) Option Premium Market Movements Daily Changes Position Profit & Loss Option Date Open Close ds dσ dt Delta Gamma Vega Theta Other Total (C17) (Open) (Close) Total 4/25/2011 0.53 0.63 0.17 1.31% 1 8.46 0.41 2.31 (1.13) (0.05) 10.00 100.00 1.00 10.00 11.00 4/26/2011 0.65 0.87 0.31 2.31% 1 17.92 1.39 3.99 (1.11) (0.19) 22.00 100.00 2.00 22.00 24.00 4/27/2011 0.95 0.69 (0.41) 0.52% 1 (27.52) 1.98 0.83 (1.21) (0.08) (26.00) 100.00 8.00 (26.00) (18.00) 4/28/2011 0.69 0.77 0.09 2.44% 1 5.15 0.11 4.04 (1.32) 0.03 8.00 100.00-8.00 8.00 4/29/2011 0.75 0.87 0.20 0.46% 1 12.09 0.53 0.74 (1.34) (0.03) 12.00 100.00 (2.00) 12.00 10.00 5/2/2011 0.89 0.90 0.07-1.35% 1 4.55 0.06 (1.95) (1.58) (0.08) 1.00 100.00 2.00 1.00 3.00 5/3/2011 0.86 0.79 (0.12) 1.72% 1 (7.96) 0.19 2.38 (1.52) (0.07) (7.00) 100.00 (4.00) (7.00) (11.00) 5/4/2011 0.77 0.81 0.10-0.52% 1 6.19 0.13 (0.71) (1.71) 0.10 4.00 100.00 (2.00) 4.00 2.00 5/5/2011 0.81 0.82-2.12% 1 - - 2.78 (1.68) (0.10) 1.00 100.00-1.00 1.00 5/6/2011 0.90 0.86 (0.07) 2.21% 1 (4.88) 0.06 2.66 (1.68) (0.16) (4.00) 100.00 8.00 (4.00) 4.00 13.99 4.87 17.06 (14.29) (0.63) 21.00 13.00 21.00 34.00 2) P16 CSCO Option Premium Market Movements Daily Changes Position Profit & Loss Option Date Open Close ds dσ dt Delta Gamma Vega Theta Other Total (P15) (Open) (Close) Total 4/25/2011 0.22 0.17 0.17-0.26% 1 2.54 (0.19) 0.22 0.56 (0.05) 3.07 (61.45) - 3.07 3.07 4/26/2011 0.15 0.11 0.31 1.48% 1 (1.27) 0.20 0.39 (0.18) (0.03) (0.90) 22.50 1.23 (0.90) 0.33 4/27/2011 0.11 0.17 (0.41) 0.17% 1 (5.24) (1.10) (0.15) 0.71 0.03 (5.75) (95.91) - (5.75) (5.75) 4/28/2011 0.17 0.14 0.09-0.39% 1 1.16 (0.05) 0.30 0.63 (0.05) 1.99 (66.44) - 1.99 1.99 4/29/2011 0.13 0.12 0.20 2.96% 1 4.20 (0.43) (3.91) 1.10 0.33 1.29 (128.99) 0.66 1.29 1.95 5/2/2011 0.12 0.12 0.07 2.31% 1 1.31 (0.05) (2.64) 1.27 0.11 - (129.44) - - - 5/3/2011 0.11 0.13 (0.12) 1.56% 1 1.24 0.08 0.97 (0.74) (0.04) 1.51 75.34 1.29 1.51 2.80 5/4/2011 0.13 0.12 0.10 2.11% 1 0.78 (0.04) (0.91) 0.55 0.11 0.49 (49.04) - 0.49 0.49 5/5/2011 0.12 0.10 - -1.26% 1 - - 0.00 0.00 (0.00) 0.00 (0.13) - 0.00 0.00 5/6/2011 0.09 0.10 (0.07) 1.72% 1 (0.33) (0.01) (0.47) 0.43 (0.01) (0.40) (40.17) 0.00 (0.40) (0.40) 4.37 (1.60) (6.20) 4.34 0.40 1.30 3.19 1.30 4.49 3) C18 CSCO Option Premium Market Movements Daily Changes Position Profit & Loss Option Date Open Close ds dσ dt Delta Gamma Vega Theta Other Total (C18) (Open) (Close) Total 4/25/2011 0.18 0.24 0.17 1.76% 1 (2.69) (0.22) (1.61) 0.56 (0.08) (4.04) (67.27) - (4.04) (4.04) 4/26/2011 0.25 0.38 0.31 2.05% 1 (12.05) (1.59) (4.08) 1.29 (0.42) (16.85) (129.58) (0.67) (16.85) (17.52) 4/27/2011 0.42 0.30 (0.41) 2.53% 1 6.74 (0.88) (1.74) 0.50 0.17 4.79 (39.92) (5.18) 4.79 (0.39) 4/28/2011 0.27 0.32 0.09 2.15% 1 (1.65) (0.06) (1.91) 0.69 (0.05) (2.99) (59.74) 1.20 (2.99) (1.79) 4/29/2011 0.30 0.38 0.20 1.35% 1 (1.25) (0.10) (0.38) 0.23 0.00 (1.49) (18.65) 1.19 (1.49) (0.30) 5/2/2011 0.36 0.39 0.07 1.30% 1 (0.53) (0.01) (0.39) 0.30 0.03 (0.60) (19.98) 0.37 (0.60) (0.23) 5/3/2011 0.38 0.32 (0.12) 0.12% 1 6.47 (0.26) (0.25) 2.26 0.07 8.28 (138.07) 0.20 8.28 8.48 5/4/2011 0.31 0.32 0.10-0.70% 1 (2.45) (0.09) 0.67 1.16 0.01 (0.71) (71.17) 1.38 (0.71) 0.67 5/5/2011 0.33 0.33-1.35% 1 - - (1.76) 1.67 0.09 - (97.70) (0.71) - (0.71) 5/6/2011 0.36 0.35 (0.07) 2.69% 1 1.97 (0.05) (2.50) 1.26 0.02 0.70 (69.53) (2.93) 0.70 (2.24) (5.46) (3.27) (13.95) 9.93 (0.15) (12.90) (5.15) (12.90) (18.05) 4) CSCO Stock Price Market Movements Daily Changes Position Profit & Loss Date Open Close ds Delta Gamma Vega Theta Other Total Stock (Open) (Close) Total 4/25/2011 16.9 17.1 0.17 (8.30) - - - - (8.30) (48.85) 0.84 (8.30) (7.47) 4/26/2011 17.2 17.5 0.31 (4.59) - - - - (4.59) (14.82) (5.37) (4.59) (9.97) 4/27/2011 17.6 17.2 (0.41) 26.02 - - - - 26.02 (63.46) (1.19) 26.02 24.83 4/28/2011 17.2 17.3 0.09 (4.65) (4.65) (51.68) (0.63) (4.65) (5.29) 4/29/2011 17.3 17.5 0.20 (15.04) (15.04) (75.22) (1.55) (15.04) (16.59) 5/2/2011 17.5 17.6 0.07 (5.32) - - - - (5.32) (76.01) 0.75 (5.32) (4.57) 5/3/2011 17.5 17.4 (0.12) 0.26 - - - - 0.26 (2.18) 3.80 0.26 4.06 5/4/2011 17.4 17.5 0.10 (4.52) - - - - (4.52) (45.19) 0.09 (4.52) (4.43) 5/5/2011 17.5 17.5 - - - - - - - (29.52) (0.45) - (0.45) 5/6/2011 17.6 17.6 (0.07) 3.25 - - - - 3.25 (46.42) (4.43) 3.25 (1.18) 50.58 - - - - (12.90) (2.44) 50.58 (21.05) 5) Portfolio Daily Changes Position Profit & Loss Option Option Option Date Delta Gamma Vega Theta Other Total (C17) (P15) (C18) Stock (Open) (Close) Total 4/25/2011-0.00 0.91 (0.00) (0.18) 0.73 100.00 (61.45) (67.27) (48.85) 1.84 0.73 2.57 4/26/2011-0.00 0.30 (0.00) (0.64) (0.34) 100.00 22.50 (129.58) (14.82) (2.82) (0.34) (3.16) 4/27/2011 - (0.00) (1.06) (0.01) 0.12 (0.95) 100.00 (95.91) (39.92) (63.46) 1.63 (0.95) 0.69 4/28/2011-0.00 2.44 (0.01) (0.08) 2.35 100.00 (66.44) (59.74) (51.68) 0.56 2.35 2.92 4/29/2011 - (0.00) (3.55) (0.01) 0.31 (3.25) 100.00 (128.99) (18.65) (75.22) (1.69) (3.25) (4.94) 5/2/2011 - (0.00) (4.97) (0.01) 0.06 (4.92) 100.00 (129.44) (19.98) (76.01) 3.13 (4.92) (1.79) 5/3/2011 - - 3.10 (0.00) (0.04) 3.05 100.00 75.34 (138.07) (2.18) 1.29 3.05 4.35 5/4/2011-0.00 (0.96) (0.00) 0.22 (0.74) 100.00 (49.04) (71.17) (45.19) (0.53) (0.74) (1.27) 5/5/2011 - - 1.02 (0.00) (0.01) 1.00 100.00 (0.13) (97.70) (29.52) (1.16) 1.00 (0.16) 5/6/2011 - - (0.32) 0.01 (0.15) (0.46) 100.00 (40.17) (69.53) (46.42) 0.64 (0.46) 0.19 - (0.00) (3.09) (0.03) (0.38) (3.51) 2.89 (3.51) (0.62) Gamma is an important hedging element that could reduce our exposure to changes
17 Option Trading and Positioning Pooya Tavana in the asset price. Particularly, if it is hedged along with delta. Theoretically, a deltagamma neutral portfolio is free of exposure to changes in the underlying asset price. In practice however, we know that is not true because the Black-Scholes mathematical formula has limitations. Moreover, there are terms with higher orders in the Taylor series that we assumed are negligible, which might not be the case in real trading practice. For hedging vega and gamma in the same time, we need to add two options to the portfolio. As I have shown, this method is easy to construct, but it does not capture the fact that implied volatility for options with different strike prices are different and they could even move in opposite directions. This phenomenon undermines the effectiveness of our approach. After going through this exercise, it is obvious that risk management is a complicated practice especially when there are options in the portfolio. A wise trader, risk manager, or treasurer should completely understand the options pricing mathematics and be able to measure Greek letters in order to successfully implement a hedging strategy.
18 Option Trading and Positioning Pooya Tavana VII. Appendix 1- Binomial Option Pricing The Binomial method could be easily implemented by using VBA in Excel. In this appendix, I demonstrate the necessary code to implement binomial option pricing model, as well as approximations for Greek letters. Please note that this code has been developed in MS Excel 2007 and might need some alteration if used in other versions of Excel. Dim s(), E_C(), E_P(), A_C(), A_P() As Variant ------------------------------------------------------------------ Function Option_pricer1(S0 As Double, E As Double, sigma As Double, rf As Double, Dev As Double, T As Double, n As Double, C_P As String, A_E As String) Dim delta_t As Double: delta_t = T / n Dim u As Double: u = Exp(sigma * (delta_t ^ 0.5)) Dim d As Double: d = 1 / u Dim p As Double: p = (Exp((rf - Dev) * delta_t) - d) / (u - d) Dim value() As Double ReDim value(1 To 5) Call Asset_Price(S0, u, d, n) If C_P = "C" And A_E = "E" Then Call European_Call(E, p, rf, delta_t, n) value(1) = E_C(0, 0) value(2) = (E_C(1, 0) - E_C(1, 1)) / (s(1, 0) - s(1, 1)) value(3) = (((E_C(2, 2) - E_C(2, 1)) / (s(2, 2) - s(2, 1))) - ((E_C(2, 1) - E_C(2, 0)) / (s(2, 1) - s(2, 0)))) / ((s(2, 2) - s(2, 0)) * 0.5) value(4) = (E_C(2, 1) - E_C(0, 0)) / (2 * delta_t) / 365 ElseIf C_P = "C" And A_E = "A" Then Call American_Call(E, p, rf, delta_t, n) value(1) = A_C(0, 0) value(2) = (A_C(1, 0) - A_C(1, 1)) / (s(1, 0) - s(1, 1)) value(3) = (((A_C(2, 2) - A_C(2, 1)) / (s(2, 2) - s(2, 1))) - ((A_C(2, 1) - A_C(2, 0)) / (s(2, 1) - s(2, 0)))) / ((s(2, 2) - s(2, 0)) * 0.5) value(4) = (A_C(2, 1) - A_C(0, 0)) / (2 * delta_t) / 365 ElseIf C_P = "P" And A_E = "E" Then Call European_Put(E, p, rf, delta_t, n) value(1) = E_P(0, 0) value(2) = (E_P(1, 0) - E_P(1, 1)) / (s(1, 0) - s(1, 1)) value(3) = (((E_P(2, 2) - E_P(2, 1)) / (s(2, 2) - s(2, 1))) - ((E_P(2, 1) - E_P(2, 0)) / (s(2, 1) - s(2, 0)))) / ((s(2, 2) - s(2, 0)) * 0.5) value(4) = (E_P(2, 1) - E_P(0, 0)) / (2 * delta_t) / 365 ElseIf C_P = "P" And A_E = "A" Then Call American_Put(E, p, rf, delta_t, n)
19 Option Trading and Positioning Pooya Tavana value(1) = A_P(0, 0) value(2) = (A_P(1, 0) - A_P(1, 1)) / (s(1, 0) - s(1, 1)) value(3) = (((A_P(2, 2) - A_P(2, 1)) / (s(2, 2) - s(2, 1))) - ((A_P(2, 1) - A_P(2, 0)) / (s(2, 1) - s(2, 0)))) / ((s(2, 2) - s(2, 0)) * 0.5) End If value(4) = (A_P(2, 1) - A_P(0, 0)) / (2 * delta_t) / 365 Option_pricer1 = value End Function ------------------------------------------------------------------ Sub Asset_Price(S0 As Double, u As Double, d As Double, n As Double) ReDim s(0 To n, 0 To n) s(0, 0) = S0 For i = 1 To n Next i End Sub For j = 0 To i Next j s(i, j) = S0 * u ^ j * d ^ (i - j) ------------------------------------------------------------------ Sub European_Call(E As Double, p As Double, rf As Double, delta_t As Double, n As Double) ReDim E_C(0 To n, 0 To n) For j = 0 To n Next j E_C(n, j) = Application.WorksheetFunction.Max(0, s(n, j) - E) For i = n - 1 To 0 Step -1 Next i End Sub For j = 0 To i Next j E_C(i, j) = (p * E_C(i + 1, j + 1) + (1 - p) * E_C(i + 1, j)) * Exp(-rf * delta_t) ------------------------------------------------------------------ Sub American_Call(E As Double, p As Double, rf As Double, delta_t As Double, n As Double) ReDim A_C(0 To n, 0 To n) For j = 0 To n Next j A_C(n, j) = Application.WorksheetFunction.Max(0, s(n, j) - E) For i = n - 1 To 0 Step -1 Next i For j = 0 To i Next j a1 = (p * A_C(i + 1, j + 1) + (1 - p) * A_C(i + 1, j)) * Exp(-rf * delta_t) a2 = Application.WorksheetFunction.Max(0, s(i, j) - E) A_C(i, j) = Application.WorksheetFunction.Max(a1, a2)
20 Option Trading and Positioning Pooya Tavana End Sub ------------------------------------------------------------------ Sub European_Put(E As Double, p As Double, rf As Double, delta_t As Double, n As Double) ReDim E_P(0 To n, 0 To n) For j = 0 To n E_P(n, j) = Application.WorksheetFunction.Max(0, -s(n, j) + E) Next j For i = n - 1 To 0 Step -1 For j = 0 To i E_P(i, j) = (p * E_P(i + 1, j + 1) + (1 - p) * E_P(i + 1, j)) * Exp(-rf * delta_t) Next j Next i End Sub ------------------------------------------------------------------ Sub American_Put(E As Double, p As Double, rf As Double, delta_t As Double, n As Double) ReDim A_P(0 To n, 0 To n) For j = 0 To n A_P(n, j) = Application.WorksheetFunction.Max(0, -s(n, j) + E) Next j For i = n - 1 To 0 Step -1 For j = 0 To i a1 = (p * A_P(i + 1, j + 1) + (1 - p) * A_P(i + 1, j)) * Exp(-rf * delta_t) a2 = Application.WorksheetFunction.Max(0, -s(i, j) + E) A_P(i, j) = Application.WorksheetFunction.Max(a1, a2) Next j Next i End Sub
21 Option Trading and Positioning Pooya Tavana VIII. Appendix 2- Numerical Approximation for Greek Letters For numeric approximation of Greek letters, I used the Binomial method. I only approximated delta and gamma and theta using this method. However, this model method could be used for approximating other Greeks too. Please refer to Hull & Basu Chapter 19 for detailed discussion. Please note that refers to the value of the option in node with n upticks in step m. Additionally, refers to the stock price in node with n upticks in step m.