Option Trading and Positioning Professor Bodurtha
|
|
- Ashley Hancock
- 6 years ago
- Views:
Transcription
1 1 Option Trading and Positioning Pooya Tavana Option Trading and Positioning Professor Bodurtha 5/7/2011 Pooya Tavana
2 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 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 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 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 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, 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 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/ % 29.21% (0.008) % (2.98) (0.80) 0.13 (1.00) 4/5/ % 31.33% (0.008) % (0.75) (0.15) /6/ % 31.98% (0.008) % (0.82) (0.64) /7/ % 32.97% (0.007) (0.25) 1.91% 1 (18.75) (0.72) 0.19 (15.00) 4/8/ % 32.61% (0.007) (0.32) 2.63% 1 (23.22) (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 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 , therefore, delta of the portfolio is ( * 100 options). In order to hedge this exposure away, I short 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 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 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/ % (2.98) (0.80) 0.13 (1.00) (1.00) (1.00) 4/5/ % (0.75) (0.15) /6/ % (0.82) (0.64) /7/ (0.25) 1.91% 1 (18.75) (0.72) 0.19 (15.00) (15.00) (10.00) 4/8/ (0.32) 1.82% 1 (23.22) (0.73) 0.32 (19.00) (2.00) (19.00) (21.00) 4/11/ (0.24) 1.65% 1 (16.30) (0.81) 0.11 (13.00) (1.00) (13.00) (14.00) (13.50) (4.65) (0.05) ) 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/ (2.62) (2.62) (52.49) - (2.62) (2.62) 4/5/ (3.35) (3.35) (55.86) (5.25) (3.35) (8.60) 4/6/ (38.81) (38.81) (61.60) (12.29) (38.81) (51.09) 4/7/ (0.25) (75.02) (5.54) /8/ (0.32) (72.57) (4.50) /11/ (0.24) (67.94) (4.35) (31.94) (18.44) 3) Portfolio Daily Changes Position Profit & Loss Date Delta Gamma Vega Theta Other Total Stock (Open) (Close) Total 4/4/ (2.98) (0.80) 0.13 (3.62) 100 (52.49) - (3.62) (3.62) 4/5/ (0.75) (0.15) (55.86) (2.25) /6/ (0.82) (0.64) (61.60) (0.29) /7/ (0.72) (75.02) (0.54) /8/ (0.73) (72.57) (6.50) 4.22 (2.28) 4/11/ (0.81) (67.94) (5.35) 3.30 (2.05) (4.65) (0.05) (14.94) 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 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 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 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/ (0.06) 3.10% 1 (3.82) (0.85) (2.00) /13/ (0.25) 1.50% 1 (15.95) (0.86) (0.02) (13.00) (3.00) (13.00) (16.00) 4/14/ % (2.15) (0.96) (0.00) (2.00) (4.00) (2.00) (6.00) 4/15/ (0.16) -1.11% 1 (9.06) 0.31 (2.33) (0.93) 0.01 (12.00) (12.00) (12.00) 4/18/ (0.15) -0.99% 1 (7.34) 0.28 (1.98) (1.00) 0.03 (10.00) (5.00) (10.00) (15.00) 4/19/ (0.04) -1.73% 1 (1.72) 0.02 (3.30) (0.99) 0.00 (6.00) (1.00) (6.00) (7.00) 4/20/ % (0.95) (0.05) /21/ % (0.02) (0.97) (0.01) (1.00) (1.00) (1.00) (2.00) (35.32) (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/ (0.06) -1.29% 1 (1.53) (0.04) (0.19) 2.89 (289.02) /13/ (0.25) -2.06% 1 (6.48) (0.70) (293.17) (2.89) - (2.89) 4/14/ % (0.00) (1.61) (244.50) /15/ (0.16) -1.24% 1 (4.10) (0.31) (265.08) /18/ (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/ (0.04) 1.98% 1 (1.00) (0.02) (3.91) (3.85) (192.69) - (3.85) (3.85) 4/20/ % (0.01) (0.05) 2.56 (255.99) /21/ % (1.46) (274.61) (15.26) (1.37) (5.34) 8.10 (5.34) ) 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/ (0.06) (89.20) (2.04) /13/ (0.25) (89.72) (5.35) /14/ (1.59) (1.59) (79.49) 8.97 (1.59) /15/ (0.16) (82.24) (1.59) /18/ (0.15) (72.23) /19/ (0.04) (68.15) /20/ (2.33) (2.33) (77.79) (19.76) (2.33) (22.10) 4/21/ (76.85) (0.78) - (0.78) (2.44) ) Portfolio Daily Changes Position Profit & Loss Option Option Date Delta Gamma Vega Theta Other Total (C17) (P15) Stock (Open) (Close) Total 4/12/ (0.14) (289.02) (89.20) (4.04) /13/ (293.17) (89.72) (11.24) 9.43 (1.81) 4/14/ (3.76) (3.59) (244.50) (79.49) 4.97 (3.59) /15/ (265.08) (82.24) (1.59) 1.16 (0.43) 4/18/ (5.64) 0.04 (0.50) (6.10) (231.20) (72.23) 9.99 (6.10) /19/ (7.22) (7.13) (192.69) (68.15) 4.78 (7.13) (2.35) 4/20/ (0.10) (255.99) (77.79) (2.98) 2.23 (0.76) 4/21/ (1.48) (1.00) (274.61) (76.85) 0.78 (1.00) (0.22)
13 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 C19 CSCO N C P15 CSCO N P 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 = , N P15 = If we buy of P15 CSCO and short 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 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/ % (1.13) (0.05) /26/ % (1.11) (0.19) /27/ (0.41) 0.52% 1 (27.52) (1.21) (0.08) (26.00) (26.00) (18.00) 4/28/ % (1.32) /29/ % (1.34) (0.03) (2.00) /2/ % (1.95) (1.58) (0.08) /3/ (0.12) 1.72% 1 (7.96) (1.52) (0.07) (7.00) (4.00) (7.00) (11.00) 5/4/ % (0.71) (1.71) (2.00) /5/ % (1.68) (0.10) /6/ (0.07) 2.21% 1 (4.88) (1.68) (0.16) (4.00) (4.00) (14.29) (0.63) ) 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/ % 1 (0.64) (0.21) (0.08) /26/ % 1 (3.93) (0.83) 0.04 (3.79) (0.83) (3.79) (4.62) 4/27/ (0.41) -1.77% 1 (2.32) (0.55) (1.10) (110.45) 1.89 (1.10) /28/ % 1 (1.84) (1.47) (0.10) /29/ % 1 (0.59) (0.24) (0.05) /2/ % 1 (0.03) (0.04) (0.01) (0.46) - (0.46) 5/3/ (0.12) 2.62% 1 (0.06) (0.00) (0.10) (0.11) (11.03) (0.07) (0.11) (0.18) 5/4/ % 1 (0.22) (0.23) (0.01) /5/ % (0.21) (0.23) 0.07 (0.37) (0.37) (0.37) 5/6/ (0.07) 0.55% 1 (0.14) (0.01) (0.08) 0.25 (0.03) - (45.18) (9.76) (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/ % 1 (4.43) (0.47) (4.28) 1.34 (0.57) (8.41) (280.45) - (8.41) (8.41) 4/26/ % 1 (12.10) (2.11) (2.56) 1.93 (0.30) (15.15) (303.05) - (15.15) (15.15) 4/27/ (0.41) 4.28% (1.44) (4.64) (91.81) /28/ % 1 (4.51) (0.21) (347.71) /29/ % 1 (5.69) (0.60) (1.01) (5.71) (190.50) - (5.71) (5.71) 5/2/ % 1 (1.84) (0.06) (141.15) (1.90) 1.41 (0.49) 5/3/ (0.12) 2.07% (0.18) (2.77) (141.43) /4/ % 1 (2.75) (0.15) (0.98) (1.82) (181.95) - (1.82) (1.82) 5/5/ % (3.78) (1.71) (170.69) 1.82 (1.71) /6/ (0.07) 2.79% (0.06) (2.96) (124.04) (20.09) (5.28) (19.52) (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/ (3.39) (3.39) (19.96) 0.77 (3.39) (2.62) 4/26/ (1.89) (1.89) (6.11) (2.20) (1.89) (4.09) 4/27/ (0.41) (55.79) (0.49) /28/ (0.56) /29/ (5.81) (5.81) (29.04) 0.40 (5.81) (5.41) 5/2/ (2.68) (2.68) (38.26) 0.29 (2.68) (2.39) 5/3/ (0.12) (43.19) /4/ (3.22) (3.22) (32.19) 1.73 (3.22) (1.49) 5/5/ (36.77) (0.32) - (0.32) 5/6/ (0.07) (51.27) (5.52) 3.59 (1.93) (2.44) ) 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/ (1.10) (0.01) (0.70) (1.81) (280.45) (19.96) 4.51 (1.81) /26/ (0.02) (0.45) (303.05) (6.11) (1.02) /27/ (2.90) (1.48) (110.45) (91.81) (55.79) 9.41 (1.48) /28/ (0.01) (347.71) (0.56) /29/ (0.01) (0.05) (190.50) (29.04) (1.60) 0.48 (1.12) 5/2/ (0.21) (0.00) (0.05) (0.27) (141.15) (38.26) (0.07) (0.27) (0.34) 5/3/ (0.50) (0.01) (0.01) (0.51) (11.03) (141.43) (43.19) 0.66 (0.51) /4/ (1.24) (0.01) 0.21 (1.04) (181.95) (32.19) (0.27) (1.04) (1.31) 5/5/ (1.22) (0.01) 0.15 (1.08) (170.69) (36.77) 1.50 (1.08) /6/ (0.38) 0.00 (0.04) (0.41) (45.18) (124.04) (51.27) 4.19 (0.41) (0.07) initial position), therefore, it is not easy to hedge vega of C17 using the other two options.
15 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 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/ % (1.13) (0.05) /26/ % (1.11) (0.19) /27/ (0.41) 0.52% 1 (27.52) (1.21) (0.08) (26.00) (26.00) (18.00) 4/28/ % (1.32) /29/ % (1.34) (0.03) (2.00) /2/ % (1.95) (1.58) (0.08) /3/ (0.12) 1.72% 1 (7.96) (1.52) (0.07) (7.00) (4.00) (7.00) (11.00) 5/4/ % (0.71) (1.71) (2.00) /5/ % (1.68) (0.10) /6/ (0.07) 2.21% 1 (4.88) (1.68) (0.16) (4.00) (4.00) (14.29) (0.63) ) 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/ % (0.19) (0.05) 3.07 (61.45) /26/ % 1 (1.27) (0.18) (0.03) (0.90) (0.90) /27/ (0.41) 0.17% 1 (5.24) (1.10) (0.15) (5.75) (95.91) - (5.75) (5.75) 4/28/ % (0.05) (0.05) 1.99 (66.44) /29/ % (0.43) (3.91) (128.99) /2/ % (0.05) (2.64) (129.44) /3/ (0.12) 1.56% (0.74) (0.04) /4/ % (0.04) (0.91) (49.04) /5/ % (0.00) 0.00 (0.13) /6/ (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) ) 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/ % 1 (2.69) (0.22) (1.61) 0.56 (0.08) (4.04) (67.27) - (4.04) (4.04) 4/26/ % 1 (12.05) (1.59) (4.08) 1.29 (0.42) (16.85) (129.58) (0.67) (16.85) (17.52) 4/27/ (0.41) 2.53% (0.88) (1.74) (39.92) (5.18) 4.79 (0.39) 4/28/ % 1 (1.65) (0.06) (1.91) 0.69 (0.05) (2.99) (59.74) 1.20 (2.99) (1.79) 4/29/ % 1 (1.25) (0.10) (0.38) (1.49) (18.65) 1.19 (1.49) (0.30) 5/2/ % 1 (0.53) (0.01) (0.39) (0.60) (19.98) 0.37 (0.60) (0.23) 5/3/ (0.12) 0.12% (0.26) (0.25) (138.07) /4/ % 1 (2.45) (0.09) (0.71) (71.17) 1.38 (0.71) /5/ % (1.76) (97.70) (0.71) - (0.71) 5/6/ (0.07) 2.69% (0.05) (2.50) (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/ (8.30) (8.30) (48.85) 0.84 (8.30) (7.47) 4/26/ (4.59) (4.59) (14.82) (5.37) (4.59) (9.97) 4/27/ (0.41) (63.46) (1.19) /28/ (4.65) (4.65) (51.68) (0.63) (4.65) (5.29) 4/29/ (15.04) (15.04) (75.22) (1.55) (15.04) (16.59) 5/2/ (5.32) (5.32) (76.01) 0.75 (5.32) (4.57) 5/3/ (0.12) (2.18) /4/ (4.52) (4.52) (45.19) 0.09 (4.52) (4.43) 5/5/ (29.52) (0.45) - (0.45) 5/6/ (0.07) (46.42) (4.43) 3.25 (1.18) (12.90) (2.44) (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/ (0.00) (0.18) (61.45) (67.27) (48.85) /26/ (0.00) (0.64) (0.34) (129.58) (14.82) (2.82) (0.34) (3.16) 4/27/ (0.00) (1.06) (0.01) 0.12 (0.95) (95.91) (39.92) (63.46) 1.63 (0.95) /28/ (0.01) (0.08) (66.44) (59.74) (51.68) /29/ (0.00) (3.55) (0.01) 0.31 (3.25) (128.99) (18.65) (75.22) (1.69) (3.25) (4.94) 5/2/ (0.00) (4.97) (0.01) 0.06 (4.92) (129.44) (19.98) (76.01) 3.13 (4.92) (1.79) 5/3/ (0.00) (0.04) (138.07) (2.18) /4/ (0.96) (0.00) 0.22 (0.74) (49.04) (71.17) (45.19) (0.53) (0.74) (1.27) 5/5/ (0.00) (0.01) (0.13) (97.70) (29.52) (1.16) 1.00 (0.16) 5/6/ (0.32) 0.01 (0.15) (0.46) (40.17) (69.53) (46.42) 0.64 (0.46) (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 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 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 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 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 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.
OPTION POSITIONING AND TRADING TUTORIAL
OPTION POSITIONING AND TRADING TUTORIAL Binomial Options Pricing, Implied Volatility and Hedging Option Underlying 5/13/2011 Professor James Bodurtha Executive Summary The following paper looks at a number
More informationFin 4200 Project. Jessi Sagner 11/15/11
Fin 4200 Project Jessi Sagner 11/15/11 All Option information is outlined in appendix A Option Strategy The strategy I chose was to go long 1 call and 1 put at the same strike price, but different times
More informationP&L Attribution and Risk Management
P&L Attribution and Risk Management Liuren Wu Options Markets (Hull chapter: 15, Greek letters) Liuren Wu ( c ) P& Attribution and Risk Management Options Markets 1 / 19 Outline 1 P&L attribution via the
More informationOPTIONS & GREEKS. Study notes. An option results in the right (but not the obligation) to buy or sell an asset, at a predetermined
OPTIONS & GREEKS Study notes 1 Options 1.1 Basic information An option results in the right (but not the obligation) to buy or sell an asset, at a predetermined price, and on or before a predetermined
More informationEvaluating Options Price Sensitivities
Evaluating Options Price Sensitivities Options Pricing Presented by Patrick Ceresna, CMT CIM DMS Montréal Exchange Instructor Disclaimer 2016 Bourse de Montréal Inc. This document is sent to you on a general
More informationFinance & Stochastic. Contents. Rossano Giandomenico. Independent Research Scientist, Chieti, Italy.
Finance & Stochastic Rossano Giandomenico Independent Research Scientist, Chieti, Italy Email: rossano1976@libero.it Contents Stochastic Differential Equations Interest Rate Models Option Pricing Models
More informationLecture Quantitative Finance Spring Term 2015
and Lecture Quantitative Finance Spring Term 2015 Prof. Dr. Erich Walter Farkas Lecture 06: March 26, 2015 1 / 47 Remember and Previous chapters: introduction to the theory of options put-call parity fundamentals
More informationA Brief Analysis of Option Implied Volatility and Strategies. Zhou Heng. University of Adelaide, Adelaide, Australia
Economics World, July-Aug. 2018, Vol. 6, No. 4, 331-336 doi: 10.17265/2328-7144/2018.04.009 D DAVID PUBLISHING A Brief Analysis of Option Implied Volatility and Strategies Zhou Heng University of Adelaide,
More informationThe Black-Scholes-Merton Model
Normal (Gaussian) Distribution Probability Density 0.5 0. 0.15 0.1 0.05 0 1.1 1 0.9 0.8 0.7 0.6? 0.5 0.4 0.3 0. 0.1 0 3.6 5. 6.8 8.4 10 11.6 13. 14.8 16.4 18 Cumulative Probability Slide 13 in this slide
More informationFIN FINANCIAL INSTRUMENTS SPRING 2008
FIN-40008 FINANCIAL INSTRUMENTS SPRING 2008 The Greeks Introduction We have studied how to price an option using the Black-Scholes formula. Now we wish to consider how the option price changes, either
More informationMath 181 Lecture 15 Hedging and the Greeks (Chap. 14, Hull)
Math 181 Lecture 15 Hedging and the Greeks (Chap. 14, Hull) One use of derivation is for investors or investment banks to manage the risk of their investments. If an investor buys a stock for price S 0,
More informationHow to Trade Options Using VantagePoint and Trade Management
How to Trade Options Using VantagePoint and Trade Management Course 3.2 + 3.3 Copyright 2016 Market Technologies, LLC. 1 Option Basics Part I Agenda Option Basics and Lingo Call and Put Attributes Profit
More informationChapter 9 - Mechanics of Options Markets
Chapter 9 - Mechanics of Options Markets Types of options Option positions and profit/loss diagrams Underlying assets Specifications Trading options Margins Taxation Warrants, employee stock options, and
More informationOPTIONS CALCULATOR QUICK GUIDE
OPTIONS CALCULATOR QUICK GUIDE Table of Contents Introduction 3 Valuing options 4 Examples 6 Valuing an American style non-dividend paying stock option 6 Valuing an American style dividend paying stock
More informationCHAPTER 10 OPTION PRICING - II. Derivatives and Risk Management By Rajiv Srivastava. Copyright Oxford University Press
CHAPTER 10 OPTION PRICING - II Options Pricing II Intrinsic Value and Time Value Boundary Conditions for Option Pricing Arbitrage Based Relationship for Option Pricing Put Call Parity 2 Binomial Option
More information2 f. f t S 2. Delta measures the sensitivityof the portfolio value to changes in the price of the underlying
Sensitivity analysis Simulating the Greeks Meet the Greeks he value of a derivative on a single underlying asset depends upon the current asset price S and its volatility Σ, the risk-free interest rate
More informationcovered warrants uncovered an explanation and the applications of covered warrants
covered warrants uncovered an explanation and the applications of covered warrants Disclaimer Whilst all reasonable care has been taken to ensure the accuracy of the information comprising this brochure,
More informationUCLA Anderson School of Management Daniel Andrei, Option Markets 232D, Fall MBA Midterm. November Date:
UCLA Anderson School of Management Daniel Andrei, Option Markets 232D, Fall 2013 MBA Midterm November 2013 Date: Your Name: Your Equiz.me email address: Your Signature: 1 This exam is open book, open notes.
More informationUnderstanding Options Gamma to Boost Returns. Maximizing Gamma. The Optimum Options for Accelerated Growth
Understanding Options Gamma to Boost Returns Maximizing Gamma The Optimum Options for Accelerated Growth Enhance Your Option Trading Returns by Maximizing Gamma Select the Optimum Options for Accelerated
More informationQueens College, CUNY, Department of Computer Science Computational Finance CSCI 365 / 765 Fall 2017 Instructor: Dr. Sateesh Mane.
Queens College, CUNY, Department of Computer Science Computational Finance CSCI 365 / 765 Fall 2017 Instructor: Dr. Sateesh Mane c Sateesh R. Mane 2017 20 Lecture 20 Implied volatility November 30, 2017
More informationOption Selection With Bill Corcoran
Presents Option Selection With Bill Corcoran I am not a registered broker-dealer or investment adviser. I will mention that I consider certain securities or positions to be good candidates for the types
More informationHomework Assignments
Homework Assignments Week 1 (p 57) #4.1, 4., 4.3 Week (pp 58-6) #4.5, 4.6, 4.8(a), 4.13, 4.0, 4.6(b), 4.8, 4.31, 4.34 Week 3 (pp 15-19) #1.9, 1.1, 1.13, 1.15, 1.18 (pp 9-31) #.,.6,.9 Week 4 (pp 36-37)
More informationValuing Put Options with Put-Call Parity S + P C = [X/(1+r f ) t ] + [D P /(1+r f ) t ] CFA Examination DERIVATIVES OPTIONS Page 1 of 6
DERIVATIVES OPTIONS A. INTRODUCTION There are 2 Types of Options Calls: give the holder the RIGHT, at his discretion, to BUY a Specified number of a Specified Asset at a Specified Price on, or until, a
More informationThe Black-Scholes Model
IEOR E4706: Foundations of Financial Engineering c 2016 by Martin Haugh The Black-Scholes Model In these notes we will use Itô s Lemma and a replicating argument to derive the famous Black-Scholes formula
More informationHedging with Options
School of Education, Culture and Communication Tutor: Jan Röman Hedging with Options (MMA707) Authors: Chiamruchikun Benchaphon 800530-49 Klongprateepphol Chutima 80708-67 Pongpala Apiwat 808-4975 Suntayodom
More informationThe objective of Part One is to provide a knowledge base for learning about the key
PART ONE Key Option Elements The objective of Part One is to provide a knowledge base for learning about the key elements of forex options. This includes a description of plain vanilla options and how
More informationTradeOptionsWithMe.com
TradeOptionsWithMe.com 1 of 18 Option Trading Glossary This is the Glossary for important option trading terms. Some of these terms are rather easy and used extremely often, but some may even be new to
More informationSimple Formulas to Option Pricing and Hedging in the Black-Scholes Model
Simple Formulas to Option Pricing and Hedging in the Black-Scholes Model Paolo PIANCA DEPARTMENT OF APPLIED MATHEMATICS University Ca Foscari of Venice pianca@unive.it http://caronte.dma.unive.it/ pianca/
More informationDerivative Securities
Derivative Securities he Black-Scholes formula and its applications. his Section deduces the Black- Scholes formula for a European call or put, as a consequence of risk-neutral valuation in the continuous
More informationCorporate Finance, Module 21: Option Valuation. Practice Problems. (The attached PDF file has better formatting.) Updated: July 7, 2005
Corporate Finance, Module 21: Option Valuation Practice Problems (The attached PDF file has better formatting.) Updated: July 7, 2005 {This posting has more information than is needed for the corporate
More informationPractical Hedging: From Theory to Practice. OSU Financial Mathematics Seminar May 5, 2008
Practical Hedging: From Theory to Practice OSU Financial Mathematics Seminar May 5, 008 Background Dynamic replication is a risk management technique used to mitigate market risk We hope to spend a certain
More informationThe Greek Letters Based on Options, Futures, and Other Derivatives, 8th Edition, Copyright John C. Hull 2012
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
More informationLecture 9: Practicalities in Using Black-Scholes. Sunday, September 23, 12
Lecture 9: Practicalities in Using Black-Scholes Major Complaints Most stocks and FX products don t have log-normal distribution Typically fat-tailed distributions are observed Constant volatility assumed,
More informationSYSM 6304: Risk and Decision Analysis Lecture 6: Pricing and Hedging Financial Derivatives
SYSM 6304: Risk and Decision Analysis Lecture 6: Pricing and Hedging Financial Derivatives M. Vidyasagar Cecil & Ida Green Chair The University of Texas at Dallas Email: M.Vidyasagar@utdallas.edu October
More informationMathematics of Finance Final Preparation December 19. To be thoroughly prepared for the final exam, you should
Mathematics of Finance Final Preparation December 19 To be thoroughly prepared for the final exam, you should 1. know how to do the homework problems. 2. be able to provide (correct and complete!) definitions
More informationLearn To Trade Stock Options
Learn To Trade Stock Options Written by: Jason Ramus www.daytradingfearless.com Copyright: 2017 Table of contents: WHAT TO EXPECT FROM THIS MANUAL WHAT IS AN OPTION BASICS OF HOW AN OPTION WORKS RECOMMENDED
More informatione.g. + 1 vol move in the 30delta Puts would be example of just a changing put skew
Calculating vol skew change risk (skew-vega) Ravi Jain 2012 Introduction An interesting and important risk in an options portfolio is the impact of a changing implied volatility skew. It is not uncommon
More informationFinancial Markets & Risk
Financial Markets & Risk Dr Cesario MATEUS Senior Lecturer in Finance and Banking Room QA259 Department of Accounting and Finance c.mateus@greenwich.ac.uk www.cesariomateus.com Session 3 Derivatives Binomial
More informationHow is an option priced and what does it mean? Patrick Ceresna, CMT Big Picture Trading Inc.
How is an option priced and what does it mean? Patrick Ceresna, CMT Big Picture Trading Inc. Limitation of liability The opinions expressed in this presentation are those of the author(s) and presenter(s)
More information1. 2 marks each True/False: briefly explain (no formal proofs/derivations are required for full mark).
The University of Toronto ACT460/STA2502 Stochastic Methods for Actuarial Science Fall 2016 Midterm Test You must show your steps or no marks will be awarded 1 Name Student # 1. 2 marks each True/False:
More informationThe Binomial Model. Chapter 3
Chapter 3 The Binomial Model In Chapter 1 the linear derivatives were considered. They were priced with static replication and payo tables. For the non-linear derivatives in Chapter 2 this will not work
More informationAdvanced Topics in Derivative Pricing Models. Topic 4 - Variance products and volatility derivatives
Advanced Topics in Derivative Pricing Models Topic 4 - Variance products and volatility derivatives 4.1 Volatility trading and replication of variance swaps 4.2 Volatility swaps 4.3 Pricing of discrete
More informationAny asset that derives its value from another underlying asset is called a derivative asset. The underlying asset could be any asset - for example, a
Options Week 7 What is a derivative asset? Any asset that derives its value from another underlying asset is called a derivative asset. The underlying asset could be any asset - for example, a stock, bond,
More informationActuarial Models : Financial Economics
` Actuarial Models : Financial Economics An Introductory Guide for Actuaries and other Business Professionals First Edition BPP Professional Education Phoenix, AZ Copyright 2010 by BPP Professional Education,
More informationAppendix to Supplement: What Determines Prices in the Futures and Options Markets?
Appendix to Supplement: What Determines Prices in the Futures and Options Markets? 0 ne probably does need to be a rocket scientist to figure out the latest wrinkles in the pricing formulas used by professionals
More informationAdvanced Equity Derivatives This course can also be presented in-house for your company or via live on-line webinar
Advanced Equity Derivatives This course can also be presented in-house for your company or via live on-line webinar The Banking and Corporate Finance Training Specialist Course Objectives The broad objectives
More informationOf Option Trading PRESENTED BY: DENNIS W. WILBORN
Of Option Trading PRESENTED BY: DENNIS W. WILBORN Disclaimer U.S. GOVERNMENT REQUIRED DISCLAIMER COMMODITY FUTURES TRADING COMMISSION FUTURES AND OPTIONS TRADING HAS LARGE POTENTIAL REWARDS, BUT ALSO LARGE
More informationCopyright 2018 Craig E. Forman All Rights Reserved. Trading Equity Options Week 2
Copyright 2018 Craig E. Forman All Rights Reserved www.tastytrader.net Trading Equity Options Week 2 Disclosure All investments involve risk and are not suitable for all investors. The past performance
More informationOption Volatility "The market can remain irrational longer than you can remain solvent"
Chapter 15 Option Volatility "The market can remain irrational longer than you can remain solvent" The word volatility, particularly to newcomers, conjures up images of wild price swings in stocks (most
More informationCompleteness and Hedging. Tomas Björk
IV Completeness and Hedging Tomas Björk 1 Problems around Standard Black-Scholes We assumed that the derivative was traded. How do we price OTC products? Why is the option price independent of the expected
More informationSkew Hedging. Szymon Borak Matthias R. Fengler Wolfgang K. Härdle. CASE-Center for Applied Statistics and Economics Humboldt-Universität zu Berlin
Szymon Borak Matthias R. Fengler Wolfgang K. Härdle CASE-Center for Applied Statistics and Economics Humboldt-Universität zu Berlin 6 4 2.22 Motivation 1-1 Barrier options Knock-out options are financial
More informationThe Black-Scholes Model
The Black-Scholes Model Liuren Wu Options Markets (Hull chapter: 12, 13, 14) Liuren Wu ( c ) The Black-Scholes Model colorhmoptions Markets 1 / 17 The Black-Scholes-Merton (BSM) model Black and Scholes
More informationDerivative Securities Fall 2012 Final Exam Guidance Extended version includes full semester
Derivative Securities Fall 2012 Final Exam Guidance Extended version includes full semester Our exam is Wednesday, December 19, at the normal class place and time. You may bring two sheets of notes (8.5
More informationEuropean call option with inflation-linked strike
Mathematical Statistics Stockholm University European call option with inflation-linked strike Ola Hammarlid Research Report 2010:2 ISSN 1650-0377 Postal address: Mathematical Statistics Dept. of Mathematics
More informationLecture 6: Option Pricing Using a One-step Binomial Tree. Thursday, September 12, 13
Lecture 6: Option Pricing Using a One-step Binomial Tree An over-simplified model with surprisingly general extensions a single time step from 0 to T two types of traded securities: stock S and a bond
More informationMathematics of Financial Derivatives
Mathematics of Financial Derivatives Lecture 8 Solesne Bourguin bourguin@math.bu.edu Boston University Department of Mathematics and Statistics Table of contents 1. The Greek letters (continued) 2. Volatility
More informationB. Combinations. 1. Synthetic Call (Put-Call Parity). 2. Writing a Covered Call. 3. Straddle, Strangle. 4. Spreads (Bull, Bear, Butterfly).
1 EG, Ch. 22; Options I. Overview. A. Definitions. 1. Option - contract in entitling holder to buy/sell a certain asset at or before a certain time at a specified price. Gives holder the right, but not
More informationA study on parameters of option pricing: The Greeks
International Journal of Academic Research and Development ISSN: 2455-4197, Impact Factor: RJIF 5.22 www.academicsjournal.com Volume 2; Issue 2; March 2017; Page No. 40-45 A study on parameters of option
More informationLecture Note 8 of Bus 41202, Spring 2017: Stochastic Diffusion Equation & Option Pricing
Lecture Note 8 of Bus 41202, Spring 2017: Stochastic Diffusion Equation & Option Pricing We shall go over this note quickly due to time constraints. Key concept: Ito s lemma Stock Options: A contract giving
More informationnon linear Payoffs Markus K. Brunnermeier
Institutional Finance Lecture 10: Dynamic Arbitrage to Replicate non linear Payoffs Markus K. Brunnermeier Preceptor: Dong Beom Choi Princeton University 1 BINOMIAL OPTION PRICING Consider a European call
More informationGLOSSARY OF OPTION TERMS
ALL OR NONE (AON) ORDER An order in which the quantity must be completely filled or it will be canceled. AMERICAN-STYLE OPTION A call or put option contract that can be exercised at any time before the
More informationHedging. MATH 472 Financial Mathematics. J. Robert Buchanan
Hedging MATH 472 Financial Mathematics J. Robert Buchanan 2018 Introduction Definition Hedging is the practice of making a portfolio of investments less sensitive to changes in market variables. There
More informationVolatility & Arbitrage Trading
2002 Market Compass, Inc. Options involve risk and are not suitable for everyone. Prior to buying or selling an option, a person must receive a copy of Characteristics and Risks of Standardized Options.
More informationAdvanced Equity Derivatives
Advanced Equity Derivatives This course can be presented in-houseor via webinar for you on a date of your choosing The Banking and Corporate Finance Training Specialist Course Overview This programme has
More informationAppendix A Financial Calculations
Derivatives Demystified: A Step-by-Step Guide to Forwards, Futures, Swaps and Options, Second Edition By Andrew M. Chisholm 010 John Wiley & Sons, Ltd. Appendix A Financial Calculations TIME VALUE OF MONEY
More informationOption Trading The Option Butterfly Spread
Option Trading The Option Butterfly Spread By Larry Gaines Butterflies provide a low risk high reward trading opportunity. Markets direction can go through months, and even years of higher than usual uncertainty.
More informationTEACHING NOTE 98-04: EXCHANGE OPTION PRICING
TEACHING NOTE 98-04: EXCHANGE OPTION PRICING Version date: June 3, 017 C:\CLASSES\TEACHING NOTES\TN98-04.WPD The exchange option, first developed by Margrabe (1978), has proven to be an extremely powerful
More informationAn Introduction to the Mathematics of Finance. Basu, Goodman, Stampfli
An Introduction to the Mathematics of Finance Basu, Goodman, Stampfli 1998 Click here to see Chapter One. Chapter 2 Binomial Trees, Replicating Portfolios, and Arbitrage 2.1 Pricing an Option A Special
More informationAdvanced Corporate Finance. 5. Options (a refresher)
Advanced Corporate Finance 5. Options (a refresher) Objectives of the session 1. Define options (calls and puts) 2. Analyze terminal payoff 3. Define basic strategies 4. Binomial option pricing model 5.
More informationThe Black-Scholes Model
The Black-Scholes Model Liuren Wu Options Markets Liuren Wu ( c ) The Black-Merton-Scholes Model colorhmoptions Markets 1 / 18 The Black-Merton-Scholes-Merton (BMS) model Black and Scholes (1973) and Merton
More informationDerivatives Analysis & Valuation (Futures)
6.1 Derivatives Analysis & Valuation (Futures) LOS 1 : Introduction Study Session 6 Define Forward Contract, Future Contract. Forward Contract, In Forward Contract one party agrees to buy, and the counterparty
More informationECON4510 Finance Theory Lecture 10
ECON4510 Finance Theory Lecture 10 Diderik Lund Department of Economics University of Oslo 11 April 2016 Diderik Lund, Dept. of Economics, UiO ECON4510 Lecture 10 11 April 2016 1 / 24 Valuation of options
More informationWebinar Presentation How Volatility & Other Important Factors Affect the Greeks
Webinar Presentation How Volatility & Other Important Factors Affect the Greeks Presented by Trading Strategy Desk 1 Fidelity Brokerage Services, Member NYSE, SIPC, 900 Salem Street, Smithfield, RI 02917.
More informationOption Volatility & Arbitrage Opportunities
Louisiana State University LSU Digital Commons LSU Master's Theses Graduate School 2016 Option Volatility & Arbitrage Opportunities Mikael Boffetti Louisiana State University and Agricultural and Mechanical
More informationTimely, insightful research and analysis from TradeStation. Options Toolkit
Timely, insightful research and analysis from TradeStation Options Toolkit Table of Contents Important Information and Disclosures... 3 Options Risk Disclosure... 4 Prologue... 5 The Benefits of Trading
More informationGlobalView Software, Inc.
GlobalView Software, Inc. MarketView Option Analytics 10/16/2007 Table of Contents 1. Introduction...1 2. Configuration Settings...2 2.1 Component Selection... 2 2.2 Edit Configuration Analytics Tab...
More informationIntroduction to Binomial Trees. Chapter 12
Introduction to Binomial Trees Chapter 12 Fundamentals of Futures and Options Markets, 8th Ed, Ch 12, Copyright John C. Hull 2013 1 A Simple Binomial Model A stock price is currently $20. In three months
More informationCFE: Level 1 Exam Sample Questions
CFE: Level 1 Exam Sample Questions he following are the sample questions that are illustrative of the questions that may be asked in a CFE Level 1 examination. hese questions are only for illustration.
More informationThe Four Basic Options Strategies
Cohen_ch01.qxd 1/12/05 10:26 PM Page 1 1 The Four Basic Options Strategies Introduction The easiest way to learn options is with pictures so that you can begin to piece together strategies step-by-step.
More informationFX Options. Outline. Part I. Chapter 1: basic FX options, standard terminology, mechanics
FX Options 1 Outline Part I Chapter 1: basic FX options, standard terminology, mechanics Chapter 2: Black-Scholes pricing model; some option pricing relationships 2 Outline Part II Chapter 3: Binomial
More informationImplied Volatility Surface
Implied Volatility Surface Liuren Wu Zicklin School of Business, Baruch College Options Markets (Hull chapter: 16) Liuren Wu Implied Volatility Surface Options Markets 1 / 1 Implied volatility Recall the
More informationquan OPTIONS ANALYTICS IN REAL-TIME PROBLEM: Industry SOLUTION: Oquant Real-time Options Pricing
OPTIONS ANALYTICS IN REAL-TIME A major aspect of Financial Mathematics is option pricing theory. Oquant provides real time option analytics in the cloud. We have developed a powerful system that utilizes
More informationPresents Mastering the Markets Trading Earnings
www.mastermindtraders.com Presents Mastering the Markets Trading Earnings 1 DISCLAIMER Neither MasterMind Traders or any of its personnel are registered broker-dealers or investment advisors. We may mention
More informationAsset-or-nothing digitals
School of Education, Culture and Communication Division of Applied Mathematics MMA707 Analytical Finance I Asset-or-nothing digitals 202-0-9 Mahamadi Ouoba Amina El Gaabiiy David Johansson Examinator:
More informationSensex Realized Volatility Index (REALVOL)
Sensex Realized Volatility Index (REALVOL) Introduction Volatility modelling has traditionally relied on complex econometric procedures in order to accommodate the inherent latent character of volatility.
More informationOptions: How About Wealth & Income?
Options: How About Wealth & Income? Disclaimer U.S. GOVERNMENT REQUIRED DISCLAIMER COMMODITY FUTURES TRADING COMMISSION FUTURES AND OPTIONS TRADING HAS LARGE POTENTIAL REW ARDS, BUT ALS O LARGE POTENTIAL
More information1 Parameterization of Binomial Models and Derivation of the Black-Scholes PDE.
1 Parameterization of Binomial Models and Derivation of the Black-Scholes PDE. Previously we treated binomial models as a pure theoretical toy model for our complete economy. We turn to the issue of how
More informationSwing Trading SMALL, MID & L ARGE CAPS STOCKS & OPTIONS
Swing Trading SMALL, MID & L ARGE CAPS STOCKS & OPTIONS Warrior Trading I m a full time trader and help run a live trading room where we trade in real time and teach people how to trade stocks. My primary
More informationMarket risk measurement in practice
Lecture notes on risk management, public policy, and the financial system Allan M. Malz Columbia University 2018 Allan M. Malz Last updated: October 23, 2018 2/32 Outline Nonlinearity in market risk Market
More informationMANAGING OPTIONS POSITIONS MARCH 2013
MANAGING OPTIONS POSITIONS MARCH 2013 AGENDA INTRODUCTION OPTION VALUATION & RISK MEASURES THE GREEKS PRE-TRADE RICH VS. CHEAP ANALYSIS SELECTING TERM STRUCTURE PORTFOLIO CONSTRUCTION CONDITIONAL RISK
More informationProfit settlement End of contract Daily Option writer collects premium on T+1
DERIVATIVES A derivative contract is a financial instrument whose payoff structure is derived from the value of the underlying asset. A forward contract is an agreement entered today under which one party
More informationSwing TradING CHAPTER 2. OPTIONS TR ADING STR ATEGIES
Swing TradING CHAPTER 2. OPTIONS TR ADING STR ATEGIES When do we want to use options? There are MANY reasons to learn options trading and MANY scenarios in which you might trade them When we want leverage
More informationLetter To Our Clients INF RMER. A Forum for Options Trading Ideas
In this issue - Letter To Our Clients - Len Yates - True Delta: Your Competitive Advantage - Steve Lentz - Butterfly Balancing with True Delta - Steve Lentz - Customer Support - Jim Graham INF RMER A Forum
More informationCourse Syllabus. [FIN 4533 FINANCIAL DERIVATIVES - (SECTION 16A9)] Fall 2015, Mod 1
Course Syllabus Course Instructor Information: Professor: Farid AitSahlia Office: Stuzin 310 Office Hours: By appointment Phone: 352-392-5058 E-mail: farid.aitsahlia@warrington.ufl.edu Class Room/Time:
More information******************************* The multi-period binomial model generalizes the single-period binomial model we considered in Section 2.
Derivative Securities Multiperiod Binomial Trees. We turn to the valuation of derivative securities in a time-dependent setting. We focus for now on multi-period binomial models, i.e. binomial trees. This
More informationCOPYRIGHTED MATERIAL. Time Value of Money Toolbox CHAPTER 1 INTRODUCTION CASH FLOWS
E1C01 12/08/2009 Page 1 CHAPTER 1 Time Value of Money Toolbox INTRODUCTION One of the most important tools used in corporate finance is present value mathematics. These techniques are used to evaluate
More informationLECTURE 2: MULTIPERIOD MODELS AND TREES
LECTURE 2: MULTIPERIOD MODELS AND TREES 1. Introduction One-period models, which were the subject of Lecture 1, are of limited usefulness in the pricing and hedging of derivative securities. In real-world
More informationThe Johns Hopkins Carey Business School. Derivatives. Spring Final Exam
The Johns Hopkins Carey Business School Derivatives Spring 2010 Instructor: Bahattin Buyuksahin Final Exam Final DUE ON WEDNESDAY, May 19th, 2010 Late submissions will not be graded. Show your calculations.
More informationGreek parameters of nonlinear Black-Scholes equation
International Journal of Mathematics and Soft Computing Vol.5, No.2 (2015), 69-74. ISSN Print : 2249-3328 ISSN Online: 2319-5215 Greek parameters of nonlinear Black-Scholes equation Purity J. Kiptum 1,
More informationNaked & Covered Positions
The Greek Letters 1 Example A bank has sold for $300,000 a European call option on 100,000 shares of a nondividend paying stock S 0 = 49, K = 50, r = 5%, σ = 20%, T = 20 weeks, μ = 13% The Black-Scholes
More information