OPTION PRICING, JAVA PROGRAMMING AND MONTE CARLO SIMULATION


 Aron Hall
 2 years ago
 Views:
Transcription
1 OPTION PRICING, JAVA PROGRAMMING AND MONTE CARLO SIMULATION NITESH AIDASANI KHYAMI Abstract. Option contracts are used by all major financial institutions and investors, either to speculate on stock market trends or to control their level of risk from other investments. Pricing options correctly is the key to success for many investment portfolios. The purpose of this project is to illustrate how Java programming and Monte Carlo simulations can be used to price options correctly for different set of assumptions concerning the behaviour of stock returns. Contents 1. Introduction: What are options? 1 2. The BlackScholes Model 3 3. Model of the Behaviour of the Stock Prices 4 4. Numerical Procedures Binomial Method Monte Carlo Method Conclusion 19 References Introduction: What are options? Throughout this project we use the term asset to describe any financial object whose value is known at present but is liable to change in the future. Typical examples are shares in a company, commodities such as gold, oil or electricity, currencies, for instance the value of 10 in US dollars. There are two basic types of options, a call option and a put option. 1
2 2 NITESH AIDASANI KHYAMI Definition 1. A call option gives the holder the right to buy the underlying asset by a certain date for a certain price. A put option gives the holder the right to sell the underlying asset by a certain date for a certain price. The price in the contract is known as the exercise price, denoted by K, and the date in the contract is known as the maturity, denoted by T. European options can be exercised only on maturity itself. American options can be exercised at any time up to maturity. To illustrate the idea, consider the situation where an investor writes you a European call option that gives you, the holder of the option, the right to buy 500 shares in Sun Microsystems for 5000 a year from now. After a year has passed, i.e. at maturity, the exercise price for the 500 shares can take the following three possible values: (1) K > (2) K = (3) K < If you are fortunate enough for the exercise price to have taken path (1), then you would exercise your right to buy the shares from the investor since you can sell them immediately at K and make a profit of (K 5000). If the exercise price has taken path (2), then you are in a no profit  no loss situation you can exercise the right to buy the shares but they can also be bought from the share market at the same price. However, if the exercise price has taken path (3), then you would not exercise your right to buy the shares since you could buy them at a cheaper price from the share market. One should note that an option gives the holder the right to buy or sell shares, and thus, the holder does not have to exercise this right if he/she does not want to. As a result, the holder of the option never loses money. However, the investor who wrote the option does not get paid at maturity and can theoretically lose an unlimited amount of money. To make the deal fairer, the holder is required to pay the investor (who wrote the option) for buying the option. The amount of money paid to purchase the option is known as value of the option. But how do we compute the fair value of an option? The answer of this question lies in the following few chapters where we will discussing the much celebrated BlackScholes formula.
3 OPTION PRICING, JAVA PROGRAMMING AND MONTE CARLO SIMULATION 3 2. The BlackScholes Model The BlackScholes Model was developed in the early 1970s by Fischer Black, Myron Scholes and Robert Merton. This model plays a vital role in the pricing and hedging of options. It has also been pivotal to the growth and success of financial engineering in the 1980s and the 1990s. In 1997, Myron Scholes and Robert Merton were awarded the Nobel prize in economics for developing the BlackScholes model. Unfortunately, Fischer Black had died in 1995, otherwise he also would have been one of the recipients. The BlackScholes formulas for the price at time zero of a European call option on a nondividendpaying stock and a European put option on a nondividendpaying stock are (2.1) c = S 0 N(d 1 ) Ke rt N(d 2 ) (2.2) p = Ke rt N( d 2 ) S 0 N( d 1 ) where d 1 = ln(s 0/K) + (r + σ 2 /2)T σ T d 2 = ln(s 0/K) + (r σ 2 /2)T σ T = d 1 σ T and N(x) is the cumulative probability distribution function for a variable that is normally distributed with a mean of zero and a standard deviation of 1. We also have that c is the price of the European call option, p is the price of the European put option, S 0 is the stock price, K is the strike price, r is the continuously compounded riskfree rate, σ is the stock price volatility, and T is the time to maturity of the option. Example 2. Consider the situation where the stock price six months from the expiration of an option is 42, the exercise price of the option is 40, the riskfree interest rate is 10% per annum, and the volatility is 0.2 per annum. What is the price of the European call and put option? We have that S 0 = 42, K = 40, r = 0.1, σ = 0.2, T = 0.5, d 1 = ln(42 /40 ) + ( /2 ) d 2 = ln(42 /40 ) + ( /2 ) = =
4 4 NITESH AIDASANI KHYAMI and K e rt = 40 e 0.05 = Now, if the option is a European call, then its value, c, is given by c = 42N ( ) N ( ) = If the option is a European put, its value, p, is given by p = N ( ) 42N ( ) = One of the aims of the project was to use Java to create a visual object to calculate the BlackScholes price of a European option. The heart of the Java program, i.e. the code that uses the BlackScholes formula for calculating the price of the option, has been shown below 1. Figure 1 shows a still image of this visual object. //Calculates d1 and d2 d1 = (Math.log(stockPrice/exercisePrice) + (interest + Math.pow(volatility, 2)/2)*maturity)/(volatility*Math.sqrt(maturity)); d2 = (Math.log(stockPrice/exercisePrice) + (interest  Math.pow(volatility, 2)/2)*maturity)/(volatility*Math.sqrt(maturity)); //Checks if the option is a call or put option and hence, calculates the value of the //option using the BlackScholes formula if(callselected) result = (stockprice*blackscholes.normalcdf(d1))  (exerciseprice*math.exp(interest*maturity)*blackscholes.normalcdf(d2)); else result = (exerciseprice*math.exp(interest*maturity)*blackscholes.normalcdf(d2))  (stockprice*blackscholes.normalcdf(d1)); //Display result output.settext(precision4.format(result)); 3. Model of the Behaviour of the Stock Prices Let s consider the following equation from Hull [1], p226: (3.1) ds(t) = rs(t)dt + σs(t)dw t for every t [0, T ] where σ is the volatility of the stock price, r is the expected rate of return, S(t) is the asset price at time t and dw t is a Wiener process. Both r and σ are assumed to 1 Matching parentheses have been coloured to improve readability of the code.
5 OPTION PRICING, JAVA PROGRAMMING AND MONTE CARLO SIMULATION 5 Figure 1. Calculation of the BlackScholes Price of European Options be constant. A Wiener process is a particular type of Markov stochastic process with a mean change of zero and a variance rate of 1.0. Expressed formally, a variable W t follows a Wiener process if it has the following two properties: (1) The change W t during a small period of time t is W t = ɛ t where ɛ is a random drawing from a standardized normal distribution, φ(0, 1). (2) The values of W t for any two different short intervals of time t are independent. Equation 3.1 is the most widely used model of stock price behaviour. This model is known as geometric Brownian motion. The solution to equation 3.1 is (3.2) σ2 (r S(t) = S(0)e 2 )t+σwt. The discretetime version of the model is S k t = rs k t t + σs k t W k t where S k t = S (k+1) t S k t 0 k N 1 t = T N. Thus, we have that our discretetime model is
6 6 NITESH AIDASANI KHYAMI (3.3) S (k+1) t = S k t + rs k t t + σs k t W k t = S k t + rs k t t + σs k t ɛ t One assumption in our model above that is clearly not true is that the volatility is constant. Consider the following model taken from Hull [1], p447, ds t = rs t dt + Y t S t dw t dy t = λ(µ Y t )dt + ξy t db t where λ, µ and ξ are constants, and dw t and db t are Wiener processes. The variable, Y t, in this model is the asset s variance rate. This is the square of its volatility. The discretetime version of the this model is S k t = rs k t t + Y k t S k t W k t where Y k t = λ(µ Y k t ) t + ξy k t B k t S k t = S (k+1) t S k t Y k t = Y (k+1) t Y k t 0 k N 1. Thus, we have that our discretetime model is (3.4) S (k+1) t = S k t + rs k t t + Y k t S k t W k t Y (k+1) t = Y k t + λ(µ Y k t ) t + ξy k t B k t One of the aims of the project was to use Java applets to create graphs of simulated paths of the processes for the price of the underlying asset, both in the case of constant volatility and when it is a stochastic process. In addition, we also had to create graphs of simulated paths for the volatility (as a stochastic process). In the limit as t 0, the discretetime model 3.3 becomes the continuoustime model 3.1. Figure 2, 3 and 4 shows still images of how the discretetime curve approaches the continuoustime curve as t 0, i.e. as N. The red curve is a sketch of the continuoustime model and the yellow curve is a sketch of the discretetime model.
7 OPTION PRICING, JAVA PROGRAMMING AND MONTE CARLO SIMULATION 7 Figure 2. N = 10 Figure 3. N = 100 Figure 4. N = 1000
8 8 NITESH AIDASANI KHYAMI 4. Numerical Procedures In this chapter we will be discussing two numerical procedures that can be used to value options when no analytical formula is available. These two procedures are called the binomial method and the Monte Carlo method Binomial Method. In this method we divide the life of an option into N intervals each of length t. Hence, we let t = T/N, where T is maturity date. In addition, one important assumption in the binomial method is that at each time interval, the stock price either moves up by a factor of u and probability p or down by a factor of d and probability 1 p. The following formulae for u, p and d are taken from Hull [1], p390, p = er t d u d u = e σ t d = 1 u = e σ t and are necessary for valuing the option. The value of the option is calculated by starting at the end of the tree, at time T, and working backwards. We will refer to the jth node at time i t as the (i, j) node (0 i N, 0 j i). Define f i,j as the value of the option at the (i, j) node. At time i t there are i + 1 possible values for the stock price. These are S 0 u j d i j j = 0, 1,..., i. For the final nodes, i.e. the nodes at time T (= N t), the value of a put option at node j is calculated by the formula max(k S 0 u j d N j, 0). Once we have calculated the value of the option at all of the N + 1 nodes at time N t, we move to time (N 1) t. Here we calculate the value of the option at all N nodes by using the formula 2 f i,j = e r t[ ] (4.1) pf i+1,j+1 + (1 p)f i+1,j 0 i N 1 0 j i. 2 This formula only applies for a European option. In the case of an American option, we also have to check at each node to see whether early exercise is preferable to holding the option for a further time period t.
9 OPTION PRICING, JAVA PROGRAMMING AND MONTE CARLO SIMULATION 9 Once we have calculated the value of the option at all of the N nodes at time (N 1) t, we move to time (N 2) t. Here we calculate the value of the option at all N 1 nodes by again using the equation 4.1. We keep repeating this algorithm until we reach time 0, where the value of the option at this time will be the actual value of the put option. The best way to understand this algorithm is to go through a worked example. Example 3. Consider a three year European put option on a nondividendpaying stock when the stock price is 9, the strike price is 10, the riskfree interest rate is 6% per annum, and the volatility is 0.3. Suppose that we divide the life of the option into four intervals of length 0.75 years. Thus, we have that S 0 = 9 K = 10 r = 0.06 T = 3 σ = 0.3 t = 0.75 and u = e σ t = e = p = er t d u d d = e σ t = e = p = = e = Figure 5 shows the binomial tree for this example. Each node has a formula, like S 0 u 4, and a number, like The formula is the stock price at that node and the number is the value of the option at that node. The value of the option at time T is calculated by the formula max(k S 0 u j d N j, 0 ). For example, in the case of node A (i = N = 4, j = 0) in Figure 5, we have that the value of the option is f 4,0 = max(k S 0 u j d N j, 0 ) = max(10 ( ), 0 ) = max( , 0 ) = max(6.816, 0 ) = 6.816
10 10 NITESH AIDASANI KHYAMI Figure 5. Tree used to value a stock option For all the other nodes, i.e. all the nodes except the final ones, the value of the option is calculated using equation 4.1. For example, in the case of node B (i = 3, j = 1) in Figure 5, we have that the value of the option is f 3,1 = e r t[ ] pf 3 +1, (1 p)f 3 +1,1 = e [ ] f 4, f 4,1 = [ ( ) + ( ) ] = [ ] = = Similarly, for the case of node C (i = j = 0), we have that the value of the option is
11 OPTION PRICING, JAVA PROGRAMMING AND MONTE CARLO SIMULATION 11 f 0,0 = e r t[ ] pf 0 +1, (1 p)f 0 +1,0 = e [ ] f 1, f 1,0 = [ ( ) + ( ) ] = [ ] = = This is a numerical estimate for the option s current value of In practice, a smaller value of t, and many more nodes, would be used. Table 1 gives all the formulae that are needed to value a European and an American, call and put option, using the binomial method. Option type Formulae for Calculating Value of Option European put f N,j = max(k S 0 u j d N j, 0) j = 0, 1,..., N f i,j = e r t[ pf i+1,j+1 + (1 p)f i+1,j ] 0 i N 1 0 j i European call f N,j = max(s 0 u j d N j K, 0) j = 0, 1,..., N f i,j = e r t[ pf i+1,j+1 + (1 p)f i+1,j ] 0 i N 1 0 j i American put f N,j = max(k S 0 u j d N j, 0) j = 0, 1,..., N f i,j = max {K S 0 u j d i j, e r t[ ] } pf i+1,j+1 + (1 p)f i+1,j 0 i N 1 0 j i American call f N,j = max(s 0 u j d N j K, 0) j = 0, 1,..., N f i,j = max {S 0 u j d i j K, e r t[ ] } pf i+1,j+1 + (1 p)f i+1,j 0 i N 1 0 j i Table 1. Table for Calculating Value of Options One of the aims of the project was to use Java to create a visual object to estimate the price of a European and an American option using the binomial method. The
12 12 NITESH AIDASANI KHYAMI heart of the Java program, i.e. the code that uses the binomial method for estimating the price of the option, has been shown below. Figure 6 shows a still image of this visual object. //Calculate the underlying asset price at each of the nodes in the binomial tree public void calculatenodeprice(double[][] nodearray){ int last = nodearray.length  1; for(int i = 0; i <= last; i++){ for(int j = 0; j <= i; j++){ nodearray[i][j] = stockprice*math.pow(up,j)*math.pow(down,ij); } } } //Calculate the option price at each of the nodes in the binomial tree public void calculateoptionvalue(double[][] optionprice){ int last = optionprice.length  1; //Calculate the option price at the final nodes in the binomial tree for(int i = 0; i <= last; i++){ if(callselected) //Call option optionprice[last][j] = Math.max(nodePrice[last][j]  exerciseprice, 0); else //Put option optionprice[last][j] = Math.max(exercisePrice  nodeprice[last][j], 0); } //Calculate the option price at the remaining nodes in the binomial tree for(int i = last  1; i >= 0; i){ for(int j = 0; j <= i; j++){ if(europeanselected){ //European put or call option optionprice[i][j] = Math.exp(interest*deltaT)* (prob*optionprice[i+1][j+1] + (1  prob)*optionprice[i+1][j]); } else if(!europeanselected && callselected){ //American call option optionprice[i][j] = Math.max(nodePrice[i][j]  exerciseprice, Math.exp(interest*deltaT) * (prob*optionprice[i+1][j+1] + (1  prob)*optionprice[i+1][j])); } else{ //American put option optionprice[i][j] = Math.max(exercisePrice  nodeprice[i][j],
13 OPTION PRICING, JAVA PROGRAMMING AND MONTE CARLO SIMULATION 13 } } } } Math.exp(interest*deltaT) * (prob*optionprice[i+1][j+1] + (1  prob)*optionprice[i+1][j])); Figure 6. Estimation of the Price of European and American Options using the Binomial Method 4.2. Monte Carlo Method. We now move on to the second numerical method called the Monte Carlo Method. We estimate the price of a Europeanstyle option by generating M independent sample values, X 1, X 2,..., X M, of the payoff from the option in a riskneutral world. We then calculate the mean of the sample payoffs to obtain an estimate of the expected payoff. Finally, we discount the expected payoff at riskfree interest rate to get an estimate of the value of the option. This method is best understood by an example. Example 4. Let us use the Monte Carlo method to estimate the value of a European call option. To do this, we need two very important formulae. The first one will tell
14 14 NITESH AIDASANI KHYAMI us the asset price at maturity, and the second one will tell us the value of the option at maturity. These two formulae are as follows (4.2) S(T ) = S(0 )exp [(r σ2 2 = S(0 )exp [(r σ2 2 )T + σw T ] ) T + σɛ ] T where ɛ N(0, 1) (4.3) value of option at maturity = max(s(t ) K, 0 ). Now our Monte Carlo algorithm for estimating the value of the option is as follows for i = 1 to M end compute an N(0,1) sample ɛ i σ2 [(r set S i = S 0 e 2 )T +σɛ i T ] set X i = max(s i K, 0) set Mean = 1 M M i=1 X i set Estimate Of Value Of Option = e rt Mean It is important to understand and realize that there is no need to apply the Monte Carlo method in this particular example since the BlackScholes formula gives us the exact solution. However, there are many complicated situations where we will not have formulae analogue to the BlackScholes formula. Such is a situation is presented in the following few paragraphs. Now before we investigate this situation, let s have a look at the error of the Monte Carlo estimate of the value of the option. We know from Hull [1], p410, that the error of this estimate is ω M where ω is the standard deviation of the sample payoffs. This means that the error in the value of the option is inversely proportional to the square root of the number of trials. Thus, if we wish to double the accuracy of our estimate, we have to increase the number of trials by a factor of 4. Thus, a very large value of M is needed to estimate the value of the option to a reasonable accuracy. This makes the Monte Carlo method computationally very time consuming. 3 However, there are several 3 Another drawback of the Monte Carlo method is that it cannot easily handle situations where there are early exercise opportunities, like for example in the case of American options.
15 OPTION PRICING, JAVA PROGRAMMING AND MONTE CARLO SIMULATION 15 methods which can be used to reduce the variance which can lead to a considerable decrease in the computation time. One such method which we will be investigating is called the Antithetic Variate Technique Antithetic Variate Technique. The algorithm for the antithetic variate technique is very similar to the Monte Carlo algorithm given above. The main difference is that before we estimated the price of a Europeanstyle option by generating M independent sample values, X 1, X 2,..., X M, of the payoff from the option by using M random N(0, 1) samples, ɛ 1, ɛ 2,..., ɛ M. In the antithetic variable technique, we will be generating 2M sample values, X 1, X 2,..., X M and X 1, X 2,..., X M. If X i is calculated using ɛ i, then X i is calculated using ɛ i. In addition, before we calculated the mean of the X i s but now, with the antithetic variates, we calculate the mean of the X i s, where X i = 1(X 2 i + X i ). Example 5. Below is the corresponding antithetic variate version of the algorithm in example 4. for i = 1 to M end compute an N(0,1) sample ɛ i σ2 [(r set S i = S 0 e 2 )T +σɛ i T ] σ2 [(r set S i = S 0 e 2 )T σɛ i T ] set X i = max(s i K, 0) set X i = max(s i K, 0) set X i = 1 2 (X i + X i ) set Mean = 1 M M i=1 X i set Estimate Value Of Option = e rt Mean Another important part of the project was to write a Java program for pricing options using Monte Carlo simulations for a specified volatility model(meanreverting). The meanreverting volatility model was shown in equation 3.4. It is as follows (4.4) S (k+1) t = S k t + rs k t t + Y k t S k t W k t Y (k+1) t = Y k t + λ(µ Y k t ) t + ξy k t B k t where λ, µ and ξ are constants, and dw t and db t are Wiener processes. The variable, Y t, in this model is the asset s variance rate. This is the square of its volatility. We
16 16 NITESH AIDASANI KHYAMI also have that 0 k N 1 t = T N. This is an example of a situation where we have no analytical formulae, but a recurrence relation. In situations alike this one, the Monte Carlo method is very helpful. The pseudocode for the meanreverting model using Monte Carlo simulations is as follows 4 for i = 1 to M end for j = 0 to N 1 end compute an N(0,1) sample ɛ 1 compute an N(0,1) sample ɛ 2 set Y j+1 = Y j + λ(µ Y j ) t + ξy j ɛ 1 t set Y j+1 = Y j + λ(µ Y j ) t ξy j ɛ 1 t set S j+1 = S j (1 + r t + Y j ɛ 2 t) set S j+1 = S j (1 + r t Y j ɛ 2 t) if option = call set X i = max(s N K) set X i = max(s N K) otherwise end if set X i = max(k S N ) set X i = max(k S N ) set X i = 1 2 (X i + X i ) set Mean = 1 M M i=1 X i set Estimate Value Of Option = e rt Mean Once the above pseudocode was implemented in Java, the program was used to predict prices of two options and hence, the results were compared with the actual 4 Note that Y 0 = µ.
17 OPTION PRICING, JAVA PROGRAMMING AND MONTE CARLO SIMULATION 17 option prices. The two call options, Barclays and Marks and Spencer, were taken from LIFFE(International Financial Futures and Options Exchange in London). Tables 2 and 3 show the data for the two call options. Tables 4 and 5 show the results produced along with the percentage error between the predicted and the actual option prices. Name of option Option type Marks & Spencer Call Option Strike Price ( ) 130 Start Date 15/11/00 Expiry Date 29/12/00 Interest rate (%) 6 µ λ ξ Table 2. Data for Marks and Spencer Call Option Name of option Option type Barclays Call Option Strike Price ( ) 1050 Start Date 9/11/00 Expiry Date 17/01/01 Interest rate (%) 6 µ λ ξ Table 3. Data for Barclays Call Option I believe that my program produced quite accurate results since the average error was less than 1% in both cases(0.856% for the Marks and Spencer option and 0.151% for the Barclays option).
18 18 NITESH AIDASANI KHYAMI Date Option Price Stock Price Days to Predicted Option error ( ) ( ) Maturity Price ( ) (%) 15/11/ /11/ /11/ /11/ /11/ /11/ /11/ /11/ /11/ /11/ /11/ /11/ /12/ /12/ /12/ /12/ /12/ /12/ /12/ /12/ /12/ /12/ /12/ /12/ /12/ /12/ /12/ /12/ /12/ Table 4. Comparison of Actual and Predicted Value of Marks and Spencer Call Option
19 OPTION PRICING, JAVA PROGRAMMING AND MONTE CARLO SIMULATION 19 Date Option Price Stock Price Days to Predicted Option error ( ) ( ) Maturity Price ( ) (%) 9/11/ /11/ /11/ /11/ /11/ /11/ /11/ /11/ /11/ /11/ /11/ /11/ /11/ /11/ /11/ /11/ /12/ /12/ /12/ /12/ /12/ /12/ /12/ /12/ /12/ /12/ Table 5. Comparison of Actual and Predicted Value of Barclays Call Option 5. Conclusion In this project we have discussed some of the basic aspects of option pricing and how we utilize the BlackScholes formula when we have an analytical formula. Analytic solutions, are, of course, only one side of the coin. If we have efficient numerical
20 20 NITESH AIDASANI KHYAMI algorithms to find the solution, the lack of an analytic solution does not constitute a serious hindrance. We showed that the Monte Carlo method using antithetic variates can produce very accurate results, despite the fact that sometimes the speed of convergence can be quite slow. There are other numerical methods which can also be used such as finite difference methods and highaccuracy PDE methods. One of the main questions that rose in mind after the completion of this project was whether the stochastic volatility processes discussed in this project can adequately describe the market. I believe that the answer to this question is much more complex than what one might think at first. Part of the difficulty is defining what the initial volatility actually means. The shortest time period over which average volatility figures are available is 10 days. This means that the initial volatility is essentially known only in a very imprecise manner. The situation gets worse when we consider that most option trading occurs just a few days before maturity. The time period over which volatility data is collected and distributed by the exchanges and financial information agencies such as Reuters and Bloomberg has been set by convention and is not really binding. Therefore, it is still possible to directly analyze the trades to obtain short term volatility information. Hence, any meaningful empirical study will necessarily be large scale and involve considerable effort.
21 OPTION PRICING, JAVA PROGRAMMING AND MONTE CARLO SIMULATION 21 References [1] John C. Hull: Options, Futures and Other Derivatives. Prentice Hall, 2000 [2] Desmond J. Higham: An Introduction to Financial Option Valuation. Cambridge University Press, 2004 [3] Edited by Bruno Dupire: MONTE CARLO, Methodologies and Applications for Pricing and Risk Management. Risk Books, 1998 [4] H. M. and P. J. Deitel: Java, How to Program. Prentice Hall, 2002 [5] Helmut Kopka and Patrick W. Daly: A Guide to L A TEX 2ε. AddisonWesley, 1995
European Options Pricing Using Monte Carlo Simulation
European Options Pricing Using Monte Carlo Simulation Alexandros Kyrtsos Division of Materials Science and Engineering, Boston University akyrtsos@bu.edu European options can be priced using the analytical
More informationOptions and Derivative Pricing. U. NaikNimbalkar, Department of Statistics, Savitribai Phule Pune University.
Options and Derivative Pricing U. NaikNimbalkar, Department of Statistics, Savitribai Phule Pune University. email: uvnaik@gmail.com The slides are based on the following: References 1. J. Hull. Options,
More information第 9 讲 : 股 票 期 权 定 价 : BS 模 型 Valuing Stock Options: The BlackScholes Model
1 第 9 讲 : 股 票 期 权 定 价 : BS 模 型 Valuing Stock Options: The BlackScholes Model Outline 有 关 股 价 的 假 设 The BS Model 隐 性 波 动 性 Implied Volatility 红 利 与 期 权 定 价 Dividends and Option Pricing 美 式 期 权 定 价 American
More informationThe BlackScholes Formula
FIN40008 FINANCIAL INSTRUMENTS SPRING 2008 The BlackScholes Formula These notes examine the BlackScholes formula for European options. The BlackScholes formula are complex as they are based on the
More informationValuing Options / Volatility
Chapter 5 Valuing Options / Volatility Measures Now that the foundation regarding the basics of futures and options contracts has been set, we now move to discuss the role of volatility in futures and
More informationThe BlackScholes Model
The BlackScholes Model Liuren Wu Zicklin School of Business, Baruch College Options Markets (Hull chapter: 12, 13, 14) Liuren Wu The BlackScholes Model Options Markets 1 / 19 The BlackScholesMerton
More information金融隨機計算 : 第一章. BlackScholesMerton Theory of Derivative Pricing and Hedging. CH Han Dept of Quantitative Finance, Natl. TsingHua Univ.
金融隨機計算 : 第一章 BlackScholesMerton Theory of Derivative Pricing and Hedging CH Han Dept of Quantitative Finance, Natl. TsingHua Univ. Derivative Contracts Derivatives, also called contingent claims, are
More informationThe BlackScholes Formula
ECO30004 OPTIONS AND FUTURES SPRING 2008 The BlackScholes Formula The BlackScholes Formula We next examine the BlackScholes formula for European options. The BlackScholes formula are complex as they
More information1. Assume that a (European) call option exists on this stock having on exercise price of $155.
MØA 155 PROBLEM SET: Binomial Option Pricing Exercise 1. Call option [4] A stock s current price is $16, and there are two possible prices that may occur next period: $15 or $175. The interest rate on
More informationLecture Note of Bus 41202, Spring 2012: Stochastic Diffusion & Option Pricing
Lecture Note of Bus 41202, Spring 2012: Stochastic Diffusion & Option Pricing Key concept: Ito s lemma Stock Options: A contract giving its holder the right, but not obligation, to trade shares of a common
More informationNumerical Methods for Option Pricing
Chapter 9 Numerical Methods for Option Pricing Equation (8.26) provides a way to evaluate option prices. For some simple options, such as the European call and put options, one can integrate (8.26) directly
More informationMATH3075/3975 Financial Mathematics
MATH3075/3975 Financial Mathematics Week 11: Solutions Exercise 1 We consider the BlackScholes model M = B, S with the initial stock price S 0 = 9, the continuously compounded interest rate r = 0.01 per
More informationFinancial Modeling. An introduction to financial modelling and financial options. Conall O Sullivan
Financial Modeling An introduction to financial modelling and financial options Conall O Sullivan Banking and Finance UCD Smurfit School of Business 31 May / UCD Maths Summer School Outline Introduction
More informationAn Introduction to Exotic Options
An Introduction to Exotic Options Jeff Casey Jeff Casey is entering his final semester of undergraduate studies at Ball State University. He is majoring in Financial Mathematics and has been a math tutor
More informationAmerican and European. Put Option
American and European Put Option Analytical Finance I Kinda Sumlaji 1 Table of Contents: 1. Introduction... 3 2. Option Style... 4 3. Put Option 4 3.1 Definition 4 3.2 Payoff at Maturity... 4 3.3 Example
More informationCaput Derivatives: October 30, 2003
Caput Derivatives: October 30, 2003 Exam + Answers Total time: 2 hours and 30 minutes. Note 1: You are allowed to use books, course notes, and a calculator. Question 1. [20 points] Consider an investor
More informationMonte Carlo Methods in Finance
Author: Yiyang Yang Advisor: Pr. Xiaolin Li, Pr. Zari Rachev Department of Applied Mathematics and Statistics State University of New York at Stony Brook October 2, 2012 Outline Introduction 1 Introduction
More informationJorge Cruz Lopez  Bus 316: Derivative Securities. Week 11. The BlackScholes Model: Hull, Ch. 13.
Week 11 The BlackScholes Model: Hull, Ch. 13. 1 The BlackScholes Model Objective: To show how the BlackScholes formula is derived and how it can be used to value options. 2 The BlackScholes Model 1.
More informationTABLE OF CONTENTS. A. PutCall Parity 1 B. Comparing Options with Respect to Style, Maturity, and Strike 13
TABLE OF CONTENTS 1. McDonald 9: "Parity and Other Option Relationships" A. PutCall Parity 1 B. Comparing Options with Respect to Style, Maturity, and Strike 13 2. McDonald 10: "Binomial Option Pricing:
More informationTwoState Option Pricing
Rendleman and Bartter [1] present a simple twostate model of option pricing. The states of the world evolve like the branches of a tree. Given the current state, there are two possible states next period.
More informationExam MFE Spring 2007 FINAL ANSWER KEY 1 B 2 A 3 C 4 E 5 D 6 C 7 E 8 C 9 A 10 B 11 D 12 A 13 E 14 E 15 C 16 D 17 B 18 A 19 D
Exam MFE Spring 2007 FINAL ANSWER KEY Question # Answer 1 B 2 A 3 C 4 E 5 D 6 C 7 E 8 C 9 A 10 B 11 D 12 A 13 E 14 E 15 C 16 D 17 B 18 A 19 D **BEGINNING OF EXAMINATION** ACTUARIAL MODELS FINANCIAL ECONOMICS
More informationOption Valuation. Chapter 21
Option Valuation Chapter 21 Intrinsic and Time Value intrinsic value of inthemoney options = the payoff that could be obtained from the immediate exercise of the option for a call option: stock price
More informationDETERMINING THE VALUE OF EMPLOYEE STOCK OPTIONS. Report Produced for the Ontario Teachers Pension Plan John Hull and Alan White August 2002
DETERMINING THE VALUE OF EMPLOYEE STOCK OPTIONS 1. Background Report Produced for the Ontario Teachers Pension Plan John Hull and Alan White August 2002 It is now becoming increasingly accepted that companies
More informationMonte Carlo simulations and option pricing
Monte Carlo simulations and option pricing by Bingqian Lu Undergraduate Mathematics Department Pennsylvania State University University Park, PA 16802 Project Supervisor: Professor Anna Mazzucato July,
More informationBinomial lattice model for stock prices
Copyright c 2007 by Karl Sigman Binomial lattice model for stock prices Here we model the price of a stock in discrete time by a Markov chain of the recursive form S n+ S n Y n+, n 0, where the {Y i }
More informationOptions 1 OPTIONS. Introduction
Options 1 OPTIONS Introduction A derivative is a financial instrument whose value is derived from the value of some underlying asset. A call option gives one the right to buy an asset at the exercise or
More informationBlackScholes Equation for Option Pricing
BlackScholes Equation for Option Pricing By Ivan Karmazin, Jiacong Li 1. Introduction In early 1970s, Black, Scholes and Merton achieved a major breakthrough in pricing of European stock options and there
More informationResearch on Option Trading Strategies
Research on Option Trading Strategies An Interactive Qualifying Project Report: Submitted to the Faculty of the WORCESTER POLYTECHNIC INSTITUTE In partial fulfillment of the requirements for the Degree
More informationA SNOWBALL CURRENCY OPTION
J. KSIAM Vol.15, No.1, 31 41, 011 A SNOWBALL CURRENCY OPTION GYOOCHEOL SHIM 1 1 GRADUATE DEPARTMENT OF FINANCIAL ENGINEERING, AJOU UNIVERSITY, SOUTH KOREA Email address: gshim@ajou.ac.kr ABSTRACT. I introduce
More informationLecture 11: RiskNeutral Valuation Steven Skiena. skiena
Lecture 11: RiskNeutral Valuation Steven Skiena Department of Computer Science State University of New York Stony Brook, NY 11794 4400 http://www.cs.sunysb.edu/ skiena RiskNeutral Probabilities We can
More informationA Comparison of Option Pricing Models
A Comparison of Option Pricing Models Ekrem Kilic 11.01.2005 Abstract Modeling a nonlinear pay o generating instrument is a challenging work. The models that are commonly used for pricing derivative might
More informationPricing Barrier Option Using Finite Difference Method and MonteCarlo Simulation
Pricing Barrier Option Using Finite Difference Method and MonteCarlo Simulation Yoon W. Kwon CIMS 1, Math. Finance Suzanne A. Lewis CIMS, Math. Finance May 9, 000 1 Courant Institue of Mathematical Science,
More informationOption pricing. Vinod Kothari
Option pricing Vinod Kothari Notation we use this Chapter will be as follows: S o : Price of the share at time 0 S T : Price of the share at time T T : time to maturity of the option r : risk free rate
More informationValuing Stock Options: The BlackScholesMerton Model. Chapter 13
Valuing Stock Options: The BlackScholesMerton Model Chapter 13 Fundamentals of Futures and Options Markets, 8th Ed, Ch 13, Copyright John C. Hull 2013 1 The BlackScholesMerton Random Walk Assumption
More informationContinuous time; continuous variable stochastic process. We assume that stock prices follow Markov processes. That is, the future movements in a
Continuous time; continuous variable stochastic process. We assume that stock prices follow Markov processes. That is, the future movements in a variable depend only on the present, and not the history
More informationMore Exotic Options. 1 Barrier Options. 2 Compound Options. 3 Gap Options
More Exotic Options 1 Barrier Options 2 Compound Options 3 Gap Options More Exotic Options 1 Barrier Options 2 Compound Options 3 Gap Options Definition; Some types The payoff of a Barrier option is path
More informationFinancial Options: Pricing and Hedging
Financial Options: Pricing and Hedging Diagrams Debt Equity Value of Firm s Assets T Value of Firm s Assets T Valuation of distressed debt and equitylinked securities requires an understanding of financial
More informationA Genetic Algorithm to Price an European Put Option Using the Geometric Mean Reverting Model
Applied Mathematical Sciences, vol 8, 14, no 143, 7157135 HIKARI Ltd, wwwmhikaricom http://dxdoiorg/11988/ams144644 A Genetic Algorithm to Price an European Put Option Using the Geometric Mean Reverting
More informationBINOMIAL OPTION PRICING
Darden Graduate School of Business Administration University of Virginia BINOMIAL OPTION PRICING Binomial option pricing is a simple but powerful technique that can be used to solve many complex optionpricing
More informationOption Values. Option Valuation. Call Option Value before Expiration. Determinants of Call Option Values
Option Values Option Valuation Intrinsic value profit that could be made if the option was immediately exercised Call: stock price exercise price : S T X i i k i X S Put: exercise price stock price : X
More informationLecture 12: The BlackScholes Model Steven Skiena. http://www.cs.sunysb.edu/ skiena
Lecture 12: The BlackScholes Model Steven Skiena Department of Computer Science State University of New York Stony Brook, NY 11794 4400 http://www.cs.sunysb.edu/ skiena The BlackScholesMerton Model
More information1 The BlackScholes Formula
1 The BlackScholes Formula In 1973 Fischer Black and Myron Scholes published a formula  the BlackScholes formula  for computing the theoretical price of a European call option on a stock. Their paper,
More informationOPTIONS and FUTURES Lecture 2: Binomial Option Pricing and Call Options
OPTIONS and FUTURES Lecture 2: Binomial Option Pricing and Call Options Philip H. Dybvig Washington University in Saint Louis binomial model replicating portfolio single period artificial (riskneutral)
More informationMathematical Finance
Mathematical Finance Option Pricing under the RiskNeutral Measure Cory Barnes Department of Mathematics University of Washington June 11, 2013 Outline 1 Probability Background 2 Black Scholes for European
More informationFINANCIAL ECONOMICS OPTION PRICING
OPTION PRICING Options are contingency contracts that specify payoffs if stock prices reach specified levels. A call option is the right to buy a stock at a specified price, X, called the strike price.
More informationUCLA Anderson School of Management Daniel Andrei, Derivative Markets 237D, Winter 2014. MFE Midterm. February 2014. Date:
UCLA Anderson School of Management Daniel Andrei, Derivative Markets 237D, Winter 2014 MFE Midterm February 2014 Date: Your Name: Your Equiz.me email address: Your Signature: 1 This exam is open book,
More informationApplication of options in hedging of crude oil price risk
AMERICAN JOURNAL OF SOCIAL AND MANAGEMEN SCIENCES ISSN rint: 156154, ISSN Online: 1511559 doi:1.551/ajsms.1.1.1.67.74 1, ScienceHuβ, http://www.scihub.org/ajsms Application of options in hedging of crude
More informationSession X: Lecturer: Dr. Jose Olmo. Module: Economics of Financial Markets. MSc. Financial Economics. Department of Economics, City University, London
Session X: Options: Hedging, Insurance and Trading Strategies Lecturer: Dr. Jose Olmo Module: Economics of Financial Markets MSc. Financial Economics Department of Economics, City University, London Option
More informationGeometric Brownian motion makes sense for the call price because call prices cannot be negative. Now Using Ito's lemma we can nd an expression for dc,
12 Option Pricing We are now going to apply our continuoustime methods to the pricing of nonstandard securities. In particular, we will consider the class of derivative securities known as options in
More informationCall Price as a Function of the Stock Price
Call Price as a Function of the Stock Price Intuitively, the call price should be an increasing function of the stock price. This relationship allows one to develop a theory of option pricing, derived
More informationCS 522 Computational Tools and Methods in Finance Robert Jarrow Lecture 1: Equity Options
CS 5 Computational Tools and Methods in Finance Robert Jarrow Lecture 1: Equity Options 1. Definitions Equity. The common stock of a corporation. Traded on organized exchanges (NYSE, AMEX, NASDAQ). A common
More informationUnderstanding Options and Their Role in Hedging via the Greeks
Understanding Options and Their Role in Hedging via the Greeks Bradley J. Wogsland Department of Physics and Astronomy, University of Tennessee, Knoxville, TN 379961200 Options are priced assuming that
More informationOn BlackScholes Equation, Black Scholes Formula and Binary Option Price
On BlackScholes Equation, Black Scholes Formula and Binary Option Price Abstract: Chi Gao 12/15/2013 I. BlackScholes Equation is derived using two methods: (1) riskneutral measure; (2)  hedge. II.
More informationPutCall Parity. chris bemis
PutCall Parity chris bemis May 22, 2006 Recall that a replicating portfolio of a contingent claim determines the claim s price. This was justified by the no arbitrage principle. Using this idea, we obtain
More informationNotes on BlackScholes Option Pricing Formula
. Notes on BlackScholes Option Pricing Formula by DeXing Guan March 2006 These notes are a brief introduction to the BlackScholes formula, which prices the European call options. The essential reading
More informationModels Used in Variance Swap Pricing
Models Used in Variance Swap Pricing Final Analysis Report Jason Vinar, Xu Li, Bowen Sun, Jingnan Zhang Qi Zhang, Tianyi Luo, Wensheng Sun, Yiming Wang Financial Modelling Workshop 2011 Presentation Jan
More informationACTS 4302 SOLUTION TO MIDTERM EXAM Derivatives Markets, Chapters 9, 10, 11, 12, 18. October 21, 2010 (Thurs)
Problem ACTS 4302 SOLUTION TO MIDTERM EXAM Derivatives Markets, Chapters 9, 0,, 2, 8. October 2, 200 (Thurs) (i) The current exchange rate is 0.0$/. (ii) A fouryear dollardenominated European put option
More informationOne Period Binomial Model
FIN40008 FINANCIAL INSTRUMENTS SPRING 2008 One Period Binomial Model These notes consider the one period binomial model to exactly price an option. We will consider three different methods of pricing
More informationAn Introduction to Modeling Stock Price Returns With a View Towards Option Pricing
An Introduction to Modeling Stock Price Returns With a View Towards Option Pricing Kyle Chauvin August 21, 2006 This work is the product of a summer research project at the University of Kansas, conducted
More informationMonte Carlo Simulation in the Pricing of Derivatives. Cara M. Marshall. Fordham University. Queens College of the City University of New York
Monte Carlo Simulation in the Pricing of Derivatives Cara M. Marshall Fordham University Queens College of the City University of New York 1 Monte Carlo Simulation in the Pricing of Derivatives Of all
More informationPractice Set #7: Binomial option pricing & Delta hedging. What to do with this practice set?
Derivatives (3 credits) Professor Michel Robe Practice Set #7: Binomial option pricing & Delta hedging. What to do with this practice set? To help students with the material, eight practice sets with solutions
More informationQUANTIZED INTEREST RATE AT THE MONEY FOR AMERICAN OPTIONS
QUANTIZED INTEREST RATE AT THE MONEY FOR AMERICAN OPTIONS L. M. Dieng ( Department of Physics, CUNY/BCC, New York, New York) Abstract: In this work, we expand the idea of Samuelson[3] and Shepp[,5,6] for
More information2. Exercising the option  buying or selling asset by using option. 3. Strike (or exercise) price  price at which asset may be bought or sold
Chapter 21 : Options1 CHAPTER 21. OPTIONS Contents I. INTRODUCTION BASIC TERMS II. VALUATION OF OPTIONS A. Minimum Values of Options B. Maximum Values of Options C. Determinants of Call Value D. BlackScholes
More informationFinancial Modeling. Class #06B. Financial Modeling MSS 2012 1
Financial Modeling Class #06B Financial Modeling MSS 2012 1 Class Overview Equity options We will cover three methods of determining an option s price 1. BlackScholesMerton formula 2. Binomial trees
More informationFinite Differences Schemes for Pricing of European and American Options
Finite Differences Schemes for Pricing of European and American Options Margarida Mirador Fernandes IST Technical University of Lisbon Lisbon, Portugal November 009 Abstract Starting with the BlackScholes
More informationCall and Put. Options. American and European Options. Option Terminology. Payoffs of European Options. Different Types of Options
Call and Put Options A call option gives its holder the right to purchase an asset for a specified price, called the strike price, on or before some specified expiration date. A put option gives its holder
More informationLecture 6: Option Pricing Using a Onestep Binomial Tree. Friday, September 14, 12
Lecture 6: Option Pricing Using a Onestep Binomial Tree An oversimplified model with surprisingly general extensions a single time step from 0 to T two types of traded securities: stock S and a bond
More informationThe Binomial Option Pricing Model André Farber
1 Solvay Business School Université Libre de Bruxelles The Binomial Option Pricing Model André Farber January 2002 Consider a nondividend paying stock whose price is initially S 0. Divide time into small
More informationInstitutional Finance 08: Dynamic Arbitrage to Replicate Nonlinear Payoffs. Binomial Option Pricing: Basics (Chapter 10 of McDonald)
Copyright 2003 Pearson Education, Inc. Slide 081 Institutional Finance 08: Dynamic Arbitrage to Replicate Nonlinear Payoffs Binomial Option Pricing: Basics (Chapter 10 of McDonald) Originally prepared
More informationBlackScholes Option Pricing Model
BlackScholes Option Pricing Model Nathan Coelen June 6, 22 1 Introduction Finance is one of the most rapidly changing and fastest growing areas in the corporate business world. Because of this rapid change,
More informationUnderlying Pays a Continuous Dividend Yield of q
Underlying Pays a Continuous Dividend Yield of q The value of a forward contract at any time prior to T is f = Se qτ Xe rτ. (33) Consider a portfolio of one long forward contract, cash amount Xe rτ, and
More informationBlackScholesMerton approach merits and shortcomings
BlackScholesMerton approach merits and shortcomings Emilia Matei 1005056 EC372 Term Paper. Topic 3 1. Introduction The BlackScholes and Merton method of modelling derivatives prices was first introduced
More informationBetting on Volatility: A Delta Hedging Approach. Liang Zhong
Betting on Volatility: A Delta Hedging Approach Liang Zhong Department of Mathematics, KTH, Stockholm, Sweden April, 211 Abstract In the financial market, investors prefer to estimate the stock price
More informationBeyond Black Scholes: Smile & Exo6c op6ons. Professor André Farber Solvay Brussels School of Economics and Management Université Libre de Bruxelles
Beyond Black Scholes: Smile & Exo6c op6ons Professor André Farber Solvay Brussels School of Economics and Management Université Libre de Bruxelles 1 What is a Vola6lity Smile? Rela6onship between implied
More informationStochastic Modelling and Forecasting
Stochastic Modelling and Forecasting Department of Statistics and Modelling Science University of Strathclyde Glasgow, G1 1XH RSE/NNSFC Workshop on Management Science and Engineering and Public Policy
More informationLecture 8: Random Walk vs. Brownian Motion, Binomial Model vs. LogNormal Distribution
Lecture 8: Random Walk vs. Brownian Motion, Binomial Model vs. Logormal Distribution October 4, 200 Limiting Distribution of the Scaled Random Walk Recall that we defined a scaled simple random walk last
More informationChapter 11 Options. Main Issues. Introduction to Options. Use of Options. Properties of Option Prices. Valuation Models of Options.
Chapter 11 Options Road Map Part A Introduction to finance. Part B Valuation of assets, given discount rates. Part C Determination of riskadjusted discount rate. Part D Introduction to derivatives. Forwards
More informationUCLA Anderson School of Management Daniel Andrei, Option Markets 232D, Fall MBA Final Exam. December Date:
UCLA Anderson School of Management Daniel Andrei, Option Markets 232D, Fall 2013 MBA Final Exam December 2013 Date: Your Name: Your Equiz.me email address: Your Signature: 1 This exam is open book, open
More informationFUNDING INVESTMENTS FINANCE 238/738, Spring 2008, Prof. Musto Class 5 Review of Option Pricing
FUNDING INVESTMENTS FINANCE 238/738, Spring 2008, Prof. Musto Class 5 Review of Option Pricing I. PutCall Parity II. OnePeriod Binomial Option Pricing III. Adding Periods to the Binomial Model IV. BlackScholes
More informationApplications of Stochastic Processes in Asset Price Modeling
Applications of Stochastic Processes in Asset Price Modeling TJHSST Computer Systems Lab Senior Research Project 20082009 Preetam D Souza November 11, 2008 Abstract Stock market forecasting and asset
More informationReturn to Risk Limited website: www.risklimited.com. Overview of Options An Introduction
Return to Risk Limited website: www.risklimited.com Overview of Options An Introduction Options Definition The right, but not the obligation, to enter into a transaction [buy or sell] at a preagreed price,
More informationHow to Value Employee Stock Options
John Hull and Alan White One of the arguments often used against expensing employee stock options is that calculating their fair value at the time they are granted is very difficult. This article presents
More informationWeek 12. Options on Stock Indices and Currencies: Hull, Ch. 15. Employee Stock Options: Hull, Ch. 14.
Week 12 Options on Stock Indices and Currencies: Hull, Ch. 15. Employee Stock Options: Hull, Ch. 14. 1 Options on Stock Indices and Currencies Objective: To explain the basic asset pricing techniques used
More informationPart V: Option Pricing Basics
erivatives & Risk Management First Week: Part A: Option Fundamentals payoffs market microstructure Next 2 Weeks: Part B: Option Pricing fundamentals: intrinsic vs. time value, putcall parity introduction
More information3. Monte Carlo Simulations. Math6911 S08, HM Zhu
3. Monte Carlo Simulations Math6911 S08, HM Zhu References 1. Chapters 4 and 8, Numerical Methods in Finance. Chapters 17.617.7, Options, Futures and Other Derivatives 3. George S. Fishman, Monte Carlo:
More informationValuing equitybased payments
E Valuing equitybased payments Executive remuneration packages generally comprise many components. While it is relatively easy to identify how much will be paid in a base salary a fixed dollar amount
More informationChapter 21 Valuing Options
Chapter 21 Valuing Options Multiple Choice Questions 1. Relative to the underlying stock, a call option always has: A) A higher beta and a higher standard deviation of return B) A lower beta and a higher
More informationPricing Options Using Trinomial Trees
Pricing Options Using Trinomial Trees Paul Clifford Oleg Zaboronski 17.11.2008 1 Introduction One of the first computational models used in the financial mathematics community was the binomial tree model.
More informationStochastic Processes and Advanced Mathematical Finance. Options and Derivatives
Steven R. Dunbar Department of Mathematics 203 Avery Hall University of NebraskaLincoln Lincoln, NE 685880130 http://www.math.unl.edu Voice: 4024723731 Fax: 4024728466 Stochastic Processes and Advanced
More informationChapter 13 The BlackScholesMerton Model
Chapter 13 The BlackScholesMerton Model March 3, 009 13.1. The BlackScholes option pricing model assumes that the probability distribution of the stock price in one year(or at any other future time)
More informationOption Values. Determinants of Call Option Values. CHAPTER 16 Option Valuation. Figure 16.1 Call Option Value Before Expiration
CHAPTER 16 Option Valuation 16.1 OPTION VALUATION: INTRODUCTION Option Values Intrinsic value  profit that could be made if the option was immediately exercised Call: stock price  exercise price Put:
More informationHedging. An Undergraduate Introduction to Financial Mathematics. J. Robert Buchanan. J. Robert Buchanan Hedging
Hedging An Undergraduate Introduction to Financial Mathematics J. Robert Buchanan 2010 Introduction Definition Hedging is the practice of making a portfolio of investments less sensitive to changes in
More informationPricing Barrier Options under Local Volatility
Abstract Pricing Barrier Options under Local Volatility Artur Sepp Mail: artursepp@hotmail.com, Web: www.hot.ee/seppar 16 November 2002 We study pricing under the local volatility. Our research is mainly
More informationFINANCIAL OPTION ANALYSIS HANDOUTS
FINANCIAL OPTION ANALYSIS HANDOUTS 1 2 FAIR PRICING There is a market for an object called S. The prevailing price today is S 0 = 100. At this price the object S can be bought or sold by anyone for any
More informationValuation of Asian Options
Valuation of Asian Options  with Levy Approximation Master thesis in Economics Jan 2014 Author: Aleksandra Mraovic, Qian Zhang Supervisor: Frederik Lundtofte Department of Economics Abstract Asian options
More informationLectures. Sergei Fedotov. 20912  Introduction to Financial Mathematics. No tutorials in the first week
Lectures Sergei Fedotov 20912  Introduction to Financial Mathematics No tutorials in the first week Sergei Fedotov (University of Manchester) 20912 2010 1 / 1 Lecture 1 1 Introduction Elementary economics
More informationReview of Basic Options Concepts and Terminology
Review of Basic Options Concepts and Terminology March 24, 2005 1 Introduction The purchase of an options contract gives the buyer the right to buy call options contract or sell put options contract some
More informationFinance 436 Futures and Options Review Notes for Final Exam. Chapter 9
Finance 436 Futures and Options Review Notes for Final Exam Chapter 9 1. Options: call options vs. put options, American options vs. European options 2. Characteristics: option premium, option type, underlying
More informationP1. Preface. Thank you for choosing ACTEX.
Preface P Preface Thank you for choosing ACTEX. Since Exam MFE was introduced in May 007, there have been quite a few changes to its syllabus and its learning objectives. To cope with these changes, ACTEX
More information