Effects of electricity price volatility and covariance on the firm s investment decisions and long-run demand for electricity
|
|
- Warren Sparks
- 8 years ago
- Views:
Transcription
1 Effects of electricity price volatility and covariance on the firm s investment decisions and long-run demand for electricity C. Brandon (cbrandon@andrew.cmu.edu) Carnegie Mellon University, Pittsburgh, PA December 31, 2004 Abstract We examine how firms investment decisions, and hence their long run electricity demand, are affected by two likely features of a deregulated electricity market price volatility and covariance with other input factor prices. Building on Abel s (1983) investment model, we show that uncertainty from price volatility leads to an increase in investment when the electricity prices are either unrelated or positively correlated with other factor prices. A negative correlation between electricity prices and other input prices, however, causes price volatility to lead to a decrease in investment. Implications for electricity market design are discussed. 1 Introduction Deregulation, by letting prices be determined in a market, may increase price volatility. Such volatility creates uncertainty for customers and may encourage suppliers to turn to dynamic pricing schemes that pass on volatility to their customers. For example, following the deregulation of the electricity markets in California, small commercial customers of San Diego Gas and Electric (SDGE) saw the charges for electricity generation fluctuate between cents per kw and cents per kw with the space of a year. In addition, the experience of deregulation in California has led to calls for dynamic prcing of electricity (Borenstein et al. 2002, Smith and Kiesling, 2003). Apart from increasing price volatility, deregulation may also cause electricity prices to become correlated with the prices of other factors of production. In Previously titled The Reaction of Small Firms to Fluctuating Electricity Prices The author gratefully acknowledges financial support from the Carnegie Mellon Electricity Industry Center. Additional thanks go to Greg Katsapis and his collegues at San Diego Gas & Electric for their valuable help. 1
2 California, for example, about a third of SDGE s small firm customers use both natural gas and electricity. Natural gas has also been the primary fuel for instate electricity generation, accounting for, by 2002, 48.7% of total generation (DOE 2004). Thus firms who use both natural gas and electricity may find their prices correlated. To date, few studies have examined the effects of electricity price volatility and covariance on firm investment decisions and long-run electricity demand. We address this gap by developing a general theoretical model. We use data from SDGE to illustrate the potential for significant correlation between wages and electricity prices and for variation in this correlation across different industries. We also discuss the potential implications of our model for market design. 2 Model Our model is a generalized version of Abel s (1983) model of investment under uncertainty, which analyses the behavior of a profit maximizing firm facing a stochastic price for its output. Although this model has been faulted for ignoring the role of irreversibility in investment decisions, it provides a convenient framework to discuss how co-movements and volatility in input prices affects investment. To simplify analysis we restrict ourselves to three inputs; a semi-fixed input, capital and two variable factors of production (x 1, x 2 ). The firm in our model has a Cobb-Douglas production function and faces exogenous, stochastic input prices for the variable factors of production. These prices follow a multi-dimensional geometric Brownian motion. The output consists of a single homogenous good sold in a competitive international market i.e. regional input price shocks do not increase the firm s output price. The firm seeks to maximize expected profit over time subject to the following constraints: dk t = (I t δk t ) dt (1) dp t /p t = σ p dz p (2) dp 1,t /p 1,t = σ p1 dz p1 (3) dp 2,t /p 2,t = σ p2 dz p2 (4) dz i dz j = ρ ij dt, i, j {p t, p 1,t, p 2,t }. (5) Subject to these constraints, firm s optimization problem is described by: V (K t, p t, p 1,t, p 2,t ) = [ ps X1,sK α s X γ 2,s p 1,sX 1,s τis φ ] p 2,s X 2,s e r(s t) ds max X 1,X 2,I t (6) Where K t, X 1,t, X 2,t denote capital and the two variable factors of production. I t is the amount of investment at time t, while the costs of the new investment 2
3 is captured by τi φ t where φ > 1, τ > 0 are both model parameters and. The unit costs of X 1 and X 2 are p 1, p 2,t respectively, while p t is the price of the firm s output. r is the time discount factor. 1 ρ i,j 1, ρ = 1 if i = j, while ρ p,p1 = ρ p,p2 = 0. φ is a constant whose value is assumed to be greater than unity due to the presence of adjustment costs. Dynamic programming methods show that the optimal solution requires: rv (K t, p t, p 1,t, p 2,t ) dt = [ ] max p t X α X 1,X 2,I t 1,tK t X γ 2,t p 1,tX 1,t τi φ t p 2,t X 2,t dt + E t dv (7) Assuming that the value function is a function of all the state variables, K t, p t, p 1,t, p 2,t we can apply the multi-dimensional Ito s lemma (see the appendix) and substitute for dk t to get the following expression for X 2,t dv. E t dv = (V K (I t δk t ) + p 1,t p 2,t V p1p 2 σ p1 σ p2 ρ p1p 2 ) dt ( p2 t V pp σp p2 1,tV p1p 1 σp ) 2 p2 2,tV p2p 2 σp 2 2 dt (8) Substituting the above for E t dv in the condition for optimality yields the following first order conditions with respect to X 1,t, X 2,t and I t : αp t X α 1 1,t K t X γ 2,t p 1,t = 0 (9) γp t X α 1,tK t X γ 1 2,t p 2,t = 0 (10) φτi φ 1 t + V K = 0 (11) Algebraic manipulation of the first order conditions yields the optimal values of the control variables solely in terms of the state variables and the parameters of the model: [( ) ( ) γ ] 1/(1 α γ) αpt γp1,t X 1,t = K t (12) p 1,t [( ) ( γpt X 2,t = p 2,t 2,t 2,t γp 1,t ) α ] 1/(1 α γ) K t (13) I t = ( ) 1/(φ 1) VK (14) φτ At the moment we do not know V K. However, by substituting for the optimal levels of the control variables in our condition for optimality, we obtain a second order differential equation that defines the value function, V (K, p, p 1, p 2 ) 1 [ (p2 ) γ (p1 ) α rv = (1 α γ) K γ α ] 1 1 α γ + (φ 1) τ ( ) φ/(φ 1) VK φτ V K δk + p 1 p 2 V p1p 2 σ p1 σ p2 ρ p1p 2 + p2 2 V ppσ 2 p + p2 1 2 V p 1p 1 σ 2 p 1 + p2 2 2 V p 2p 2 σ 2 p 2 (15) 1 We dropped the time subscript to aid clarity, since no meaning is lost by doing so 3
4 To get a closed form solution for the value function, we now need to assume constant returns to scale technology. In doing so, the above expression simplifies to: rv = ( ) γ p2 γ ( p1 ) α K + (φ 1) τ α ( ) φ/(φ 1) VK V K δk φτ +p 1 p 2 V p1p 2 σ p1 σ p2 ρ p1p 2 + p2 2 V ppσ 2 p + p2 1 2 V p 1p 1 σ 2 p 1 + p2 2 2 V p 2p 2 σ 2 p 2 (16) It can be readily checked that the solution to this equation takes the following form: where (1 ) x = 2 2 σp 2 α (α + ) 2 2 σp 2 1 ( φ z = φ 1 V (K, p, p 1, p 2 ) = mh y + nhk (17) h = p 1 α p y = 1 p γ φ φ 1 2 (18) (19) n = γ γ α α r + δ + x (20) m = 1 (φ 1) r + z (21) σp 2 2 αγ 2 σ p 1 σ p2 ρ p1p 2 (22) γ (γ + ) 2 2 ) 2 x + φ ( ασp γσp 2 2 σp) (φ 1) 2 (23) By substituting these values into (12-14) we obtain the optimal values for X 1, X 2 demand and the level of investment. X 1,t = p 1 t X 2,t = p 1 t ( α p 1,t ( γ p 2,t ) 1 γ ) 1 α I t = ( γ p 2,t ( α ( nh φτ p 1,t ) γ Kt (24) ) α Kt (25) ) 1/(φ 1) (26) Note that since 1 γ = α +, inspection reveals that the relationship between optimal X 2 demand and the demand for X 1 is what we would expect, namely X 2,t = γp 1,t αp 2 X 1,t The reader is welcome to check that when we set σ p1, σ p2, γ = 0 and = 1 α our answers for optimal X 1 and investment match those in Abel s (1983) model. 4
5 Effects of input price volatility on investment and the long run demand for X 2 Having derived the expression for optimal investment above we see that volatility affects investment by affecting the value of n. Since φ > 1 we know that investment is increasing in n. In turn, since input price volatility only enters the expression for n through the value of x, we know that anything that increases the value of x will decrease the optimal level of investment (and vice versa). Thus the sign of the partial derivatives of σ p1 and σ p2 determine whether an increase in input price volatility will increase or decrease investment. x α (α + ) = σ p1 2 σ p1 αγ 2 σ p 2 ρ p1p 2 (27) x γ (γ + ) = σ p2 2 σ p2 αγ 2 σ p 1 ρ p1p 2 (28) These results are consistent with those of the original model. So long as the correlation between the price of X 2 and the price of X 1 is positive, an increase in input price volatility reduces the value of x, increases the risk to expected profits and thus boosts investment. However, a negative relationship between the price of X 1 and the price of X 2 could potentially reverse this effect. For example investment decreases with increased X 2 price volatility if the covariance between p 1 and p 2 is negative so that: 1 γ γ ( σp1 σ p2 ) < ρ p1p 2 (29) In other words, if the X 2 price becomes significantly more volatile than the price of X 1, and the two are negatively correlated, the effect will be a reduction in investment. This is quite different to the results of the original model where increased price volatility always increased investment. Similarly, investment decreases with increased volatility in p 1 if the covariance between p 1 and p 2 is negative enough that: 1 α α ( σp2 σ p1 ) < ρ p1p 2 (30) But what might cause a negative correlation between p 1 and p 2? In our model the firm competes in an international market, thus if the costs of the firm s foreign competitors remain unchanged any increase in X 2 price will force firms to cut costs or go out of business, either could depress p 1. Alternatively, if X 2 became very cheap, firms would exploit the cost advantages to boost output and thus would bid up p 1. 2 Abel uses where we have used φ and γ where we have used τ 5
6 3 An empirical example: correlation between electricity prices and wages To see if this perverse effect could be anything beyond a theoretical anomaly, and to see if the effect of input price covariance could have a significant effect, we decided to look at the labor costs and electricity prices faced by firms in deregulated electricity markets. We obtained data from SDGE on their small non-residential customers for the period from September 1999 to September We examined the correlations between wages and the commodity cost of electricity paid by manufacturing firms. We looked at different two digit SIC codes to see if there were differences in the relationship between electricity prices and wages across industries. While most industries have positive correlations between wages and electricity prices in this period, firms in SIC 28 (chemical products) experienced strong negative correlation. Depending on when they were billed each month, the correlation between wages and electricity prices ranged from to , while the standard deviations in wages ranged from The range for the deviation in electricity price was from Thus, even if we use the most conservative values α > is enough to ensure the second inequality holds true. If α > is a fair characterization for these firms, then our model suggests that any increase in wage volatility would result in a decrease in investment. In our model, electricity price volatility affects electricity demand only in the long run, through its effect on investment. Looking at (25) reveals that electricity consumption is linear in the capital stock. Thus any increase in the capital stock increases electricity consumption while anything to decrease it leads to a drop in electricity demand. Thus the long term effects of a permanent increase in electricity price volatility on electricity consumption will depend on whether it increases investment (in which case demand will rise) or not. The above results may be modified by the inclusion of irreversibility in investment. Irreversibility dampens the positive impact of profit uncertainty on investment through the value of the option to delay investment. The inclusion of input price variance and co-variance, however, affects the magnitude of the uncertainty associated with profits. Therefore the inclusion of irreversible investment does not alter the arguement that input price covariance and volatility must be considered when determining the effects of uncertainty on investment. 4 Implications Real time pricing (RTP), where retail prices reflect changes in the wholesale price in real time, could involve dramatic price volatility for consumers. While the California experience was unusual and partly due to price manipulation, the SDGE tariff wasn t a RTP, but rather a weighted average of the RTP. Yet the commodity costs per kw still fluctuated from a little over 3 cents to 23 cents a kw. Without providing easy and accessible hedging options for small customers (which would effectively undermine the very incentives RTP is trying to create) 6
7 our model shows that this could cause significant problems for some firms. RTP may be an attractive option only for firms who can easily smooth consumption. Our model also suggests an interesting trade-off. Deregulating electricity is expected to reduce the average price consumers pay for their power. On the other hand, in the absence of accessible hedging options for small businesses, it also exposes these firms to greater electricity price uncertainty. Our model suggests that how firms respond may vary across industries and that if electricity prices fail to fall much, not all electricity consumers will be better off if they are exposed to fluctuating prices. The key implication is that different market designs can affect not just the average market clearing price but also price volatility and how it may co-vary with other factor prices. This in turn can have a significant impact on the purchasing firms investment plans and thus its long run demand for different inputs. We believe that this implies that the context of the market becomes important. Different industries will have different technological options and different market structures for their input and output markets, so that any changes to the market structure of one input will affect different industries in different ways. 5 References Abel, Andrew B. (1983): Optimal Investment Under Uncertainty, American Economic Review, 13: Borenstein, Severin, Michael Jaske, and Arthur Rosenfeld, (2002): Dynamic Pricing, Advanced Metering, and Demand Response in Electricity Markets, Hewlett Foundation Energy Series, available at DOE (2004) profiles/california.pdf Smith, Vernon L. and Lynne Kiesling: Demand, Not Supply Wall Street Journal, New York, N.Y.: pg. A 10. August 20, 2003 Zarnikau, Jay (1990): Customer Responsiveness to Real-Time Pricing of Electricity, Energy Journal, 11(4), A Multidimensional Ito s lemma applied to a geometric Brownian motion Let f (X 1, X 2,..., X n ) be a function where the variables X 1, X 2,..., X n follow GBM of the form: dx i (t) = µ i X i (t) dt + σ i X i (t) dw i (t), (31) 7
8 with the correlation coefficients defined as: dw i dw j = ρ ij dt, i, j = 1, 2,..., n. (32) 1 ρ ij 1 and ρ ij = 1 when i = j. Then applying Ito s lemma we get: n df = f µ i + f X i t + 1 n σ 2 2 f i 2 i=1 i=1 X 2 i + 2 f ρ ij σ i σ j dt + X i X j i j n i=1 σ i f X i dw i (33) B A note on the data The manufacturing wage data came from the Labor Market Information for California website. Wages were calculated as average hours * average hourly earnings. We used information for firms located in the San Diego MSA from archived hours and earnings data files. These files are available to the public at The firms in our sample did not all face the same electricity prices. This is because SDGE bills its customers based on a bill cycle. The bill cycle number determines what day the customer s meter will be read for each month. The result of this arrangement is that different firms have bills that cover different periods. As a result, all variables (including wages) have to be weighted accordingly to ensure a fair comparison. We include a table with the meter reading dates so that readers can see how the data was adjusted. Information on the commodity charge (PX) that customers paid came from SDGE s website and is currently available at Finally, we chose to ignore two additional complicating factors that were not essential to our arguements. Firstly, SDGE actually uses 22 bill cycle numbers, the last code represents customized billing arrangements, which were not observable. Secondly, customers located within the city limits of San Diego also had to pay a 1.9% tax so their actual PX costs were higher. 8
9 Table 1: Dates Meters Read Month Bill Group 8/1999 8/3 8/4 8/5 8/6 8/9 8/10 8/11 9/1999 9/1 9/2 9/7 9/8 9/9 9/10 9/13 10/ /4 10/5 10/6 10/7 10/8 10/11 10/12 11/ /2 11/3 11/4 11/5 11/8 11/9 11/10 12/ /2 12/3 12/6 12/7 12/8 12/9 12/10 1/2000 1/3 1/4 1/5 1/6 1/7 1/10 1/11 2/2000 2/2 2/3 2/4 2/7 2/8 2/9 2/10 3/2000 3/3 3/6 3/7 3/8 3/9 3/10 3/13 4/2000 4/3 4/4 4/5 4/6 4/7 4/10 4/11 5/2000 5/2 5/3 5/4 5/5 5/8 5/9 5/10 6/2000 6/1 6/2 6/5 6/6 6/7 6/8 6/9 7/2000 6/30 7/5 7/6 7/7 7/10 7/11 7/12 8/2000 8/2 8/3 8/4 8/7 8/8 8/9 8/10 9/2000 8/31 9/1 9/5 9/6 9/7 9/8 9/11 10/ /2 10/3 10/4 10/5 10/6 10/9 10/10 11/ /31 11/1 11/2 11/3 11/6 11/7 11/8 12/ /30 12/1 12/4 12/5 12/6 12/7 12/8 1/2001 1/2 1/3 1/4 1/5 1/8 1/9 1/10 2/2001 1/31 2/1 2/2 2/5 2/6 2/7 2/8 3/2001 3/2 3/5 3/6 3/7 3/8 3/9 3/12 4/2001 4/2 4/3 4/4 4/5 4/6 4/9 4/10 5/2001 5/1 5/2 5/3 5/4 5/7 5/8 5/9 6/2001 5/31 6/1 6/4 6/5 6/6 6/7 6/8 7/2001 6/29 7/2 7/3 7/5 7/6 7/9 7/10 8/2001 7/31 8/1 8/2 8/3 8/6 8/7 8/8 9/2001 8/29 8/30 9/4 9/5 9/6 9/7 9/10 10/ /1 10/2 10/3 10/4 10/5 10/8 10/9 11/ /30 10/31 11/1 11/2 11/5 11/6 11/7 12/ /30 12/3 12/4 12/5 12/6 12/7 12/10 1/2002 1/2 1/3 1/4 1/7 1/8 1/9 1/10 2/2002 1/31 2/1 2/4 2/5 2/6 2/7 2/8 3/2002 3/4 3/5 3/6 3/7 3/8 3/11 3/12 4/2002 4/2 4/3 4/4 4/5 4/8 4/9 4/10 5/2002 5/1 5/2 5/3 5/6 5/7 5/8 5/9 6/2002 5/31 6/3 6/4 6/5 6/6 6/7 6/10 7/2002 7/1 7/2 7/3 7/8 7/9 7/10 7/11 8/2002 8/1 8/2 8/5 8/6 8/7 8/8 8/9 9/2002 8/30 9/3 9/4 9/5 9/6 9/9 9/10 9
10 Table 1 continued Month Bill Group 8/1999 8/12 8/13 8/16 8/17 8/18 8/19 8/20 9/1999 9/14 9/15 9/16 9/17 9/20 9/21 9/22 10/ /13 10/14 10/15 10/18 10/19 10/20 10/21 11/ /11 11/12 11/15 11/16 11/17 11/18 11/19 12/ /13 12/14 12/15 12/16 12/17 12/20 12/21 1/2000 1/12 1/13 1/14 1/18 1/19 1/20 1/21 2/2000 2/11 2/14 2/15 2/16 2/17 2/18 2/22 3/2000 3/14 3/15 3/16 3/17 3/20 3/21 3/22 4/2000 4/12 4/13 4/14 4/17 4/18 4/19 4/20 5/2000 5/11 5/12 5/15 5/16 5/17 5/18 5/19 6/2000 6/12 6/13 6/14 6/15 6/16 6/19 6/20 7/2000 7/13 7/14 7/17 7/18 7/19 7/20 7/21 8/2000 8/11 8/14 8/15 8/16 8/17 8/18 8/21 9/2000 9/12 9/13 9/14 9/15 9/18 9/19 9/20 10/ /11 10/12 10/13 10/16 10/17 10/18 10/19 11/ /9 11/10 11/13 11/14 11/15 11/16 11/17 12/ /11 12/12 12/13 12/14 12/15 12/18 12/19 1/2001 1/11 1/12 1/15 1/16 1/17 1/18 1/19 2/2001 2/9 2/12 2/13 2/14 2/15 2/16 2/20 3/2001 3/13 3/14 3/15 3/16 3/19 3/20 3/21 4/2001 4/11 4/12 4/13 4/16 4/17 4/18 4/19 5/2001 5/10 5/11 5/14 5/15 5/16 5/17 5/18 6/2001 6/11 6/12 6/13 6/14 6/15 6/18 6/19 7/2001 7/11 7/12 7/13 7/16 7/17 7/18 7/19 8/2001 8/9 8/10 8/13 8/14 8/15 8/16 8/17 9/2001 9/11 9/12 9/13 9/14 9/17 9/18 9/19 10/ /10 10/11 10/12 10/15 10/16 10/17 10/18 11/ /8 11/9 11/12 11/13 11/14 11/15 11/16 12/ /11 12/12 12/13 12/14 12/17 12/18 12/19 1/2002 1/11 1/14 1/15 1/16 1/17 1/18 1/21 2/2002 2/11 2/12 2/13 2/14 2/15 2/19 2/20 3/2002 3/13 3/14 3/15 3/18 3/19 3/20 3/21 4/2002 4/11 4/12 4/15 4/16 4/17 4/18 4/19 5/2002 5/10 5/13 5/14 5/15 5/16 5/17 5/20 6/2002 6/11 6/12 6/13 6/14 6/17 6/18 6/19 7/2002 7/12 7/15 7/16 7/17 7/18 7/19 7/22 8/2002 8/12 8/13 8/14 8/15 8/16 8/19 8/20 9/2002 9/11 9/12 9/13 9/16 9/17 9/18 9/19 10
11 Table 1 continued Month Bill Group 8/1999 8/23 8/24 8/25 8/26 8/27 8/30 8/31 9/1999 9/23 9/24 9/27 9/28 9/29 9/30 10/1 10/ /22 10/25 10/26 10/27 10/28 10/29 11/1 11/ /22 11/23 11/24 11/26 11/29 11/30 12/1 12/ /22 12/23 12/27 12/28 12/29 12/30 12/31 1/2000 1/24 1/25 1/26 1/27 1/28 1/31 2/1 2/2000 2/23 2/24 2/25 2/28 2/29 3/1 3/2 3/2000 3/23 3/24 3/27 3/28 3/29 3/30 3/31 4/2000 4/21 4/24 4/25 4/26 4/27 4/28 5/1 5/2000 5/22 5/23 5/24 5/25 5/26 5/30 5/31 6/2000 6/21 6/22 6/23 6/26 6/27 6/28 6/29 7/2000 7/24 7/25 7/26 7/27 7/28 7/31 8/1 8/2000 8/22 8/23 8/24 8/25 8/28 8/29 8/30 9/2000 9/21 9/22 9/25 9/26 9/27 9/28 9/29 10/ /20 10/23 10/24 10/25 10/26 10/27 10/30 11/ /20 11/21 11/22 11/24 11/27 11/28 11/29 12/ /20 12/21 12/22 12/26 12/27 12/28 12/29 1/2001 1/22 1/23 1/24 1/25 1/26 1/29 1/30 2/2001 2/21 2/22 2/23 2/26 2/27 2/28 3/1 3/2001 3/22 3/23 3/26 3/27 3/28 3/29 3/30 4/2001 4/20 4/23 4/24 4/25 4/26 4/27 4/30 5/2001 5/21 5/22 5/23 5/24 5/25 5/29 5/30 6/2001 6/20 6/21 6/22 6/25 6/26 6/27 6/28 7/2001 7/20 7/23 7/24 7/25 7/26 7/27 7/30 8/2001 8/20 8/21 8/22 8/23 8/24 8/27 8/28 9/2001 9/20 9/21 9/24 9/25 9/26 9/27 9/28 10/ /19 10/22 10/23 10/24 10/25 10/26 10/29 11/ /19 11/20 11/21 11/26 11/27 11/28 11/29 12/ /20 12/21 12/24 12/26 12/27 12/28 12/31 1/2002 1/22 1/23 1/24 1/25 1/28 1/29 1/30 2/2002 2/21 2/22 2/25 2/26 2/27 2/28 3/1 3/2002 3/22 3/25 3/26 3/27 3/28 3/29 4/1 4/2002 4/22 4/23 4/24 4/25 4/26 4/29 4/30 5/2002 5/21 5/22 5/23 5/24 5/28 5/29 5/30 6/2002 6/20 6/21 6/24 6/25 6/26 6/27 6/28 7/2002 7/23 7/24 7/25 7/26 7/29 7/30 7/31 8/2002 8/21 8/22 8/23 8/26 8/27 8/28 8/29 9/2002 9/20 9/23 9/24 9/25 9/26 9/27 9/30 11
Current Accounts in Open Economies Obstfeld and Rogoff, Chapter 2
Current Accounts in Open Economies Obstfeld and Rogoff, Chapter 2 1 Consumption with many periods 1.1 Finite horizon of T Optimization problem maximize U t = u (c t ) + β (c t+1 ) + β 2 u (c t+2 ) +...
More informationOn Market-Making and Delta-Hedging
On Market-Making and Delta-Hedging 1 Market Makers 2 Market-Making and Bond-Pricing On Market-Making and Delta-Hedging 1 Market Makers 2 Market-Making and Bond-Pricing What to market makers do? Provide
More informationThe Behavior of Bonds and Interest Rates. An Impossible Bond Pricing Model. 780 w Interest Rate Models
780 w Interest Rate Models The Behavior of Bonds and Interest Rates Before discussing how a bond market-maker would delta-hedge, we first need to specify how bonds behave. Suppose we try to model a zero-coupon
More informationECON20310 LECTURE SYNOPSIS REAL BUSINESS CYCLE
ECON20310 LECTURE SYNOPSIS REAL BUSINESS CYCLE YUAN TIAN This synopsis is designed merely for keep a record of the materials covered in lectures. Please refer to your own lecture notes for all proofs.
More informationBlack-Scholes Equation for Option Pricing
Black-Scholes 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 informationEfficient Retail Pricing in Electricity and Natural Gas Markets
Efficient Retail Pricing in Electricity and Natural Gas Markets Steven L. Puller Department of Economics Texas A&M University and NBER Jeremy West Department of Economics Texas A&M University January 2013
More informationVolatility at Karachi Stock Exchange
The Pakistan Development Review 34 : 4 Part II (Winter 1995) pp. 651 657 Volatility at Karachi Stock Exchange ASLAM FARID and JAVED ASHRAF INTRODUCTION Frequent crashes of the stock market reported during
More informationConvenience Yield-Based Pricing of Commodity Futures
Convenience Yield-Based Pricing of Commodity Futures Takashi Kanamura, J-POWER BFS2010 6th World Congress in Toronto, Canada June 26th, 2010 1 Agenda 1. The objectives and results 2. The convenience yield-based
More informationThe RBC methodology also comes down to two principles:
Chapter 5 Real business cycles 5.1 Real business cycles The most well known paper in the Real Business Cycles (RBC) literature is Kydland and Prescott (1982). That paper introduces both a specific theory
More informationThe Valuation of Currency Options
The Valuation of Currency Options Nahum Biger and John Hull Both Nahum Biger and John Hull are Associate Professors of Finance in the Faculty of Administrative Studies, York University, Canada. Introduction
More informationLecture. S t = S t δ[s t ].
Lecture In real life the vast majority of all traded options are written on stocks having at least one dividend left before the date of expiration of the option. Thus the study of dividends is important
More informationThis PDF is a selection from a published volume from the National Bureau of Economic Research
This PDF is a selection from a published volume from the National Bureau of Economic Research Volume Title: Fiscal Policy and Management in East Asia, NBER-EASE, Volume 16 Volume Author/Editor: Takatoshi
More informationLecture 14 More on Real Business Cycles. Noah Williams
Lecture 14 More on Real Business Cycles Noah Williams University of Wisconsin - Madison Economics 312 Optimality Conditions Euler equation under uncertainty: u C (C t, 1 N t) = βe t [u C (C t+1, 1 N t+1)
More informationWhen to Refinance Mortgage Loans in a Stochastic Interest Rate Environment
When to Refinance Mortgage Loans in a Stochastic Interest Rate Environment Siwei Gan, Jin Zheng, Xiaoxia Feng, and Dejun Xie Abstract Refinancing refers to the replacement of an existing debt obligation
More informationMathematical Finance
Mathematical Finance Option Pricing under the Risk-Neutral Measure Cory Barnes Department of Mathematics University of Washington June 11, 2013 Outline 1 Probability Background 2 Black Scholes for European
More informationELECTRICITY REAL OPTIONS VALUATION
Vol. 37 (6) ACTA PHYSICA POLONICA B No 11 ELECTRICITY REAL OPTIONS VALUATION Ewa Broszkiewicz-Suwaj Hugo Steinhaus Center, Institute of Mathematics and Computer Science Wrocław University of Technology
More information第 9 讲 : 股 票 期 权 定 价 : B-S 模 型 Valuing Stock Options: The Black-Scholes Model
1 第 9 讲 : 股 票 期 权 定 价 : B-S 模 型 Valuing Stock Options: The Black-Scholes Model Outline 有 关 股 价 的 假 设 The B-S 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 informationMarkups and Firm-Level Export Status: Appendix
Markups and Firm-Level Export Status: Appendix De Loecker Jan - Warzynski Frederic Princeton University, NBER and CEPR - Aarhus School of Business Forthcoming American Economic Review Abstract This is
More informationThe VAR models discussed so fare are appropriate for modeling I(0) data, like asset returns or growth rates of macroeconomic time series.
Cointegration The VAR models discussed so fare are appropriate for modeling I(0) data, like asset returns or growth rates of macroeconomic time series. Economic theory, however, often implies equilibrium
More informationInvestment in deepwater oil and gas exploration projects: A. multi-factor analysis with a real options model
Investment in deepwater oil and gas exploration projects: A multi-factor analysis with a real options model Abstract Deepwater oil and gas projects embody high risks from geologic and engineering aspects,
More informationDoes Black-Scholes framework for Option Pricing use Constant Volatilities and Interest Rates? New Solution for a New Problem
Does Black-Scholes framework for Option Pricing use Constant Volatilities and Interest Rates? New Solution for a New Problem Gagan Deep Singh Assistant Vice President Genpact Smart Decision Services Financial
More informationWhat is the (Real Option) Value of a College Degree?
What is the (Real Option) Value of a College Degree? JEFFREY R. STOKES 313 Curris Business Building Department of Finance College of Business Administration University of Northern Iowa, Cedar Falls, IA
More informationThe Black-Scholes-Merton Approach to Pricing Options
he Black-Scholes-Merton Approach to Pricing Options Paul J Atzberger Comments should be sent to: atzberg@mathucsbedu Introduction In this article we shall discuss the Black-Scholes-Merton approach to determining
More informationIntermediate Macroeconomics: The Real Business Cycle Model
Intermediate Macroeconomics: The Real Business Cycle Model Eric Sims University of Notre Dame Fall 2012 1 Introduction Having developed an operational model of the economy, we want to ask ourselves the
More informationOptimal proportional reinsurance and dividend pay-out for insurance companies with switching reserves
Optimal proportional reinsurance and dividend pay-out for insurance companies with switching reserves Abstract: This paper presents a model for an insurance company that controls its risk and dividend
More informationInternational Stock Market Integration: A Dynamic General Equilibrium Approach
International Stock Market Integration: A Dynamic General Equilibrium Approach Harjoat S. Bhamra London Business School 2003 Outline of talk 1 Introduction......................... 1 2 Economy...........................
More informationShort-Run Drivers Response to Un unexpected Gasoline Price Spike
Short-run Driver Response to a Gasoline Price Spike: Evidence from San Diego, CA Andrew Narwold University of San Diego Dirk Yandell University of San Diego Drivers response to an unexpected gasoline price
More informationLecture 6 Black-Scholes PDE
Lecture 6 Black-Scholes PDE Lecture Notes by Andrzej Palczewski Computational Finance p. 1 Pricing function Let the dynamics of underlining S t be given in the risk-neutral measure Q by If the contingent
More informationMaximum likelihood estimation of mean reverting processes
Maximum likelihood estimation of mean reverting processes José Carlos García Franco Onward, Inc. jcpollo@onwardinc.com Abstract Mean reverting processes are frequently used models in real options. For
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 informationJoint Product Signals of Quality. James E. McClure and Lee C. Spector. Ball State University Muncie, Indiana. September 1991.
Joint Product Signals of Quality By James E. McClure and Lee C. Spector Ball State University Muncie, Indiana September 1991 Abstract In the absence of other information about the quality of an experience
More information1 National Income and Product Accounts
Espen Henriksen econ249 UCSB 1 National Income and Product Accounts 11 Gross Domestic Product (GDP) Can be measured in three different but equivalent ways: 1 Production Approach 2 Expenditure Approach
More informationFour Derivations of the Black Scholes PDE by Fabrice Douglas Rouah www.frouah.com www.volopta.com
Four Derivations of the Black Scholes PDE by Fabrice Douglas Rouah www.frouah.com www.volopta.com In this Note we derive the Black Scholes PDE for an option V, given by @t + 1 + rs @S2 @S We derive the
More information2. Real Business Cycle Theory (June 25, 2013)
Prof. Dr. Thomas Steger Advanced Macroeconomics II Lecture SS 13 2. Real Business Cycle Theory (June 25, 2013) Introduction Simplistic RBC Model Simple stochastic growth model Baseline RBC model Introduction
More informationThe Black-Scholes Formula
FIN-40008 FINANCIAL INSTRUMENTS SPRING 2008 The Black-Scholes Formula These notes examine the Black-Scholes formula for European options. The Black-Scholes formula are complex as they are based on the
More informationHow the Greeks would have hedged correlation risk of foreign exchange options
How the Greeks would have hedged correlation risk of foreign exchange options Uwe Wystup Commerzbank Treasury and Financial Products Neue Mainzer Strasse 32 36 60261 Frankfurt am Main GERMANY wystup@mathfinance.de
More informationwhere N is the standard normal distribution function,
The Black-Scholes-Merton formula (Hull 13.5 13.8) Assume S t is a geometric Brownian motion w/drift. Want market value at t = 0 of call option. European call option with expiration at time T. Payout at
More informationUniversidad de Montevideo Macroeconomia II. The Ramsey-Cass-Koopmans Model
Universidad de Montevideo Macroeconomia II Danilo R. Trupkin Class Notes (very preliminar) The Ramsey-Cass-Koopmans Model 1 Introduction One shortcoming of the Solow model is that the saving rate is exogenous
More informationLIBRARY OF THE MASSACHUSETTS INSTITUTE OF TECHNOLOGY
LIBRARY OF THE MASSACHUSETTS INSTITUTE OF TECHNOLOGY Digitized by the Internet Archive in 2011 with funding from Boston Library Consortium IVIember Libraries http://www.archive.org/details/calloptionpricinoofisc
More informationVI. Real Business Cycles Models
VI. Real Business Cycles Models Introduction Business cycle research studies the causes and consequences of the recurrent expansions and contractions in aggregate economic activity that occur in most industrialized
More informationReal Estate Investments with Stochastic Cash Flows
Real Estate Investments with Stochastic Cash Flows Riaz Hussain Kania School of Management University of Scranton Scranton, PA 18510 hussain@scranton.edu 570-941-7497 April 2006 JEL classification: G12
More informationReal Business Cycle Models
Real Business Cycle Models Lecture 2 Nicola Viegi April 2015 Basic RBC Model Claim: Stochastic General Equlibrium Model Is Enough to Explain The Business cycle Behaviour of the Economy Money is of little
More informationImport Prices and Inflation
Import Prices and Inflation James D. Hamilton Department of Economics, University of California, San Diego Understanding the consequences of international developments for domestic inflation is an extremely
More informationDerivation of Local Volatility by Fabrice Douglas Rouah www.frouah.com www.volopta.com
Derivation of Local Volatility by Fabrice Douglas Rouah www.frouah.com www.volopta.com The derivation of local volatility is outlined in many papers and textbooks (such as the one by Jim Gatheral []),
More informationLecture 15. Sergei Fedotov. 20912 - Introduction to Financial Mathematics. Sergei Fedotov (University of Manchester) 20912 2010 1 / 6
Lecture 15 Sergei Fedotov 20912 - Introduction to Financial Mathematics Sergei Fedotov (University of Manchester) 20912 2010 1 / 6 Lecture 15 1 Black-Scholes Equation and Replicating Portfolio 2 Static
More informationThe Black-Scholes pricing formulas
The Black-Scholes pricing formulas Moty Katzman September 19, 2014 The Black-Scholes differential equation Aim: Find a formula for the price of European options on stock. Lemma 6.1: Assume that a stock
More informationVanna-Volga Method for Foreign Exchange Implied Volatility Smile. Copyright Changwei Xiong 2011. January 2011. last update: Nov 27, 2013
Vanna-Volga Method for Foreign Exchange Implied Volatility Smile Copyright Changwei Xiong 011 January 011 last update: Nov 7, 01 TABLE OF CONTENTS TABLE OF CONTENTS...1 1. Trading Strategies of Vanilla
More informationMechanics 1: Conservation of Energy and Momentum
Mechanics : Conservation of Energy and Momentum If a certain quantity associated with a system does not change in time. We say that it is conserved, and the system possesses a conservation law. Conservation
More informationMA Macroeconomics 10. Growth Accounting
MA Macroeconomics 10. Growth Accounting Karl Whelan School of Economics, UCD Autumn 2014 Karl Whelan (UCD) Growth Accounting Autumn 2014 1 / 20 Growth Accounting The final part of this course will focus
More informationVolatility Models for Commodity Markets. by Paul L. Fackler and Yanjun Tian
Volatility Models for Commodity Markets by Paul L. Fackler and Yanjun Tian Suggested citation format: Fackler, P. L., and Y. Tian. 1999. Volatility Models for Commodity Markets. Proceedings of the NCR-134
More informationSTATE OF NEW YORK PUBLIC SERVICE COMMISSION. At a session of the Public Service Commission held in the City of Albany on February 13, 2008
STATE OF NEW YORK PUBLIC SERVICE COMMISSION At a session of the Public Service Commission held in the City of Albany on February 13, 2008 COMMISSIONERS PRESENT: Garry A. Brown, Chairman Patricia L. Acampora
More informationChapter 21: Consumer Behavior and Utility Maximization
Chapter : Consumer Behavior and Utility Maximization ANSWERS TO END-OF-CHAPTER QUESTIONS - Explain the law of demand through the income and substitution effects, using a price increase as a point of departure
More informationCHAPTER 8 FACTOR EXTRACTION BY MATRIX FACTORING TECHNIQUES. From Exploratory Factor Analysis Ledyard R Tucker and Robert C.
CHAPTER 8 FACTOR EXTRACTION BY MATRIX FACTORING TECHNIQUES From Exploratory Factor Analysis Ledyard R Tucker and Robert C MacCallum 1997 180 CHAPTER 8 FACTOR EXTRACTION BY MATRIX FACTORING TECHNIQUES In
More informationImplications of Intellectual Property Rights for Dynamic Gains from Trade
FEDERAL RESERVE BANK OF SAN FRANCISCO WORKING PAPER SERIES Implications of Intellectual Property Rights for Dynamic Gains from Trade Michele Connolly Duke University and Diego Valderrama Federal Reserve
More informationhttp://www.jstor.org This content downloaded on Tue, 19 Feb 2013 17:28:43 PM All use subject to JSTOR Terms and Conditions
A Significance Test for Time Series Analysis Author(s): W. Allen Wallis and Geoffrey H. Moore Reviewed work(s): Source: Journal of the American Statistical Association, Vol. 36, No. 215 (Sep., 1941), pp.
More informationTechnical Efficiency Accounting for Environmental Influence in the Japanese Gas Market
Technical Efficiency Accounting for Environmental Influence in the Japanese Gas Market Sumiko Asai Otsuma Women s University 2-7-1, Karakida, Tama City, Tokyo, 26-854, Japan asai@otsuma.ac.jp Abstract:
More informationA Simple Model of Price Dispersion *
Federal Reserve Bank of Dallas Globalization and Monetary Policy Institute Working Paper No. 112 http://www.dallasfed.org/assets/documents/institute/wpapers/2012/0112.pdf A Simple Model of Price Dispersion
More informationWinter Impacts of Energy Efficiency In New England
Winter Impacts of Energy Efficiency In New England April 2015 Investments in electric efficiency since 2000 reduced electric demand in New England by over 2 gigawatts. 1 These savings provide significant
More informationThe Heston Model. Hui Gong, UCL http://www.homepages.ucl.ac.uk/ ucahgon/ May 6, 2014
Hui Gong, UCL http://www.homepages.ucl.ac.uk/ ucahgon/ May 6, 2014 Generalized SV models Vanilla Call Option via Heston Itô s lemma for variance process Euler-Maruyama scheme Implement in Excel&VBA 1.
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 informationQuarterly Wholesale/Retail Price Report
Quarterly Wholesale/Retail Price Report February 29 Contents Overview 3 Summary of analysis 1. Customer bills, wholesale costs and margins 5 Electricity Gas 2. Scenario analysis: Impact of retail price
More informationHow credit analysts view and use the financial statements
How credit analysts view and use the financial statements Introduction Traditionally it is viewed that equity investment is high risk and bond investment low risk. Bondholders look at companies for creditworthiness,
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 Black-Scholes
More informationLecture 11: The Greeks and Risk Management
Lecture 11: The Greeks and Risk Management This lecture studies market risk management from the perspective of an options trader. First, we show how to describe the risk characteristics of derivatives.
More informationAdoption of Residential Solar Power Under Uncertainty: Implications for Renewable Energy Incentives
Adoption of Residential Solar Power Under Uncertainty: Implications for Renewable Energy Incentives Christoph Bauner Postdoctoral research associate, Department of Resource Economics, University of Massachusetts
More informationAuctioning Keywords in Online Search
Auctioning Keywords in Online Search Jianqing Chen The Uniersity of Calgary iachen@ucalgary.ca De Liu Uniersity of Kentucky de.liu@uky.edu Andrew B. Whinston Uniersity of Texas at Austin abw@uts.cc.utexas.edu
More informationEXP 481 -- Capital Markets Option Pricing. Options: Definitions. Arbitrage Restrictions on Call Prices. Arbitrage Restrictions on Call Prices 1) C > 0
EXP 481 -- Capital Markets Option Pricing imple arbitrage relations Payoffs to call options Black-choles model Put-Call Parity Implied Volatility Options: Definitions A call option gives the buyer the
More information. In this case the leakage effect of tax increases is mitigated because some of the reduction in disposable income would have otherwise been saved.
Chapter 4 Review Questions. Explain how an increase in government spending and an equal increase in lump sum taxes can generate an increase in equilibrium output. Under what conditions will a balanced
More informationOnline Appendix for Demand for Crash Insurance, Intermediary Constraints, and Risk Premia in Financial Markets
Online Appendix for Demand for Crash Insurance, Intermediary Constraints, and Risk Premia in Financial Markets Hui Chen Scott Joslin Sophie Ni August 3, 2015 1 An Extension of the Dynamic Model Our model
More informationVolatility, Productivity Correlations and Measures of. International Consumption Risk Sharing.
Volatility, Productivity Correlations and Measures of International Consumption Risk Sharing. Ergys Islamaj June 2014 Abstract This paper investigates how output volatility and productivity correlations
More informationSECOND-DEGREE PRICE DISCRIMINATION
SECOND-DEGREE PRICE DISCRIMINATION FIRST Degree: The firm knows that it faces different individuals with different demand functions and furthermore the firm can tell who is who. In this case the firm extracts
More informationMeasurement and Mitigation of Market Power in Wholesale Electricity Markets
Measurement and Mitigation of Market Power in Wholesale Electricity Markets Frank A. Wolak Department of Economics Stanford University Stanford, CA 94305-6072 wolak@zia.stanford.edu http://www.stanford.edu/~wolak
More informationCapital Constraints, Lending over the Cycle and the Precautionary Motive: A Quantitative Exploration (Working Paper)
Capital Constraints, Lending over the Cycle and the Precautionary Motive: A Quantitative Exploration (Working Paper) Angus Armstrong and Monique Ebell National Institute of Economic and Social Research
More informationSome Practical Issues in FX and Equity Derivatives
Some Practical Issues in FX and Equity Derivatives Phenomenology of the Volatility Surface The volatility matrix is the map of the implied volatilities quoted by the market for options of different strikes
More informationThe Effects ofVariation Between Jain Mirman and JMC
MARKET STRUCTURE AND INSIDER TRADING WASSIM DAHER AND LEONARD J. MIRMAN Abstract. In this paper we examine the real and financial effects of two insiders trading in a static Jain Mirman model (Henceforth
More informationThe theory of storage and the convenience yield. 2008 Summer School - UBC 1
The theory of storage and the convenience yield 2008 Summer School - UBC 1 The theory of storage and the normal backwardation theory explain the relationship between the spot and futures prices in commodity
More informationPricing in a Competitive Market with a Common Network Resource
Pricing in a Competitive Market with a Common Network Resource Daniel McFadden Department of Economics, University of California, Berkeley April 8, 2002 I. This note is concerned with the economics of
More informationBrownian Motion and Stochastic Flow Systems. J.M Harrison
Brownian Motion and Stochastic Flow Systems 1 J.M Harrison Report written by Siva K. Gorantla I. INTRODUCTION Brownian motion is the seemingly random movement of particles suspended in a fluid or a mathematical
More informationThe Golden Rule. Where investment I is equal to the savings rate s times total production Y: So consumption per worker C/L is equal to:
The Golden Rule Choosing a National Savings Rate What can we say about economic policy and long-run growth? To keep matters simple, let us assume that the government can by proper fiscal and monetary policies
More informationAsset Management Contracts and Equilibrium Prices
Asset Management Contracts and Equilibrium Prices ANDREA M. BUFFA DIMITRI VAYANOS PAUL WOOLLEY Boston University London School of Economics London School of Economics September, 2013 Abstract We study
More informationValuation of Lease Contracts In Continuous Time With Stochastic Asset Values
Valuation of Lease Contracts In Continuous Time With Stochastic Asset Values Riaz Hussain *+ * The University of Scranton Abstract A lease is a derivative security the value of which depends upon the value
More informationAffine-structure models and the pricing of energy commodity derivatives
Affine-structure models and the pricing of energy commodity derivatives Nikos K Nomikos n.nomikos@city.ac.uk Cass Business School, City University London Joint work with: Ioannis Kyriakou, Panos Pouliasis
More informationGeneration Asset Valuation with Operational Constraints A Trinomial Tree Approach
Generation Asset Valuation with Operational Constraints A Trinomial Tree Approach Andrew L. Liu ICF International September 17, 2008 1 Outline Power Plants Optionality -- Intrinsic vs. Extrinsic Values
More informationInflation. Chapter 8. 8.1 Money Supply and Demand
Chapter 8 Inflation This chapter examines the causes and consequences of inflation. Sections 8.1 and 8.2 relate inflation to money supply and demand. Although the presentation differs somewhat from that
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 informationKeynesian Macroeconomic Theory
2 Keynesian Macroeconomic Theory 2.1. The Keynesian Consumption Function 2.2. The Complete Keynesian Model 2.3. The Keynesian-Cross Model 2.4. The IS-LM Model 2.5. The Keynesian AD-AS Model 2.6. Conclusion
More informationMacroeconomics Lecture 1: The Solow Growth Model
Macroeconomics Lecture 1: The Solow Growth Model Richard G. Pierse 1 Introduction One of the most important long-run issues in macroeconomics is understanding growth. Why do economies grow and what determines
More informationMoreover, under the risk neutral measure, it must be the case that (5) r t = µ t.
LECTURE 7: BLACK SCHOLES THEORY 1. Introduction: The Black Scholes Model In 1973 Fisher Black and Myron Scholes ushered in the modern era of derivative securities with a seminal paper 1 on the pricing
More informationFIN 411 -- Investments Option Pricing. Options: Definitions. Arbitrage Restrictions on Call Prices. Arbitrage Restrictions on Call Prices
FIN 411 -- Investments Option Pricing imple arbitrage relations s to call options Black-choles model Put-Call Parity Implied Volatility Options: Definitions A call option gives the buyer the right, but
More informationDetermination of Optimum Fair Premiums in Property-Liability Insurance: An Optimal Control Theoretic Approach
Determination of Optimum Fair Premiums in Property-Liability Insurance: An Optimal Control Theoretic Approach by Amin Ussif and Gregory Jones ABSTRACT Dynamic valuation models for the computation of optimum
More informationA.2 The Prevalence of Transfer Pricing in International Trade
19. Transfer Prices A. The Transfer Price Problem A.1 What is a Transfer Price? 19.1 When there is a international transaction between say two divisions of a multinational enterprise that has establishments
More informationChapter 22 Real Options
Chapter 22 Real Options Multiple Choice Questions 1. The following are the main types of real options: (I) The option to expand if the immediate investment project succeeds (II) The option to wait (and
More informationMerton-Black-Scholes model for option pricing. Peter Denteneer. 22 oktober 2009
Merton-Black-Scholes model for option pricing Instituut{Lorentz voor Theoretische Natuurkunde, LION, Universiteit Leiden 22 oktober 2009 With inspiration from: J. Tinbergen, T.C. Koopmans, E. Majorana,
More information3 Results. σdx. df =[µ 1 2 σ 2 ]dt+ σdx. Integration both sides will form
Appl. Math. Inf. Sci. 8, No. 1, 107-112 (2014) 107 Applied Mathematics & Information Sciences An International Journal http://dx.doi.org/10.12785/amis/080112 Forecasting Share Prices of Small Size Companies
More information3 Introduction to Assessing Risk
3 Introduction to Assessing Risk Important Question. How do we assess the risk an investor faces when choosing among assets? In this discussion we examine how an investor would assess the risk associated
More informationAssessing the Value of Flexibility in the Dutch Office Sector using Real Option Analysis
Assessing the Value of Flexibility in the Dutch Office Sector using Real Option Analysis Joost P. Poort, Jun Hoo Abstract Flexibility is one of the key aspects of the Zuidas project, a major construction
More informationGENERAL EQUILIBRIUM WITH BANKS AND THE FACTOR-INTENSITY CONDITION
GENERAL EQUILIBRIUM WITH BANKS AND THE FACTOR-INTENSITY CONDITION Emanuel R. Leão Pedro R. Leão Junho 2008 WP nº 2008/63 DOCUMENTO DE TRABALHO WORKING PAPER General Equilibrium with Banks and the Factor-Intensity
More informationEnvelope Theorem. Kevin Wainwright. Mar 22, 2004
Envelope Theorem Kevin Wainwright Mar 22, 2004 1 Maximum Value Functions A maximum (or minimum) value function is an objective function where the choice variables have been assigned their optimal values.
More informationBINOMIAL OPTIONS PRICING MODEL. Mark Ioffe. Abstract
BINOMIAL OPTIONS PRICING MODEL Mark Ioffe Abstract Binomial option pricing model is a widespread numerical method of calculating price of American options. In terms of applied mathematics this is simple
More informationRelative prices and Balassa Samuleson e ect
Relative prices and Balassa Samuleson e ect Prof. Ester Faia, Ph.D. Johann Wolfgang Goethe Universität Frankfurt a.m. March 2009 rof. Ester Faia, Ph.D. (Johann Wolfgang Goethe Relative Universität prices
More information