Mathematical Appendix

Mathematical Appendix There are libraries full of textbooks on applied mathematics and there is no point trying to replicate these here. On the other hand, it can be very frustrating for a reader to spend a lot of time digging out the necessary mathematics when his objective is to understand options as fast as possible. We therefore quickly skim through a few areas which are essential for an understanding of option theory, and present the mathematical tools in a format which is immediately applicable. Many of the mathematical problems of option theory were first solved as physics problems, and the physics vernacular has crept into the options literature. We follow this practice and make no attempt to present the material in a pure or abstract form; in any case, intuitive understanding is often increased by an appreciation of the underlying physical process. A.1 DISTRIBUTIONS AND INTEGRALS (i) Probability Distribution Functions: If F(x) is the probability distribution function for a ran- dom variable x, the probability P[x < a]isgivenby Px < a=  a −∞ F(x)dx or F(x) = ∂P[x < a] ∂a  a→x For two random variables, similar results hold: Px < a; y < b=  a −∞  b −∞ F(x, y)dx dy or F(x, y)= ∂ 2 P [ x < a; y < b ] ∂a ∂b  a→x ; b→y (A1.1) (ii) Normal Distribution: The expression x ∼ N (µ, σ 2 ) means that x is a random variable (variate), normally distributed with mean µ and variance σ 2 . A special case of the normal distribution is the standard normal distribution which has mean 0 and variance 1. The probability density function for the standard normal variate z is n(z) = 1 √ 2π e − 1 2 z 2 which is displayed in Figure A1.1. Mathematical Appendix n(z )area = N[Z ] 0 Z -Z Figure A1.1 Normal distribution function The cumulative distribution function is the shaded area in Figure A1.1: N[Z] = 1 √ 2π  Z −∞ e − 1 2 z 2 dz There is no closed form expression for this integral, which must be solved by numerical methods. We will not give an evaluation method for N[Z] here, as it is included as a standard function in spread sheets such as Excel. The converse formula is also used in this book: ∂N[z] ∂z = 1 √ 2π e − 1 2 z 2 = n(z) (A1.2) (iii) From the symmetry of the normal distribution function about the y-axis, we can write N[−Z ] = 1 √ 2π  −Z −∞ e − 1 2 z 2 dz = 1 √ 2π  +∞ Z e − 1 2 z 2 dz (A1.3) Given that the area under the curve must be 1, symmetry also allows us to write N[Z ] + N[−Z ] = 1 (A1.4) (iv) Lognormal Distribution: If S t is a random variable and x t = ln S t is normally distributed, then S t is said to be lognormally distributed; this is assumed to be the case for most securities, exchange and commodity prices. The well-known normal distribution of x is symmetrical about the mean, and x can take either positive or negative values. The position of the normal distribution function is determined by the mean while its shape (tall and thin vs. short and fat) is determined by the variance. However, ln S is not defined for negative S so that the lognormal distribution is taken as zero for negative values of S. This fits rather well with securities which cannot have negative prices. The precise shape of the lognormal distribution function depends on both its mean and variance: a sample of normal distributions with different means (but the same variance) is shown in Figure A1.2, together with their associated lognormal distribution functions. 300 A.1 DISTRIBUTIONS AND INTEGRALS Normal Mean : negative Normal Mean : zero Normal Mean : positive 00 0 Figure A1.2 Normal and lognormal distributions (v) Some Useful Integrals: A number of integrals occur repeatedly in option theory and the most important are given in this Appendix. (A) I Z −∞ (a) =  Z −∞ e az n(z)dz = 1 √ 2π  Z −∞ e az− 1 2 z 2 dz = e 1 2 a 2 1 √ 2π  Z −∞ e − 1 2 (z−a) 2 dz = e 1 2 a 2 1 √ 2π  Z−a −∞ e − 1 2 y 2 dy = e 1 2 a 2 N[Z − a] (A1.5) (B) The same factorization of terms in the exponential is used in the following: I +∞ Z (a) =  +∞ Z e az n(z)dz = e 1 2 a 2  +∞ Z−a n(y)dy = e 1 2 a 2 N[a − Z] (A1.6) where we have also used equation (A1.1). (C) Commonly used integrals in option theory are used to evaluate conditional expecta- tions such as E[S T − X : X < S T ], where z T = [ln(S T /S 0 ) − mT]/σ √ T and m = r − q − 1 2 σ 2 . Four results are given here which come directly from (A) and (B) above r E[K : S T < X] = K P[S T < X] = K P[z T < Z X ] = K  Z X −∞ n(z T )dz T = K N[Z X ] where Z X = [ln(X/S 0 ) − mT]/σ √ T . r E[K : X < S T ] = K P[X < S T ] = K P[Z X < z T ] = K  +∞ Z X n(z T )dz T = K N[−Z X ] 301 Mathematical Appendix r E[S T : S T < X] = E[S T : x T < Z X ] =  Z X −∞ S 0 e mT+σ √ Tz T n(z T )dz T = S 0 e mT+ 1 2 σ 2 N[Z X − σ √ T ] r E[S T : X < S T ] = E[S T : Z X < x T ] =  ∞ Z X S 0 e mT+σ √ Tz T n(z T )dz T = S 0 e mT+ 1 2 σ 2 N[σ √ T − Z X ] (A1.7) Our notation uses Z X which illustrates the origin of the term in square brackets as a limit of integration. A more common notation in the literature uses d 1 and d 2 where d 1 = σ √ T − Z X and d 2 = d 1 − σ √ T (=−Z X ). (D) Using the definition z T = (ln S T /S 0 − mT)/σ √ T (or more precisely its equivalent S T = S 0 e mT e σ √ Tz T ) yields the following frequently used result: E  S λ T  = S λ 0 e λmT  +∞ −∞ e λσ √ Tz T n(z T )dz T = S λ 0 e λmT I +∞ −∞ (λσ √ T ) = S λ 0 e λmT+ 1 2 λ 2 σ 2 T = F λ 0T e 1 2 λ(λ−1)σ 2 T (A1.8) Where F 0T is the forward price of the stock. (E) A related, but slightly more tricky pair of integrals are used in the investigation of lookback options; the first is I Z −∞ (a, b) =  Z −∞ e az N  φ (z − b) σ √ T  dz =  1 a e az N  φ (z − b) σ √ T  +Z −∞ − φ aσ √ 2π T  Z −∞ e az exp  − (z − b) 2 2σ 2 T  dz = 1 a e aZ N  φ (Z − b) σ √ T  − φ a e ab+ 1 2 a 2 σ 2 T N  (Z − b − aσ 2 T ) σ √ T  (A1.9) where we have first integrated by parts and then used equation (A1.5). The same approach gives I ∞ Z (a, b) =  Z −∞ e az N  φ (z − b) σ √ T  dz =  1 a e az N  φ (z − b) σ √ T  ∞ Z − φ aσ √ 2π T  ∞ Z e az exp  − (z − b) 2 2σ 2 T  dz =− 1 a e aZ N  φ (Z − b) σ √ T  − φ a e ab+ 1 2 a 2 σ 2 T N  − (Z − b − aσ 2 T ) σ √ T  (A1.10) (vi) Bivariate Normal Variables: Suppose y and z are two independent, standard, normal variates. By definition, these have the following properties: • Standard E[y] = E[z] = 0; var[y] = var[z] = 1 • Independent cov[y, z] = E[yz] = 0 302 A.1 DISTRIBUTIONS AND INTEGRALS Let us define another random variable x by the equation x = ρy +  1 − ρ 2 z, where ρ is a constant and x has the following properties: r In general, the sum of two normal variates is itself a normal variate. Thus x is normally distributed. r E[x] = 0; var[x] = ρ 2 var[y] + (1 − ρ 2 )var[z] = 1. r Correlation [x, y] = cov[x, y] √ var[x]var[y] = E[xy] = E[ρy 2 +  1 − ρ 2 yz] = ρ. t W T W T t Figure A1.3 Brownian path Thus x is a standard normal variate which has correlation ρ with y. Alternatively ex- pressed, any two correlated standard normal variates x and y can be decomposed into independent standard normal variates. Consider the single Brownian path shown in Figure A1.3. The distance W τ moved be- tween time 0 and time τ is independent of the distance W T−τ = W T − W τ moved be- tween time τ and time T. On the other hand, W T and W τ are obviously not independent since they overlap. From the definition of a Brownian motion as W t = √ tz t , where z t is a standard normal variate, we have W T = W τ + W T−τ √ Tz T = √ τ z τ + √ T − τ z T−τ z T =  τ T z τ +  1 − τ T z T−τ Comparing this with the decomposition we examined immediately before shows that z T and z τ are standard normal variates with correlation ρ = √ τ/T . (vii) Bivariate Normal Distribution: Suppose two standard normal variates z 1 and z 2 have correla- tion ρ. Their joint distribution function is written n 2 (z 1 , z 2 ; ρ) = 1 2π  1 − ρ 2 exp  − 1 2(1 − ρ 2 )  z 2 1 − 2ρz 1 z 2 + z 2 2   (A1.11) In general terms, n 2 (z 1 , z 2 ; ρ) can be represented as a bell-shaped hill. The contour lines of this hill are shown in Figure A1.4. If the correlation ρ is zero, this bell is perfectly symmet- rical with a circular mouth. If, however, ρ has non-zero value, then the bell is elongated to an ellipse, along an axis at 45 ◦ to z 1 and z 2 as shown in the second two graphs. The 45 ◦ axis used depends on the sign of the correlation: positive slope for positive correlation, and negative slope for negative correlation. The flatness of the ellipse depends on the degree of correlation. 303 Mathematical Appendix The volume under the bell-shaped hill is unity. The cumulative density function is the volume under the shaded part shown in the first graph of Figure A1.5. It is defined by N 2 [a, b; ρ] =  a −∞  b −∞ n 2 (z 1 , z 2 ; ρ)dz 1 dz 2 (A1.12) (viii) Symmetry Properties of N 2 [a, b; ρ]: The properties below follow from the symmetry of Figure A1.4. r = 0 r negative r positive 1 z 2 z 2 z 2 z 1 z 1 z Figure A1.4 Contours of n 2 (z 1 , z 2 ; ρ) (A) Given the symmetry of z 1 and z 2 in equations (A1.11) and (A1.12), it follows that N 2 [a, b; ρ] = N 2 [b, a; ρ] (A1.13) (B) Referring to the second graph of Figure A1.5 N 2 [∞, b; ρ] =  b −∞ dz 2  +∞ −∞ n 2 (z 1 , z 2 ; ρ)dz 1 = 1 2π  1 − ρ 2  b −∞  ∞ −∞ exp  − 1 2(1 − ρ 2 )  z 2 1 − 2ρz 1 z 2 + z 2 2   dz 1 dz 2 = 1 √ 2π  b −∞ e − 1 2 z 2 2 dz 2 = N[b] (A1.14) where we have made the change of variable z 1 =  1 − ρ 2 y + ρz 2 and slogged out the integral with respect to y, holding z 2 constant (i.e. dz 1 =  1 − ρ 2 dy). z 2 X 1 N 2 a, b; r z 2 z 1 NN 2 ∞ , b; r = b b b a z 1 X 1 Figure A1.5 Cumulative bivariate normal function 304 A.1 DISTRIBUTIONS AND INTEGRALS N 2 z 2 X 1 b a z 1 X 1 z 2 z 1 X 1 z 2 z 1 Rotate 90 0 Rotate 180 A B C B B A A C C N 2 z 2 X 1 X 1 z 1 X 1 X 1 z 2 z 2 z 1 z 1 X 1 X 1 z 2 z 2 z 1 z 1 -a -a -b b Rotate 180 0 A B C B B A A C C a, b; r Figure A1.6 Cumulative bivariate normal identities (C) Comparing the first and third graphs of Figure A1.6 shows that  ∞ −a dz 1  ∞ −b n 2 (z 1 , z 2 ; ρ)dz 2 =  a −∞ dz 1  b −∞ n 2 (z 1 , z 2 ; ρ)dz 2 = N 2 [a, b; ρ] (A1.15) (D) Referring to the first graph and using the fact that the volume of the elliptical bell-shaped “hill” is 1: Shaded volume = 1 − volume (A + B) − volume C N 2 [a, b; ρ] = 1 − (1 − N[a]) − volume C Volume C =  a −∞  ∞ b n 2 (z 1 , z 2 ; ρ)dz 1 dz 2 = N[a] − N 2 [a, b; ρ] (A1.16) (E) The second graph is just the first rotated through 90 ◦ . Given that the volume of the hill is unity and from property (c) above, we have Shaded volume = 1 − volume A − volume (B + C) N 2 [a, b; ρ] = 1 − N 2 [b, −a; −ρ] − ( 1 − N[b] ) N 2 [a, b; ρ] + N 2 [−a, b; −ρ] = N[b] (A1.17) (F) The third graph is the first rotated through 180 ◦ . Symmetry and previous results allow us to write Shaded volume = 1 − volume (A + B + C) = 1 −{volume ( A + B) + volume (B + C) − volume B} = 1 −{N[−a] + N[−b] − N 2 [−a, −b; ρ]} N 2 [a, b; ρ] = N[a] + N[b] − 1 + N 2 [−a, −b; ρ] (A1.18) (ix) More Useful Results: (A)  ∞ Z 2  ∞ Z 1 e az 1 n 2 (z 1 , z 2 ; ρ)dz 1 dz 2 = 1 2π  1 − ρ 2  ∞ Z 2  ∞ Z 1 exp  az 1 − 1 2(1 − ρ 2 )  z 2 1 − 2ρz 1 z 2 + z 2 2   dz 1 dz 2 305 Mathematical Appendix = e 1 2 a 2 1 2π  1 − ρ 2  ∞ Z 2 −ρa  ∞ Z 1 −a exp  − 1 2(1 − ρ 2 )  y 2 1 − 2ρy 1 y 2 + y 2 2   dy 1 dy 2 = e 1 2 a 2 N 2 [a − Z 1 ,ρa − Z 2 ; ρ] (A1.19) where we have made the substitutions z 1 = y 1 + a, z 2 = y 2 + ρa and slogged through the algebra in the exponential. The final result relies on equation (A1.15). (B)  +∞ Z 1  Z 2 −∞ e az 1 n 2 (z 1 , z 2 ; ρ)dz 1 dz 2 =  +∞ Z 1 e az 1 dz 1   +∞ −∞ −  +∞ Z 2  n 2 (z 1 , z 2 ; ρ)dz 2 =  +∞ Z 1 e az 1 n ( z 1 )dz 1 −  +∞ Z 1  +∞ Z 2 e az 1 n 2 (z 1 , z 2 ; ρ)dz 1 dz 2 = e 1 2 a 2 {N[a − Z 1 ] − N 2 [a − Z 1 ,ρa − Z 2 ; ρ]} (A1.20) where we have used equations (A1.6) and (A1.19) for the last step. (C) In order to evaluate equation (14.1) for the value of a compound call option (call on a call) or equation (14.5) for an extendible option, we need to evaluate E[S T − X : S ∗ τ < S τ ; X < S T ]. As in Section A.1(v), item (C) for the univariate case, we write m = r − q − 1 2 σ 2 and switch to the more convenient standard normal variates z T = [ln(S T /S 0 ) − mT]/σ √ T and z τ = [ln(S τ /S 0 ) − mτ ]/σ √ τ : E[S T − X : S ∗ τ < S τ ; X < S T ] =  ∞ Z ∗  ∞ Z X (S 0 e mT+σ √ Tz T − X )n 2 (z τ , z T ; ρ)dz τ dz T = S 0 e (r−q)T N 2 [σ √ τ − Z ∗ ,σ √ T − Z X ; ρ] − X N 2 [−Z ∗ ,−Z X ; ρ] (A1.21) where we have used the integral results of (A) above with Z X = [ln(X/S 0 ) − mT]/σ √ T , Z ∗ = [ln(S ∗ τ /S 0 ) − mτ ]/σ √ τ and ρ = √ τ/T . More common notation uses d 1 = σ √ T − Z X , d 2 =−Z X , b 1 = σ √ τ − Z ∗ and b 2 =−Z ∗ . (D) A general result for bivariate distributions is f (z 1 , z 2 ) = f z 1 | z 2  f (z 2 ) where the three terms are the joint, the conditional and the simple probability density functions of the random variable z 1 . From equation (A1.11), we may therefore write for two standard normal variables z 1 and z 2 : nz 1 | z 2 = n 2 (z 1 , z 2 ; ρ) n(z 2 ) = 1 2π  1 − ρ 2 exp  − 1 2(1 − ρ 2 ) {z 1 − ρz 2 } 2  ∼ N (ρz 2 , (1 − ρ 2 )) (A1.22) i.e. the conditional distribution of z 1 is normal with mean ρz 2 and variance (1 − ρ 2 ). (x) Numerical Approximations for the Cumulative Bivariate Normal Function: Standard spread sheets do not have add-in functions for calculating bivariate cumulative normal functions. A simple algorithm follows (Drezner, 1978). 306 A.1 DISTRIBUTIONS AND INTEGRALS (A) We start with some definitions: let a  = a/  2(1 − ρ 2 ), b  = b/  2(1 − ρ 2 ) and the func- tion (a, b; ρ) be defined in the region a, b and ρ all ≤ 0by (a, b; ρ) =  1 − ρ 2 π 5  i=1 A i 5  j=1 A j f i, j f i, j = exp{a  (2x i − a  ) + b  (2x j − b  ) + 2ρ(x i − a  )b  (x j − b  )} where the values of A i and x i are as follows: iA i x i 1 0.24840615 0.10024215 2 0.39233107 0.48281397 3 0.21141819 1.0609498 4 0.03324666 1.7797294 5 0.00082485334 2.6697604 (B) In the region a ≤ 0, b ≤ 0 and ρ ≤ 0, N 2 [a, b; ρ] is closely approximated by (a, b; ρ). If these conditions on a, b and ρ do not hold, N 2 [a, b; ρ] is obtained by manipulation: r If 0 < a × b × ρ use the relationship N 2 [a, b; ρ] = N 2 [a, 0; ρ ab ] + N 2 [0, b; ρ ba ] − δ ab where ρ ab = (ρa − b)sign[a]  a 2 − 2ρab − b 2 ; δ ab = 1 + sign[a]sign[b] 4 sign[a] = 1 (if 0 ≤ x) =−1 (if x < 0) r If a × b × ρ ≤ 0 and r a ≤ 0, 0 ≤ b, 0 ≤ ρ use N 2 [a, b; ρ] = N[a] − N 2 [a, −b; −ρ] r a ≤ 0, b ≤ 0, 0 ≤ ρ N 2 [a, b; ρ] = N[b] − N 2 [−a, b; −ρ] r 0 ≤ a, 0 ≤ b,ρ ≤ 0 use N 2 [a, b; ρ] = N[a] + N[b] − 1 + N 2 [−a, −b; ρ] r a ≤ 0, b ≤ 0,ρ ≤ 0 use N 2 [a, b; ρ] = (a, b; ρ) (xi) Product of Two Securities Prices: S t is an asset price (e.g. an equity stock) which we assume to be lognormally distributed, i.e. x t = ln S t is normally distributed. It is shown in Section 3.2 that E[S T ] = S 0 e (µ−q)T = S 0 e mT+ 1 2 σ 2 T (A1.23) where µ and q are the continuous (exponential) growth rate and dividend yield of the asset; m = E[x T ]; σ 2 = var[x T ]. 307 Mathematical Appendix We now examine the behavior of a quantity defined by Q t = S (1) t S (2) t , where S (1) t and S (2) t are the prices of two lognormally distributed assets. Writing y t = ln Q t , the following general results are evoked: r σ 2 Q = vary t =var  x (1) t + x (2) t  = σ 2 1 + σ 2 2 + 2ρ 12 σ 1 σ 2 . r Ey t =E  x (1) t + x (2) t  = m 1 T + m 2 T = m Q T. r It is a specific property of normal distributions that y t is also normally distributed. From the first two of these relationships, an expression for E[Q T ] corresponding to equa- tion (A1.23) is now written as E[Q T ] = Q 0 e (µ Q −q Q )T = Q 0 e m Q T + 1 2 σ 2 Q T = Q 0 e (m 1 +m 2 + 1 2 σ 2 1 + 1 2 σ 2 2 +ρ 12 σ 1 σ 2 )T = Q 0 e (µ 1 −q 1 )T +(µ 2 −q 2 )T +ρ 12 σ 1 σ 2 T which is equivalent to E  S (1) T S (2) T  = E  S (1) T  E  S (2) T  e ρ 12 σ 1 σ 2 T (A1.24) Alternatively, we could write µ Q − q Q = (µ 1 + µ 2 ) − (q 1 + q 2 ) + ρ 12 σ 1 σ 2 . In the risk-neutral environment in which most of our calculations are performed, each of the “assets” S (1) t , S (2) t and Q t enjoys the risk-free return, i.e. µ 1 = µ 2 = µ Q = r; therefore q Q = q 1 + q 2 − r − ρ 12 σ 1 σ 2 (A1.25) It follows from the above analysis that any composite price, made up of the product or quo- tient of lognormally distributed prices, is itself lognormally distributed. The various formulas developed for single prices are therefore easily adapted to describe the behavior of such com- posite prices; Chapters 12 and 13 are largely based on this technique. By contrast, the sum or difference of two lognormally distributed prices does not have a well-defined distribution and is therefore analytically intractable. (xii) Covariances and Correlations of Stock Prices: It is worth giving some standard definitions and results as referred to in various chapters. (A) If x (1) t = ln S (1) t ,wedefineσ 1 the volatility of S (1) t as the square root of the variance of x (1) t : σ 2 1 = var  x (1) t  = E  x (1) t − ¯ x (1)  2  = E  x (1) t  2  −  ¯ x (1)  2 ; ¯ x (1) = E  x (1) t  The covariance of two variables x (1) t and x (2) t is defined by cov  x (1) t , x (2) t  = E  x (1) t − ¯ x (1)  x (2) t − ¯ x (2)  = E  x (1) t x (2) t  − ¯ x (1) ¯ x (2) and the correlation between the two stocks is defined by ρ 12 = cov  x (1) t , x (2) t  σ 1 σ 2 , (B) The volatility of AS (1) t where A is a constant is given by σ 2 A1 = var  ln  AS (1) t  = var  const. + x (1) t  = var  x (1) t  = σ 2 1 (Note the radical difference from the result var[Ax] = A 2 var[x].) 308 [...]... position of the drunk after the first step is E[x1 ] = pU − (1 − p)D (A2.1) and the variance is 2 var[x1 ] = E x1 − E2 [x1 ] = pU 2 + (1 − p)D 2 − { pU − (1 − p)D}2 = p(1 − p)(U + D)2 309 (A2.2) Mathematical Appendix 3U xn 2U U 2U-D U-D 0 U-2D n −D −2D −3D Figure A2.1 Random walk grid Referring to Figure A2.1, consider the probability of reaching the point “2U − D” after three steps This could be achieved... small, so that terms O[(δt)2 ] can be safely ignored Let us repeat the derivation of equation (A2.2) in the present format: σ 2 δt = var[x] = E[x 2 ] − E2 [x] = pU 2 + (1 − p)D 2 − µ2 (δt)2 311 Mathematical Appendix But the last term in this equation is O[(δt)2 ] and may be dropped, leaving us with the relationships { pU − (1 − p)D} = µ δt; { pU 2 + (1 − p)D 2 } = σ 2 δt (A2.6) The second of these... the drunk has traveled and the time he has been going Two pronged Three pronged 2 ¥ Two pronged = three pronged 2U p 1- p U D p1 U 1 - p1 - p2 M p2 D U-D 2D Figure A2.3 Binomial vs trinomial 313 Mathematical Appendix When we say flexibility we really mean greater ability to choose parameters Thus in equation (A2.6) we have two equations for three unknowns (U, D and p) ; this gives us the flexibility to... then be written f x T , T | xt , t = p f x T , T | xt + U, t + δt + (1 − p) f x T , T | xt − D, t + δt We simplify the notation by writing f x T , T | xt , t = f and use the following Taylor 315 Mathematical Appendix expansion up to O[δt]: f x T , T | xt + δxt , t + δt = f + ∂f 1 ∂2 f 2 ∂f δxt + δx δt + ∂t ∂ xt 2 ∂ xt2 t Our equation then becomes 0= ∂f ∂f 1 ∂2 f δt + { pU − (1 − p)D} + { pU 2 + (1 −... probability density function given in Section A.3(i) shows that this function is a solution of both the backward and forward equations In later sections of this Appendix we will solve the backward equation with other boundary conditions In Appendix A.4 it is shown that the backward equation is in fact just two steps away from the Black Scholes equation (v) Variable µ and σ 2 : In the early part of... p f , to give ∂ {(1 − p) f } δT f = (1 − p) f − ∂T ∂ 1 ∂2 + {(1 − p) f }D + + {(1 − p) f }(D + )2 ∂ xt 2 ∂ xt2 ∂ ∂ 1 ∂2 { p f } δT − + pf − { p f }U − + { p f }(U − )2 ∂T ∂ xt 2 ∂ xt2 x 317 xN j Mathematical Appendix or collecting terms − ∂ 1 ∂2 ∂f − {( pU − − (1 − p)D + ) f } + {( p(U − )2 − (1 − p)(D + )2 ) f } = 0 2 ∂T ∂ xT 2 ∂ xT Using equations (A2.6) for the instantaneous drift and variance, with... second Fourier’s law of heat flow states that the rate of flow of heat is proportional to the temperature gradient in the wire, i.e ϕ(x, T ) ∝ ∂θ(x, T )/∂ x Consider the increase over time δT 319 Mathematical Appendix q(x,T) j(x + dx, T) j(x, T) x x +dx Figure A4.1 Heat flow in a wire of the heat (thermal energy) δE within a small element of length δx This may be written in two ways: ∂ 2θ ∂ϕ δx δT ∝ 2... by ∂θ ∂ x ∂θ ∂θ δx = δT = v δT ∂x ∂x ∂T ∂x The diffusion equation with convection can therefore be written δθ = ∂θ ∂ 2θ ∂θ =a 2 +b ∂T ∂x ∂x Fluid velocity = v x + dx x Figure A4.4 Convection 321 Mathematical Appendix In summary, a general form of the heat/diffusion equation can be written ∂θ ∂θ ∂ 2θ =a 2 +b + cθ + Q(x, T ) ∂T ∂x ∂x where b, c and Q can be positive or negative but a must be positive... if our domain of interest is infinite, although the theory can be pushed further to yield the Fourier integral, which is the continuous limit as we allow the periodic distance L to approach ∞ 323 Mathematical Appendix The analog of the Fourier series for a non-periodic function is the following Fourier integral: ∞ f (x) = {a(ω) cos ωx + b(ω) sin ωx} dω 0 where the coefficients a(ω) and b(ω) are given... as a periodic function for which we ignore values outside the region 0 to L It may therefore be represented by the Fourier series θ(x, T ) = ∞ a0 nπ nπ + x + bn sin x an cos 2 L L n=1 325 (A6.1) Mathematical Appendix We assume that the dependence on T is confined to the coefficients an and bn This is broadly equivalent to saying that the temperature profile remains roughly the same as it decays to an . Mathematical Appendix There are libraries full of textbooks on applied mathematics. understanding of option theory, and present the mathematical tools in a format which is immediately applicable. Many of the mathematical problems of option theory