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Financial Engineering Zvi Wiener [email protected] tel: 02-588-3049 Zvi Wiener ContTimeFin - 1 slide 1 Main Books Shimko D. Finance in Continuous Time, A Primer. Kolb Publishing Company, 1992, ISBN 1-878975-07-2 Wilmott P., S. Howison, J. Dewynne, The Mathematics of Financial Derivatives, A Student Introduction, Cambridge University Press, 1996, ISBN 0-521-49789-2 Zvi Wiener ContTimeFin - 1 slide 2 Useful Books Duffie D., Dynamic Asset Pricing Theory. Duffie D., Security Markets, Stochastic Models. Neftci S., An Introduction to the Mathematics of Financial Derivatives. Steele M., Invitation to Stochastic Differential Equations and Financial Applications. Karatzas I., and S. Shreve, BM and Stochastic Calculus. Zvi Wiener ContTimeFin - 1 slide 3 Primary Asset Valuation Discrete-time random walk: W(t+1) = W(t) + e(t+1); W(0) = W0; e~i.i.d. N(0,1) Zvi Wiener ContTimeFin - 1 slide 4 Primary Asset Valuation N(0,1) Zvi Wiener ContTimeFin - 1 slide 5 Primary Asset Valuation Discrete-time random walk refinement: W(t+) = W(t) + e(t+); W(0) = W0; e~i.i.d. N(0, ) This process has the same expected drift and variance over n periods as the initial process had in one period. Zvi Wiener ContTimeFin - 1 slide 6 Primary Asset Valuation t=0 Zvi Wiener 0.25 0.5 ContTimeFin - 1 0.75 N(0,1) 1 slide 7 Primary Asset Valuation Set dt W(t+ dt) = W(t) + e(t+ dt); W(0) = W0; e~i.i.d. N(0, dt) Define dW(t) = W(t+dt) - W(t) white noise, (dt)a = 0 for any a > 1 Zvi Wiener ContTimeFin - 1 slide 8 Primary Asset Valuation Needs["Statistics`NormalDistribution`"] nor[mu_,sig_]:=Random[NormalDistribution[mu,sig]]; tt=NestList[ (#+nor[0, 0.1])&, 0, 300]; ListPlot[tt,PlotJoined->True,PlotLabel->"Random Walk"]; Zvi Wiener ContTimeFin - 1 slide 9 Primary Asset Valuation Random Walk 0.5 50 100 150 200 250 300 -0.5 -1 Zvi Wiener ContTimeFin - 1 slide 10 Main Properties 1. E[dW(t)] = 0 2. E[dW(t) dt] = E[dW(t)] dt = 0 3. E[dW(t)2] = dt Zvi Wiener ContTimeFin - 1 slide 11 Main Properties (cont.) 4. Var[dW(t)2] = E[dW(t)4] - E2[dW(t)2] = 3 dt2 - dt2 = 0 5. E[(dW(t)dt)2] = E [dW(t)2] (dt)2 = 0 6. Var[dW(t)dt] = E[(dW(t)dt)4] - E2[dW(t)dt] = 0 Zvi Wiener ContTimeFin - 1 slide 12 E[dW(t)] = 0 By definition the mean of the normally distributed variable is zero. Zvi Wiener ContTimeFin - 1 slide 13 E[dW(t) dt] = E[dW(t)] dt = 0 The expectation of the product of a random variable (dW) and a constant (dt) equals the constant times the expected value of the random variable. Zvi Wiener ContTimeFin - 1 slide 14 E[dW(t)2] = dt For any distribution with zero mean the expected value of the squared random variable is the same as its variance. Zvi Wiener ContTimeFin - 1 slide 15 2 Var[dW(t) ] 4 E[dW(t) ] - = 2 2 E [dW(t) ] = 3 dt2 - dt2 = 0 The fourth central moment of the standard normal distribution is 3, and (dt)2 = 0. Zvi Wiener ContTimeFin - 1 slide 16 E[(dW(t)dt)2] = E [dW(t)2] (dt)2 = 0 Follows from properties 2 and 3. Zvi Wiener ContTimeFin - 1 slide 17 Var[dW(t)dt] = E[(dW(t)dt)4] - E2[dW(t)dt] = 0 Follows from properties 2 and 5. Zvi Wiener ContTimeFin - 1 slide 18 Important Property if Var[f(dW)] = 0 then E[f(dW)] = f(dW) Zvi Wiener ContTimeFin - 1 slide 19 Multiplication Rules Rule 1. (dW(t))2 = dt Rule 2. (dW(t)) dt = 0 Rule 3. dt2 = 0 Zvi Wiener ContTimeFin - 1 slide 20 W(t) is called a standard Wiener process, or a Brownian motion. W (0) W0 t W (t ) W0 dW ( ) 0 Zvi Wiener ContTimeFin - 1 slide 21 Major Properties of W 1. W(t) is continuous in t. 2. W(t) is nowhere differentiable. 3. W(t) is a process of unbounded variation. 4. W(t) is a process of bounded quadratic variation. Zvi Wiener ContTimeFin - 1 slide 22 Major Properties of W 5. The conditional distribution of W(u) given W(t), for u > t, is normal with mean W(t) and variance (u-t). 6. The variance of a forecast W(u) increases indefinitely as u . Zvi Wiener ContTimeFin - 1 slide 23 Brownian Motion - BM The standard BM is useful since many general stochastic processes can be written in terms of W. X(t+1) = X(t) + (X(t),t) + (X(t),t) e(t+1) X(0) = X0, e~i.i.d. N(0,1) generalized drift Zvi Wiener heteroscedastisity (changing variance) ContTimeFin - 1 slide 24 Brownian Motion - BM Choose a shorter time interval : X(t+) = X(t) + (X(t),t) + (X(t),t) e(t+ ) X(0) = X0, e~i.i.d. N(0, ) generalized drift Zvi Wiener heteroscedastisity (changing variance) ContTimeFin - 1 slide 25 Brownian Motion - BM As we let dt we see that dX(t) = (X(t),t) dt + (X(t),t) dW(t) X(0) = X0 generalized univariate Wiener process, (diffusion). dX = (X,t) dt + (X,t) dW, Zvi Wiener ContTimeFin - 1 X(0) = X0 slide 26 Interpretation How can we interpret the fact that dX = dt + dW the random variable dX has local mean dt and local variance 2dt. A discrete analogy is X = + z. Zvi Wiener ContTimeFin - 1 slide 27 Arithmetic BM dX = dt + dW Let (X,t) = , and (X,t) = two constants. Then X follows an arithmetic Brownian Motion with drift and volatility . This is an appropriate specification for a process that grows at a linear rate and exhibits an increasing uncertainty. Zvi Wiener ContTimeFin - 1 slide 28 Arithmetic BM dX = dt + dW 1. X may be positive or negative. 2. If u > t, then Xu is a future value of the process relative to time t. The distribution of Xu given Xt is normal with mean Xt+ (u-t) and standard deviation (u-t)1/2. 3. The variance of a forecast Xu tends to infinity as u does (for fixed t and Xt). Zvi Wiener ContTimeFin - 1 slide 29 Arithmetic BM dX = dt + dW X time Zvi Wiener ContTimeFin - 1 slide 30 Arithmetic BM dX = dt + dW X time Zvi Wiener ContTimeFin - 1 slide 31 Arithmetic BM dX = dt + dW Is appropriate for variables that can be positive and negative, have normally distributed forecast errors, and have forecast variance increasing linearly in time. Example: net cash flows. Is inappropriate for stock price. Zvi Wiener ContTimeFin - 1 slide 32 Geometric BM dX = Xdt + XdW Let (X,t) = X, and (X,t) = X. Then X follows an geometric Brownian Motion. Zvi Wiener ContTimeFin - 1 slide 33 Geometric BM dX = Xdt + XdW This is an appropriate specification for a process that grows exponentially at an average rate of has volatility proportional to the level of the variable. It also exhibits an increasing uncertainty. Zvi Wiener ContTimeFin - 1 slide 34 Geometric BM dX = Xdt + XdW 1. If X(0) > 0, it will always be positive. 2. X has an absorbing barrier at X = 0. Thus if X hits zero (a zero probability event) it will remain there forever. Zvi Wiener ContTimeFin - 1 slide 35 Geometric BM dX = Xdt + XdW 3. The conditional distribution of Xu given Xt is lognormal. The conditional mean of ln(Xt) is ln(Xt) + (u-t) - 0.5 2(u-t) and conditional standard deviation of ln(Xt) is (u-t)1/2. ln(Xt) is normally distributed. Zvi Wiener ContTimeFin - 1 slide 36 Geometric BM dX = Xdt + XdW 4. The conditional expected value of Xu is Et[Xu] = Xtexp[ (u-t)] 5. The variance of a forecast Xu tends to infinity as u does (for fixed t and Xt). Zvi Wiener ContTimeFin - 1 slide 37 Geometric BM dX = Xdt + XdW X time Zvi Wiener ContTimeFin - 1 slide 38 Geometric BM dX = Xdt + XdW X time Zvi Wiener ContTimeFin - 1 slide 39 Geometric BM dX = Xdt + XdW Is often used to model security values, since the proportional changes in security price are independent and identically normally distributed (sometimes). Example: currency price, stocks. Is inappropriate for dividends, interest rates. Zvi Wiener ContTimeFin - 1 slide 40 Mean Reverting Process dX = (-X)dt + X dW Ornstein-Uhlenbeck when = 1 Let (X,t) = (-X), and (X,t) = X , where 0 - speed of adjustment - long run mean - volatility Zvi Wiener ContTimeFin - 1 slide 41 Mean Reverting Process dX = (-X)dt + X dW This is an appropriate specification for a process that has a long run value but may be beset by short-term disturbances. We assume that , , and are positive for simplicity. Zvi Wiener ContTimeFin - 1 slide 42 Mean Reverting Process dX = (-X)dt + X dW 1. If X(0) > 0, it will always be positive. 2. As X approaches zero, the drift is positive and volatility vanishes. 3. As u becomes infinite, the variance of a forecast Xu is finite. Zvi Wiener ContTimeFin - 1 slide 43 Mean Reverting Process dX = (-X)dt + X dW 4. If = 0.5, the distribution of Xu given Xt for u > t is non-central chi-squared, the mean of the distribution is: (Xt- ) exp[-(u-t)] + the variance of the distribution is (CIR 85): 2 (u t ) 2 (u t ) 2 ( u t ) 2 Xt e e 1 e 2 Zvi Wiener ContTimeFin - 1 slide 44 Mean Reverting Process dX = (-X)dt + X dW X time Zvi Wiener ContTimeFin - 1 slide 45 Mean Reverting Process dX = (-X)dt + X dW X time Zvi Wiener ContTimeFin - 1 slide 46 Mean Reverting Process dX = (-X)dt + X dW SeedRandom[2] tt=NestList[ (#+0.3(1-#)+0.1*#*nor[0,0.1])&, 1.01,130]; ListPlot[tt,PlotJoined->True, Axes->False]; Zvi Wiener ContTimeFin - 1 slide 47 Mean Reverting Process dX = (-X)dt + X dW Is often used to model economic variables and do not represent traded assets. Example: interest rates, volatility. Zvi Wiener ContTimeFin - 1 slide 48 Ito’s lemma Consider a real valued function f(X): RR. Taylor series expansion: 1 2 f ( X ) f ( X ) f X ( X ) f XX ( X ) 2 1 3 3 f XXX ( X ) o( ) 6 Zvi Wiener ContTimeFin - 1 slide 49 Ito’s lemma If X is a “standard” variable, then 2 is o() f ( X ) f ( X ) f X ( X ) o() f ( X dX ) f ( X ) f X ( X )dX df ( X ) f X ( X )dX Zvi Wiener ContTimeFin - 1 slide 50 Ito’s lemma If X is a stochastic variable (following diffusion) then the term dX2 does NOT vanish. dX dt dW dX (dt ) 2 2 dX dt 2 Zvi Wiener 2 2 2dtdW (dW ) 2 ContTimeFin - 1 2 slide 51 Ito’s lemma 1 2 f ( X dX ) f ( X ) f X ( X )dX f XX ( X )dX 2 1 2 df ( X ) f X ( X )dX f XX ( X )dX 2 1 2 df f X dX f XX dX 2 Zvi Wiener ContTimeFin - 1 slide 52 Ito’s lemma If f = f(X,t) and dX = dt + dW, then df f X 0.5 f XX f t dt f X dW Zvi Wiener 2 ContTimeFin - 1 slide 53 Financial Applications A Suppose that a security with value V guarantees $1dt every instant of time forever. This is the continuous time equivalent of a risk-free perpetuity of $1. If the risk-free interest rate is constant r, what is the (discounted) value of the security? Zvi Wiener ContTimeFin - 1 slide 54 Financial Applications A 1. V = V(t), there are NO stochastic variables. dV = Vtdt 2. The expected capital gain on V is ECG = E[dV] = Vtdt 3. The expected cash flows to V is ECF = 1 dt 4. The total return on V is ECG + ECF = (Vt+1)dt Zvi Wiener ContTimeFin - 1 slide 55 Financial Applications A 5. Since there is no risk, the total return must be equal to the risk-free return on V, or rVdt. (Vt+1) dt = r V dt 6. Divide both sides by dt: Vt = rV - 1 Zvi Wiener ContTimeFin - 1 slide 56 Financial Applications A Vt = rV - 1 DSolve[ V'[t]==r*V[t]-1, V[t], t ] V(t) = c Exp[r t] + 1/r given V(0) one can find c Zvi Wiener ContTimeFin - 1 slide 57 Financial Applications B Suppose that X follows a geometric Brownian motion with drift and volatility . A security with value V collects Xdt continuously forever. V represents a perpetuity that grows at an average exponential rate of , but whose risks in cash flow variations are considered diversificable. The economy is risk-neutral, and the risk-free interest rate is constant at r. What is the value of this security? Zvi Wiener ContTimeFin - 1 slide 58 Financial Applications B 1. V = V(X), since V is a perpetual claim, its price does not depend on time. dV = VxdX + 0.5 VxxdX2, dX = Xdt + XdW, dX2= 2X2dt dV = [XVx+0.5 2X2Vxx]dt +XVxdW Zvi Wiener ContTimeFin - 1 slide 59 Financial Applications B 2. The expected capital gain: ECG = E[dV] = [XVx+0.5 2X2Vxx]dt since E[dW] = 0 3. The Expected cash flow: ECF = X dt Zvi Wiener ContTimeFin - 1 slide 60 Financial Applications B 4. Total return: TR = ECG + ECF = [XVx+X+0.52X2Vxx]dt 5. But the return must be equal to the risk free return on the same investment V. rVdt = [XVx+X+0.52X2Vxx]dt 6. Thus the PDE: rV = XVx+X+0.52X2Vxx Zvi Wiener ContTimeFin - 1 slide 61 Financial Applications B rV = XVx+X+0.52X2Vxx there are several ways to solve it. One can guess that doubling X will double the price V. If V is proportional to X, then V = X, Vx= , and Vxx=0, then the equation becomes r X= X+X = 1/(r- ) V(X) = X/(r- ) Zvi Wiener ContTimeFin - 1 slide 62

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