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Financial Engineering

Financial Engineering. Zvi Wiener mswiener@mscc.huji.ac.il tel: 02-588-3049. W - Wiener Process = Brownian Motion dW ~ N(0, dt) (dW(t)) 2 = dt (dW(t)) dt = 0 dt 2 = 0. W - Wiener Process = Brownian Motion dW ~ N(0, dt). dX = dt + dW Arithmetical BM

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Financial Engineering

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  1. Financial Engineering Zvi Wiener mswiener@mscc.huji.ac.il tel: 02-588-3049 ContTimeFin - 2

  2. W - Wiener Process = Brownian Motion dW ~ N(0, dt) (dW(t))2 = dt (dW(t)) dt = 0 dt2 = 0 ContTimeFin - 2

  3. W - Wiener Process = Brownian MotiondW ~ N(0, dt) dX = dt + dW Arithmetical BM dX = Xdt + XdW Geometrical BM dX = (-X)dt + XdW Mean reverting dX = (X,t)dt + (X,t)dW diffusion ContTimeFin - 2

  4. X   time Arithmetic BMdX =  dt +  dW ContTimeFin - 2

  5. X time Geometric BMdX = Xdt + XdW ContTimeFin - 2

  6. Mean Reverting ProcessdX = (-X)dt + XdW X  time ContTimeFin - 2

  7. Ito’s lemma If f = f(X) and dX = dt + dW, then ContTimeFin - 2

  8. Multivariate Ito’s Lemma Introduce a second variable Y to the system that follows a diffusion dX = (X,Y,t)dt + (X,Y,t)dW dY = (X,Y,t)dt + (X,Y,t)dZ where Z is another standard Wiener process. We define dZdW = dt as the correlation between the two processes. ContTimeFin - 2

  9. Multivariate Ito’s Lemma It can be shown that E[dZdW] = dt (dZdW)2 = 0 Probabilistically dZ can be projected on dW dZ = dW + (1- 2)1/2de where de is a standard Wiener process uncorrelated with dW, de dW = 0 ContTimeFin - 2

  10. Multivariate Ito’s Lemma The multivariate Ito’s Lemma: f = f(X,Y,t) df = fxdX + fydY + ftdt + 0.5(fxxdX2 + 2fxydXdY + fyydY2) ContTimeFin - 2

  11. Multiplication Table dW dZ dt dW dt dt 0 dZ dt dt 0 dt 0 0 0 Note: terms of higher order (dt)a with a>1 we set to be zero, since we work with the first order terms only. ContTimeFin - 2

  12. Multiplication Table dX2 = (dt + dW)2 = 2dt dY2 = (dt + dZ)2 = 2dt dXdY = (dt + dW) (dt + dZ) =    dt ContTimeFin - 2

  13. Jump Processes Diffusion processes are continuous. In order to include jumps we use Poisson processes. Define q(t) such that q(0) = 0 and is constant until a Poisson event occurs. When there is a Poisson event value of q increases by 1. ContTimeFin - 2

  14. Jump Processes In its simplest form, a Poisson process with a constant intensity parameter  is: dq(t) = 1 with probability dt = 0 with probability 1-dt at every moment in time, where dq(t) is the instantaneous change in q in moment t. ContTimeFin - 2

  15. Standard Poisson Process q t ContTimeFin - 2

  16. Standard Poisson Process jump[lam_, dt_]:=If[ Random[ ] < lam*dt, 1, 0]; tt = NestList[(# + jump[0.5, 0.1])&, 0, 300]; ListPlot[tt, PlotJoined->True]; ContTimeFin - 2

  17. A Random Variable with Compact Support A random variable has compact support if the domain over which the random variable has positive probability measure is a compact set. Compact set - closed and bounded. In any infinity sequence of points there is a convergent subsequence. ContTimeFin - 2

  18. Diffusion with Jumps dX = dt + dW + dq This means that X has jumps by an amount  whenever a Poisson event occurs (with intensity ). ContTimeFin - 2

  19. Default Event A default event can be modeled as a jump to zero value. If X is the value of the security, then  = -X can be interpreted as a default event - price drops to zero and remains there forever. dX = dt + dW - Xdq ContTimeFin - 2

  20. Default Event dX = dt + dW - Xdq mean continuous changes dt continuous variance 2dt occasional default (probability dt) A general model of default allows a spectrum of levels (see D. Duffie). ContTimeFin - 2

  21. Jump Diffusion For a real valued function f(X) the change in function value conditional on the occurrence of an event is f(X+)-f(X). Therefore the expected change in function is: dt E[f(X+)-f(X)] + (1- dt) [0] = E[f(X+)-f(X)]dt ContTimeFin - 2

  22. Residual Risk In financial models, we often assume that residual risk is diversifiable. That is, no investor cares about this risk in the pricing of securities. We also assume that the timing of the jumps and the level of X are independent of each other. However, we may allow  to depend on X or follow its own stochastic process. ContTimeFin - 2

  23. Ito’s Lemma with jumps ContTimeFin - 2

  24. 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? ContTimeFin - 2

  25. 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 ContTimeFin - 2

  26. 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 ContTimeFin - 2

  27. 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 ContTimeFin - 2

  28. 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? ContTimeFin - 2

  29. 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 ContTimeFin - 2

  30. 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 ContTimeFin - 2

  31. Financial Applications B 4. Total return: TR = ECG + ECF = [XVx+X+0.52X2Vxx]dt 5. But the return must be equal to the risk free return on the same investment V. rVdt = [XVx+X+0.52X2Vxx]dt 6. Thus the PDE: rV = XVx+X+0.52X2Vxx ContTimeFin - 2

  32. Financial Applications B rV = XVx+X+0.52X2Vxx 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- ) ContTimeFin - 2

  33. Financial Applications C Modify example B to provide for a sudden possible drop to zero in the value of V. If a Poisson event occurs, one gives up V in exchange for nothing. The gain is zero and the loss is V, so the change in the value of V is 0-V, or -V. The possibility of this jump in any instant is dt. ContTimeFin - 2

  34. Financial Applications C 1. V = V(X, q), there is no t because of the perpetual nature of V. dV = VxdX + 0.5 VxxdX2 +[0-V]dq ContTimeFin - 2

  35. Financial Applications C 2. Expected Capital Gain: ECG = E[dV] = [XVx+0.5 2X2Vxx-V]dt 3. Expected Cash Flow: ECF = Xdt ContTimeFin - 2

  36. Financial Applications C 4. Total Return: TR = ECG + ECF = [XVx + 0.5 2X2Vxx - V + X]dt 5. Return on an alternative investment: rVdt = [XVx + 0.5 2X2Vxx - V + X]dt ContTimeFin - 2

  37. Financial Applications C 6. PDE: rV = XVx + 0.5 2X2Vxx - V + X This is the same equation as in B, but with (r+) instead of r. Note that one can NOT solve it as a standard Cauchy problem because of a singularity at the origin (X=0). ContTimeFin - 2

  38. Financial Applications C V = X/(r +  - ) We discount the cash flow at a higher rate (r + ) to compensate for the probability of full default. Alternatively we can see this as an adjustment of the growth rate to ( - ) and discount at the risk free rate. ( - ) is the certainty-equivalent growth rate. ContTimeFin - 2

  39. Financial Applications D Suppose X follows geometric Brownian motion, and an independent Poisson process determines the timing of cash payments equal to the contemporaneous value of X. Let V represent the claim to the first cash flow in this stochastic perpetuity. What is the value of V? ContTimeFin - 2

  40. Financial Applications D 1. V = V(X,q), since V is a perpetual claim, its price does not depend on time. dV = VxdX + 0.5 VxxdX2 + [X-V]dt Note: we give up the asset V to receive the payment X. ContTimeFin - 2

  41. Financial Applications D 2. The expected capital gain: ECG = E[dV] = [XVx+0.5 2X2Vxx+ (X-V)]dt 3. The Expected cash flow: ECF = 0 There are no continuous cash payments. ContTimeFin - 2

  42. Financial Applications D 4. Total return TR = ECG + ECF = [XVx+0.5 2X2Vxx+ X - V]dt 5. Return on an alternative investment: rVdt = [XVx+0.5 2X2Vxx+ X - V]dt ContTimeFin - 2

  43. Financial Applications D 6. PDE rV = XVx+0.5 2X2Vxx+ X - V In Example B we had: rV = XVx+X+0.52X2Vxx ContTimeFin - 2

  44. Financial Applications D (D) rV = XVx+0.5 2X2Vxx+ X - V (B) rV = XVx+X+0.52X2Vxx This is the same equation as in Example B, except for two substitutions: 1. (r + ) instead of r 2. The value of V is multiplied by  (the cash flow term is X instead of X). ContTimeFin - 2

  45. Financial Applications D (D) rV = XVx+0.5 2X2Vxx+ X - V V = X/(r +  - ) Check it! ContTimeFin - 2

  46. Conclusions We have studied Present Value (PV) calculations in continuous time settings. We have received ODE, since all our models were perpetual (no explicit time dependence). ContTimeFin - 2

  47. Conclusions In a risk-neutral economy: 1. Calculate the expected capital gain on an asset from Ito’s lemma. 2. Add the expected cash flows to get the total return. 3. Set the total return equal to the risk-free return. 4. Solve the appropriate DE. ContTimeFin - 2

  48. Exercise 1.1 Assume X follows geometric BM with drift  and volatility . Let Y = ln(X). a. What process does Y follow? b. What is the distribution of Yu, given Yt (t<u)? c. What is the expected value of Xu, given Xt? Hint: if z ~N(, 2), then E[ez] = exp[ + 0.5 2] ContTimeFin - 2

  49. Exercise 1.1 - Solution dX = Xdt + XdW Y = ln(X) dY = (ln(X))’dX + 0.5 (ln(X))”(dX)2 = (Xdt + XdW)/X + 0.5 (-1/X2) 2X2dt = ( - 0.5 2)dt + dW - arithmetic BM Yu ~ N(Yt + ( - 0.5 2)(u-t), 2(u-t)) E[eYu] = Xte(u-t) ContTimeFin - 2

  50. Exercise 1.2 Assume X follows arithmetic BM with drift  and volatility . A security V pays Xdt forever. If X becomes negative, the holder of the asset must make payments to the security issuer. The economy is risk-neutral, and the risk-free discount rate is r. ContTimeFin - 2

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