Zvi WienerContTimeFin - 2 slide 1 Financial Engineering Zvi Wiener [email protected] tel:...
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Transcript of Zvi WienerContTimeFin - 2 slide 1 Financial Engineering Zvi Wiener [email protected] tel:...
![Page 2: Zvi WienerContTimeFin - 2 slide 1 Financial Engineering Zvi Wiener mswiener@mscc.huji.ac.il tel: 02-588-3049.](https://reader035.fdocuments.net/reader035/viewer/2022062715/56649d7d5503460f94a5fa7f/html5/thumbnails/2.jpg)
Zvi Wiener ContTimeFin - 2 slide 2
W - Wiener Process = Brownian Motion
dW ~ N(0, dt)
(dW(t))2 = dt
(dW(t)) dt = 0
dt2 = 0
![Page 3: Zvi WienerContTimeFin - 2 slide 1 Financial Engineering Zvi Wiener mswiener@mscc.huji.ac.il tel: 02-588-3049.](https://reader035.fdocuments.net/reader035/viewer/2022062715/56649d7d5503460f94a5fa7f/html5/thumbnails/3.jpg)
Zvi Wiener ContTimeFin - 2 slide 3
W - Wiener Process = Brownian MotiondW ~ N(0, dt)
dX = dt + dW Arithmetical BM
dX = Xdt + XdW Geometrical BM
dX = (-X)dt + XdW Mean reverting
dX = (X,t)dt + (X,t)dW diffusion
![Page 4: Zvi WienerContTimeFin - 2 slide 1 Financial Engineering Zvi Wiener mswiener@mscc.huji.ac.il tel: 02-588-3049.](https://reader035.fdocuments.net/reader035/viewer/2022062715/56649d7d5503460f94a5fa7f/html5/thumbnails/4.jpg)
Zvi Wiener ContTimeFin - 2 slide 4
Arithmetic BM dX = dt + dW
time
X
![Page 5: Zvi WienerContTimeFin - 2 slide 1 Financial Engineering Zvi Wiener mswiener@mscc.huji.ac.il tel: 02-588-3049.](https://reader035.fdocuments.net/reader035/viewer/2022062715/56649d7d5503460f94a5fa7f/html5/thumbnails/5.jpg)
Zvi Wiener ContTimeFin - 2 slide 5
Geometric BM dX = Xdt + XdW
time
X
![Page 6: Zvi WienerContTimeFin - 2 slide 1 Financial Engineering Zvi Wiener mswiener@mscc.huji.ac.il tel: 02-588-3049.](https://reader035.fdocuments.net/reader035/viewer/2022062715/56649d7d5503460f94a5fa7f/html5/thumbnails/6.jpg)
Zvi Wiener ContTimeFin - 2 slide 6
Mean Reverting Process
dX = (-X)dt + XdW
time
X
![Page 7: Zvi WienerContTimeFin - 2 slide 1 Financial Engineering Zvi Wiener mswiener@mscc.huji.ac.il tel: 02-588-3049.](https://reader035.fdocuments.net/reader035/viewer/2022062715/56649d7d5503460f94a5fa7f/html5/thumbnails/7.jpg)
Zvi Wiener ContTimeFin - 2 slide 7
Ito’s lemma
If f = f(X) and dX = dt + dW, then
dWfdtffdf XXXX
2
2
![Page 8: Zvi WienerContTimeFin - 2 slide 1 Financial Engineering Zvi Wiener mswiener@mscc.huji.ac.il tel: 02-588-3049.](https://reader035.fdocuments.net/reader035/viewer/2022062715/56649d7d5503460f94a5fa7f/html5/thumbnails/8.jpg)
Zvi Wiener ContTimeFin - 2 slide 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.
![Page 9: Zvi WienerContTimeFin - 2 slide 1 Financial Engineering Zvi Wiener mswiener@mscc.huji.ac.il tel: 02-588-3049.](https://reader035.fdocuments.net/reader035/viewer/2022062715/56649d7d5503460f94a5fa7f/html5/thumbnails/9.jpg)
Zvi Wiener ContTimeFin - 2 slide 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
![Page 10: Zvi WienerContTimeFin - 2 slide 1 Financial Engineering Zvi Wiener mswiener@mscc.huji.ac.il tel: 02-588-3049.](https://reader035.fdocuments.net/reader035/viewer/2022062715/56649d7d5503460f94a5fa7f/html5/thumbnails/10.jpg)
Zvi Wiener ContTimeFin - 2 slide 10
Multivariate Ito’s Lemma
The multivariate Ito’s Lemma:
f = f(X,Y,t)
df = fxdX + fydY + ftdt +
0.5(fxxdX2 + 2fxydXdY + fyydY2)
![Page 11: Zvi WienerContTimeFin - 2 slide 1 Financial Engineering Zvi Wiener mswiener@mscc.huji.ac.il tel: 02-588-3049.](https://reader035.fdocuments.net/reader035/viewer/2022062715/56649d7d5503460f94a5fa7f/html5/thumbnails/11.jpg)
Zvi Wiener ContTimeFin - 2 slide 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.
![Page 12: Zvi WienerContTimeFin - 2 slide 1 Financial Engineering Zvi Wiener mswiener@mscc.huji.ac.il tel: 02-588-3049.](https://reader035.fdocuments.net/reader035/viewer/2022062715/56649d7d5503460f94a5fa7f/html5/thumbnails/12.jpg)
Zvi Wiener ContTimeFin - 2 slide 12
Multiplication Table
dX2 = (dt + dW)2 = 2dt
dY2 = (dt + dZ)2 = 2dt
dXdY = (dt + dW) (dt + dZ) = dt
22 5.05.0 dYfdXdYfdXfdYfdXfdf yyxyxxyx
dZfdWf
dtffffff
df
yx
yyxyxxtyx
22 5.05.0
![Page 13: Zvi WienerContTimeFin - 2 slide 1 Financial Engineering Zvi Wiener mswiener@mscc.huji.ac.il tel: 02-588-3049.](https://reader035.fdocuments.net/reader035/viewer/2022062715/56649d7d5503460f94a5fa7f/html5/thumbnails/13.jpg)
Zvi Wiener ContTimeFin - 2 slide 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.
![Page 14: Zvi WienerContTimeFin - 2 slide 1 Financial Engineering Zvi Wiener mswiener@mscc.huji.ac.il tel: 02-588-3049.](https://reader035.fdocuments.net/reader035/viewer/2022062715/56649d7d5503460f94a5fa7f/html5/thumbnails/14.jpg)
Zvi Wiener ContTimeFin - 2 slide 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.
![Page 15: Zvi WienerContTimeFin - 2 slide 1 Financial Engineering Zvi Wiener mswiener@mscc.huji.ac.il tel: 02-588-3049.](https://reader035.fdocuments.net/reader035/viewer/2022062715/56649d7d5503460f94a5fa7f/html5/thumbnails/15.jpg)
Zvi Wiener ContTimeFin - 2 slide 15
Standard Poisson Processq
t
![Page 16: Zvi WienerContTimeFin - 2 slide 1 Financial Engineering Zvi Wiener mswiener@mscc.huji.ac.il tel: 02-588-3049.](https://reader035.fdocuments.net/reader035/viewer/2022062715/56649d7d5503460f94a5fa7f/html5/thumbnails/16.jpg)
Zvi Wiener ContTimeFin - 2 slide 16
Standard Poisson Processjump[lam_, dt_]:=If[ Random[ ] < lam*dt, 1, 0];
tt = NestList[(# + jump[0.5, 0.1])&, 0, 300];
ListPlot[tt, PlotJoined->True];
50 100 150 200 250 300
5
10
15
20Jump Process
![Page 17: Zvi WienerContTimeFin - 2 slide 1 Financial Engineering Zvi Wiener mswiener@mscc.huji.ac.il tel: 02-588-3049.](https://reader035.fdocuments.net/reader035/viewer/2022062715/56649d7d5503460f94a5fa7f/html5/thumbnails/17.jpg)
Zvi Wiener ContTimeFin - 2 slide 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.
![Page 18: Zvi WienerContTimeFin - 2 slide 1 Financial Engineering Zvi Wiener mswiener@mscc.huji.ac.il tel: 02-588-3049.](https://reader035.fdocuments.net/reader035/viewer/2022062715/56649d7d5503460f94a5fa7f/html5/thumbnails/18.jpg)
Zvi Wiener ContTimeFin - 2 slide 18
Diffusion with Jumps
dX = dt + dW + dq
This means that X has jumps by an amount whenever a Poisson event occurs (with intensity ).
![Page 19: Zvi WienerContTimeFin - 2 slide 1 Financial Engineering Zvi Wiener mswiener@mscc.huji.ac.il tel: 02-588-3049.](https://reader035.fdocuments.net/reader035/viewer/2022062715/56649d7d5503460f94a5fa7f/html5/thumbnails/19.jpg)
Zvi Wiener ContTimeFin - 2 slide 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
![Page 20: Zvi WienerContTimeFin - 2 slide 1 Financial Engineering Zvi Wiener mswiener@mscc.huji.ac.il tel: 02-588-3049.](https://reader035.fdocuments.net/reader035/viewer/2022062715/56649d7d5503460f94a5fa7f/html5/thumbnails/20.jpg)
Zvi Wiener ContTimeFin - 2 slide 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).
![Page 21: Zvi WienerContTimeFin - 2 slide 1 Financial Engineering Zvi Wiener mswiener@mscc.huji.ac.il tel: 02-588-3049.](https://reader035.fdocuments.net/reader035/viewer/2022062715/56649d7d5503460f94a5fa7f/html5/thumbnails/21.jpg)
Zvi Wiener ContTimeFin - 2 slide 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
![Page 22: Zvi WienerContTimeFin - 2 slide 1 Financial Engineering Zvi Wiener mswiener@mscc.huji.ac.il tel: 02-588-3049.](https://reader035.fdocuments.net/reader035/viewer/2022062715/56649d7d5503460f94a5fa7f/html5/thumbnails/22.jpg)
Zvi Wiener ContTimeFin - 2 slide 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.
![Page 23: Zvi WienerContTimeFin - 2 slide 1 Financial Engineering Zvi Wiener mswiener@mscc.huji.ac.il tel: 02-588-3049.](https://reader035.fdocuments.net/reader035/viewer/2022062715/56649d7d5503460f94a5fa7f/html5/thumbnails/23.jpg)
Zvi Wiener ContTimeFin - 2 slide 23
Ito’s Lemma with jumps
dqXfXfdtfdXfdXfdf
qtXff
txxx )()(5.0
),,(2
![Page 24: Zvi WienerContTimeFin - 2 slide 1 Financial Engineering Zvi Wiener mswiener@mscc.huji.ac.il tel: 02-588-3049.](https://reader035.fdocuments.net/reader035/viewer/2022062715/56649d7d5503460f94a5fa7f/html5/thumbnails/24.jpg)
Zvi Wiener ContTimeFin - 2 slide 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?
![Page 25: Zvi WienerContTimeFin - 2 slide 1 Financial Engineering Zvi Wiener mswiener@mscc.huji.ac.il tel: 02-588-3049.](https://reader035.fdocuments.net/reader035/viewer/2022062715/56649d7d5503460f94a5fa7f/html5/thumbnails/25.jpg)
Zvi Wiener ContTimeFin - 2 slide 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
![Page 26: Zvi WienerContTimeFin - 2 slide 1 Financial Engineering Zvi Wiener mswiener@mscc.huji.ac.il tel: 02-588-3049.](https://reader035.fdocuments.net/reader035/viewer/2022062715/56649d7d5503460f94a5fa7f/html5/thumbnails/26.jpg)
Zvi Wiener ContTimeFin - 2 slide 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
![Page 27: Zvi WienerContTimeFin - 2 slide 1 Financial Engineering Zvi Wiener mswiener@mscc.huji.ac.il tel: 02-588-3049.](https://reader035.fdocuments.net/reader035/viewer/2022062715/56649d7d5503460f94a5fa7f/html5/thumbnails/27.jpg)
Zvi Wiener ContTimeFin - 2 slide 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
![Page 28: Zvi WienerContTimeFin - 2 slide 1 Financial Engineering Zvi Wiener mswiener@mscc.huji.ac.il tel: 02-588-3049.](https://reader035.fdocuments.net/reader035/viewer/2022062715/56649d7d5503460f94a5fa7f/html5/thumbnails/28.jpg)
Zvi Wiener ContTimeFin - 2 slide 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?
![Page 29: Zvi WienerContTimeFin - 2 slide 1 Financial Engineering Zvi Wiener mswiener@mscc.huji.ac.il tel: 02-588-3049.](https://reader035.fdocuments.net/reader035/viewer/2022062715/56649d7d5503460f94a5fa7f/html5/thumbnails/29.jpg)
Zvi Wiener ContTimeFin - 2 slide 29
Financial Applications B1. 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
![Page 30: Zvi WienerContTimeFin - 2 slide 1 Financial Engineering Zvi Wiener mswiener@mscc.huji.ac.il tel: 02-588-3049.](https://reader035.fdocuments.net/reader035/viewer/2022062715/56649d7d5503460f94a5fa7f/html5/thumbnails/30.jpg)
Zvi Wiener ContTimeFin - 2 slide 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
![Page 31: Zvi WienerContTimeFin - 2 slide 1 Financial Engineering Zvi Wiener mswiener@mscc.huji.ac.il tel: 02-588-3049.](https://reader035.fdocuments.net/reader035/viewer/2022062715/56649d7d5503460f94a5fa7f/html5/thumbnails/31.jpg)
Zvi Wiener ContTimeFin - 2 slide 31
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
![Page 32: Zvi WienerContTimeFin - 2 slide 1 Financial Engineering Zvi Wiener mswiener@mscc.huji.ac.il tel: 02-588-3049.](https://reader035.fdocuments.net/reader035/viewer/2022062715/56649d7d5503460f94a5fa7f/html5/thumbnails/32.jpg)
Zvi Wiener ContTimeFin - 2 slide 32
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- )
![Page 33: Zvi WienerContTimeFin - 2 slide 1 Financial Engineering Zvi Wiener mswiener@mscc.huji.ac.il tel: 02-588-3049.](https://reader035.fdocuments.net/reader035/viewer/2022062715/56649d7d5503460f94a5fa7f/html5/thumbnails/33.jpg)
Zvi Wiener ContTimeFin - 2 slide 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.
![Page 34: Zvi WienerContTimeFin - 2 slide 1 Financial Engineering Zvi Wiener mswiener@mscc.huji.ac.il tel: 02-588-3049.](https://reader035.fdocuments.net/reader035/viewer/2022062715/56649d7d5503460f94a5fa7f/html5/thumbnails/34.jpg)
Zvi Wiener ContTimeFin - 2 slide 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
![Page 35: Zvi WienerContTimeFin - 2 slide 1 Financial Engineering Zvi Wiener mswiener@mscc.huji.ac.il tel: 02-588-3049.](https://reader035.fdocuments.net/reader035/viewer/2022062715/56649d7d5503460f94a5fa7f/html5/thumbnails/35.jpg)
Zvi Wiener ContTimeFin - 2 slide 35
Financial Applications C
2. Expected Capital Gain:
ECG = E[dV] = [XVx+0.5 2X2Vxx-V]dt
3. Expected Cash Flow:
ECF = Xdt
![Page 36: Zvi WienerContTimeFin - 2 slide 1 Financial Engineering Zvi Wiener mswiener@mscc.huji.ac.il tel: 02-588-3049.](https://reader035.fdocuments.net/reader035/viewer/2022062715/56649d7d5503460f94a5fa7f/html5/thumbnails/36.jpg)
Zvi Wiener ContTimeFin - 2 slide 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
![Page 37: Zvi WienerContTimeFin - 2 slide 1 Financial Engineering Zvi Wiener mswiener@mscc.huji.ac.il tel: 02-588-3049.](https://reader035.fdocuments.net/reader035/viewer/2022062715/56649d7d5503460f94a5fa7f/html5/thumbnails/37.jpg)
Zvi Wiener ContTimeFin - 2 slide 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).
![Page 38: Zvi WienerContTimeFin - 2 slide 1 Financial Engineering Zvi Wiener mswiener@mscc.huji.ac.il tel: 02-588-3049.](https://reader035.fdocuments.net/reader035/viewer/2022062715/56649d7d5503460f94a5fa7f/html5/thumbnails/38.jpg)
Zvi Wiener ContTimeFin - 2 slide 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.
![Page 39: Zvi WienerContTimeFin - 2 slide 1 Financial Engineering Zvi Wiener mswiener@mscc.huji.ac.il tel: 02-588-3049.](https://reader035.fdocuments.net/reader035/viewer/2022062715/56649d7d5503460f94a5fa7f/html5/thumbnails/39.jpg)
Zvi Wiener ContTimeFin - 2 slide 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?
![Page 40: Zvi WienerContTimeFin - 2 slide 1 Financial Engineering Zvi Wiener mswiener@mscc.huji.ac.il tel: 02-588-3049.](https://reader035.fdocuments.net/reader035/viewer/2022062715/56649d7d5503460f94a5fa7f/html5/thumbnails/40.jpg)
Zvi Wiener ContTimeFin - 2 slide 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.
![Page 41: Zvi WienerContTimeFin - 2 slide 1 Financial Engineering Zvi Wiener mswiener@mscc.huji.ac.il tel: 02-588-3049.](https://reader035.fdocuments.net/reader035/viewer/2022062715/56649d7d5503460f94a5fa7f/html5/thumbnails/41.jpg)
Zvi Wiener ContTimeFin - 2 slide 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.
![Page 42: Zvi WienerContTimeFin - 2 slide 1 Financial Engineering Zvi Wiener mswiener@mscc.huji.ac.il tel: 02-588-3049.](https://reader035.fdocuments.net/reader035/viewer/2022062715/56649d7d5503460f94a5fa7f/html5/thumbnails/42.jpg)
Zvi Wiener ContTimeFin - 2 slide 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
![Page 43: Zvi WienerContTimeFin - 2 slide 1 Financial Engineering Zvi Wiener mswiener@mscc.huji.ac.il tel: 02-588-3049.](https://reader035.fdocuments.net/reader035/viewer/2022062715/56649d7d5503460f94a5fa7f/html5/thumbnails/43.jpg)
Zvi Wiener ContTimeFin - 2 slide 43
Financial Applications D
6. PDE
rV = XVx+0.5 2X2Vxx+ X - V
In Example B we had:
rV = XVx+X+0.52X2Vxx
![Page 44: Zvi WienerContTimeFin - 2 slide 1 Financial Engineering Zvi Wiener mswiener@mscc.huji.ac.il tel: 02-588-3049.](https://reader035.fdocuments.net/reader035/viewer/2022062715/56649d7d5503460f94a5fa7f/html5/thumbnails/44.jpg)
Zvi Wiener ContTimeFin - 2 slide 44
Financial Applications D
(D) rV = XVx+0.5 2X2Vxx+ X - V
(B) rV = XVx+X+0.52X2Vxx
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).
![Page 45: Zvi WienerContTimeFin - 2 slide 1 Financial Engineering Zvi Wiener mswiener@mscc.huji.ac.il tel: 02-588-3049.](https://reader035.fdocuments.net/reader035/viewer/2022062715/56649d7d5503460f94a5fa7f/html5/thumbnails/45.jpg)
Zvi Wiener ContTimeFin - 2 slide 45
Financial Applications D
(D) rV = XVx+0.5 2X2Vxx+ X - V
V = X/(r + - )
Check it!
![Page 46: Zvi WienerContTimeFin - 2 slide 1 Financial Engineering Zvi Wiener mswiener@mscc.huji.ac.il tel: 02-588-3049.](https://reader035.fdocuments.net/reader035/viewer/2022062715/56649d7d5503460f94a5fa7f/html5/thumbnails/46.jpg)
Zvi Wiener ContTimeFin - 2 slide 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).
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Zvi Wiener ContTimeFin - 2 slide 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.
![Page 48: Zvi WienerContTimeFin - 2 slide 1 Financial Engineering Zvi Wiener mswiener@mscc.huji.ac.il tel: 02-588-3049.](https://reader035.fdocuments.net/reader035/viewer/2022062715/56649d7d5503460f94a5fa7f/html5/thumbnails/48.jpg)
Zvi Wiener ContTimeFin - 2 slide 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]
![Page 49: Zvi WienerContTimeFin - 2 slide 1 Financial Engineering Zvi Wiener mswiener@mscc.huji.ac.il tel: 02-588-3049.](https://reader035.fdocuments.net/reader035/viewer/2022062715/56649d7d5503460f94a5fa7f/html5/thumbnails/49.jpg)
Zvi Wiener ContTimeFin - 2 slide 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)
![Page 50: Zvi WienerContTimeFin - 2 slide 1 Financial Engineering Zvi Wiener mswiener@mscc.huji.ac.il tel: 02-588-3049.](https://reader035.fdocuments.net/reader035/viewer/2022062715/56649d7d5503460f94a5fa7f/html5/thumbnails/50.jpg)
Zvi Wiener ContTimeFin - 2 slide 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.
![Page 51: Zvi WienerContTimeFin - 2 slide 1 Financial Engineering Zvi Wiener mswiener@mscc.huji.ac.il tel: 02-588-3049.](https://reader035.fdocuments.net/reader035/viewer/2022062715/56649d7d5503460f94a5fa7f/html5/thumbnails/51.jpg)
Zvi Wiener ContTimeFin - 2 slide 51
Exercise 1.2
a. What is the value of V? (Hint: V is linear in X)
b. Suppose the security holder has the right to
abandon the asset if cash flows become
sufficiently negative, i.e., when X=q (q<0). What
is the value of V?
Hint: V=k1exp[k2(X-q)] + k3X + k4. Note that
k2<0. Also, when X=q, V(X)=0. Check that the
ODE is satisfied.
![Page 52: Zvi WienerContTimeFin - 2 slide 1 Financial Engineering Zvi Wiener mswiener@mscc.huji.ac.il tel: 02-588-3049.](https://reader035.fdocuments.net/reader035/viewer/2022062715/56649d7d5503460f94a5fa7f/html5/thumbnails/52.jpg)
Zvi Wiener ContTimeFin - 2 slide 52
Exercise 1.2
c. If q can be chosen optimally, what is the value-maximizing choice? Verify the second order conditions.
d. What is the value of the abandonment optimal?
Hint: look at parts a and b.
![Page 53: Zvi WienerContTimeFin - 2 slide 1 Financial Engineering Zvi Wiener mswiener@mscc.huji.ac.il tel: 02-588-3049.](https://reader035.fdocuments.net/reader035/viewer/2022062715/56649d7d5503460f94a5fa7f/html5/thumbnails/53.jpg)
Zvi Wiener ContTimeFin - 2 slide 53
Exercise 1.2 - Solution
dX = dt + dW
ECG = E[dV] = Vxdt + 0.52Vxxdt + VxE[dW]
ECF = Xdt, if X > q, otherwise 0.
TR = (Vx + 0.52Vxx + X IX>q)
Vx + 0.52Vxx + X IX>q = rV
![Page 54: Zvi WienerContTimeFin - 2 slide 1 Financial Engineering Zvi Wiener mswiener@mscc.huji.ac.il tel: 02-588-3049.](https://reader035.fdocuments.net/reader035/viewer/2022062715/56649d7d5503460f94a5fa7f/html5/thumbnails/54.jpg)
Zvi Wiener ContTimeFin - 2 slide 54
Exercise 1.2 - SolutionVx + 0.52Vxx + X IX>q = rV
a. V(X) = x/r + /r2
b. k1= - q/r - /r2, k3= 1/r, k4= /r2
the sign is minus, since k2 must be negative.
2
22
2
2
r
k
V=k1exp[k2(X-q)] + k3X + k4
![Page 55: Zvi WienerContTimeFin - 2 slide 1 Financial Engineering Zvi Wiener mswiener@mscc.huji.ac.il tel: 02-588-3049.](https://reader035.fdocuments.net/reader035/viewer/2022062715/56649d7d5503460f94a5fa7f/html5/thumbnails/55.jpg)
Zvi Wiener ContTimeFin - 2 slide 55
Exercise 1.2 - Solution
Vx + 0.52Vxx + X IX>q = rV
2
1*0
rkq
q
V
d. Value of the abandonment option is the difference between values with and without the option, when q is chosen optimally.
*)(
2
21
' qXkerk
valuesoption
![Page 56: Zvi WienerContTimeFin - 2 slide 1 Financial Engineering Zvi Wiener mswiener@mscc.huji.ac.il tel: 02-588-3049.](https://reader035.fdocuments.net/reader035/viewer/2022062715/56649d7d5503460f94a5fa7f/html5/thumbnails/56.jpg)
Zvi Wiener ContTimeFin - 2 slide 56
Exercise 1.3
Assume that X follows geometric BM, with drift and volatility . The economy is risk-neutral, and the risk-free discount rate is r. A machine prints a certificate worth X(t) at random times t generated by a Poisson arrival process with intensity .
a. What is the value of this machine?
b. What is the value of a contingent claim to the
first certificate printed by the machine?
![Page 57: Zvi WienerContTimeFin - 2 slide 1 Financial Engineering Zvi Wiener mswiener@mscc.huji.ac.il tel: 02-588-3049.](https://reader035.fdocuments.net/reader035/viewer/2022062715/56649d7d5503460f94a5fa7f/html5/thumbnails/57.jpg)
Zvi Wiener ContTimeFin - 2 slide 57
Exercise 1.3
c. Assume Y follows geometric BM with drift and volatility . The correlation between X and Y is 0. What is the value of a certificate produced by X, if it lets its bearer (only) have X certificates printed by machine Y (worth Y at the time of printing)? Y prints at the same average rate, and the number of certificates is determined by the first arrival time.
![Page 58: Zvi WienerContTimeFin - 2 slide 1 Financial Engineering Zvi Wiener mswiener@mscc.huji.ac.il tel: 02-588-3049.](https://reader035.fdocuments.net/reader035/viewer/2022062715/56649d7d5503460f94a5fa7f/html5/thumbnails/58.jpg)
Zvi Wiener ContTimeFin - 2 slide 58
Exercise 1.3 - Solution a.
dX = Xdt + XdW
ECG = E[dV] = XVxdt + 0.52X2Vxxdt
ECF = Xdt
TR = (XVx + 0.5X22Vxx + X)dt
XVx + 0.5X22Vxx + X = rV
V(X)= X/(r - )
![Page 59: Zvi WienerContTimeFin - 2 slide 1 Financial Engineering Zvi Wiener mswiener@mscc.huji.ac.il tel: 02-588-3049.](https://reader035.fdocuments.net/reader035/viewer/2022062715/56649d7d5503460f94a5fa7f/html5/thumbnails/59.jpg)
Zvi Wiener ContTimeFin - 2 slide 59
Exercise 1.3 - Solution b.
dX = Xdt + XdW
ECG = E[dV] = XVxdt + 0.52X2Vxxdt
ECF = (X-V)dt
TR = (XVx + 0.5X22Vxx + (X-V))dt
XVx + 0.5X22Vxx + (X-V) = rV
V(X)= X/(r + - )
![Page 60: Zvi WienerContTimeFin - 2 slide 1 Financial Engineering Zvi Wiener mswiener@mscc.huji.ac.il tel: 02-588-3049.](https://reader035.fdocuments.net/reader035/viewer/2022062715/56649d7d5503460f94a5fa7f/html5/thumbnails/60.jpg)
Zvi Wiener ContTimeFin - 2 slide 60
Exercise 1.4
A low-risk health insurance policy holder realizes medical losses at random times according to a Poisson arrival process. The level of the loss is given by X, a process which follows GBM with drift and volatility . The economy is risk-neutral, and the risk-free discount rate is r. Medical expenses occur at a rate dt. There is an additional possibility that the claimant will suddenly become a high-risk claimant.
![Page 61: Zvi WienerContTimeFin - 2 slide 1 Financial Engineering Zvi Wiener mswiener@mscc.huji.ac.il tel: 02-588-3049.](https://reader035.fdocuments.net/reader035/viewer/2022062715/56649d7d5503460f94a5fa7f/html5/thumbnails/61.jpg)
Zvi Wiener ContTimeFin - 2 slide 61
Exercise 1.4
High-risk claimants experience the same possible losses X, but at a higher frequency dt. The timing of the switch from a low-risk to a high-risk is governed by a Poisson process with intensity parameter .
a. If the policy holder stays low-risk forever, what is the value of the policy today?
b. What is the value of a high-risk policy today?
c. What is the value of a low-risk policy today?
![Page 62: Zvi WienerContTimeFin - 2 slide 1 Financial Engineering Zvi Wiener mswiener@mscc.huji.ac.il tel: 02-588-3049.](https://reader035.fdocuments.net/reader035/viewer/2022062715/56649d7d5503460f94a5fa7f/html5/thumbnails/62.jpg)
Zvi Wiener ContTimeFin - 2 slide 62
Exercise 1.4 - Solution a.
dX = Xdt + XdW
ECG = E[dV] = XVxdt + 0.52X2Vxxdt
ECF = Xdt
TR = (XVx + 0.5X22Vxx + X)dt
XVx + 0.5X22Vxx + X = rV
V(X)= X/(r - )
![Page 63: Zvi WienerContTimeFin - 2 slide 1 Financial Engineering Zvi Wiener mswiener@mscc.huji.ac.il tel: 02-588-3049.](https://reader035.fdocuments.net/reader035/viewer/2022062715/56649d7d5503460f94a5fa7f/html5/thumbnails/63.jpg)
Zvi Wiener ContTimeFin - 2 slide 63
Exercise 1.4 - Solution b.
dX = Xdt + XdW
ECG = E[dV] = XVxdt + 0.52X2Vxxdt
ECF = Xdt
TR = (XVx + 0.5X22Vxx + X)dt
XVx + 0.5X22Vxx + X = rV
V(X)= X/(r - )
![Page 64: Zvi WienerContTimeFin - 2 slide 1 Financial Engineering Zvi Wiener mswiener@mscc.huji.ac.il tel: 02-588-3049.](https://reader035.fdocuments.net/reader035/viewer/2022062715/56649d7d5503460f94a5fa7f/html5/thumbnails/64.jpg)
Zvi Wiener ContTimeFin - 2 slide 64
Exercise 1.4 - Solution c.dX = Xdt + XdW
ECG = E[dV] = XVxdt + 0.52X2Vxxdt
ECF = Xdt + (X/(r - )-V)dt
TR = XVx + 0.5X22Vxx + X + (X/(r - )-V)
XVx + 0.5X22Vxx + X + (X/(r - )-V) = rV
rr
XXV )(
![Page 65: Zvi WienerContTimeFin - 2 slide 1 Financial Engineering Zvi Wiener mswiener@mscc.huji.ac.il tel: 02-588-3049.](https://reader035.fdocuments.net/reader035/viewer/2022062715/56649d7d5503460f94a5fa7f/html5/thumbnails/65.jpg)
Zvi Wiener ContTimeFin - 2 slide 65
Exercise 1.5
Assume that the value of an index X follows GBM with drift and volatility . An asset V promises that, when X reaches Q, the bearer will be paid R and the asset will be retired. The economy is risk-neutral, and the risk-free rate is r.
a. What is the value of the asset (Hint: V=AX).
b. What are sufficient conditions for > 0?
![Page 66: Zvi WienerContTimeFin - 2 slide 1 Financial Engineering Zvi Wiener mswiener@mscc.huji.ac.il tel: 02-588-3049.](https://reader035.fdocuments.net/reader035/viewer/2022062715/56649d7d5503460f94a5fa7f/html5/thumbnails/66.jpg)
Zvi Wiener ContTimeFin - 2 slide 66
Exercise 1.5 - Solution
dX = Xdt + XdW
ECG = E[dV] = XVxdt + 0.52X2Vxxdt
ECF = 0
TR = (XVx + 0.5X22Vxx)dt
XVx + 0.5X22Vxx = rV
V(Q)=R
![Page 67: Zvi WienerContTimeFin - 2 slide 1 Financial Engineering Zvi Wiener mswiener@mscc.huji.ac.il tel: 02-588-3049.](https://reader035.fdocuments.net/reader035/viewer/2022062715/56649d7d5503460f94a5fa7f/html5/thumbnails/67.jpg)
Zvi Wiener ContTimeFin - 2 slide 67
Exercise 1.5 - Solution
V
Q X
R
![Page 68: Zvi WienerContTimeFin - 2 slide 1 Financial Engineering Zvi Wiener mswiener@mscc.huji.ac.il tel: 02-588-3049.](https://reader035.fdocuments.net/reader035/viewer/2022062715/56649d7d5503460f94a5fa7f/html5/thumbnails/68.jpg)
Zvi Wiener ContTimeFin - 2 slide 68
Exercise 1.5 - SolutionXVx + 0.5X22Vxx = rV
AQ = R
V(X) = AX
AX + 0.52 (-1)AX = rAX
+ 0.52 (-1) = r
0.52 2 + ( - 0.52) - r = 0
2
2222 25.05.0
r
Q
XRXV )(
![Page 69: Zvi WienerContTimeFin - 2 slide 1 Financial Engineering Zvi Wiener mswiener@mscc.huji.ac.il tel: 02-588-3049.](https://reader035.fdocuments.net/reader035/viewer/2022062715/56649d7d5503460f94a5fa7f/html5/thumbnails/69.jpg)
Zvi Wiener ContTimeFin - 2 slide 69
Exercise 1.6
Assume that the value of an index X follows GBM with drift and volatility . A perpetual call option is written such that when it is exercised (at X=Q), the holder receives Q-E (E - is the exercise price). The economy is risk-neutral, and the risk-free rate is r.
a. What is the value of the option, assuming it is exercised when X = Q? (Hint: V=AX, do not forget the boundary condition V(Q)=Q-E)
![Page 70: Zvi WienerContTimeFin - 2 slide 1 Financial Engineering Zvi Wiener mswiener@mscc.huji.ac.il tel: 02-588-3049.](https://reader035.fdocuments.net/reader035/viewer/2022062715/56649d7d5503460f94a5fa7f/html5/thumbnails/70.jpg)
Zvi Wiener ContTimeFin - 2 slide 70
Exercise 1.6
b. Assuming that the holder of the call option will act to maximize the current value of the option, what Q will he choose?
c. What are sufficient conditions for > 0?
d. Verify that the value satisfies the DE you derived.
e. What are the comparative static properties of the model?
![Page 71: Zvi WienerContTimeFin - 2 slide 1 Financial Engineering Zvi Wiener mswiener@mscc.huji.ac.il tel: 02-588-3049.](https://reader035.fdocuments.net/reader035/viewer/2022062715/56649d7d5503460f94a5fa7f/html5/thumbnails/71.jpg)
Zvi Wiener ContTimeFin - 2 slide 71
Exercise 1.6 - Solution a.
dX = Xdt + XdW
ECG = E[dV] = XVxdt + 0.52X2Vxxdt
ECF = 0
TR = (XVx + 0.5X22Vxx)dt
XVx + 0.5X22Vxx = rV
V(Q)=Q-E
![Page 72: Zvi WienerContTimeFin - 2 slide 1 Financial Engineering Zvi Wiener mswiener@mscc.huji.ac.il tel: 02-588-3049.](https://reader035.fdocuments.net/reader035/viewer/2022062715/56649d7d5503460f94a5fa7f/html5/thumbnails/72.jpg)
Zvi Wiener ContTimeFin - 2 slide 72
Exercise 1.6 - Solution b.
V(X) = (Q-E)(X/Q)
Optimal exercise is such that maximizes V(X):
1)1( QEQXQ
V
1*
E
Q
![Page 73: Zvi WienerContTimeFin - 2 slide 1 Financial Engineering Zvi Wiener mswiener@mscc.huji.ac.il tel: 02-588-3049.](https://reader035.fdocuments.net/reader035/viewer/2022062715/56649d7d5503460f94a5fa7f/html5/thumbnails/73.jpg)
Zvi Wiener ContTimeFin - 2 slide 73
Exercise 1.6 - Solution c, e.
V(X) = (Q-E)(X/Q)
Sufficient condition for > 1, is > 0 and r > .
V increases in X
V decreases in E
V increases in r and
V depends on through
![Page 74: Zvi WienerContTimeFin - 2 slide 1 Financial Engineering Zvi Wiener mswiener@mscc.huji.ac.il tel: 02-588-3049.](https://reader035.fdocuments.net/reader035/viewer/2022062715/56649d7d5503460f94a5fa7f/html5/thumbnails/74.jpg)
Zvi Wiener ContTimeFin - 2 slide 74
Exercise 1.7
Assume X follows GBM with drift and volatility . Assume Y follows GBM with drift and volatility . The correlation between the Wiener components of the two processes is dZxdZy=dt.
a. Write down the laws of motion of the system.
b. Let V = XY. What process does V follow? Define a new process (v, v and dZv), so that dV/V= vdt + vdZv.
![Page 75: Zvi WienerContTimeFin - 2 slide 1 Financial Engineering Zvi Wiener mswiener@mscc.huji.ac.il tel: 02-588-3049.](https://reader035.fdocuments.net/reader035/viewer/2022062715/56649d7d5503460f94a5fa7f/html5/thumbnails/75.jpg)
Zvi Wiener ContTimeFin - 2 slide 75
Exercise 1.7
c. What are the correlations of dV with dX, dY?
d. Let W = X/Y. What process does W follow? Organize your results as in b.
e. What are the correlations of dW with dX, dY?
f. Run a theoretical regression of dY/Y on dX/X. What are your coefficients? What is the standard error of the regression? What are the time series properties of the volatility of the projection (i.e., the error term)? What is the theoretical R2?
![Page 76: Zvi WienerContTimeFin - 2 slide 1 Financial Engineering Zvi Wiener mswiener@mscc.huji.ac.il tel: 02-588-3049.](https://reader035.fdocuments.net/reader035/viewer/2022062715/56649d7d5503460f94a5fa7f/html5/thumbnails/76.jpg)
Zvi Wiener ContTimeFin - 2 slide 76
Exercise 1.8
A security with value V pays ydt continuously until x reaches the point q. y follows ABM with drift and volatility , and x follows ABM with drift and volatility w. The correlation between the Wiener components of the two processes is dZxdZy=dt
a. What DE must V satisfy?
b. What are the boundary conditions?
c. Value the asset.
![Page 77: Zvi WienerContTimeFin - 2 slide 1 Financial Engineering Zvi Wiener mswiener@mscc.huji.ac.il tel: 02-588-3049.](https://reader035.fdocuments.net/reader035/viewer/2022062715/56649d7d5503460f94a5fa7f/html5/thumbnails/77.jpg)
Zvi Wiener ContTimeFin - 2 slide 77
Exercise 1.8 - SolutionV(X,Y), dWdZ=dt
dY = dt + dW dX = dt + wdZ
ECG = (Vy+0.52Vyy+Vx+0.5w2Vxx+Vxy) dt
ECF = Ydt, if X > q, otherwise 0.
TR = ECG + ECF = alternative return (rV)
Vy+0.52Vyy+Vx+0.5w2Vxx+Vxy + Y= rV
V(q,Y)=0
![Page 78: Zvi WienerContTimeFin - 2 slide 1 Financial Engineering Zvi Wiener mswiener@mscc.huji.ac.il tel: 02-588-3049.](https://reader035.fdocuments.net/reader035/viewer/2022062715/56649d7d5503460f94a5fa7f/html5/thumbnails/78.jpg)
Zvi Wiener ContTimeFin - 2 slide 78
Exercise 1.9
A firm earns Xdt continuously, where X follows ABM with drift and volatility . This is the only asset of the firm. If X becomes negative, then the firm must decide whether to honor its obligations or abandon its operations. We assume it is optimal to abandon its operations. We assume it is optimal to abandon operations when earnings fall below a constant level q. The firm wishes to sell contingent claims against its earnings.
![Page 79: Zvi WienerContTimeFin - 2 slide 1 Financial Engineering Zvi Wiener mswiener@mscc.huji.ac.il tel: 02-588-3049.](https://reader035.fdocuments.net/reader035/viewer/2022062715/56649d7d5503460f94a5fa7f/html5/thumbnails/79.jpg)
Zvi Wiener ContTimeFin - 2 slide 79
Exercise 1.9
To value an arbitrary contingent claim, we first value four primitive contingent claims with the following cash flows:
g1(X) = 1
g2(X) = X
g3(X) = I{X>c}
g4(X) = X I{X>c}
Here I is the indicator function. g1 receives $1dt until X=q, then he receives nothing.
![Page 80: Zvi WienerContTimeFin - 2 slide 1 Financial Engineering Zvi Wiener mswiener@mscc.huji.ac.il tel: 02-588-3049.](https://reader035.fdocuments.net/reader035/viewer/2022062715/56649d7d5503460f94a5fa7f/html5/thumbnails/80.jpg)
Zvi Wiener ContTimeFin - 2 slide 80
Exercise 1.9
g2 receives $Xdt until X=q; if X < 0, the cash is paid instead of received.
g3 receives $1dt if X is above c, and is worthless when X reaches q.
g4 receives $Xdt if X is above c, and is worthless when X reaches q.
![Page 81: Zvi WienerContTimeFin - 2 slide 1 Financial Engineering Zvi Wiener mswiener@mscc.huji.ac.il tel: 02-588-3049.](https://reader035.fdocuments.net/reader035/viewer/2022062715/56649d7d5503460f94a5fa7f/html5/thumbnails/81.jpg)
Zvi Wiener ContTimeFin - 2 slide 81
Exercise 1.9
Let Yi(X) represent the value of a claim giving rise to cash flows of gi(X). We explicitly allow for the optimal abandonment of cash flows; Yi must satisfy the boundary condition Yi(q) = 0 when X reaches the abandonment point q.
We also require YiX< for all X>q.
![Page 82: Zvi WienerContTimeFin - 2 slide 1 Financial Engineering Zvi Wiener mswiener@mscc.huji.ac.il tel: 02-588-3049.](https://reader035.fdocuments.net/reader035/viewer/2022062715/56649d7d5503460f94a5fa7f/html5/thumbnails/82.jpg)
Zvi Wiener ContTimeFin - 2 slide 82
Exercise 1.9
a. What is the value of each of the Yi(X)?
Hint 1: V = A1exp(k1X) + A2exp(k2X) + A3X + A4
Hint 2: Assume different forms for X<c and X>c.
Hint 3: The solution must be continuous and differentiable at c.
![Page 83: Zvi WienerContTimeFin - 2 slide 1 Financial Engineering Zvi Wiener mswiener@mscc.huji.ac.il tel: 02-588-3049.](https://reader035.fdocuments.net/reader035/viewer/2022062715/56649d7d5503460f94a5fa7f/html5/thumbnails/83.jpg)
Zvi Wiener ContTimeFin - 2 slide 83
Exercise 1.9b. The contingent claim holders are stock holders, bond holders, government, third parties. The marginal tax rate is . The distribution of earnings occurs instantaneously and is as follows:
Case Debt Equity Gov. 3rd P.
c<X c (1- )(X-c) (X-c) 0
q<X<c X-k 0 0 k
X<q 0 0 0 0
Value each of the claims using the primitive derivative claims against the earnings of the firm.
![Page 84: Zvi WienerContTimeFin - 2 slide 1 Financial Engineering Zvi Wiener mswiener@mscc.huji.ac.il tel: 02-588-3049.](https://reader035.fdocuments.net/reader035/viewer/2022062715/56649d7d5503460f94a5fa7f/html5/thumbnails/84.jpg)
Zvi Wiener ContTimeFin - 2 slide 84
Exercise 1.9
c. Find the operating and capital structure policy
(levels of q and c) that maximizes the sum of
debt and equity values. Write down the first
order conditions for an interior maximum only.
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Zvi Wiener ContTimeFin - 2 slide 85
Exercise 1.9 - Solution a.
V(X) X = dt + dW
ECG = Vxdt + 0.52Vxxdt
ECF = g(X)dt
TR = ECG + ECF = Vx + 0.52Vxx + g(X)
Vx+ 0.52Vxx+ g(X) = rV
V(q) = 0
![Page 86: Zvi WienerContTimeFin - 2 slide 1 Financial Engineering Zvi Wiener mswiener@mscc.huji.ac.il tel: 02-588-3049.](https://reader035.fdocuments.net/reader035/viewer/2022062715/56649d7d5503460f94a5fa7f/html5/thumbnails/86.jpg)
Zvi Wiener ContTimeFin - 2 slide 86
Exercise 1.9 - Solution b.
Debt: D = c Y3 + Y2 - kY1 - Y4 + kY3
Equity: E = (1- )(Y4 - cY3)
Government: G = (Y4 - c Y3)
Others: T = k(Y1 - Y3)