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DERIVATIVES: ASSIGNMENT 3, PART B

HARJOAT S. BHAMRA

Part B needs to be written up and submitted online. Part B is partly a computational exercise, so submit

your code. The are Bonus Parts which you do not need to do, but doing them can boost your overall mark by

compensating for errors and omissions elsewhere.

(1) Consider an investor who does not consume or work to earn any labour income. All she does is invest her

financial wealth in N + 1 assets. The first asset is risk-free with return rtdt over the interval [t, t + dt).

The remaining N assets are risky and risky asset n ∈ {1, . . . , N} has return dRn,t, where

dRn,t = µn,tdt + σn,tdZn,t.

Zn is a standard Brownian motion under the physical probability measure P, such that Et[dZi,tdZj,t] =

ρij,tdt for i 6= j.

The investor invests the fraction φn,t of her date-t wealth in risky asset n.

(a) Explain why the return on the investor’s portfolio over the interval [t, t + dt) is given by

dRp,t =

1 −

XN

n=1

φn,t!

dt +

XN

n=1

φn,tdRn,t.

(b) Hence, show that

dRp,t = rdt + φ

>

t

(µ − r1)dt + φ

t

ΣdZt

,

where

µ

t

= (µ1,t, . . . , µN,t)

>,

1 = (1, . . . , 1)>,

Σt = diag(σ1,t, . . . , σN,t),

φ

t

= (φ1,t, . . . , φN,t)

>,

dZt = (dZ1,t, . . . , dZN,t)

>.

(c) The investor has the following objective function

Et[dRp,t] −

1

2

γV art[dRp,t],

where γ > 0. Explain the ideas behind the design of this objective function.

(d) Show that the optimal portfolio is obtained by solving the following linear-quadratic optimization

problem:

max

φt

φ

>

t

(µ

t

− 1rt) −

1

2

γφ>

t

V tφ

t

,

where V t is an N by N matrix such that

[V t]ij =

ρij,tσi,tσj,t , i 6= j

σ

2

i,t , i = j

,

and ρij,t = ρji,t.

(e) Show that the optimal portfolio vector is

φ

t

= V

−1

t

(µ

t

− 1rt).

(f) Use a computer algebra package such as Mathematica to calculate V

−1

t when N = 3 for special

cases of your choice. The obvious one to start with is ρ12,t = ρ13,t = ρ23,t = 0.

1

2 HARJOAT S. BHAMRA

(2) Suppose the risk-free rate r is constant and the cum-dividend return on the SP500 over the interval

[t, t + dt) is given by

dRt = (r + λvt)dt +

√

vtdZt,

where

dvt = κ(θ − vt)dt + σ

√

vtdZv,t,

where Z and Zv are standard Brownian motions under P such that

Et[dZtdZv,t] = −ρdt, ρ > 0.

A household’s optimal consumption-portfolio choice problem is given by the following optimal stochastic

control problem

Jt = sup

(Cs)s≥t

,(φs)s≥t

Et

Z ∞

t

e

−δ(s−t)u(Cs)ds

s.t.

dWt = WtdRp,t − Ctdt,

where φt is the fraction of date-t wealth invested in the SP500,

dRp,t = (r + φtλvt)dt + φt

√

vtdZt,

and

u(x) = x

1−γ

1 − γ

.

The supremized objective function is known as the value function, which we denote via J. The

date-t value function depends on date-t wealth and the date-t instantaneous variance vt via

Jt = H(vt)

γu(Wt),

where H(v) satisfies the following ordinary differential equation (ode)

0 = 1 −

”

1

γ

δ +

1 −

1

γ

r +

1

2

γv

λ

2 − γ

2

2

(1 − ρ

2

)

H0

(v)

H(v)

2

!!# H(v) + κ

0

(θ

0 − v)H0

(v)

+

1

2

2

vH00 (1) (v),

and

κ

0 = κ −

(γ − 1)λρ

γ

,

θ

0 =

κ

κ0

θ.

The optimal portfolio and consumption expenditure choices are given by

φt =

1

γ

λ − ρ

H0

(vt)

H(vt)

Ct =

1

H(vt)

Wt.

(a) For the special case γ = 1, solve the ordinary differential equation. Ensure your solution remains

finite as v → ∞. Hence, find the household’s optimal portfolio and consumption expenditure choices.

Explain the economics underlying your solution.

(b) Use your solution to provide wealth and expenditure management advice for a client with current

wealth of 1M GBP.

(c) By defining

k(v) = 1

γ

δ +

1 −

1

γ

r +

1

2

γv

λ

2 − γ

2

2

(1 − ρ

2

)

H0

(v)

H(v)

2

!! ,

µ

0

v

(v) = κ

0

(θ

0 − v),

σ

2

v

(v) =

2

v,

we can write the ode (1) as

0 = 1 − k(v)H(v) + µ

0

v

(v)H0

(v) + 1

2

σ

2

v

(v)H00 (2) (v).

DERIVATIVES: ASSIGNMENT 3, PART B 3

To solve (2) we shall assume that v can take N + 1 equally spaced values: v1, . . . vN+1, which the

coder needs to choose. Define ∆v = v2 − v1. We have thereby discretized the state space of the

stochastic process v.

Define the N + 1 by N + 1 matrix S via

S =

−p1,2 p1,2 0 · · · · · · · · · 0

p2,1 −(p2,1 + p2,3) p2,3 0 · · · · · · 0

0 p3,2 −(p3,2 + p3,4) p3,4 0 · · · 0

.

.

. · · · · · · · · · · · · · · ·

.

.

.

0 · · · 0 pN−1,N−2 −(pN−1,N−2 + pN−1,N ) pN−1,N 0

0 · · · · · · 0 pN,N−1 −(pN,N−1 + pN,N+1) pN,N+1

0 · · · · · · · · · 0 pN+1,N −pN+1,N

,

where

pn,n+1 =

µ

0

v

(vn)

∆v

I{µ0

v

(vn)>0} +

1

2

σ

2

v

(vn)

(∆v)

2

pn,n−1 = −

µ

0

v

(vn)

∆v

I{µ0

v

(vn)<0} +

1

2

σ

2

v

(vn)

(∆v)

2

.

Define the N + 1 by N + 1 matrix K via

K = diag(k(v1), . . . , k(vN+1)),

where

k(vn) = 1

γ

δ +

1 −

1

γ

r +

1

2

γvn

λ

2 − γ

2

2

(1 − ρ

2

)

H(vn+1)−H(vn)

∆v

I{µ0

v

(vn)>0} +

H(vn)−H(vn−1)

∆v

I{µ0

v

(vn)<0}

H(vn)

2

.

Define the 1 by N + 1 vector of ones 1 = (1, . . . , 1)>.

We can find H = (H(v1), . . . , H(vN+1))> via the following recursive procedure

Hk+1 = (I − S∆t)

−1

[1∆t + (I − K∆t)Hk

(3) ].

The above equation starts with Hk and then provides an updated version of H, labelled as Hk+1

.

We start with an initial guess, labelled as H1 and then use the above equation to generate H2

, H3

,

and so on until ||Hk+1 − Hk

|| = maxn∈{1,…,N+1} |Hk+1(vn) − Hk

(vn)| < for some > 0 which

the coder needs to choose.

Use the above iterative procedure to write code that finds the vector H.

(d) Check your code converges for the special case ρ = ±1. You will need to play with ∆t and N.

(e) Bonus Part: If you are familiar with the Method of Variation of Parameters, you can derive an

integral expression for the solution of (2) for the special case ρ = ±1. Use this expression to check

your code.

(f) Make sure your code converges for the general case ρ

2 6= 1. You will need to play around some more

with ∆t and N.

(g) Assume the stochastic discount factor process Λ is given by

dΛt

Λt

= −rdt − λ

√

vtdZt,

where λ > 0 is a constant. Derive an expression for dvt under the risk-neutral measure Q.

(h) Bonus Part: Find a way to calibrate your model. You can use two sets of data: the SP500 and

dividends (you can find monthly data on Robert Shiller’s website) and vanilla options on the SP500.

You can price the options using Heston’s model.

(i) Find the myopic and hedging demand components of the optimal portfolio and plot them as functions

of √

vt

for the parameter values you have chosen (look online!) or calibrated.

(j) Use your results from the previous subquestion to provide wealth and expenditure management

advice for a client with current wealth of 1M GBP.

(k) Bonus Part: Use the reinforcement learning algorithm described in Section 3 of ‘Reinforcement

Learning for Continuous Stochastic Control Problems’ by Munos and Bourgine (on the Hub with

the assignment) to solve for the optimal policies without assuming a particular stochastic process

for v. Compare the optimal policies with those you found in the previous subquestion.

4 HARJOAT S. BHAMRA

Additional Notes

From the ode and the Feynman-Kac Theorem, it follows that

(4) H(vt) = E

P

0

t

Z ∞

t

e

−

R u

t

ksdsdu

,

where P

0

is some probability measure we need to determine. Observe that

dvt = µ

0

v

(vt)dt + σv(vt)dZP

0

v,t,

where ZP

0

v

is a standard Brownian motion under P. We also know that

dvt = µv(vt)dt + σv(vt)dZv,t.

From Girsanov’s Theorem

µ

0

v

(vt)dt = µv(vt)dt + Et

dM0

t

M0

t

dvt

,

where M0

is an exponential martingale under P, which we shall now identify. From the above equation,

we obtain

κ

0

(θ

0 − vt)dt = κ(θ − vt)dt + Et

dM0

t

M0

t

dvt

.

Hence,

Et

dM0

t

M0

t

dvt

= −(κ

0 − κ)vtdt,

because κ

0θ

0 = κθ. Therefore,

Et

dM0

t

M0

t

dvt

=

(γ − 1)λρ

γ

vtdt.

Therefore, one possibility for M0

is

dM0

t

M0

t

= −

(γ − 1)λ

γ

√

vtdZt.

We can use M0

to define P

0 via

E

P

0

t

[IA] = E

P

t

M0

T

M0

t

IA

,

where A is an event realized at date-T.

We shall derive the recursive scheme for finding H via (4). Note that (4) implies

H(vt) = dt + e

−ktdtE

P

0

t

[H(vt+dt)]

= E

P

0

t

(5) [dt + (1 − ktdt)H(vt+dt)].

We now discretize the state space for the stochastic process v. This means using a continuous time

Markov chain under P

0

to approximate v under P

0

. We assume the Markov chain can take values in the

set {v1, . . . , vN+1}. The Markov chain which approximates v has transition matrix S, where pn,n+1dt

the probability under P

0 of the Markov chain taking the value vn at the end of the interval [t, t + dt),

conditional on the current value being vn. Hence, we obtain the discretized state-space version of (5):

Hi =

NX

+1

j=1

[e

Sdt]ij [dt + (1 − kidt)Hj ],

where Hi = H(vi), ki = k(vi) and [e

Sdt]ij is the probability under P

0 of transitioning from state i (where

the chain takes value vi) to state j (where the chain takes value vj ) over the interval [t, t + dt).

In vector-matrix form, we have

H = e

Sdt[1dt + (I − Kdt)]H

Observe that e

Sdt[1dt + (I − Kdt)] is an operator, but is not a linear operator, because K depends

on H. The above equation tells us that H is a fixed point of this nonlinear operator.

Now observe that e

Sdt = (e−Sdt)−1

. Hence, when we discretize time, we obtain In vector-matrix

form, we have

H = (e

−S∆t

)

−1

[1∆t + (I − K∆t)]H.

DERIVATIVES: ASSIGNMENT 3, PART B 5

Using the expansion e−S∆t = I +

P

n=1(−)

n S

n(∆t)

n

n!

, we obtain

H = (I − S∆t)

−1

[1∆t + (I − K∆t)]H + o(∆t).

We can start with a guess for H and thereby obtain the recursive scheme (3).

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