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Notation here follows Shreve, Stochastic Calculus for Finance, vols. I & II, Springer, 2004.

Theoretical Problems

1. Let Ft be the Öltration generated by a Wiener process W (t). Let R (t) be the interest

rate process used to deÖne the discount process D (t). Assume there exists a unique

risk-neutral measure, leading to the Wiener process W~ (t) with respect to P~. If V (T)

is a random variable that is FT -measurable, and V (t) is deÖned via:

V (t) = 1

D (t)

E~ (D (T) V (T)jFt)

then D (t) V (t) is a martingale.

(a) Suppose V (T) > 0 a.s. Show that V (t) > 0 a.s. (from its deÖnition above).

(b) Show that there exists an adapted process ( ~ t) such that:

dV (t) = R (t) V (t) dt +

( ~ t)

D (t)

dW~ (t)

Hint: Start with a formula for d (D (t) V (t)) as per the martingale representation

theorem, then as you expand this out recognize that dV dt = 0.

(c) Show that there exists an adapted process (t) such that we can write:

dV (t) = R (t) V (t) dt + (t) V (t) dW~ (t)

By the way, (t) can be random and in particular it is Öne if the formula for

(t) you derived involves V (t). The point is there is SOME process you can

put there that works! How did we use strict positivity? (Think of D (t), V (t)

as continuous processes; D (t) is intrinsically positive, but what happens if V (t)

can take on negative as well as positive values?) This shows that V (t) is a

generalized geometric Brownian motion process. The point of this problem is

that every strictly positive asset is a generalized geometric Brownian motion.

2. Let X (t); Y (t) be ItÙ processes given by:

dX (t) = a (t) dt + b (t) dW (t)

dY (t) = c (t) dt + d (t) dW (t)

where a; b; c; d are adapted processes. Assume Y (t) > 0 a.s., and V (t) = X (t) =Y (t).

Obtain an SDE satisÖed by V (t), simpliÖed so it has the above form (i.e., in the form

of an ItÙ process). Note that X (t); Y (t), but not dX (t) or dY (t), can appear in your

Önal expression for dV (t).

Simulating Stochastic Di§erential Equations

We are going to be viewing continuous-time deterministic functions and stochastic processes

in discretized time. For simplicity, we will take equally spaced time intervals, say t = n,

0 n N, with Önal time T = N. (Careful, including t = 0, this is N + 1 points). We

will use the notation:

x [n] = x (n) = xn

Consider a stochastic di§erential equation of the form:

dX (t) = (t; X (t)) dt + (t; X (t)) dW (t)

where (); () are deterministic functions, and as usual W (t) is a Wiener process. The

parameter controls the variance of the increment dW; speciÖcally, in the discretization, let

us use the notation:

dW [n] = W [n + 1] W [n]

where fdW [n]g are iid N (0; ). The SDE actually means:

X (t) = X (0) + Z t

0

(u; X (u)) du +

Z t

0

(u; X (u)) dW (u)

In discretized form, with t = n this becomes:

X [n] = X [0] + Xn1

m=0

(m; X [m]) +

Xn1

m=0

(m; X [m]) dW [m]

Letís default with = 0:01 and N = 250 (which is approximately the number of days per

year for a typical Önancial instrument).

1. Start with basic geometric Brownian motion:

dSt = Stdt + StdWt

with S (0) = 1, = 0:1, = 0:2. Also assume a constant underlying interest rate

r = 0:05. Under the risk-neutral measure, dSt satisÖes a modiÖed SDE involving dt

and dW~

t

. See Problem 3 below: If you ever detect St 0 trap that condition,

note that it occurred, and discard that path (donít use it).

(a) Write the modiÖed SDE involving dW~

t

. SpeciÖcally, calculate the coe¢ cients that

appear in this SDE from the speciÖc values of ; ; r provided. Unless speciÖed

otherwise, however, grow your paths using the ORIGINAL formulation (i.e., the

ACTUAL probabilities).

(b) Generate 1000 paths of S [n].

(c) Use a Monte Carlo approach to estimate E (S [N=2]) and E (S [N]) directly.

(d) We now want to connect this to the Black-Scholes model. Let V (T) = (S (T) K)

+

,

the payout of a European call option with strike price K. In our discrete notation,

V [N] = (S [N] K)

+

. Use your estimate for E (S [N]) as your value for K. Use

the Black-Scholes model to obtain a formula for V [N=2] in terms of S [N=2], and

graph it (for the speciÖed parameters ; ; r

(e) Now take the Örst 10 paths you generated, S

(i)

[n], 1 i 10. For each S

(i)

[N=2],

you can compute V

(i)

[N=2] from the Black-Scholes formula. On the other hand,

you can use a Monte Carlo approach to compute V

(i)

[N=2] using the martingale

property of the discounted stock price. So, for each i, grow 1000 paths from N=2

to N and average to estimate V

(i)

[N=2]. Compare these estimated values with the

exact values in these cases. Report the results how you see Öt: a table; you could

superimpose two scatter plots (S

(i)

[N=2] versus actual V

(i)

[N=2], and S

(i)

[N=2]

versus actual V

(i)

[N=2]). Comments: Careful. First, you need to grow NEW

paths, emanating from time t = T=2, towards t = T, taking S

(i)

[N=2] as an initial

condition. Also, you are computing V

(i)

[N=2] as an expectation by virtue of the

martingale property, BUT the question is WHICH SDE FOR St DO YOU

USE?

2. Cox-Ingersoll-Ross Interest Rate Model

dR (t) = (a R (t)) dt +

p

R (t)dW (t)

with ; ; positive, and let R (0) = r > 0. This model for interest rate R (t) guarantees R (t) > 0. This model is typically used for short term interest rates, which tend to

exhibit volatility. Although no closed form solution exists, it is possible to determine

certain properties for it. In particular the mean and variance of R (t) are given by:

E (R (t)) = e

tr +

1 e

t

!

var (R (t)) =

2

r

e

t e

2t

+

2

2

2

1 2e

t e

2t

!

2

2

2

Discretizing this SDE for simulation purposes is very tricky because the property that

R (t) > 0 does not carry over if you use the simple discretization as in the previous problem! This actually brings up a larger questionñ does the discretized system

accurately reáect properties of the original SDE?

(a) Select: = 1, = 0:10, r = 0:05, = 0:5. Generate 1000 paths for R (t) over

a span 0 t 10, using = 0:01. In your code, trap the condition that R 0

(if this occurs, your code should display an exception, should halt that one path,

but should continue computing the other paths). So the goal is to generate 1000

valid paths (with R (t) never reaching 0). How many ìbadî paths do you get

in order to reach 1000?

(b) Graph the Örst 10 paths for R (t), just to see what it looks like.

(c) Use a Monte Carlo approach to estimate the mean and variance of R (t) at t = 1

and t = 10, and compare with the exact formulas given above. Here you should

use the valid paths only.

3. Go back to Problem 1 again. We know the actual St (in continuous-time) is geometric

Brownian motion so we necessarily have St > 0 a.s., and yet is that property guaranteed in your simulation? On the other hand, with all your simulations, did you ever

encounter simulated St 0? I expect not. Why not? [Even if you got a negative

answer, why did I guess that you wouldnít? The hint is that it is a guess

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