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python代写-7CCMFM02T

时间：2020-12-30

7CCMFM02T (CMFM02)

King’s College London

University Of London

This paper is part of an examination of the College counting towards the award of a degree.

Examinations are governed by the College Regulations under the authority of the Academic

Board.

FOLLOW the instructions you have been given on how to upload your solutions

MSc Examination

7CCMFM02T (CMFM02) Risk Neutral Valuation

January 2021

Time Allowed: Two Hours

Full marks will be awarded for complete answers to all FOUR questions.

Within a given question, the relative weights of the different parts are

indicated by a percentage figure.

You may consult lecture notes.

2021 ©King’s College London

7CCMFM02T (CMFM02)

1. a. Construct a n-dimensional Gaussian copula with correlation matrix ρij

and show how we can use this simulate n correlated standard Exponential

random variables E1, ..., En.

Solution. Recall that the joint density of n Normal random variables

X1, ..., Xn is

1√

(2pi)ndet(Σ)

e−

1

2

xTΣ−1x, where here Σi,j = E(XiXj), but in

this case here the Zi’s will have variance of 1, so Σi,j = ρij = E(ZiZj).

Σ is assumed to be given here, but has to positive definite to ensure that

the Var of c1Z1 + ... + cnZn is non-negative for any vector c = (c1, .., cn)

with ci ∈ R (this variance is given by cTΣc). We know from the Applied

Probability Revision notes that Ui := Φ(Zi) ∼ U([0, 1]). We then set

Ei = F

−1(Ui) where F (x) = 1 − e−x is the distribution function of an

Exp(1) random variable.

- 2 - See Next Page

7CCMFM02T (CMFM02)

2. a. Consider a jump-diffusion model for a log stock price process Xt as in the

lecture notes with negative only jumps. Explain how we can price a call

option on S¯T , where S¯t = max0≤u≤tSu is the running maximum process of

S (example of a lookback option). [30%]

Solution. For pricing options, recall we first have to choose µ so that

E(St) = E(eXt) = eV (1)t = ert (so as to make the discounted stock price a

martingale) so we must impose that V (1) = r. To see this note that

E(e−rtSt|Fs) = e−rtE(eXt |Fs) = e−rtE(e(Xs+Xt−Xs)|Fs)

= e−rteXsE(eXt−Xs|Fs)

= e−rteXsE(eXt−Xs)

= e−rteXsE(eXt−s)

= Sse

−rteV (1)(t−s) = e−rsSs

where we have used that X has independent increments i.e. Xt − Xs is

independent of (Xu)0≤u≤s, and X is a stationary process, i.e. Xt−Xs has

the same distribution as Xt−s.

From the notes we also know that E(e−qτa) = e−aΦ(q) for q > 0, where Φ(q)

is the largest inverse of V (p) = logE(epX1) = 1

t

logE(epXt). To compute

the density of τa, recall that we set −q = ik for k ∈ R, and compute the

inverse Fourier transform of E(eikτa) as

fτa(t) =

1

2pi

∫ ∞

k=−∞

e−ikt E(eikτa)dk =

1

2pi

∫ ∞

k=−∞

e−ikte−aΦ(−ik)dk .

Then

P(X¯t ≥ a) = P(τa ≤ t)

so the density fX¯t(a) of X¯t is given by

fX¯t(a) = −

d

da

P(X¯t ≥ a) = − d

da

P(τa ≤ t) = − d

da

∫ t

0

fτa(s)ds

e−rTE((S¯T −K)+) = e−rTE((eX¯T −K)+) = e−rT

∫ ∞

a=0

fX¯T (a)(a−K)+da

assuming X0 = 0.

- 3 - Final Page

King’s College London

University Of London

This paper is part of an examination of the College counting towards the award of a degree.

Examinations are governed by the College Regulations under the authority of the Academic

Board.

FOLLOW the instructions you have been given on how to upload your solutions

MSc Examination

7CCMFM02T (CMFM02) Risk Neutral Valuation

January 2021

Time Allowed: Two Hours

Full marks will be awarded for complete answers to all FOUR questions.

Within a given question, the relative weights of the different parts are

indicated by a percentage figure.

You may consult lecture notes.

2021 ©King’s College London

7CCMFM02T (CMFM02)

1. a. Construct a n-dimensional Gaussian copula with correlation matrix ρij

and show how we can use this simulate n correlated standard Exponential

random variables E1, ..., En.

Solution. Recall that the joint density of n Normal random variables

X1, ..., Xn is

1√

(2pi)ndet(Σ)

e−

1

2

xTΣ−1x, where here Σi,j = E(XiXj), but in

this case here the Zi’s will have variance of 1, so Σi,j = ρij = E(ZiZj).

Σ is assumed to be given here, but has to positive definite to ensure that

the Var of c1Z1 + ... + cnZn is non-negative for any vector c = (c1, .., cn)

with ci ∈ R (this variance is given by cTΣc). We know from the Applied

Probability Revision notes that Ui := Φ(Zi) ∼ U([0, 1]). We then set

Ei = F

−1(Ui) where F (x) = 1 − e−x is the distribution function of an

Exp(1) random variable.

- 2 - See Next Page

7CCMFM02T (CMFM02)

2. a. Consider a jump-diffusion model for a log stock price process Xt as in the

lecture notes with negative only jumps. Explain how we can price a call

option on S¯T , where S¯t = max0≤u≤tSu is the running maximum process of

S (example of a lookback option). [30%]

Solution. For pricing options, recall we first have to choose µ so that

E(St) = E(eXt) = eV (1)t = ert (so as to make the discounted stock price a

martingale) so we must impose that V (1) = r. To see this note that

E(e−rtSt|Fs) = e−rtE(eXt |Fs) = e−rtE(e(Xs+Xt−Xs)|Fs)

= e−rteXsE(eXt−Xs|Fs)

= e−rteXsE(eXt−Xs)

= e−rteXsE(eXt−s)

= Sse

−rteV (1)(t−s) = e−rsSs

where we have used that X has independent increments i.e. Xt − Xs is

independent of (Xu)0≤u≤s, and X is a stationary process, i.e. Xt−Xs has

the same distribution as Xt−s.

From the notes we also know that E(e−qτa) = e−aΦ(q) for q > 0, where Φ(q)

is the largest inverse of V (p) = logE(epX1) = 1

t

logE(epXt). To compute

the density of τa, recall that we set −q = ik for k ∈ R, and compute the

inverse Fourier transform of E(eikτa) as

fτa(t) =

1

2pi

∫ ∞

k=−∞

e−ikt E(eikτa)dk =

1

2pi

∫ ∞

k=−∞

e−ikte−aΦ(−ik)dk .

Then

P(X¯t ≥ a) = P(τa ≤ t)

so the density fX¯t(a) of X¯t is given by

fX¯t(a) = −

d

da

P(X¯t ≥ a) = − d

da

P(τa ≤ t) = − d

da

∫ t

0

fτa(s)ds

e−rTE((S¯T −K)+) = e−rTE((eX¯T −K)+) = e−rT

∫ ∞

a=0

fX¯T (a)(a−K)+da

assuming X0 = 0.

- 3 - Final Page