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python代寫-7CCMFM02T

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.
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.
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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 SˉT , where Sˉt = 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
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(Xˉt ≥ a) = P(τa ≤ t)
so the density fXˉt(a) of Xˉt is given by
fXˉt(a) = ?
d
da
P(Xˉt ≥ a) = ? d
da
P(τa ≤ t) = ? d
da
∫ t
0
fτa(s)ds
e?rTE((SˉT ?K)+) = e?rTE((eXˉT ?K)+) = e?rT
∫ ∞
a=0
fXˉT (a)(a?K)+da
assuming X0 = 0.
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