Stochastic Analysis in Finance
MATH11154 Solutions and comments May 2020
1. Let (Wt)t≥0 be a Wiener martingale with respect to a filtration (Ft)t≥0 on a probability space (Ω,F,P).
(a) Prove that for any constant λ ̸= 0 the process V = (λ−1Wλ2t)t≥0 is a Wiener martingale with respect to the filtration (Fλ2t)t≥0. [6 marks]
(b) Prove that M = (Wt2)t≥0 is a submartingale with respect to a filtration (Ft)t≥0. (c) Consider the process U = (σWt)t∈[0,T], where σ ∈ R is a constant such that |σ| ≠ 1.
(i) Determine [U ]t , the quadratic variation of U over the interval [0, t] for any t ∈ [0, T ].
[7 marks]
(You may use without proof what you know about the quadratic variation of Wiener processes.)
(ii) Prove there is no probability measure Q, equivalent to P, such that under Q the process (Ut)t∈[0,T ] is a Wiener process. [6 marks]
Solution:
(a) Vt is Fλ2t-measurable for every t ≥ 0; Vt − Vs = λ−1(Wλ2t − Wλ2s) ∼ N(0,t − s) and it is independent of Fλ2s for 0 ≤ s ≤ t. [6 marks]
(b) E|Mt|=EWt2 =t<∞andMt isFt-measurableforeveryt≥0;for0≤s≤tbyJensen’s inequality E(W2|Fs) ≥ (E(Wt|Fs))2 = Ws2, since W is a martingale with respect to (Ft)t≥0.
N (n)−1
Sn := |Utni+1 −Utni |2 →t inL2(Ω,F,Q)asn→∞.
i=0
Hence limk→∞ Snk = t for Q-almost every ω ∈ Ω, for a subsequence nk → ∞. Since P and Q are equivalent, it follows that limk→∞ Snk = t for P-almost every ω ∈ Ω. Consequently, σ2t = [U]t = t, which means |σ| = 1. [6 marks]
2. Let (Wt)t≥0 be a Wiener martingale with respect to a filtration (Ft)t≥0, let (bt)t∈[0,T] be an Ft-adapted process such that almost surely
T
b2t dt<∞ foragivenT >0,
0
and define the stochastic process
γt(b) = exp bs dWs − 21 b2s ds for t ∈ [0,T].
[6 marks]
Clearly, [U]t = [cW]t = σ2[W]t = σ2t.
partitions of [0, t] such that limn→∞ maxi |tni+1 − tni | = 0, we have
[6 marks] [7 marks]
(c) (i)
(ii) IfU isaWienerprocessunderQ,thenforasequence0=tn0
(a) Using the Main Theorem on Pricing European type options prove that C(K,T)=E(S0eσWT−Tσ2/2 −Ke−rT)+, P(K,T)=E(Ke−rT −S0eσWT−Tσ2/2)+,
where the notation a+ := max(a, 0) is used. [5 marks]
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Stochastic Analysis in Finance
MATH11154 Solutions and comments May 2020
(b) Prove that the process
Xt = (S0eσWt−σ2t/2 − Ke−rt)+, t ≥ 0
is a submartingale with respect to the history Ft = σ(Ws,s ≤ t), t ≥ 0, of the Wiener
process W. [5 marks]
(c) Using (a) and (b) prove that C(K,T) is an increasing function of T. [5 marks]
(d) CalculateP(K,T)−C(K,T)from(a),anddeterminelimT→∞P(K,T)whenr>0.
[5 marks]
(e) Using (c) and (d) prove that P (K, T ) is an increasing function of T if and only if r = 0.
[5 marks]
Solution:
(a) By the Main Theorem on Pricing European we have C(K,T)=e−rTEQ(ST −K)+ =EQ(S ̃T −e−rTK)+
=EQ(S0eσW ̃T−Tσ2/2 −e−rTK)+ =E(S0eσWT−Tσ2/2 −Ke−rT)+. In the same way we get
P(K,T)=E(Ke−rT −S0eσWT−Tσ2/2)+.
(b) Xt is Ft-measurable for every t, E|Xt| ≤ ES0eσWt−σ2t/2 = S0 and for 0 ≤ s ≤ t by Jensen’s inequality we have
E(Xt|Fs) ≥
σWt−σ2t/2 −rt E(S0e − Ke
+
σWt−σ2t/2 −rt+ |Fs) − Ke
σWs−σ2s/2 −rt+ −Ke
≥ S0e
σWs−σ2s/2
−rs+
−Ke =Xs.
= S0e
(c) By (b) C(K, T ) = EXT is an increasing function of T .
[5 marks]
|Fs)
=
E(S0e
[5 marks] (d) By (a) P(K,T) − C(K,T) = E(Ke−rT − S0eσWT −Tσ2/2) = Ke−rT − S0EeσWT −Tσ2/2 =
Ke−rT −S0. SinceP(K,T)≥0andP(K,T)≤Ke−rT,wehave
0 ≤ limsupP(K,T) ≤ limsupKe−rT = 0,
T→∞ T→∞ which implies limT →∞ P (K, T ) = 0.
[5 marks]
(e) If r = 0 then P(K,T) = C(K,T) + K − S0, which shows that P(K,T) is increasing in T because C(K,T) is increasing in T. If r > 0 then limT→∞ P(K,T) = 0, which implies that P(K,T) cannot be increasing in T.
[5 marks] 4. Consider again the Black-Scholes market with bond and stock prices as in Question 3. We
[5 marks]
want to compute the price V at t = 0 of the European type option with payoff L if mint∈[0,T]St≤K
h := 0 otherwise atexpirydateT,whereL>0andK>0aresomeconstantssuchthatS0 >K.
3
Stochastic Analysis in Finance
MATH11154 Solutions and comments May 2020
(a) Using the main theorem on pricing European type options, show that
V =Le−rTP wherea:=σr −12σ,b:=σ−1ln(K/S0).
min (Wt +at)≤b , t∈[0,T ]
(b) Using Girsanov’s theorem show that
V = Le−rT E1[mint∈[0,T ] Wt≤b]eaWT − 12 a2T ,
where a and b are the constants defined in (a).
[5 marks] (c) Denote the event [mT ≤ b,WT ≤ x] by Ax for every x ∈ R, where mT := mint∈[0,T] Wt.
Using the reflection principle for the Wiener process W, prove that
P(WT ≤x) if xb. Then
P(Ax)=P(mT ≤b)−P(mT ≤b,WT >x), andbythereflectionprincipleP(mT ≤b,WT >x)=P(WT ≤2b−x).
[5marks]
(d) Since
(e) From (d)
[3 marks]
b x2 ∞ (2b−x)2 eaxe−2T dx+ eaxe− 2T dx
d dx
Le−rT∞ 12
if x≤b if x > b
−∞
g(y)dy for all x ∈ (∞,∞).
P(A )= 2πT
x2
√1
e−2T (2b−x)
=:g(x) is nonnegative and continuous at every x ∈ R such that ∞ g(x) dx < ∞,
x
√ 1 2πT
e− 2T
2
P(Ax) =
x −∞
V = √ eax−2a Tg(x)dx=C 2πT −∞
−∞
b
5
Stochastic Analysis in Finance
MATH11154 Solutions and comments May 2020
√2√
withC:=Le−rTe−1a2T =Le−(r+12a2)T.Bythechangeofvariablez:=2b−x
Thus
b −∞
2πT 2πT
∞ (2b−x)2 b z2
eaxe− 2T dx = ea(2b−z)e− 2T dz. b −∞
V=C
a(2b−x) e
ax − x2
+e e2Tdx.
[2 marks]
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