Tutorials Advanced Financial Mathematics
Summer Term 2023
PD Dr. V. Paulsen Series 02 12.04.2023
The exercises of this series deepens the contents of the chapter Black-Scholes model.
Written solutions should be handed over to the tutor until 19th of April.
Ex 1:
Let (Wt )t≥0 be a Wiener-Process and (Lt )0≤t≥T be defined by
Lt = E(e−rT STα |Ft )
for all 0 ≤ t ≤ T , whereat St = exp(rt) exp(σWt − 21 σ 2 t).
Determine a representation of L as exponential martingale.
Ex 2: Power Call
We consider a Black-Scholes Modell with volatility σ, initial stock price x > 0 and interest
rate r > 0.. According the equivalent martingale measure P∗
1
St = xert exp(σWt − σ 2 t), t ≥ 0
2
with a Wiener-process W w.r.t. P∗ . Show
1. For 0 ≤ t ≤ T and α > 0
1 2 1
E∗ (exp(−rT )STα |Ft ) = xα e(α−1)T (r+ 2 ασ ) exp(ασWt − (ασ)2 t).
2
2. For α > 0
1 2
E∗ e−rT STα = xα e(α−1)T (r+ 2 ασ ) .
3. The arbitrage-free initial price p0 (C) of a derivative with payoff
C = (STα − K)+
at T is given by
α α
log xK + αT (r + σ 2 (α − 21 )) log xK + αT (r − 21 σ 2 )
α −rT
p0 (C) = x h(T )Φ √ −Ke Φ √
ασ T ασ T
1 2
with h(T ) = e(α−1)T (r+ 2 ασ ) .
Hint: p(C) = E∗ e−rT (STα − K)+ .
Summer Term 2023
PD Dr. V. Paulsen Series 02 12.04.2023
The exercises of this series deepens the contents of the chapter Black-Scholes model.
Written solutions should be handed over to the tutor until 19th of April.
Ex 1:
Let (Wt )t≥0 be a Wiener-Process and (Lt )0≤t≥T be defined by
Lt = E(e−rT STα |Ft )
for all 0 ≤ t ≤ T , whereat St = exp(rt) exp(σWt − 21 σ 2 t).
Determine a representation of L as exponential martingale.
Ex 2: Power Call
We consider a Black-Scholes Modell with volatility σ, initial stock price x > 0 and interest
rate r > 0.. According the equivalent martingale measure P∗
1
St = xert exp(σWt − σ 2 t), t ≥ 0
2
with a Wiener-process W w.r.t. P∗ . Show
1. For 0 ≤ t ≤ T and α > 0
1 2 1
E∗ (exp(−rT )STα |Ft ) = xα e(α−1)T (r+ 2 ασ ) exp(ασWt − (ασ)2 t).
2
2. For α > 0
1 2
E∗ e−rT STα = xα e(α−1)T (r+ 2 ασ ) .
3. The arbitrage-free initial price p0 (C) of a derivative with payoff
C = (STα − K)+
at T is given by
α α
log xK + αT (r + σ 2 (α − 21 )) log xK + αT (r − 21 σ 2 )
α −rT
p0 (C) = x h(T )Φ √ −Ke Φ √
ασ T ασ T
1 2
with h(T ) = e(α−1)T (r+ 2 ασ ) .
Hint: p(C) = E∗ e−rT (STα − K)+ .
