THE AUSTRALIAN NATIONAL UNIVERSITY
RESEARCH SCHOOL OF FINANCE, ACTUARIAL STUDIES
AND STATISTICS
FINM3007: Advanced Derivatives Pricing and Applications
Semester 2, 2024
Assignment 2
Due: 11 October 2024, 11:59:59 PM
Total Marks: 32
General instructions. Submit your answers as a single PDF
file on Wattle. To ensure full marks, include working out and
explanations. For some questions, you may want to use computer
software. When this occurs, your written explanation should be
sufficient to explain your answer without having to read code.
Attach the cover sheet with your submission. Please check that
your submission on Wattle can be opened and the content of the
file is complete and correct.
Question 1. Let c(K,T ) be the price of a European call with strike price
K and maturity time T on a stock. By the definition of the derivative,
∂2c(K,T )
∂K2
= lim
δ→0
c(K − δ, T )− 2c(K,T ) + c(K + δ, T )
δ2
.
Suppose that call options with all strike prices K > 0 are traded and assume
the risk-free rate r is constant.
(a) Consider a portfolio that holds 1
δ2
, − 2
δ2
, 1
δ2
units of European calls with
strike pricesK−δ, K, K+δ and maturity time T , respectively. By con-
sidering the payoff, show that the value of the portfolio is nonnegative
at time T . [2 marks]
(b) Suppose that the implied risk-neutral distribution of the stock price
at time T calculated by the Breeden-Litzenberger formula is negative
for some values. Using (a), explain how an arbitrage strategy can be
constructed. [3 marks]
1
Question 2. Suppose the risk-free rate is r = 0.05 and the implied volatili-
ties on a stock with current price S0 = 100 with strike prices K and maturity
times T satisfy
σI(K,T ) = (0.1 + 0.4T )e
−(K/100)+1.
In the local volatility model, the stock price S = (St)t∈[0,T ∗] satisfies
dSt
St
= rdt+ σ(St, t)dZt, t ∈ [0, T ∗],
under the risk-neutral measure P̂, where σ is the local volatility function.
Note that the finite difference approximation with stepsize δ for the first and
second order derivatives of f are
∂f(x)
∂x
≈ f(x+
δ
2
)− f(x− δ
2
)
δ
,
∂2f(x)
∂x2
≈ f(x− δ)− 2f(x) + f(x+ δ)
δ2
,
respectively.
(a) Using the Dupire formula and finite difference approximations with
stepsize 0.1 for derivatives with respect to K and with stepsize 0.001
for derivatives with respect to T , find the value σ(109.5, 0.125) of the
local volatility surface σ(K,T ). [4 marks]
(b) Recall that we can approximate S by the Euler scheme
S(i+1)∆t − Si∆t = rSi∆t∆t+ σ(Si∆t, i∆t)Si∆t∆Zi, i = 0, 1, . . . , n− 1,
where ∆Zi ∼ N(0,∆t) are iid and ∆t = T ∗/n. We set n = 5. In order
to approximately evaluate the local volatility surface σ in the Euler
scheme, the following table gives σ(K,T ) computed using the Dupire
formula in the same way as (a) on a grid where the values are computed
at the midpoints of the grid (except at the endpoints of K where the
grid has been extended to 0 and ∞), and where A = σ(109.5, 0.125)
from (a).
T K
(0, 92) [92, 99) [99, 106) [106, 113) [113,∞)
[0, 0.05) 0.152799 0.130798 0.113750 0.099506 0.095336
[0.05, 0.1) 0.197973 0.172181 0.149667 0.130093 0.113211
[0.1, 0.15) 0.244338 0.212461 0.184647 A 0.139612
[0.15, 0.2) 0.290171 0.252262 0.219200 0.190515 0.165706
[0.2, 0.25] 0.335778 0.291842 0.253544 0.220332 0.191618
2
Now consider a fixed-strike lookback call on the stock with strike price
K = 110 and maturity time T ∗ = 0.25, which pays off
cfix-lT ∗ = (MT ∗ −K)+,
where MT ∗ = maxs∈[0,T ∗] Ss is the maximum of the stock price. Find
the payoff of the option for the sample path of S approximated by the
above Euler scheme with
∆Z0 = −0.069474, ∆Z1 = −0.403733, ∆Z2 = −0.224149,
∆Z3 = 0.169977, ∆Z4 = 0.416593.
[3 marks]
(c) Repeat (b) using the random variables
∆Z0 = −0.247006, ∆Z1 = −0.092446, ∆Z2 = 0.022566,
∆Z3 = 0.154839, ∆Z4 = −0.079953.
instead. [1 mark]
(d) Suppose that by repeating the procedure in (b) and (c) 8 more times,
we obtain the following payoffs:
3.551903, 0, 0, 0, 14.186861, 11.632845, 0, 0.
On the basis of these 10 simulation results (1 from (b), 1 from (c), 8
from (d)), approximate the price of the lookback call at time 0.
[2 marks]
Question 3. Let V be the price process of a ¥ asset which pays dividend
yield q, and S be the ¥/$ exchange rate process. Assume the bivariate Black-
Scholes model holds so that (V, S) follows bivariate GBM with correlation
ρV S. Let σV and σS be the volatility of V and S, respectively. The domestic
currency is $ with risk-free rate rd, and the foreign currency is ¥ with risk-free
rate rf . Both risk-free rates are constant. A foreign currency European call
option struck in the domestic currency with $ strike price Kd and maturity
time T pays off
cT = $(VT/ST −Kf )+.
Note that VT/ST is the value of the ¥ asset converted into $.
3
(a) By using either the fact that V/S is a $ traded asset, or Tutorial 6
Solutions, Question 3 (b) (you do not need to use both), or otherwise,
show that under the $ risk-neutral measure P̂$,
d(Vt/St)
Vt/St
= (rd − q)dt+ σdZt,
where σ =
√
σ2V − 2ρV SσV σS + σ2S and Zt is a P̂$-SBM. [3 marks]
(b) By using the Black-Scholes formula with appropriate substitutions or
otherwise, find the $ price of the foreign currency call struck in the
domestic currency at time 0. [3 marks]
Question 4. Suppose that under the risk-neutral measure P̂, a stock price
S = (St)t∈[0,T ] follows
St = S0e
mt+V (t),
where V ∼ VGP(b∗, µ∗, σ∗) is a variance gamma process. Suppose the risk-
free rate is r and S0.
(a) Consider a European call on the stock with strike price K and ma-
turity time T . Assuming that differentiation and integration can be
interchanged, show that the delta at time 0 is
∆ = e(m−r)T
(b∗)b
∗T
Γ(b∗T )
∫ ∞
0
N(d1(S
′
0, r
′, g, σ∗))gb
∗T−1e−(b
∗−r′)g dg,
where
S ′0 = S0e
mT ,
r′ = µ∗ +
1
2
(σ∗)2,
d1(S0, r, T, σ) =
log(S0/K) + (r +
1
2
σ2)T
σ
√
T
.
[2 marks]
(b) Suppose that r = 0.05, S0 = 100, and the risk-neutral parameters are
b∗ = 0.6, µ∗ = −0.05, σ∗ = 0.2. Consider a European call on this
4
stock with maturity T = 2 and strike K = 105. By approximating the
integral in (a) using the Riemann sum∫ ∞
0
f(g) dg ≈
n−1∑
i=0
f
(
xi + xi+1
2
)
(xi+1 − xi),
where n = 10, xi = i, i = 0, 1, . . . , n, find the delta of the call ∆.
[4 marks]
Question 5. Let (Ft)t∈[0,T ] be a filtration on the σ-field F , and let Z =
(Zt)t∈[0,T ] be a SBM under P such that Zt is Ft-measurable. We want to
prove a version of Girsanov’s theorem which is used to obtain the EMM
for the standard Black-Scholes model. In this case, it is enough to consider
γ ∈ R as a constant.
(a) By computing the cumulant generating function of Z under the Esscher
measure Qh, show that Z ∼ BM(h, 1) under Qh. [3 marks]
(b) Define a new process ZQ = (ZQt )t∈[0,T ] by
ZQt = γt+ Zt.
Define a new probability measure by
Q(A) = E[e−γZT−
1
2
γ2T I(A)], A ∈ F .
By applying (a) in the case where h = −γ and writing out the definition
of the corresponding Esscher measure, show that ZQ is a SBM under
Q. [2 marks]