Description
1. Let (W1
t
)t≥0, and (Z)t≥0 be Brownian motions and define the stochastic process (W2
t
)t≥0
by
W2
t = ρW1
t +
p
1 − ρ
2Zt
(a) Show that (W2
t
)t≥0 is itself a Brownian motion.
(b) Calculate the covariance and the correlation between W1
t
and W2
t
.
(c) Plot one sample path for each of the two Brownian motions for the cases where
(i)ρ = 0.9, (ii)ρ = 0.3, (iii)ρ = 0, (iv)ρ = −0.3, (v)ρ = −0.9 (one plot per case).
Context: Correlated Brownian motions are often used in modeling assets with stochastic
volatility. Specifically, equities like stock show usually a highly negative correlation
between asset and volatility.
2. Assume that stock prices follow under the risk-neutral measure the dynamics of Scott’s
exponential Ornstein-Ulenbeck process model,
dSt = rSt dt + e
YtSt dW1
t
, S0 = s
dYt = λ
κ − Yt
dt + ξ dW2
t
, Y0 = y
E[W1
t W2
t
] = ρt
with parameters s = 100, y = −1, λ = 5, κ = −1.5, ρ = −0.2 and ξ = 0.25 and the
market contains a money market account with constant interest rate r = 3%. Calculate
the price of a lookback put option
max
t∈[0,T]
St − ST
with maturity T = 0.3 using the Milstein scheme to create the sample paths
2
3. Assume that stock prices follow under the risk-neutral measure the CEV model,
dSt = rSt dt + σSβ
t St dW1
t
, S0 = s
with parameters s = 100 and σ = 0.2 and that the market contains a money market
account with constant interest rate r = 3%.
(a) Calculate the price of European call options with maturity T = 1 and strikes K
every $5 between $85 and $125 for β = 0, −0.3, −0.5, −0.7, −1.
(b) Calculate the implied volatility of the options of (a). Describe the role of β on the
volatility smile.
4. Assume that stock prices follow under the risk-neutral measure the Heston model,
dSt = rSt dt +
p
YtSt dW1
t
, S0 = s
dYt = λ
κ − Yt
dt + ξ
p
Yt dW2
t
, Y0 = y
E[W1
t W2
t
] = ρt
with parameters s = 100, λ = 3, κ = 0.1 ξ = 0.25, y = 0.08, ρ = −0.8 and that the
market contains a money market account with constant interest rate r = 3%.
(a) Calculate the price of European call options with maturity T = 1 and strikes K
every $5 between $85 and $125.
(b) Calculate the implies volatility of these options. Check how the volatility changes
with changes in different parameters, notably ρ and ξ.
Context: One of the main reason to develop more sophisticated models then
Black-Scholes is that the Black-Scholes model is unable to capture the volatility
smile/skew of the markets. The two problems above investigate how well CEV model
and Heston model are capable to model the implied volatility and which flexibility they
offer.
5. We have seen in class that in a numerical implementation of the CIR process via Euler
scheme, the process may become negative. Instead of simply cut it off at zero, there
exist refined techniques. One of them should be developed in the following steps: Let Xt
be the solution of the SDE, dXt = µ(Xt) dt + σ(Xt) dWt
, then one can define the
generalized Stratonovich Integral
Z T
0
f(Xt) ◦1 dWt = limn→∞
Xn−1
i=0
ti∈[0,T]
f(Xti+1 )
Wti+1 − Wti
and one can show that it relates to the Itˆo integral as follows,
Z T
0
f(Xt) ◦1 dWt =
Z T
0
f(Xt) dWt +
Z T
0
f
0
(Xt)σ(Xt)
3
(a) Use the concept of generalized Stratonovich integration to derive an implicit
recursion scheme for the SDE
dYt = a(b − Yt) dt + ξ
p
Yt dWt
; Y0 = y > 0; a, b, ξ > 0.
Finally you should get for the partition ti = iT/n of [0, T] the scheme
Yti+1 = Yti + (ab −
ξ
2
2
− aYti+1 )
T
n
+ ξ
p
Yti+1 (Wti+1 − Wti
).
(b) Derive out of the implicit scheme an explicit representation of Yti+1 . This should be
possible at least in the case that ξ
2 ≤ 2ab.
(c) Simulate ten sample paths of the process (Yt) on the interval [0, 1] with step size
0.01 and plot the result.
6. Calibrate the Heston Stochastic volatility model
dSt = rSt dt +
p
YtSt dW1
t
, S0 = s
dYt = λ
κ − Yt
dt + ξ
p
Yt dW2
t
, Y0 = y
E[W1
t W2
t
] = ρt
to market data (choose a realistic risk-free rate from data). Use as data options on
Apple stock with maturity May 19, 2017. Discard all strikes that have a trading volume
of less then 10 and use always the marked mid-price (average between bid and ask).
Calibrate Θ = {λ, κ, ξ, y, ρ} by minimizing the sum of the squared errors
argmin
Θ
X
i
C
Θ
i − Ci
2
.
This can be done numerically, using a numerical optimization procedure (you might use
as initial data λ = 3, κ = 0.1 ξ = 0.25, y = 0.08, ρ = −0.8) Do the procedure twice, once
using put and once using call data and compare the results.
Note: All programming problems should be either in Python 2.7 or Python 3.5. Matlab
and R are accepted, but no support for these languages is provided. Please comment the
programs extensively and send them in a .zip file with title Lastname HW6.zip and
suject line ”MA 573 HW6 Lastname” to Qingyun Ren qren@wpi.edu before the due
date of the homework (replacing the bold words by your actual last name). Please
provide printouts of programs amd plots that one can comment on them.
4 points per problem
