Description
1. Consider the following finite difference schemes for some sufficiently smooth function
f : (t, x, y) 7→ f(t, x, y):
3f(t, x, y) − 4f(t − ∆t, x, y) + f(t − 2∆t, x, y)
2∆t
(1)
−f(t, x + 3∆x, y) + 4f(t, x + 2∆x, y) − 5f(t, x + ∆x, y) + 2f(t, x, y)
∆x
2
(2)
f(t, x + ∆x, y) − f(t, x + ∆x, y − ∆y) − f(t, x, y) + f(t, x, y − ∆y)
∆x∆y
(3)
(a) Find for every scheme to which limit it converges when the increments (∆-terms)
tend to zero.
(b) Find for each scheme its accuracy, i.e., determine the order of the error term.
Context: Such more complicated schemes are also sometimes used, e.g., when
implementing boundary conditions.
2. Derive a system of ordinary differential equations (in vector-matrix form v
0 = Av) that
approximates the Black–Scholes PDE
−vt(t, s) + 1
2
σ
2
s
2
vss(t, s) + rsvs(t, s) = rv(t, s)
by using a space discretization with finite differences
vs(t, si) ≈
v(t, si+1) − v(t, si)
∆s
vss(t, si) ≈
v(t, si+1) − 2v(t, si) + v(t, si−1)
∆s
2
2
on the equally spaced grid {s1, s2, . . . , sI}, si+1 − si = ∆s. Specifically, derive (the
coefficients of) the matrix A explicitly. (You may ignore the issue of the boundary
condition, i.e., the first and last row of the matrix for the moment).
3. Calibrate the Black–Scholes model
dSt = rSt dt + σSt dW1
t
, S0 = s
to market data (choose a realistic risk-free rate from data and use call options). 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 = {σ} 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 , e.g.,
curve fit from scipy.optimize,
https://python4mpia.github.io/fitting_data/least-squares-fitting.html
(a) Calculate the Black-Scholes price simulating the path of the stock price and using
Monte-Carlo integration. Use different starting values for the least square fit, e.g.,
σ = 0.2, 0.3, 0.02, 3.
(b) Do know the same as in (a), but use know the Black-Scholes formula instead of
Monte Carlo integration.
(c) Compare the results from (a) and (b) and try to understand the difference.
Explore the reasons why one of the techniques works well, and one not. To do this,
it might be helpful to know that curve fit uses internally the
Levenberg-Marquardt algorithm and do an online search to understand how this
algorithm works and why it might not work well with one of the pricing techniques.
Note: This problem intends to clarify the issues that lead to the problems encountered
in problem 6 on assignment 6.
8 points per problems
