STA2570-W21 Assignment 1
Due on Friday February 5 at 11:59 pm
1. The Black-Scholes PDE for determining the value V of an American put option is given by
max
{
Vt +
1
2
σ2S2VSS + rSVS − rV = 0, (K − S)+ − V
}
, S > 0, t < T,
where S is the stock price (in dollars), t is time (in years), K is the strike price of the option, T is the
expiration date, σ is the stock volatility per annum, and r is the annual continuous rate of interest. We
will compute the value of an American put with K = 10 and T = 1 by numerically solving the PDE.
(a) Take σ = 0.25 and r = 0.05. Implement the explicit finite difference scheme discussed in class with
∆x = (b− a)/26 (where (a, b) is the spacial domain) and two values of α := ∆t/(∆x)2: α = 15 and
α = 17. In both cases plot the graph of the computed value of the option against the stock price.
Explain your observation.
(b) Fix r = 0.05. Implement the explicit finite difference scheme with ∆x = (b − a)/26, α =
min(10, 0.5/σ2) and σ varying in the interval 0 ≤ σ ≤ 3, taking ∆σ = 3/20. Assuming the current
value of the stock is 10, plot the graph of the computed value of the option against volatility.
(c) Fix σ = 0.25. Implement the scheme with ∆x = (b−a)/26 and α = 10. For a given value of r, plot
the graphs of the computed value of the American put, and the exact value of the corresponding
European put against stock price on the same axis. Compare the two graphs for the three different
values of r: r = 0, 0.05, 0.1, and give an explanation.
(d) For σ = 0.25 and r = 0.05, plot the early-exercising boundary in the (t, S)-plane. Again use
∆x = (b− a)/26 and α = 10 here.
2. This problem deals with a continuously sampled, arithmetically averaged floating strike Asian option.
Today is t = 0 and the option expires in T years. The rate of interest is r and the stock volatility is σ.
Write It =
∫ t
0 Sudu where St is the stock price at time t. The pricing function V (t, S, I) of the Asian
option satisfies the following PDE:
Vt +
1
2
σ2S2VSS + rSVS + SVI − rV = 0,
with terminal data V (T, S, I) = (S − I/T )+ for a call and V (T, S, I) = (I/T − S)+ for a put. We
consider call option in part (a)-(c) and put option in part (d).
(a) Exploit the homogeneity in the problem by writing ξ = I/S and V (t, S, I) = Sw(t, ξ). Derive a
PDE for w and write down the corresponding terminal condition.
(b) Truncate the domain of w to [0, T ]× [0, ξmax]. What is a reasonable value of ξmax and why? Also
give the boundary condition for w at ξ = ξmax and ξ = 0.
(c) Write down an explicit finite difference scheme for the PDE in part (a)-(b) in which the first
derivative is approximated by the central difference for moderate and large ξ and by the forward
difference for small ξ. Implement the scheme with T = 1, σ = 0.25, r = 0.05 and ∆ξ = ξmax/2
9.
Find the corresponding value of the option when the stock price today is 20.
(d) To compute the price of an arithmetically averaged floating strike Asian put option, we can use
the following put-call parity:(
S − I
T
)+
−
(
I
T
− S
)+
= S − I
T
.
Compute, in closed-form, the time-zero price of the contingent claim with terminal payoff ST−IT /T .
(Hint: solve a PDE for w(t, ξ) with appropriate terminal condition; the solution is linear in ξ.)
3. In this problem you are asked to find the value of a one-touch knock-out using finite difference methods.
This is a double-barrier option in the FX market. There is a knock-in barrier L and a knock-out barrier
U . The option pays 1 at maturity T if and only if the underlying price process S hits the knock-in
barrier first, i.e. the payoff is 1{τL(a) The option can be valued by solving the transformed Black-Scholes PDE
ut +
1
2
σ2uxx +
(
r − 1
2
σ2
)
ux − ru = 0.
Write down the terminal and boundary conditions for the PDE problem.
(b) Write down the Crank-Nicolson scheme for solving the PDE problem in part (a). Identify the
numerical linear algebra problem to be solved in each time step.
(c) Implement the Crank-Nicolson scheme in part (b) with L = 18, U = 22, σ = 0.25, r = 0.05
and T = 0.5. Take ∆x = (b − a)/26 where (a, b) is the spatial domain of the PDE. Choose an
appropriate α := ∆t/(∆x)2 and explain your choice. Plot the graph of the option value at time
zero against the spot price, and print out the option value corresponding to S0 = 20. Use LU
decomposition to solve the numerical linear algebra problem.
(d) Suppose you want to implement the Crank-Nicolson scheme with S0 = 20 and L,U, σ, r, T given in
part (c). Assume you take ∆x = (b − a)/26 and α = ∆t/(∆x)2 = 100, and you use SOR method
in the numerical linear algebra step. What is the optimal relaxation parameter? Show details of
your calculation.
(e) Implement part (d) using the optimal relaxation parameter you found, and give your option values
when the number of SOR iterations used is 5, 10, 20, 30.


































































(a) The option can be valued by solving the transformed Black-Scholes PDE
ut +
1
2
σ2uxx +
(
r − 1
2
σ2
)
ux − ru = 0.
Write down the terminal and boundary conditions for the PDE problem.
(b) Write down the Crank-Nicolson scheme for solving the PDE problem in part (a). Identify the
numerical linear algebra problem to be solved in each time step.
(c) Implement the Crank-Nicolson scheme in part (b) with L = 18, U = 22, σ = 0.25, r = 0.05
and T = 0.5. Take ∆x = (b − a)/26 where (a, b) is the spatial domain of the PDE. Choose an
appropriate α := ∆t/(∆x)2 and explain your choice. Plot the graph of the option value at time
zero against the spot price, and print out the option value corresponding to S0 = 20. Use LU
decomposition to solve the numerical linear algebra problem.
(d) Suppose you want to implement the Crank-Nicolson scheme with S0 = 20 and L,U, σ, r, T given in
part (c). Assume you take ∆x = (b − a)/26 and α = ∆t/(∆x)2 = 100, and you use SOR method
in the numerical linear algebra step. What is the optimal relaxation parameter? Show details of
your calculation.
(e) Implement part (d) using the optimal relaxation parameter you found, and give your option values
when the number of SOR iterations used is 5, 10, 20, 30.

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