The stock price process St for times t 2 [0, T ] is known to follow the
stochastic di↵erential equation
dSt = St(µdt+ (t) dWt)
where Wt is a Brownian motion, S0 = 140.0000, µ = 0.0425 and
(t) = 0.19⇥ (1.4286⇥ t)1.5.
A trader may also invest in a risk-free bank account that grows at the
continuously compounded rate r = 0.0075. The units of time for this
question are years and all prices are in dollars.
The contract of this derivative is defined as follows. If the stock ever hits
the barrier level B = 162.0000 on or before the maturity T = 1.0000 of
the option then the payo↵ of the derivative will be 0. Otherwise the payo↵
of the option will be f(ST ) where ST is the stock price at maturity and
the function f is defined by
f(ST ) = max{exp(0.0062⇥ ST )⇥ ST 49, 0}.
You should price this option using the Crank-Nicolson finite di↵erence
method and may assume that the price V (t, S) satisfies the partial di↵er-
ential equation
@V
@t
+
1
2
(t)2S2
@2V
@S2
+ rS
@V
@S
rV = 0
with boundary condition V (t, B) = 0, plus additional boundary conditions
you should identify. You should use the Crank-Nicolson finite di↵erence
method with a uniform grid containing 1001 time points and 1001 stock
price values. x
Record the price of the derivative at time 0 and the delta of the derivative
at time 0 in the table of results at the end of your essay. [20%]
2. Suppose that a trader attempts to replicate this option using a discrete-
time version of the delta-hedging trading strategy, rehedging at the same
time-points you used when implementing the finite di↵erence method. The
trader uses the value for the price and the value of delta computed by the
finite di↵erence method, using linear interpolation to calculate values at
points which are not in the grid.
Calculate an SDE satisfied by Zt = logSt and simulate Zt using the
Euler scheme for this SDE using the same time points as above. Using
the stock prices arising from this simulation, perform 1000 simulations of
the trader’s strategy and plot a histogram of the error in this replication
strategy. You should compute the 25th and 75th percentiles of the error
and record these values in the table at the end of your essay.[20%]
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3. Interpret your results. This means that you should make some financially
relevant observations based on your simulation, plotting charts you find
interesting. In order to make interesting observations you might want to
run variations on the simulation, for example varying the barrier. Please
remember to edit your findings and to only comment on the most inter-
esting points. [15%]
4. Describe how you have tested your code is correct, for example explaining
how you have checked that you have priced the derivative correctly. [15%]
5. You must finish your coursework by completing the following table of
results:
Price
Delta
25th percentile
75th percentile
The remaining 10% of your marks will be allocated based on the quality of
your code and your write-up.
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