In the slides present a structured description of the methods that can be used to calculate the delta for an asian option. Only European options are considered. The reference list has added as the last slide. Enjoy the presentation!

1. THE DELTA OF AN
ARITHMETIC ASIAN OPTION
VIA THE PATHWISE METHOD
Anna Borisova
University of Bocconi
1/12/2014

2. Assignment
1. Compute the Monte Carlo simulation for the
price of an Asian option on a lognormal
asset (with descrete monitoring at dates
t1, t2, … , tM);
2. Provide the pathwise estimate of the Delta of
these options.

3. Structure
❏ Mathematical model;
❏ The code features;
❏ Output.

4. The price of an Asian option by MCS
An Asian option (call) has discounted payoff:
Y = e-rT
[S - K]+
S =
1
m
S(ti )
i=1
m
å For fixed dates 0<t1<…<tm<T
Since for Monte Carlo simulation we describe the risk-neutral dynamics of the stock price, we
need to use the stochastic differential equation for modeling the price movement of the
underlying asset
S(T) = S(0)exp([r -
1
2
s 2
]T +sW(T))
Where W(T) is the random variable,
normally distributed with mean 0 and
variance T.

5. S(T) = S(0)exp([r -
1
2
s 2
]T +sW(T))
Monte Carlo simulation of the
lognormal asset price movement
Model:
S = 100, r = 0.045, σ=15%,
trials = 100000
For fixed dates 0<t1<…<tm<T

6. Constructing the Brownian motion
The properties of Brownian motion:
• W(0) = 0;
• The increments {W(t1)-W(t0), W(t2)-W(t1), … , W(tk)-W(tk-1)} are independent;
• W(t) – W(s) is normally distributed N(0, t-s) for any 0 ≤ s < t ≤ T.
S(t) = S(0)exp([r -0.5s 2
)t + ts Zi
i=0
t
å

7. 0.5697
-1.0916
-1.3929
-0.1540
-1.7215
-0.1277
-0.9524
-1.9553
-0.5648
Numberofsimulations Number of averaging periods
+
+
+
+
+
+
-0.5367
-4.7684
-2.0854
0.4157
-2.8131
-1.5206
0.5697
-1.0916
-1.3929
S(t) = S(0)exp([r -0.5s 2
)t + ts Zi
i=0
t
å

8. Matlab code

9. Model:
S = 100, r = 0.045, σ=15%
12 averaging points
100simulations5000simulations
1000simulations10000simulations

10. Option value estimation and its
confidence level
The sample standard deviation
sC =
1
n-1
(Yi - ˆYn )2
i=1
n
å
1-δ quantile of the standard normal distribution zdConfidence interval:
ˆYn ± zd/2
sC
n
ˆYn =
1
n
Yi
1
n
å
E[ ˆYn ]= Y
ˆYn -Y
sC / n
Þ N(0,1)
The estimation of the option value is unbiased
As number of replications increases, the standardized estimator converges in
distribution to the standard normal

11. Matlab code

12. TRADE OFF: The value of the option, confidence interval and
computational time
Number of
simulations
100 1000 3000 5000 7000 10000
Value of the
option
0.9306 1.3199
1.125
6
1.3533 1.3347 1.2666 1.1797 1.1634 1.2084 1.2256 1.2575 1.1560
Comput.
time
0.15 0.11 0.13 0.17 0.42 0.47 0.76 0.85 1.13 1.16 1.67 2.14
95% c.l.
lower bound
0.4123
0.929
8
1.2043 1.0898 1.1318 1.1917
95% c.l.
upper bound
1.4490
1.321
4
1.4652 1.2696 1.2850 1.3233
99% c.l.
lower bound
0.3496 1.1336 1.1454 1.0755 1.1472 1.0932
99% c.l.
upper bound
2.2901 1.5730 1.3878 1.2512 1.3040 1.2188

13. The pathwise estimator of the option
delta
This estimator has great practical value
• This estimator is unbiased;
• mean(S) is simulated in estimating the price of the option also, so finding the delta
requires just a little additional effort;
• This method reduces variance and computing time compared to finite-difference.
dY
dS(0)
=
dY
dS
dS
dS(0)
= e-rT
1{S > K}
S
S(0)
dS
dS(0)
=
1
m
dS(ti )
dS(0)
=
1
m
S(ti )
S(0)
=
i=1
m
å
i=1
m
å
S
S(0)
dY
dS(0)
= e-rT
1{S > K}
S
S(0)

14. Matlab code

15. Model:
Asian call option
S(0) = from 0 to 200
with step 1;
r = 4,5%;
sigma = 1;
K=50;
T = 1 with 24 averaging
points (two times a
month);
Trials = 10000.

16. Matlab code

17. References:
• Glasserman “Monte Carlo Methods in Financial
Engineering”
• Mark Broadie, Paul Glasserman “Estimating Security
Price Derivatives Using Simulations”
• John C. Hull “Options, futures and other derivatives”
• Huu Tue Huynh, Van Son Lai, Issouf Sourmare
“Stochastic Simulation and Applications in Finance with
MATLAB® Programs”