Calculating Value at Risk using
Monte Carlo Simulation
(Futures, options &Equity)
Group members
Najat Mohammed
James Okemwa
Mohamed Osman
“Regular naps prevent old age,
especially if you take them while
driving ”
Introduction
The most common measure of risk is volatility. However, volatility does not care about the
direction of an investment’s movement. For instance a stock can be volatile because it
suddenly jumps higher. Investors however, are concerned about the odds of losing money.
Value at Risk (VaR) tackles the investors concern by providing answers to the following
questions.
 What is my worse case scenario?
 How much can I really lose in a bad month?
As such VaR must have three components.
 The time Horizon to be analyzed i.e. the time period over which a portfolio is held. In
the following pages we have assumed our time horizon to be 30 days.
 The confidence interval i.e. the interval estimate in which the VaR would not be
expected to exceed the maximum loss. In our illustration we use several just to
compare the results.
 The Loss amount. This can also be expressed as a percentage.
Monte Carlo simulation is a method of generating scenarios that mimic a process and then
finding the average values of the scenarios to approximate the outcome of the process. In
principle it utilizes the law of large numbers i.e the mean value of a sample chosen from a
finite population, is equal to the true mean of the population when the sample is considerably
large.
In our assignment we use Monte Carlo simulation to generate terminal prices of a stock whose
underlying asset is potatoes. Then we calculate the related payoffs and price series. The prices
series is then used to determine the return series which is in turn used to calculate the
Volatility and Value at Risk.
Stock Price at time t
We are going to use the Black-Scholes formula to simulate the prices.
St=S0*exp[(r – q - 0.52)t + tzt]
where
S0 = initial stock price at time zero,
r = risk free rate
 = volatility
q = dividend yield
t = time
zt = random sample from a normal distribution, with 𝜇 = 0 and 𝜎 = 1.
Using Excel zt can be obtained in these models by normally scaling the
random numbers generated. The formula used in Excel is
NORMINV(RAND()).
Step 1:Construct a Monte Carlo Simulator for prices of the underlying
So
r
q
Q
Q^2
t
K
Initial Stock price
Risk free interest rate
dividend yield
Volatility
Time to maturity
Strike Price
Runs
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Parameters declaration
180
0.50%
0.01%
14.00%
0.0196
0.082191781
181
Normal Random variable
Zt
0.0264308
-0.7306616
-0.3943008
0.0605489
1.1382286
-0.7034839
0.8912692
-1.4414165
0.4434502
-0.6407301
1.0717440
0.4472491
-1.1548099
0.3771981
0.3518195
Deterministic component Random Component
(r-q-0.5Q^2)t
(Q*t*Zt)
0.0040945
0.0003041
0.0040945
-0.0084076
0.0040945
-0.0045372
0.0040945
0.0006967
0.0040945
0.0130974
0.0040945
-0.0080949
0.0040945
0.0102557
0.0040945
-0.0165862
0.0040945
0.0051027
0.0040945
-0.0073728
0.0040945
0.0123324
0.0040945
0.0051464
0.0040945
-0.0132882
0.0040945
0.0043404
0.0040945
0.0040483
Stock Price
(So*Exp(((r-q-0.5Q^2)t+(Q*t*Zt)))
180.79
179.23
179.92
180.86
183.12
179.28
182.60
177.77
181.66
179.41
182.98
181.67
178.35
181.52
181.47
Step 2: Expand the Monte Carlo Simulator
In order to calculate the Value at Risk (VaR) measure we require a series of
returns which in turn requires time-series price data. To simulate this
particular environment we assume that we have a series of similar option
contracts that commence and expire on a one-day roll-forward basis.
We assume that time to maturity for our portfolio is one month(30 days).
Suppose that an option commences at time 0 and expires at time 30. The next
commences a time 1 and expires at time 31, the next at time 2 and expires at
time 32 and so on.
Based on this premise we will obtain a time series of daily terminal prices. In
our illustration we have repeated this process in order to generate time-series
data for terminal prices for a period of 180 days(half a year) as shown here.
Runs
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
Normal Random variable
Deterministic component
Random Part
Stock Price
Zt
(r-q-0.5Q^2)t
(Q*t*Zt)
(So*Exp(((r-q-0.5Q^2)t+(Q*t*Zt)))
-1.347451108
0.004094521
-0.015504917
177.96
-0.130610508
0.004094521
-0.001502915
180.47
1.031253946
0.004094521
0.011866484
182.90
0.098540463
0.004094521
0.00113389
180.94
0.054440527
0.004094521
0.000626439
180.85
-0.082145813
0.004094521
-0.000945239
180.57
1.432573228
0.004094521
0.016484404
183.74
-1.183339405
0.004094521
-0.013616508
178.29
-0.039509899
0.004094521
-0.000454634
180.66
-1.353207298
0.004094521
-0.015571152
177.95
-0.797648033
0.004094521
-0.009178416
179.09
1.203659477
0.004094521
0.013850328
183.26
-0.380368986
0.004094521
-0.004376849
179.95
-1.592325097
0.004094521
-0.018322645
177.46
-1.407569031
0.004094521
-0.016196685
177.83
1.461169186
0.004094521
0.016813454
183.80
-0.823354701
0.004094521
-0.009474218
179.03
-0.485952773
0.004094521
-0.005591785
179.73
-1.385176037
0.004094521
-0.015939012
177.88
-0.565489369
0.004094521
-0.006507001
179.57
-1.976297172
0.004094521
-0.022740954
176.67
1.53797826
0.004094521
0.017697284
183.97
-1.12362048
0.004094521
-0.012929332
178.42
-0.612291725
0.004094521
-0.007045549
179.47
0.132068416
0.004094521
0.001519691
181.01
0.723195857
0.004094521
0.008321706
182.25
-0.713504286
0.004094521
-0.008210186
179.26
-0.268014233
0.004094521
-0.003083999
180.18
-0.70611559
0.004094521
-0.008125166
179.28
Date Terminal Prices
0.430880824
0.004094521
0.004958081
181.64
1
181.64
0.166415087
0.004094521
0.001914913
181.08
2
181.08
0.085873491
0.004094521
0.000988133
180.92
3
180.92
-1.445653704
0.004094521
-0.016634919
177.76
4
177.76
0.540559266
0.004094521
0.006220134
181.87
5
181.87
0.503162108
0.004094521
0.005789811
181.79
6
181.79
0.498751335
0.004094521
0.005739056
181.78
7
181.78
1.639513795
0.004094521
0.018865638
184.18
8
184.18
-1.075302551
0.004094521
-0.012373344
178.52
9
178.52
0.33823693
0.004094521
0.003892041
181.44
10
181.44
Step 3: Run scenarios
Step 2 above generates a 180-day terminal price series under a single scenario. The
process now needs to be repeated several times (in our illustration we have used 1000
simulation runs). After running 1000 scenarios take a simple average across all the
180 days for each data point.
Terminal Prices
Scenario
Date
1
2
3
4
..
..
997
998
999
1000
Average Terminal
Price
1
180.46
180.84
179.19
181.69
..
..
179.31
182.78
185.06
180.06
2
180.08
176.70
182.80
180.17
..
..
180.41
180.14
183.47
179.36
3
184.34
179.33
182.25
178.10
..
..
181.11
180.32
180.92
178.16
...
...
...
...
...
...
...
...
...
...
180.65
180.78
180.84
...
...
...
...
...
...
...
...
...
...
...
178
181.54
181.50
183.39
178.85
..
..
178.62
178.61
181.39
180.81
179
181.05
182.19
176.84
182.43
..
..
179.42
181.36
181.80
180.54
180
181.84
180.82
182.37
181.44
..
..
178.33
178.74
181.15
182.54
...
180.69
180.76
180.76
Step 4: Calculate the intrinsic value or payoffs then calculate the
discounted value of the payoffs.
This two steps combined give us the prices for our instruments(futures, options and equity)
Payoff for a long futures = Terminal Price – Strike
Price=Payoff*e-rT
Where r is the risk free rate and
T is the time to maturity of the option, future or equity i.e. 30 days.
Step 5: Calculate the return series
Now that we have the derivatives average price series we will determine the return series by taking the natural
logarithm of successive prices.
The return on Date 3 for a futures contract will therefore be ln((0.37)/(0.32)) =12%.
Dates
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Call
Prices
Futures
0.738
0.725
0.716
0.645
0.661
0.756
0.704
0.785
0.690
0.711
0.748
0.698
0.74
0.673
0.687
-0.21
-0.32
-0.37
-0.27
-0.34
-0.14
-0.30
-0.23
-0.19
-0.16
-0.30
-0.25
-0.30
-0.30
-0.44
Put
0.96
1.00
1.03
0.99
1.01
0.92
0.98
0.94
0.91
0.88
0.97
0.97
0.98
0.97
1.07
Equity
Dates Call
Returns
Futures Put
180.79
180.67
180.63
180.73
180.66
180.86
180.70
180.77
180.81
180.84
180.70
180.74
180.70
180.70
180.56
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
45%
12%
-29%
21%
-91%
79%
-27%
-19%
-21%
65%
-15%
17%
0%
38%
-2%
-1%
-10%
3%
13%
-7%
11%
-13%
3%
5%
-7%
5%
-9%
2%
5%
3%
-4%
1%
-9%
6%
-4%
-3%
-4%
10%
0%
0%
-1%
9%
Equity
-0.065%
-0.024%
0.053%
-0.035%
0.109%
-0.091%
0.041%
0.021%
0.020%
-0.078%
0.023%
-0.025%
0.000%
-0.077%
Step 6: Calculate the VaR measure
In this step we calculate the Value at Risk Using the Historical method where we refer to the simulated daily return
processes as our historical Returns. As such VaR is calculated by computing a percentile over the range of the return
data set under consideration. This is easily calculated in Excel by the function =PERCENTILE (array, (100-X)%) Where X
is the confidence interval.
In our illustration, the 1day Value at Risk at different confidence levels, using the Historical returns is calculated as
shown below. These are the 30-day holding VaR’s for the various instruments.
Confidence Level
99%
95%
90%
80%
60%
50%
Call
-17%
-12%
-9%
-7%
-1%
0%
Futures Put
-79%
-65%
-55%
-30%
-14%
-3%
Equity
-13%
-8%
-7%
-5%
-2%
0%
-0.131%
-0.077%
-0.063%
-0.040%
-0.007%
0.003%
To interpret this table for example it tells us that we are 95% confident that if we held put options our loss would not
exceed 8%. You can also tell that it is risky to invest in futures contract since the loss percentages are quite high.
To calculate the N-day VaR we use the formula,
√
N-day VaR99%=1-day VaR99%* N
Value at Risk, Histograms and risk management in Excel
Finally we seek to answer three important questions
a)
b)
c)
How bad can things get when they really get bad?
What is the most that you can lose on a really bad day?
What is the worst that can happen?
We can summarize this in one sentence as;
What is the worst that can happen and over what period and with what odds?
Ordered Returns
Call
futures
put
Equity
2
-23%
-22%
-16%
-0.151%
3
-23%
-17%
-10%
-0.126%
4
-21%
-17%
-9%
-0.123%
5
-21%
-15%
-9%
-0.118%
6
-20%
-15%
-9%
-0.113%
7
-20%
-14%
-7%
-0.105%
8
-20%
-14%
-7%
-0.105%
9
-19%
-11%
-7%
-0.098%
10
-19%
-11%
-7%
-0.097%
10
-18%
-11%
-7%
-0.097%
11
-16%
-11%
-6%
-0.090%
12
-16%
-10%
-6%
-0.080%
13
-15%
-10%
-6%
-0.080%
14
-15%
-10%
-6%
-0.077%
15
-15%
-10%
-6%
-0.073%
Frequency
Histogram
30
120.00%
25
100.00%
20
80.00%
15
60.00%
Frequency
Cumulative %
10
40.00%
5
20.00%
0
0.00%
Bin
Probabilities of losses occuring
For instance there is a 0.56% chance that the worst scenario (23.43% loss) would happen
Bin
-23.43%
-19.68%
-15.93%
-12.17%
-8.42%
-4.66%
-0.91%
2.84%
6.60%
10.35%
14.11%
17.86%
21.61%
More
Frequency
1
5
4
14
13
22
25
21
25
18
16
8
4
3
Cumulative %
0.56%
3.35%
5.59%
13.41%
20.67%
32.96%
46.93%
58.66%
72.63%
82.68%
91.62%
96.09%
98.32%
100.00%
CONCLUSION
Putting all this information together we can say that; if one should invest 100
SEK in long call options then the maximum they can lose is 23.43 SEK, in any
given day with a probability of 0.56%.
Value at Risk is a widely used risk measure of the risk of loss on a specific
portfolio of financial assets. For a given portfolio, probability and time
horizon, Value at Risk is a threshold value such that the probability that the
market to market loss on the portfolio over the given time horizon exceeds this
value in a given probability level. Value at Risk and volatility are the most
commonly used risk measurements. Value at Risk is easy to calculate and can
be used in many fields.
“Always borrow money from a
pessimist. He won’t expect it back”Oscar Wilde
You have been a nice audience Thank you!