Transcript Regular naps prevent old age, especially if you take them
Calculating Value at Risk using Monte Carlo Simulation
(Futures, options &Equity) Group members Najat Mohammed James Okemwa Mohamed Osman
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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.
S t =S 0 *exp[(r – q - 0.5
2 )t +
tz t ] where
S 0 = initial stock price at time zero, r = risk free rate = volatility q = dividend yield t = time z t = random sample from a normal distribution, with 𝜇 = 0 and 𝜎 = 1. Using Excel z t 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
Parameters declaration
180 0.50% 0.01% 14.00% 0.0196
0.082191781
181
Runs
8 9 10 11 12 4 5 6 7 1 2 3
Normal Random variable Z t
0.0264308
-0.7306616
-0.3943008
0.0605489
1.1382286
-0.7034839
0.8912692
-1.4414165
0.4434502
-0.6407301
1.0717440
0.4472491
13 14 15 -1.1548099
0.3771981
0.3518195
Deterministic component Random Component (r-q-0.5Q^2)t (Q*t*Z t)
0.0040945
0.0003041
0.0040945
0.0040945
0.0040945
0.0040945
-0.0084076
-0.0045372
0.0006967
0.0130974
0.0040945
0.0040945
0.0040945
0.0040945
0.0040945
0.0040945
0.0040945
0.0040945
0.0040945
0.0040945
-0.0080949
0.0102557
-0.0165862
0.0051027
-0.0073728
0.0123324
0.0051464
-0.0132882
0.0043404
0.0040483
Stock Price (S o*Exp(((r-q-0.5Q^2)t+(Q*t*Z t)))
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
17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Normal Random variable Z t
-1.347451108
-0.130610508
Deterministic component (r-q-0.5Q^2)t
0.004094521
0.004094521
Random Part (Q*t*Z t)
-0.015504917
-0.001502915
Stock Price (S o*Exp(((r-q-0.5Q^2)t+(Q*t*Z t)))
177.96
180.47
1.031253946
0.098540463
0.054440527
-0.082145813
1.432573228
-1.183339405
-0.039509899
-1.353207298
-0.797648033
1.203659477
-0.380368986
-1.592325097
-1.407569031
1.461169186
-0.823354701
-0.485952773
0.004094521
0.004094521
0.004094521
0.004094521
0.004094521
0.004094521
0.004094521
0.004094521
0.004094521
0.004094521
0.004094521
0.004094521
0.004094521
0.004094521
0.004094521
0.004094521
0.011866484
0.00113389
0.000626439
-0.000945239
0.016484404
-0.013616508
-0.000454634
-0.015571152
-0.009178416
0.013850328
-0.004376849
-0.018322645
-0.016196685
0.016813454
-0.009474218
-0.005591785
182.90
180.94
180.85
180.57
183.74
178.29
180.66
177.95
179.09
183.26
179.95
177.46
177.83
183.80
179.03
179.73
-1.385176037
-0.565489369
-1.976297172
1.53797826
-1.12362048
-0.612291725
0.132068416
0.723195857
-0.713504286
-0.268014233
-0.70611559
0.430880824
0.166415087
0.085873491
-1.445653704
0.540559266
0.503162108
0.498751335
1.639513795
-1.075302551
0.33823693
0.004094521
0.004094521
0.004094521
0.004094521
0.004094521
0.004094521
0.004094521
0.004094521
0.004094521
0.004094521
0.004094521
0.004094521
0.004094521
0.004094521
0.004094521
0.004094521
0.004094521
0.004094521
0.004094521
0.004094521
0.004094521
-0.015939012
-0.006507001
-0.022740954
0.017697284
-0.012929332
-0.007045549
0.001519691
0.008321706
-0.008210186
-0.003083999
-0.008125166
0.004958081
0.001914913
0.000988133
-0.016634919
0.006220134
0.005789811
0.005739056
0.018865638
-0.012373344
0.003892041
177.88
179.57
176.67
183.97
178.42
179.47
181.01
182.25
179.26
180.18
179.28
181.64
181.08
180.92
177.76
181.87
181.79
181.78
184.18
178.52
181.44
Date Terminal Prices
1 181.64
2 3 181.08
180.92
4 5 6 7 177.76
181.87
181.79
181.78
8 9 10 184.18
178.52
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 1 2 3 4 .. .. 997 998 999 Average Terminal Price 1000 1 180.46 180.84 179.19 181.69 .. .. 179.31 182.78 185.06 180.06 180.65 2 180.08 176.70 182.80 180.17 .. .. 180.41 180.14 183.47 179.36 180.78 Date 3 184.34 ... ... 179.33 ... ... 182.25 ... ... 178.10 ... ... .. ... ... .. ... ... 181.11 ... ... 180.32 ... ... 180.92 ... ... 178.16 ... ... 180.84 ... ... 178 181.54 181.50 183.39 178.85 .. .. 178.62 178.61 181.39 180.81 180.69 179 181.05 182.19 176.84 182.43 .. .. 179.42 181.36 181.80 180.54 180.76 180 181.84 180.82 182.37 181.44 .. .. 178.33 178.74 181.15 182.54 180.76
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
9 10 11 12 6 7 8 13 14 15 1 2 3 4 5
Prices Call
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
Futures
-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
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
Returns Dates
9 10 11 12 6 7 8 13 14 15 1 2 3 4 5
Call
-2% -1% -10% 3% 13% -7% 11% -13% 3% 5% -7% 5% -9% 2%
Futures Put
45% 12% -29% 21% -91% 79% -27% -19% -21% 65% -15% 17% 0% 38% 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% -13% -8% -7% -5% -2% 0%
Equity
-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 VaR 99% =1-day VaR 99% * √ 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?
6 7 8 9 10 10 11 12 13 14 2 3 4 5 15
Call Ordered Returns futures put
-23% -23% -21% -21% -22% -17% -17% -15% -20% -20% -20% -19% -19% -18% -16% -16% -15% -15% -15% -14% -14% -11% -11% -11% -11% -10% -10% -10% -15% -10% -16% -10% -9% -9% -9% -7% -7% -7% -7% -7% -6% -6% -6% -6% -6%
Equity
-0.151% -0.126% -0.123% -0.118% -0.113% -0.105% -0.105% -0.098% -0.097% -0.097% -0.090% -0.080% -0.080% -0.077% -0.073%
10 5 0 20 15 30 25
Bin Histogram
120.00% 100.00% 80.00% 60.00% 40.00% 20.00% 0.00% Frequency Cumulative %
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.