1. Mälardalens Högskola
Course: MMA707
Teacher: Jan Röman
Date: 09/10/2008
Pricing European Call Options with
Monte Carlo Simulation and
Stochastic Volatility
By
Marcelo Tames
Yoseph Ghebreyesus
Diego R. Calvo
Table of Contents
Introduction ....................................................................................................................... 3
The idea of Monte Carlo ................................................................................................ 4
Implementation in Excel ............................................................................................... 6
Black-Scholes Comparison ........................................................................................... 8
Histogram Representation ........................................................................................... 9
Conclusion ........................................................................................................................ 10
References ........................................................................................................................ 11
Appendix ........................................................................................................................... 12
Visual Basic code ....................................................................................................... 12
2
Monte-Carlo Simulation
TASK: Build an application in Excel/VBA to solve option prices (European options) with Monte-Carlo.
Also include a stochastic volatility in the model. Plot the histogram of the outcome and calculate the
probability to reach the strike.
Introduction
Monte Carlo simulation treats randomness by selecting variable values from a certain stochastic
model. This method has become increasingly popular due to the computational power of our current
technology which allows us to implement complex mathematical experiments intended to imitate
real-life systems in areas such as finance, physics, computer science, etc.
Our Monte Carlo Simulation uses geometric Brownian motion as the building block for modeling
stock prices considering stochastic volatility. Thus, the periodic return which is expressed in
continuous compounding is a function of two components:
1. Constant drift
2. Random shock, i.e. volatility multiplied by a randomized critical z value
To this model we added the Euler Scheme in the Heston model type for stochastic volatility and as a
result the drift which was supposed to be constant becomes a random variable of time. After
performing 𝑚 simulations, with the same process, we evaluate the payoff of the European call option
at maturity for each simulation. The next step is to obtain an average payoff and discount it into the
present to obtain today’s fair option price. Once the calculations are over we can then proceed to
compare our price against the price suggested by the with the Black and Scholes formula. Finally, we
plot the corresponding histogram for our simulated stock prices at time n to see the distribution.
Based on that, we calculate the probabilities for the price to be in certain intervals to compare by
simple sight whether it´s close to the strike price or not.
3
The idea of Monte Carlo
The Monte Carlo simulation is a well-known method to estimate Value at Risk (VaR) in regard to the
asset class (i.e. basic equity or stocks). This basic simulation is just an application of the geometric
Brownian motion which is a particular type of Wiener process. This stochastic process, in turn,
models the random behavior of a stock price in continuous time. This geometric Brownian motion
satisfies the following stochastic differential equation for continuous time and random variable:
𝑑𝑆𝑡 = 𝜇𝑆𝑡 𝑑𝑡 + 𝜎𝑆𝑡 𝑑𝑊𝑡
(1)
Where:
𝑆𝑡 = Stock price at time t
𝜇= drift
𝜎= volatility
𝑑𝑊𝑡 = Wiener process
However, our work in Excel simulates the future stock price measured in trading days. Therefore we
adjust this equation for the discrete time case:
∆𝑆𝑡 = 𝑆𝑡−1 (𝜇∆𝑡 + 𝜎𝜀√∆𝑡) (2)
Where the new terms are:
∆𝑆𝑡 = Change in the stock price per unit of time
∆𝑡= Time interval. In our case one day
𝜀= Standard normal random number
As one can note, in the right side of the equation, we have factored the stock price from the previous
day letting the terms in parenthesis be the drift and shock successively.
It is important to remark that this geometric Brownian motion is a Markov process since the price of
tomorrow only depends on today´s price and not the past. This property gives us some insight about
market efficiency (even in its weak form), which tells that all the past information is already
impounded in today´s price and the future price will depend solely on current and future events.
John Hull, in his book Options, futures, and other derivative securities, expresses our last formula as a
lognormal continuously compounded periodic return of the stock (𝑙𝑛(𝑆𝑡 / 𝑆𝑡−1 )), which is verified by
the Ito´s Lemma. Consequently, this lognormal random variable will be approximately normally
𝜎2
distributed with mean (𝜇 − 2 )𝑡 and variance 𝜎 2 𝑡 .That´s why we say that the price ratios are
lognormal making our geometric Brownian motion a lognormal diffusion process. The equivalent
(and simpler) formula that we implement in our model is as follows:
𝑙𝑛(𝑆𝑡 / 𝑆𝑡−1 ) = 𝛼𝑡 + 𝑧𝑡 𝜎𝑡
(3)
Where:
𝛼= The deterministic component, drift
𝑧𝑡 𝜎= The stochastic component, where 𝑧𝑡 will be our generated random variable for the stock price
which will be scaled by its correspondent stochastic volatility at time t.
This formula represents the core of our simulation. In this case, we set the daily expected return 𝛼𝑡
to be positive because we expect the stock price to drift upward, entailing a positive expected return
4
over time. In a basic model 𝛼𝑡 is completely deterministic but it becomes a random variable as well
since it depends on the stochastic volatility. Furthermore, we set the initial values for the stock price
𝑆0 and volatility 𝜎1 before starting the simulation. To calculate the stochastic volatility, we decided to
implement the Heston model. The main reason for this choice is that there is an algorithm for timediscretization and Monte Carlo simulation of a Heston-type stochastic volatility model.
This variant is called the Euler scheme, which allows us to calculate the stochastic volatility for the
period in which the random stock price is generated. The formula is as follows:
𝜎𝑡 = 𝜎𝑡−1 + 𝜅(𝜃 − 𝜎𝑡−1 )∆𝑡 + 𝜀𝑡 √𝜎𝑡−1 √∆𝑡
(4)
Where:
𝜅, 𝜃 = preset drifts
𝜀𝑡 = Standard normal random number at time t
The algorithm is based on a careful analysis of the properties of affine stochastic volatility diffusions
and is straightforward and quick to implement and execute. Tests on realistic model
parameterizations reveal that the computational efficiency and robustness of the simulation scheme
compares very favorably to existing methods.
Coming back to (3), the stochastic component is what we´ve been calling the random shock which
will be a function of both random stock price and random volatility. That is what allows us to model
the stochastic process getting different paths each time.
5
Implementation in Excel
To start with, we need to show how each term of (3) is set as an input in the Excel sheet.
Expected Daily Drift (mean), 𝛼𝑡 : First, we set the value of the annual drift or expected return of the
stock in one cell, then, in another cell we set the value of the daily drift which is simply calculated by
dividing the annual drift by 252, considering that a year has 252 trading days. Finally, in another cell
we convert this daily drift into an “expected” daily drift by subtracting one half of the variance at
𝜎2
time t. In simple math: daily drift - 2𝑡 , this is some sort of geometric average and since we are using
stochastic volatility, 𝛼 can´t be a constant but now becomes a function of t as well, beginning with 𝛼1
as the initial value. The reason for this calculation is that the stochastic volatility is eroding returns.
This is due to the key assumption of the random walk idea: that returns are expressed in
percentages. For example, let´s imagine that 𝑆0 is $100 and then goes up 10% to $110. Then goes
back 10%, which nets 𝑆2 to $99 ($110 x 90% = $99). Then it goes up 10% again, to net 𝑆3 to $108.90
($99 x 110% = $108.9). Finally, it goes down again 10% to net 𝑆4 to $98.01. It may be counterintuitive, but the principal is slowly eroding even though the average gain is 0%. That difference is
about half of the variance of 𝑆𝑡 .
Expected Daily volatility, 𝜎𝑡 : First, we set the value of the annual volatility or expected risk of the
stock in one cell, then; in another cell we set the value of the initial daily volatility by dividing the
annual volatility by the square root of 252 trading days. The square root is because volatility scales
with the square root of time while variance scales with time directly. Next, we start running (4) to get
the successive volatilities but before we set the deterministic drifts in (4) 𝜅 and 𝜃. To generate the
standard normal random number 𝜀𝑡 at time t i.e. we use an especial Excel function called:
NORMSINV (RAND ()). Just RAND provides the probability between 0 and 1 and NORMSINV translates
that into the inverse standard normal cumulative distribution, that will give generally a value
between -3 and 3, this number appears in the cell named “N (0, 1) Vol”, the idea of this is that we get
to randomize the next´s period volatility which will result after multiplying this number by √𝜎𝑡−1 √∆𝑡
and later adding the remaining terms in (4) that are known. This process runs n times according to
the number of nodes (time steps in trading days) we choose to have. Observe that all this is for just
one simulation (iteration or single path).
Generated random variable for the stock price, 𝑧𝑡 : The process to generate 𝑧𝑡 is using NORMSINV
(RAND ()) once again. We should usually get a random value between -3 and 3 as well and this
number appears in the cell named “N (0, 1) Price”. Like 𝜀𝑡 this process runs n times according to the
number of nodes we choose to have. Observe that all this is for just one simulation.
Once we get these three variables for a single node t, we can apply (3) to get the lognormal return
(𝑙𝑛(𝑆𝑡 / 𝑆𝑡−1 )). Finally to get the simulated stock price for the next period, we simply multiply the
current price at time by the exponential of the lognormal return. i.e. 𝑆𝑡 = 𝑆𝑡−1 ∗ 𝑒 𝑙𝑛(𝑆𝑡 / 𝑆𝑡−1 )
The routine generates m simulations which can be thought of as paths formed by n amount of nodes.
Each of these nodes will have stochastic volatility inbuilt on to them.
Since the task is to obtain the payoff of a European call option today (t=0) our next step is to evaluate
every 𝑛𝑡ℎ stock price that came out as a final outcome of each simulation. The evaluation is based
on the formula for the European call option payoff:
𝐶( 𝑛𝑡ℎ , 𝑚) = max{𝑆 𝑛𝑡ℎ − 𝐾, 0}
(5)
6
Where:
𝑛𝑡ℎ = Last node or time step.
𝑚 = Number of simulations.
𝐶( 𝑛𝑡ℎ , 𝑚) = The value of the call option at the last node which resulted from each of the 𝑚
simulations or paths.
𝑆 𝑛𝑡ℎ = The final stock price i.e. at the last node for each of the 𝑚 simulations.
𝐾= The strike price which is a given constant.
As a result, we will have obtained 𝑚 different payoffs. Consequently, we calculate the arithmetic
mean of all the payoffs. The formula is as follows:
𝐶̅ = ∑𝑚
𝑖=1
𝐶( 𝑛𝑡ℎ ,𝑚)
𝑛
(6)
Since that result is at time n in the future we discount it assuming a continuously compounded
interest rate. The formula is as follows:
𝐶0̅ = 𝐶̅ ∗ 𝑒 −𝑟𝑛
(7)
Where:
𝐶0̅ = The European call option value at time 0 i.e. today.
𝑟 = Risk-free interest rate
𝑛 = Total number of nodes or time steps in each simulation.
𝐶0̅ represents our final result. The main purpose of the whole simulation was to come up with that
option value at time 0. Therefore, this should be the “fair” price to pay for such security. To have a
basis for comparison, we use the Black-Scholes formula to calculate the option price for the same
stock with the same strike price and time n. Therefore we can see whether our result converges with
Black-Scholes or not.
7
Black-Scholes Comparison
We have also implemented the Black-Scholes model to calculate the fair price of the stock to
compare it with Monte Carlo simulation.
The formula is for the Black Scholes Model is given by:
𝐶0 = 𝑆0 𝑁(𝑑1 ) − 𝐾𝑒 −𝑟𝑡 𝑁(𝑑2 )
𝑑1 =
2
𝑆
𝐿𝑛 ( 0⁄𝐾 ) + ((𝑟 − 𝑞) + 𝜎 ⁄2) 𝑇
𝜎√𝑇
𝑑2 = 𝑑1 − 𝜎√𝑇
Where:
S0 = The initial stock price
K = The strike price
σ = Volatility
q= Dividend
The Black-Scholes formula is an equation of the market for an equity, in which the equity’s price is
determined via a stochastic or random process. A Monte Carlo model is a method of estimating a
value by the random generation of numbers. When used in conjunction random numbers are
generated and fed into the Black-Scholes model. This model is executed repeatedly and the
aggregate results are used to price the option.
8
Histogram Representation
The construction of both Probability Distribution and Stock Price Distribution histograms is
very straight forward in Excel.
Since the prices are calculated with the code in the first sheet; all the histogram sheet does
is receive the data and filter it allowing us to count the frequency by which the stock prices
show up. Knowing the frequency and the amount of stock prices calculated, the next step
would be to determine the probabilities and plot them using the intervals selected.
The page refreshes itself each time the simulation is executed and the histograms' speak for
themselves showing the distribution of the simulated stock prices at time n. The page has
the limitation of being customized for accepting 125 prices but it would only take the user a
couple of adjustments to take under account as many inputs as desired.
9
Conclusion
In this project we have taken to the next level the basic Monte Carlo simulation to simulate stock
prices. By considering stochastic volatility we have seen that the prices can really jump higher or
lower and consequently give us a tremendous difference in payoffs for the call option.
Also in this experiment we have noticed that there is a large probability for such contract to go
default. Because of that, the histogram doesn´t show a definite distribution.
When comparing the call option´s fair price calculated using the Black-Scholes formula with our
method can deviate quite a bit and sometimes get very close to Black-Scholes which may mean just a
shot of luck.
We have addressed this problem for a long time but concluded that it may be because of two
reasons:


First, since the volatility is stochastic, it makes the lognormal return of the stock completely
random even randomizing the drift 𝛼 which is constant in the simple model. Therefore the
more nodes a single path takes, the closer to zero the stock price tends to be. This is because
at some point the lognormal return is exceedingly negative making the stock price lose most
of its value and for the next period it´s difficult that the simulation takes it back to a normal
level since it depends on the price today. We thought that this was caused by negative
random volatilities, but after making a restriction to that, we still came up with the same
problem.
Second, in the Euler scheme, to calculate the stochastic volatility we introduced two new
constants; Kappa and Theta; which we assigned values, but even trying with lower or higher
values the volatility ends up at some point extremely high causing the phenomenon
explained above.
In conclusion, the combination of three random values obtained for (3) ;which is our main equation;
returns a very volatile and sometime even incoherent lognormal return at some point in the stock´s
price path.
10
References

Hull, John C. Options, futures, and other derivative securities, 6th ed. Chapter 12: Wiener
Processes and Ito's Lemma

http://www.javaquant.net/papers/LeifAndersenHeston.pdf

http://www.bionicturtle.com/learn/article/monte_carlo_simulation_with_gbm_frm_lo_141/

http://www.investopedia.com/articles/04/021804.asp

http://www.riskglossary.com/link/black_scholes_1973.htm

http://www.investopedia.com

http://www.wikipedia.org
11
Appendix
Visual Basic code
Sub CalculateButton_Click()
Dim Simulations As Integer
Dim Drift As Double
Dim Volatility As Double
Dim RandomNumberVolatility As Double
Dim RandomNumberPrice As Double
Dim i As Integer 'Simulations, days
Dim j As Integer 'Nodes'
Dim Gotoy As Integer
Dim Gotox As Integer
Dim StockPrice As Double
Dim Payoff As Double
Dim Stike As Double
Dim IniDailyDrift As Double
Dim ConstTheta As Double
Dim ConstK As Double
Dim FairPrice As Double
Dim RiskFreeInterestRate As Double
Sheets("sheet1").Activate
FairPrice = 0
'-----------Inputs-----------------------------'
Simulations = Range("D3").Value
Periods = Range("E3").Value
StockPrice = Range("F3").Value
Strike = Range("G3").Value
Volatility = Range("H6").Value 'initial volatility t=1
DailyDrift = Range("I6").Value ' FIXED
ExpectedDailyDrift = Range("J6").Value ' VARIABLE
ConstTheta = Range("J3").Value
ConstK = Range("K3").Value
'--Main Funtion, Begin-------------------------'
Gotoy = 10
Gotox = 12
If Simulations <= 5000 Then
For i = 1 To Simulations
For j = 1 To Periods
RandomNumberPrice = Application.WorksheetFunction.NormSInv(Rnd()) 'Zt
RandomNumberVolatility = Application.WorksheetFunction.NormSInv(Rnd()) 'Epsilont
If (Volatility < 0) Or (Volatility > 0.2) Then
Volatility = 0.01
End If
Volatility = Volatility + ConstK * (ConstTheta - Volatility) + RandomNumberVolatility *
Sqr(Volatility)
Returni = ExpectedDailyDrift + Volatility * RandomNumberPrice ' Core formula logreturn
ExpectedDailyDrift = DailyDrift - 0.5 * (Volatility) ^ 2 'alpha(t)= daily drift - 0,5*(volatility(t)^2)
volatility must change
StockPrice = StockPrice * Exp(Returni) 'Stock price for the next period
Cells(Gotoy, Gotox) = StockPrice
12
Gotox = Gotox + 1
Next j
Gotox = 12
Payoff = StockPrice - Strike ' call option payoff max{Sn-K,0}
If Payoff < 0 Then
Payoff = 0
End If
FairPrice = Payoff + FairPrice
Cells(Gotoy, 4).Value = i
Cells(Gotoy, 5).Value = RandomNumberVolatility
Cells(Gotoy, 6).Value = RandomNumberPrice
Cells(Gotoy, 7).Value = Volatility
Cells(Gotoy, 8).Value = Returni
Cells(Gotoy, 9).Value = StockPrice
Cells(Gotoy, 10).Value = Payoff
Gotoy = Gotoy + 1
'-----------Restart Inputs For the New Simulation-----------------------------'
Volatility = Range("H6").Value 'initial volatility t=1
DailyDrift = Range("I6").Value ' FIXED
ExpectedDailyDrift = Range("J6").Value ' VARIABLE
'-----------------------------------------------------------------------------'
Next i
'-----------Option Price Calculation-------------------------------------------'
FairPrice = (FairPrice / Simulations) * e ^ (-Periods * RiskFreeInterestRate)
Range("B5").Value = FairPrice
'-----------------------------------------------------------------------------'
Else
MsgBox "It will only run for 5000"
End If
End Sub
Sub ClearButton_Click()
Dim AmountRandomNumber As Double
Dim Gotoy As Integer
Dim Gotox As Integer
Simulations = Range("D3")
Periods = 12 + Range("E3").Value
Gotoy = 10
Sheets("sheet1").Activate
Gotox = 4
Range("B5") = ""
For x = 4 To Periods
For i = 1 To Simulations
Cells(Gotoy, Gotox) = ""
Gotoy = Gotoy + 1
Next i
Gotoy = 10
Gotox = Gotox + 1
Next x
End Sub
13