WSZiB : Lectures of Prof. dr Peter Sloot

7 Special Topic in Stochastic Finance

7.3 Stochastic Modeling and Simulation in Finance

III.1 Stochastic Modeling

III.1.1 The Random Nature of the Financial Market
The values of the major indices and the graphs of them are quoted frequently on newspapers, television news bulletins. To many people these 'mountain ranges' showing the variation of the value of an asset or index with time is an excellent example of the 'random walk'.
We cannot predict tomorrow's values of asset prices. The past history of the asset value is there as a financial time series for us to examine as much as we want: but we cannot use it to forecast the next move that the asset will make. This does not mean that it tells us nothing. We know from a statistical analysis of it what are the likely jumps in asset price, what are their mean and variance and, generally, what is the likely distribution of future asset prices.
According to the efficient market hypothesis, which states that market prices reflect the knowledge and expectations of all investors, asset prices must move randomly. This basically tells two things:
     • The past history is fully reflected in the present price, which does not hold any further information;
     • Makets respond immediately to any new information about an asset price.
Thus the modeling of asset prices is really about modeling the arrival of new information which affects the price. With the two assumptions above, changes in the asset price are a Markov process.
Now suppose that at time t the asset price is S. Let us consider a small subsequent time interval dt, during which S changes to S + dS. We can model the corresponding return on the asset, dS / S , into two parts. One is a predictable, deterministic return akin to the return on money invested in a risk-free bank. It gives a contribution

          μdt

to the return dS / S, where μ is a measure of the average rate of growth of the asset price, also known as the drift. Normally, μ is taken to be a constant.
The second contribution to dS / S models the random change in the asset price in response to external effects, such as unexpected news. It is represented by a random sample drawn from a normal distribution with mean zero and adds a term

       σdz

to dS / S . Here σ is a number called the volatility, which measures the standard deviation of the returns. The quantity dz is the sample from a normal distribution, which is discussed further below.
Putting these contributions together, we obtain the stochastic differential equation

         = σdz + μdt          (III.1)
which is the mathematical representation of our simple recipe for generating asset prices.
The only symbol in the above equation whose role is not yet entirely clear is dz. If we were to cross out the term involving dz, by taking σ = 0, we would be left with the ordinary differential equation

   =  μdt          (III.2)

          or           =  μS;
When μ is constant this can be solved exactly to give exponential growth in the value of the asset, i.e.

        S = S          (III.4)
where Sis the value of the asset at t = tThus if σ = 0 the asset price is totally deterministic and we can predict the future price of the asset with certainty.
The term dz, which contains the randomness that is certainly a feature of asset prices, is known as a Wiener process. It has the following properties:
     • dz is a random variable, drawn from a normal distribution;
     • the mean of dz is zero;
     • the variance of dz is dt.
One way of writing this is

   dz = φ
where φ is a random variable drawn from a standardized normal distribution. The standardized distribution has zero mean, unit variance and a probability density function given by


for -∞ < φ < ∞. If we define the expectation operator ξ by

  ξ [F(.)] =
for any function F, then
       ξ [φ] = 0           (III.7)

    and      ξ [] = 1          (III.8)

Equation (III.1) is a particular example of a random walk. It does not refer to the past history of the asset price; the next asset price ( S + dS ) depends solely on today's price and hence it is a Markov process.

III.1.2  Modeling with Stochastic Calculus
In modeling, analyzing and predicting activities in finance, greater and greater emphasis has been placed upon stochastic methods: Ito's lemma, stochastic differential equations, stochastic stability and stochastic control. Such methods are expected to capture the various complexities, measurement errors and uncertainties that are associated with financial reality. Among them, Ito's lemma is the basic stochastic calculus rule for computing stochastic differentials of composite random functions.

• Ito's Lemma
Let u(X,t) be a continuous, non-random function with continuous partial derivatives and X(t) a stochastic process defined by

       dX (t) = a ( X, t )dt+b ( X, t )dz(t)
where dz(t) is the standard Wiener process. Then the stochastic process Y(t) = u (X (t),t) has the following form of stochastic differential

                 dY(t) = ()dt + b(X,t)dz(t)        (III.10)
The derivation of Ito's Lemma is given in Appendix. Here we simply use it to model stochastic processes. In the following we show two examples of stochastic modeling:

(1). Geometric Brownian Motion model for asset price dynamics
The asset model we discussed above is

;
With the help of the correspondent expression of Ito's lemma                 df=          (III.12)
we can quickly derive a probability density function for the asset price, S(t). Setting       f = lnS          (III.13)
we have
Substituting them into the expression of Ito's Lemma III.12 results in                     d ln(S) = σ dz + ()dt          (III.14)

i.e     & ln()- ln()=  σ dz + ()dt          (III.15)
Finally we get   &
  = Exp( σ dz + () dt )          (III.16);
It shows that the asset price follows a lognormal distribution.  &

(2). Derivative securities pricing model
Again we start at the asset model

   &                  (III.17)   &
Suppose that f is the price of a derivative contingent on S, which must be a function of S and t. Ito's lemma says

     &  df=          (III.18)

An important step in derivatives valuation is the elimination of randomness: the two random walks in S (equation III.17) and f (equation III.18) are both driven by the single random variable dz. We can exploit this fact to construct a third variable Π whose variation dΠ is wholly deterministic during the small time period dt.
Let Δ be a constant number during the timestep dt and let

                 &       Π = -f  + ΔS          (III.19)
We can write

     & dΠ= -df  + ΔdS
   &
    & = -σS
        &  +Δ( σSdz + μSdt )
  &
     &           = - σS ( )dz
       &    -( μS()+ )dt          (III.20) ;
Now, by choosing Δ = /, we can make the coefficient of dX vanish. So, It follows that by choosing a portfolio of the stock and the derivative, the Wiener process can be eliminated
The appropriate portfolio is          &       - 1: derivative

&;+: shares of stock
The holder of this portfolio is short (agreeing to sell in the future) one derivative and long (agreeing to buy in the future) an amount / of shares. Defining Π as the value of the portfolio, we have     &     Π = -f  +S          (III.21)   nbsp;

          and         & dΠ = - df  +dS          (III.22)
Substituting equations (III.17) and (III.18) into equation (III.22) yields

   &             dΠ =  (-- ) dt          (III.23)
Because this equation does not involve dz, the portfolio must be riskless during time Δt. The assumptions listed in the preceding section imply that the portfolio must instantaneously earn the same rate of return as other short-term  risk-free securities. If it earned more than this return, arbitrageurs could make a riskless profit by shorting the risk-free  securities and using the proceeds to buy the portfolio; if it earned less, they could make a riskless profit by shorting the portfolio and buying risk-free  securities. It follows that

          &;dΠ =  rΠdt          (III.24)
where r is the risk-free  interest rate. Substituting from equation (III.21) and (III.23), this becomes&

  &        (-- )dt = r (-f + S )dt          (III.25)

          So that           &        + + r S - rf = 0          (III.26)  &            This parabolic partial differential equation is called the Black-Scholes-Merton   differential equation. Combining it with particular boundary conditions of European call, we can derive Black-Scholes formula as shown in Chapter I. The complete derivation is in Appendix.

III.2  Stochastic Simulation

III.2.1  The Principle of Monte Carlo Simulation for Finance
The Monte Carlo method simulates the random movement of the asset prices and provides a probabilistic solution to the option pricing problems. Since most derivative pricing problems can be formulated as the valuation of the discounted expectation of the terminal payoff function, the Monte Carlo simulation becomes naturally an effective numerical tool for pricing derivative securities whose analytic closed form solutions do not exist, e.g. Asian option pricing.
Consider a derivative dependent on a single underlying asset S that provides a payoff at time T. Assuming that interest rates are constant, we can value the derivative by following the steps below:
1. Sample a random path for S in a risk-neutral  world ( the risk preferences of the investors do not affect the price): = Exp( σdz  + (μ - 1/) dt ).
2. Calculate the payoff, for an European call option, it is .
3. Discount the expected payoff at the risk-free rate to get an estimate of the value of the derivative c = .
4. Repeat steps 1 and 2 to get many sample values of the payoff.
5. Calculate the mean of the values to get the price of the derivative: = .
The major advantage  of the Monte Carlo approach is its ease to accommodate complicated terminal payoff function in a derivative pricing model. For example, the terminal payoff may depend on the average of the asset price over certain time interval (Asian options) or the extremum value of the asset price over some period of time (Lookback options). It is quite straightforward to obtain the average or extremum value in the simulated price path in individual simulation runs, and this represents a instinctive advantage. The main drawback of the Monte Carlo simulation is the demand for a large number of simulation trials in order to achieve a high level of accuracy. However, viewing from another perspective, practitioners dealing with a newly invented option may obtain an estimate of its price using the Monte Carlo approach through brute force simulation, rather than risking themselves in the construction of an analytic pricing model for the new option.
For the same reasons, Monte Carlo is also widely used in calculation of VaR (value at risk). Under Monte Carlo method, possible futures of assets values are constructed and the value change of the portfolio under those asset value changes are remembered. The resulting distribution of portfolio value changes can then be statistically analyzed. To calculate, say 5% VaR, it is only necessary to sort the value changes in ascending order and select the change that is 5% of the way through the list.
The main steps in a basic Monte Carlo approach to estimating VaR are as follows:
1. Value the portfolio using the current values of assets.
2. Sample every asset once to obtain its change in value at the end of horizon Δt.
3. Calculate the value change of the portfolio ΔP, at the end of the horizon using the result of step2.
4. Repeat step 2 and 3 many times to build up a probability distribution for ΔP.
5. Fix the required VaR from the distribution.
Here we give a example to explain this process. Consider a portfolio with initial value $100. We run Monte Carlo simulation following the above steps for 100 times and obtain a series of value change of the portfolio (see the table below). This series is then ranked best-case-to-worst    and the number corresponding to the chosen level of confidence is picked from the series. In the case, for the 97.5% confidence level one merely averages outcomes number 97 and 98 to come up with the VaR: -$3.30.



The Monte Carlo prices for two assets are generated according to the formulae

         &
              &                    (III.27)
The correlation of the assets is imposed by requiring that the random numbers be correlated. Specifically, a vector of two uncorrelated random number, is constructed, and the correlation is effected by transformation

        &           = .
                                  (III.28)
where ρ is the correlation of the two assets. So we can have



The derivation of A refers to [7].  The A matrix for portfolio with n assets can also be derived.

III.2.2  Application Examples Implemented in Mathematica

III.2.2.1  Simulation of Stock Price Behavior
Here we simulate stock price behavior formulated by (Refer to III.1.1)

   &   σ dz + μ dt

          or               σSdz + μSdt,   t = 0, 1, . . . n          &;

The above code is for only one path of the random stock movement. Repeating the procedure for many times and putting the results together we get the following multiple-path  graph. The principle behind this is important for Monte Carlo simulation for finance. As that we stated in III.2.1 and that we will see late, Monte Carlo simulation needs a number of times of random procedure followed by somehow diverse results, of which the mean is what we seek after.

III.2.2.2  Monte Carlo European Option Pricing

In the method of Monte Carlo simulation for pricing plain vanilla (or standard) European option, instead of adopting a closed form equation, like Black-Scholes formula, we simply take use of the fundamental equation for stock price (III.16). With the stock price at expiry date, we can calculate the payoff . Then, by discounting the payoff to the beginning date we can get the price of the option. Repeating many times we can obtain the approximated option price by averaging the results of the sample values.
At first, we define the simulation procedure:

Notice that to improve the accuracy of the result, a method called the antithetic variable technique is used in this code, i.e using one set of random number to generate two sample sets: tb1 and 2m-tb1 .  Now let's run, for example, 1000 samples:

The simulation result is: the mean of option price is around 6.1; the deviation of the samples is around 0.2.  By the principle of Monte Carlo simulation, we expect that the more times we sample, the more accurately the final result converges to the analytic result. For comparison, here we calculate the option price using the code for Black-Scholes formula in II.3.2.1. The result is assigned to BStheory.

In the following graph, we put together all the results of Monte Carlo simulation with different sample numbers:

n mean stderrr

1000 6.349394258264993` 0.21353360107220595`

2000 5.994455556362885` 0.14783970850539316`

4000 6.233084890903034` 0.10871592559304973`

8000 6.1371137050672715` 0.07521829152616075`

16000 6.114214865629044` 0.05251337021063431`

A plotting function ShowPrice is defined below to illustrate these results comparing with the theoretic solution.

The red vertical bars represent the results of Monte Carlo simulation with different sample numbers, the black horizontal line is the theoretical result. Notice that the height of a red  bar represents the standard deviation of a simulation result, and the central point of it represents the mean.  From the graph, we see that when the sampling number becomes large enough, e.g, 16000 times here, the simulation result is very closed to the analytic one, and the correspondent stand deviation becomes much smaller.

III.2.2.3  Monte Carlo Asian Option Pricing

An Asian option is an option of which the payoff is calculated by using the average price of the underlying asset during at least some part of the life of the option. For a call option, the payoff is max(, 0); for a put, it is max (, 0).
Asian options are considered to be without analytic closed form solutions. To price Asian options, people resort to Monte Carlo methods. The basic procedure of Monte Carlo Asian call option pricing is the same as that of standard European call option. The main difference lies in the calculation of payoff. In the following code, we calculate the price of Asian option whose average prices of the underlying asset  are taken along the horizon between certain time T1 and the expiration.

Notice that the price of the underlying asset at T1 corresponds to the formula ; the prices in the path between T1 and T use .
Because we lack an analytic solution for Asian options, simply for the comparison of the results with different sample numbers, one of the running results which is with 100000 samples is given here. It is time consuming and you don't have to run it yourself.

Now we can run this Monte Carlo program with some reasonable sample numbers and show the results in a same graph.

n mean stderrr

1000 5.999242108313521` 0.28023365130781813`

2000 5.750609478007954` 0.19360006085122533`

3000 5.6111255196411305` 0.15664927444121413`

4000 5.691260103115371` 0.13782827097780526`

REQUIRED
III.2.a Define a function MCLookbackCall[s_, σ_, τ_, r_, n_, navg_] for a lookback call option. The payoff from a lookback call is the amount that the final stock price exceeds the minimum stock price achieved during the life of the option. The above code for Asian option pricing can be used here after some modification. Notice that here path is the whole range of the option's life and strike price is not used. Calculate the option price MCLookbackCall[50, 0.4, 5/12, 0.1, 1000, 30].
Hint: use the Mathematica functions Last[] and Min[].

III.2.2.4  Monte Carlo Value at Risk

Monte Carlo simulation provides a brute force method of calculating VaR for portfolio with nonlinearity of instruments and can handle virtually any type of portfolio, however complex or exotic. Like the above cases of the option pricing simulation, the fundamental factor of Monte Carlo simulation of VaR is the stochastic process of asset prices, which is presented here by a function GeoBrownianMotion.

Here we deal with a portfolio contain two assets. The function MonteCarloVAR is constructed below to project the prices of two random assets undergoing stochastic motion:

As a first example, we consider a portfolio of two stocks, S1and S2. These equities have identical volatilities but initial values (or prices, if the volume of the assets are fixed) are S1=100 and S2=50.

In the following calculation, the function MonteCarloVAR returns a list of the price changes for the two instruments of a time horizon of one day.

Below we define a second function ShowVaR that takes lists of pairs and constructs distribution functions and VaRs of each of the two instruments, as well as the sum of them -  the value change of the overall portfolio (Refer to III.2.1).

Now, We carry out a simulation of 2000 iterations.

Then we can take 20 pairs of data to construct the distribution of VARs of the two assets and the portfolio, and show the portfolio's 5%VAR.

In this figure, the red curve is the distribution of price changes of the first instrument 1, the blue curve is for the instrument 2 and the black curve is for the portfolio. The vertical bars show the respective VaRs. The portfolio's VaR, S1 + S2, is printed in the graph.

REQUIRED
III.2.b Change the value of the correlation of the two assets to ρ = -1  and 1,  respectively, and run the above simulation. How do you explain the results?

To illustrate the effect of nonlinear instruments, we consider, for the sake of simplicity, a portfolio of two instruments that depend only on the asset S1. We initially consider a simple portfolio consisting of two equal amount of (S1)/2.

Here, the blue curve exactly overlays the red curve. In the following example we give a comparatively more sophisticated nonlinear example.

As explained above, hedging is a widely used strategy for risk reduction. Here we give a simple example of hedging, in which the portfolio contains only two assets: a underlying asset, i.e a share of stock,  and a derivative asset, i.e a standard European call on the stock. When the delta is approximately 0.5, by writing (i.e selling) the call option at the price near the strike price, the holder of the portfolio can hedge half of the position in S1, but has to pay the price that there is no longer the possibility of substantial positive gains.

So, suppose the value of the stock is S1, the value of the call option can be expressed as "-EuroCall[S1, 100, 0.05, 0.0, 0.20,0.01]", where the function EuroCall is exactly the function Black-Scholes in II.3.2 and K = 100, σ = 0.2, r = 0.05, t = 0.0. Notice that he is the writer or seller, but not the buyer or holder of the call option, the same role as the car dealer above. Hence a " - ' is added in front of the function.

The portfolio is constructed:

Now, We carry out a simulation of 2000 iterations.

Here, the red curve is again the nearly normal distribution associated with the linear position in the underlying asset S1. The blue curve is the payoff distribution of the short position in the call option. The skewness of the distribution describes the highly nonlinear price change behavior of the call option near expiry. The black curve is the net position of the portfolio. In this case the VaR has been reduced, comparing with previous example. So writing the call as a hedge has indeed reduced the amount that can be lost under the same condition.

This example illustrates an important point about the use of Monte Carlo simulation to estimate VaR. Although it is much more computation intensive than some other methods, we have  much more available information about the price behavior of the portfolio:  indeed, we have the distribution and sub-distributions of prices changes, and much more other information relevant to financial decision making.

REQUIRED
III.2.c Now change the above portfolio to contain two assets: the same underlying asset and a put option on the asset. This time, however, the holder of the underlying asset buy the option. The pricing formula for a put option is presented in section I.3.3. Use the parametric values in the above hedging example to calculate the 5%VAR of the portfolio. What is your  comment on your result?

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