Chapter 24 - Monte Carlo Valuation

Question 1

Motivation behind using monte carlo valuation

Answer 1

When the other options are very complex or difficult to use. Monte carlo performs simulations instead of relying on some expclicit formula

Question 2

What distrirbution is used with the monte carlo valuation?

Answer 2

Risk neutral distribution, where we assuem that assets earn risk free returns on average, and thus allow us to discount using the risk free rate.

Question 3

why dont we use the actual distribution when doing monte carlo?

Answer 3

It would generate a discounting nightmare

Question 4

WHen we simulate possible future values, what do we get

Answer 4

as a byproduct, we get the distribution of payoffs.

Question 5

how do we generate random numbers, and elaobrate on what kind of random numbers we usually are interested in

Answer 5

We usually want some distribution.

We first start iwth the uniform distribution.

we make a random draw from uniform distribuiton, and treat it as a quantile. Fori nstance, drawing 0.7 is interperted as drawing 70% quantile.

Then we use the inverse of the cumulative distribution of whatever distribution we wish to draw from. For instance the inverse normal distribution.

This works because quantiles are uniformly distributed.

So, we are basically saying “what value gives us a quantile of x”. And since we use cumulative distribution, quantiles are basically the same as probability.

Question 6

elaborate on simulating something across multiple periods (stock prices)

Answer 6

Generally, there is no difference in results whether we simulate the periods directly, or if we take it sequentially one period at a time, and then just use the simulated stock price result after 1 period as the starting point for simulating the next period and so on.

However, if we want the path, we need to do it sequantially.

Question 7

what is the expression for simulating stock prices T periods into the future

Answer 7

Question 8

when is path simulating perhaps important?

Answer 8

Asian options and barrier options. This is becasue these are dependent on the path.

Question 9

in a multi period binomial tree, what is the expression for the call price

Answer 9

Cosnsits os 2 parts. One is the probability of reaching a specific node, and the other is the value at this node. Then we take expected value and ultimately discount it.

Question 10

elaborate on the monte carlo value, or time zero monte carlo price of some option

Answer 10

We sim many times, and take the average. In fact, we take the average discounted payoff.

Denote the payoff as a function of the stock price, V(S_t, T)

Note that all S variables are at time T. Thus, we have a sum that sums over “draws” where each draw try to do the same thing.

Question 11

Consider the formula for time zero monte carlo price.

What could the call option function be? Instead of the abstract function V(…)

Answer 11

V(S_T^{i}, T) = max(0, S_T^{i} - K)

Question 12

Recall how we can model stock price development as log normal

Answer 12

Question 13

how do we assess accuracy of some monte carlo method?

Answer 13

We need to measure the standard deviation.

the motne carlo valiue is given by:

C = 1/n ∑c(i) (sort of )

Takign variance of this, we get:

1/n^2 ∑var(c(i))

1/n^2 nvar(c(i))

= var(ci) / n

This means that the standard deviaiton of the monte carlo estimator is given by SD(ci) / sqrt(n)

SO: We have the the standard deviation of the monte carlo estimator is equal to the standard deviation of the single monte carlo DRAW, divided by the square root of the number of draws.

Since the standard deviation of the single draw is eqwual to the standard deviation of the stock, our estimator will have a lower standard deviation than the single value. This simply means that when we run the trial many times and take the average, we should see values that are less spread than the indidividuals.

Question 14

under what conditions is monte carlo especially useful

Answer 14

1) When the number of random elements is too great to permit direct numerical solution

2) When underlying variables are distributed in a way that makes direct solutions difficult

3) Where options are path dependent: Payoff at expiration depends on the path taken as well as the final outcome.

Question 15

elaborate on asian options

Answer 15

asian options have payoffs that are based on the average stock price during the holding period.

Question 16

what is arithmetic asian options

Answer 16

asian options where the payofgf is calculated using the arithmetic average

Question 17

in general, what can we say about the returns of an asian option vs regular ones

Answer 17

Because it average out the returns, we should see a more tightly distributed series of payoffs. Averaging will reduce the likelihood of large gains and large losses.

Question 18

How do we compute a 3 month stock price when we want to include the path?

Answer 18

Make the same procedure 3 times. We therefore have 3 different draws from the standard normal distribution.

In the context of monte carlo, these 3 draws together make one trial.

Question 19

how do we use monte carlo to value an asian option

Answer 19

We perform the path dependent stock price trial many times.
Then we include each trial’s results in this formula:

C_asian = e^(-rt) E[max(0, (S1+S2+S3)/3 - K)]

The arithmetic average computaiton happen inside of the max() function. The monte carlo average (which we always include) happen in the expected value function.

Question 20

what is an interesting property regarding Monte carlo valuation and asian options?

Answer 20

Keeping the number of trials the same, as we increase the number of averages (periods), the arithmetic, geometric, and exact geometric price, and the standard deviation of result will decrease.

so it is simply that if we reduce the period, so that we include more checkpoints in the average-computation, we get lower means. This makes sense, as we will include more points early on, and shift the relative weighting away from the final outcome.

Question 21

why do we call regualr monte carlo “naive”

Answer 21

Because it makes no attempt to reduce the variance of the simualted answer for a given number of trials.

Question 22

what is the motivation behind looking into other variations of monte carlo?

Answer 22

Naive monte carlo require a lot of simulations and will still have kind of bad standard deviation results.

For instance, if we have a standard deviation of a couple percent, this is too large in the context of stock prices and option prices.

Question 23

is it easy to use monte carlo for american options

Answer 23

No because american options involve us working backwards from the termination date to see if early exercise is ever beneficial at any point. The monte carlo simulation does not generally provide for this.

However, recently there have been variations that allow for american options to be valued using monte carlo methods.

Question 24

elaborate on using monte carlo for american options

Answer 24

Firstly, we need to include the paths.

The ITM nodes are the candidates for early exercise. Recall why this is. OTM and ATM are not candidates because we’d never exercise them early. IT would be cheaper to obtain the shares using a simple stock purchase (generally speaking).

Then we are looking into the idea of comparing immediate value against continuation value. The immediate value is easy, it is simply the exercise now benefits. However, computing the continuation value is more difficult.

The problem is associated with “lookahead bias”. Since we dont know whether we should exercise or not (since we dont know the path the stock price take in the future), we need to remove this bias.

To do this, we base our decision on average outcome from some point looking forward. To do this, we could use regression.
Or we can do a branching technique. We look into these later

Question 25

elaborate on the regression approach to valuing american options

Answer 25

We generate a bunch of paths, also identifying the candidates for early exercise as those being ITM. Then we set them up like a matrix where each row represent a path, and each column is one price record at some point along the path. Then our goal is to run regression to compute the continuation value. The book suggests a regression model of "y = a + bS + cS^2" where S is the stock price. We get a lot of paths, so I suppose one path gives one training instance. they use regression to model continuation value. But they sort of only use one independent variable which is the stock price. And then they basically generate many paths, and for each path we have one training instance, or data point, in the regression model. Actually, it looks like they use the same model at each time step, so we fit a model based on the number of paths AND the number of steps in each path. However, the relationship try to say that the continuation value is a function of the current stock price and the current stock price squared, which might be too simple. However, assuming that the functional form is decent enough, the idea is to fit a model based on every path-step in our simulations, and then use this estimator to run estimates, and compare this against the intrinsic value at some point in time to figure out whether early exercise is beneficial or not. IMPORTANT: THey use a new fitted model for each step. This is because they use the results from the next layer to compute the value of the current layer. This allows us to propagate the continuation value backwards through the paths.

Question 26

problem with the regression approach

Answer 26

Difficult to select functional form of the regrssion

Question 27

What can we say about the log normal distribution and large stock price movements?

Answer 27

Log normal distribution will assign low probabilities for large stock price movements.

Question 28

Log normal distribution will assign low probabilities for large stock price movements. Is this good?

Answer 28

No. We typically observe that a lot of action (price movement) happen at the same time. Therefore, it is beneficial to model in jumps, or large moves, in the stock price.

Question 29

what are the poisson process assumptions

Answer 29

1) The probability of an event occurring is porportional to the lenght of the interval 2) The propbability that more than one event will occur in a small interval h is substantially smaller than the probability of one event ocurring. 3) The number of events in non-overlapping intervals are indepednent 4) The expected number of events between time t and t+h is independent of t.

Question 30

Elaborate on the Poisson distribution

Answer 30

the poisson distribuiton gives the probability of some number of poisson events occurring during some period of time/interval. "During an interval of size t, what is the probability that 'm' events will occur when the average number of events is lambda x t".

Question 31

can we use Poisson to model jumps

Answer 31

yes, but poisson only accounts for the timing. The magnitude is something that we need to model as well.

Question 32

what is typically used to model the size of the jump?

Answer 32

A draw from the log-normal distribution

Question 33

what is an assumption made regarding jumps that we should be careful about

Answer 33

The assumption that jumps are idiosyncratic and can be diverisifed. Large market moves are by definition systematic. The reason why it is common to include the assumption of idiosyncratic jumps, is that since they are diverisifed, they will not affect the risk premium of the assets, and we dont have to extend our modeling much.

Question 34

Using the log normal distribution, how do we represent the size of a jump. assume the poisson process has been used to say that a jump should occur now. How can we determine its size?

Answer 34

Question 35

elaborate on simulating stock price over a period of time h when jumps are a possibility

Answer 35

We need to pick 2 random numbers (uniform): 1) The number of jumps 2) ordinary (non jump) log normal return Then for each jump, we pick one new random variable to determine the magnitude of the jumps.

Question 36

elaborate on the effect that jumps has on the stock price

Answer 36

Jumps have a multiplicative effect on the stock price. This is nice, because we can then model the jumps together as a single jump that accounts for all the effects. We refer to this as the cumulative jump.

Question 37

elaborate on the alpha parameter of the stock price movement when we consider jumps

Answer 37

Since we assume that jumps are idiosyncratic, it means that the jumps will not affect our return or risk premium. However, the jump return alpha parameter (not the one the qiuestion asked about) will have an effect, depending on whether it goes up or down on average. Therefore, we need to balance out this effect by adjusting the original stock price alpha. We want to maintain the no-jump drift. The jumps should not affect the expected return. If the jump is 10% down on average, we need to adjust the average expected return by raising it to soem degree. Iti s very simple: We subtract from the original alpha "lambda k". k is the shit we get from "e^{alpha_j} - 1 = k" which represent the expected percentage jump. So we basically just insert "â = a - lambda k" where we originally had the â parameter. OF course, lambda is the poisson shite.

Question 38

how can we generate 2 correlated log normal variables+

Answer 38

determine p, the degree of correlation. Assume the log nromal variables are given as the image. Then we say: we generate 2 iid N(0,1) variables epsilon_1 and epsilon_2. W = epsilon_1 Z = p epsilon_1 + epsilon_2 sqrt(1 - p^2) When we enter this into the model, we get that the variables following log normal distribution are correlated with coefficient equal to p. It also retains Z as stnadard normal.

Question 39

what is the