In this second edition of Frequently Asked Questions in Quantitative Finance I continue in my mission to pull quant finance up from the dumbed-down depths. The CQF program. Contribute to ranzhaocgu/CQF development by creating an account on GitHub. Designations used by companies to distinguish their products are often claimed as trademarks. All brand II Title: Quantitative finance. HG Definition of common terms. .. answers to the end-of-chapter questions in this book.
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2 Frequently Asked Questions In Quantitative Finance. There follows a www- pflegeelternnetz.info Artzner, P. HYDROPONICS By Dudly Harris ISBN pflegeelternnetz.info tells you everything you need to know about Hydroponics Online - Hydropon. Frequently Asked Questions In Quantitative Finance including key models, important formulæ, popular contracts, essays and opinions, a history of quantitative.
Long answer A common criticism of traditional VaR has been that it does not satisfy all of certain commonsense criteria. These are genuine exact arbitrages albeit the latter being model dependent. For this exercise f x is x2 a convex function. Money is made by portfolios, not by individual contracts. He did some other clever stuff as well. Usually these cannot be solved analytically and so they must be solved numerically.
I hope you enjoy this book, and that it shows you how interesting this important subject can be. And I hope you'll join me and others in this industry on the discussion forum on wilmott. See you there! Request permission to reuse content from this site. Undetected country. NO YES. Description About the Author Permissions Table of contents. Selected type: Added to Your Shopping Cart. Getting agreement between finance theory and finance practice is important like never before. In the last decade the derivatives business has grown to a staggering size, such that the outstanding notional of all contracts is now many multiples of the underlying world economy.
No longer are derivatives for helping people control and manage their financial risks from other business and industries, no, it seems that the people are toiling away in the fields to keep the derivatives market afloat! Apologies for the mixed metaphor!
Modern Portfolio Theory addresses this question and provides a framework for quantifying and understanding risk and return. Markowitz showed how it might be possible to better both of these simplistic portfolios by taking into account the correlation between the returns on these stocks.
To Markowitz all investments and all portfolios should be compared and contrasted via a plot of expected return versus risk, as measured by standard deviation. The mathematics of risk and return is very simple.
This line is called the Capital Market Line and the portfolio at the point at which it is tangential is called the Market Portfolio. Now, again according to the theory, no one ought to hold any portfolio of assets other than the risk-free investment and the Market Portfolio. But how much of that risk and return are related to the market as a whole? Whereas MPT has arbitrary correlation between all investments, CAPM, in its basic form, only links investments via the market as a whole.
CAPM is an example of an equilibrium model, as opposed to a no-arbitrage model such as Black—Scholes. The mathematics of CAPM is very simple. In this representation we can see that the return on an asset can be decomposed into three parts: Notice how all the assets are related to the index but are otherwise completely uncorrelated.
Multi-index versions of CAPM can be constructed. The parameters alpha and beta are also commonly referred to in the hedge-fund world. Performance reports for trading strategies will often quote the alpha and beta of the strategy. A good strategy will have a high, positive alpha with a beta close to zero. With beta being small you would expect performance to be unrelated to the market as a whole and with large, positive alpha you would expect good returns whichever way the market was moving.
References and Further Reading Lintner, J The valuation of risk assets and the selection of risky investments in stock portfolios and capital budgets. Review of Economics and Statistics 47 Chapter 2: Econometrica 34 — Sharpe, WF Capital asset prices: Journal of Finance 19 3 — Tobin, J Liquidity preference as behavior towards risk.
These random factors can be fundamental factors or statistical. For there to be no arbitrage opportunities there must be restrictions on the investment processes.
Long answer Modern Portfolio Theory represents each asset by its own random return and then links the returns on different assets via a correlation matrix. In Arbitrage Pricing Theory returns on investments are represented by a linear combination of multiple random factors, with associated factor weighting. Portfolios of assets can also be decomposed in this way.
In practice we can choose the factors to be macroeconomic or statistical. Here are some possible macroeconomic variables. Statistical variables come from an analysis of a covariance of asset returns. From this one extracts the factors by some suitable decomposition.
Although APT talks about arbitrage, this must be contrasted with the arbitrage arguments we see in spot versus forward and in option pricing. These are genuine exact arbitrages albeit the latter being model dependent. In APT the arbitrage is only approximate. Journal of Economic Theory 13 — Chapter 2: Short answer Maximum Likelihood Estimation MLE is a statistical technique for estimating parameters in a probability distribution.
Example You have three hats containing normally distributed random numbers. This is hat A. This is hat B. This is hat C. You pick a number out of one hat. Which hat do you think it came from? MLE can help you answer this question.
One popular way of doing this is Maximum Likelihood Estimation. The method is easily explained by a very simple example. You are attending a maths conference. You arrive by train at the city hosting the event. You take a taxi from the train station to the conference venue. The taxi number is 20, How many taxis are there in the city?
This is a parameter estimation problem. Estimating the number of taxis in the city from that event is a question of assumptions and statistical methodology. Taxi numbers are strictly positive integers Numbering starts at 1 No number is repeated No number is skipped.
We will look at the probability of getting into taxi number 20, when there are N taxis in the city. N Which N maximizes the probability of getting into taxi number 20,?
This example explains the concept of MLE: Choose parameters that maximize the probability of the outcome actually happening. You have three hats containing normally distributed random numbers. FAQs 67 N. We now pick a second number from the same hat. It is 0. This looks more likely to have come from hat B. Probabilities and hats. This is easy. That is: Last year I earned: Logic and Practice. Sage Chapter 2: FAQs 71 What is Cointegration? Short answer Two time series are cointegrated if a linear combination has constant mean and standard deviation.
In other words, the two series never stray too far from one another. This can be the basis for pairs trading. An alternative statistical measure to correlation is cointegration. Two stocks may be perfectly correlated over short timescales yet diverge in the long run, with one growing and the other decaying.
Conversely, two stocks may follow each other, never being more than a certain distance apart, but with any correlation, positive, negative or varying. If we are delta hedging then maybe the short timescale correlation matters, but not if we are holding stocks for a long time in an unhedged portfolio.
Stocks, which tend to grow, are not stationary. In a sense, stationary series do not wander too far from their mean. If we can, then we say that the stocks are cointegrated. The error in this tracking portfolio will have constant mean and standard deviation, so should not wander too far from its average. This is clearly easier than using all stocks for the tracking when, of course, the tracking error would be zero.
We could analyse the cointegration properties of two related stocks, Nike and Reebok, for example, to look for relationships. This would be pairs trading. The important difference is that cointegration assumes far fewer properties for the individual time series. Most importantly, volatility and correlation do not appear explicitly. Another feature of cointegration is Granger causality which is where one variable leads and another lags.
The idea has long been used in the world of gambling. They are willing to bet any number of times. Clearly you can make a lot of money with this special coin. How much of this should you bet? As you win and your wealth grows you will bet a larger amount. If you bet too little then it will take a long time for you to make a decent amount. The Kelly criterion is to bet a certain fraction of your wealth so as to maximize your expected growth of wealth.
This function is plotted in Figure 2. FAQs 75 0. Expected return versus betting fraction. This is the Kelly fraction. A betting fraction of less than this would be a conservative strategy. Anything to the right will add volatility to returns, and decrease the expected returns. Too far to the right and the expected return becomes negative. This money management principle can be applied to any bet or investment, not just the coin toss. A common choice is half Kelly. Other money management strategies are, of course, possible, involving target wealth, probability of ruin, etc.
FAQs 77 Why Hedge? Clearly, if you have two portfolios with the same expected return the one with the lower risk is the better investment. Example You buy a call option, it could go up or down in value depending on whether the underlying go up or down.
So now sell some stock short. If you sell the right amount short then any rises or falls in the stock position will balance the falls or rises in the option, reducing risk. Long answer To help to understand why one might hedge it is useful to look at the different types of hedging.
An example of such hedging is put—call parity. There is a simple relationship between calls and puts on an asset when they are both European and with the same strikes and expiries , the underlying stock and a zero-coupon bond with the same maturity.
This relationship is completely independent of how the underlying asset changes in value. Another example is spot-forward parity. Such model-independent hedges are few and far between. The obvious example is the hedging used in the Black—Scholes analysis that leads to a whole theory for the value of derivatives.
In pricing derivatives we typically need to at least know the volatility of the underlying asset. If the model is wrong then the option value and any hedging strategy could also be wrong. Delta hedging One of the building blocks of derivatives theory is delta hedging.
This is the theoretically perfect elimination of all risk by using a very clever hedge between the option and its underlying. Delta hedging exploits the perfect correlation between the changes in the option value and the changes in the stock price. Because of the frequent rehedging, any dynamic hedging strategy is going to result in losses due to transaction costs.
In some markets this can be very important. If you have two instruments depending on the same risk of default, you can calculate the sensitivities, the deltas, of their prices to this quantity and then buy the two instruments in amounts inversely proportional to these deltas one long, one short.
This is also delta hedging. If two underlyings are very highly correlated you can use one as a proxy for the other for hedging purposes. You would then only be exposed to basis risk. Be careful with this because there may be times when the close relationship breaks down. With such a large portfolio you can theoretically reduce risk to negligible levels. A portfolio that is delta hedged is insensitive to movements in the underlying as long as those movements are quite small.
There is a small error in this due to the convexity of the portfolio with respect to the underlying. Gamma hedging is a more accurate form of hedging that theoretically eliminates these second-order effects. Typically, one hedges one, exotic, say, contract with a vanilla contract and the underlying. The quantities of the vanilla and the underlying are chosen so as to make both the portfolio delta and the portfolio gamma instantaneously zero. Vega hedging The prices and hedging strategies are only as good as the model for the underlying.
The key parameter that determines the value of a contract is the volatility of the underlying asset. Nor is it usually a constant as assumed in the simple theories. Obviously, the value of a contract depends on this parameter, and so to ensure that a portfolio value is insensitive to this parameter we can vega hedge.
This means that we hedge one option with both the underlying and another option in such a way that both the delta and the vega, the sensitivity of the portfolio value to volatility, are zero.
This is often quite satisfactory in practice but is usually theoretically inconsistent; we should not use a constant volatility basic Black—Scholes model to calculate sensitivities to parameters that are assumed not to vary.
The distinction between variables underlying asset price 80 Frequently Asked Questions in Quantitative Finance and time and parameters volatility, dividend yield, interest rate is extremely important here.
To get around this problem it is possible to independently model volatility, etc. In such a way it is possible to build up a consistent theory. Static hedging There are quite a few problems with delta hedging, on both the practical and the theoretical side. In practice, hedging must be done at discrete times and is costly. Sometimes one has to buy or sell a prohibitively large number of the underlying in order to follow the theory.
This is a problem with barrier options and options with discontinuous payoff. On the theoretical side, the model for the underlying is not perfect, at the very least we do not know parameter values accurately.
Delta hedging alone leaves us very exposed to the model, this is model risk. The static hedge is put into place now, and left until expiry. Superhedging In incomplete markets you cannot eliminate all risk by classical dynamic delta hedging.
But sometimes you can superhedge meaning that you construct a portfolio that has a positive payoff whatever happens to the market. A simple example of this would be to superhedge a short call position by buying one of the stock, and never rebalancing.
Unfortunately, as you can probably imagine, and certainly as in this example, superhedging might give you prices that differ vastly from the market. FAQs 81 Margin hedging Often what causes banks, and other institutions, to suffer during volatile markets is not the change in the paper value of their assets but the requirement to suddenly come up with a large amount of cash to cover an unexpected margin call.
Writing options is very risky. The downside of buying an option is just the initial premium, the upside may be unlimited. The upside of writing an option is limited, but the downside could be huge.
For this reason, to cover the risk of default in the event of an unfavourable outcome, the clearing houses that register and settle options insist on the deposit of a margin by the writers of options.
Margin comes in two forms: The initial margin is the amount deposited at the initiation of the contract. The total amount held as margin must stay above a prescribed maintenance margin. If it ever falls below this level then more money or equivalent in bonds, stocks, etc. The amount of margin that must be deposited depends on the particular contract. To prevent this situation it is possible to margin hedge.
That is, set up a portfolio such that a margin calls on one part of the portfolio are balanced by refunds from other parts. Market crashes have at least two obvious effects on our hedging. First of all, the moves are so large and rapid that they cannot be traditionally delta hedged.
The convexity effect is not small. Second, normal market correlations become meaningless. Typically all correlations become one or minus one. Crash or Platinum hedging exploits the latter effect in such a way as to minimize the worst possible outcome for the portfolio.
The method, called CrashMetrics, does not rely on parameters such as volatilities and so is 82 Frequently Asked Questions in Quantitative Finance a very robust hedge. Platinum hedging comes in two types: Short answer Marking to market means valuing an instrument at the price at which it is currently trading in the market.
With an option this may not happen until expiration. When you hedge options you have to choose whether to use a delta based on the implied volatility or your own estimate of volatility. You buy the stock. Obviously this is open to serious abuse and so it is usual, and often a regulatory requirement, to quote the mark-to-market value. Patience, my son. Long answer If instruments are liquid, exchange traded, then marking to market is straightforward.
You just need to know the most recent market-traded price. After all, you presumably entered the trade because you thought you would make a gain. With futures and short options there are also margins to be paid, usually daily, to a clearing house as a safeguard against credit risk.
So if prices move against you, you may have to pay a maintenance margin. This will be based on the prevailing market values of the futures and short options. There is no margin on long options positions because they are paid for up front, from which point the only way is up.
Marking to market of exchange-traded instruments is clearly very straightforward. But what about exotic or over-the-counter OTC contracts? These are not traded actively, they may be unique to you and your counterparty. These instruments have to be marked to model. And this obviously raises the question of which model to use. So the question about which model to use becomes a question about which volatility to use.
With credit instruments the model often boils down to a number for risk of default. Here are some possible ways of marking OTC contracts. Perhaps his best forecast going forward. This has the advantage of being real, tradeable prices, and unprejudiced.
They get very annoyed. This has the advantage of giving prices that are consistent with the information in the market, and are therefore arbitrage free. Although there is always the question of which volatility model to use, deterministic, stochastic, etc. It can also be time consuming to have to crunch prices frequently.
One subtlety concerns the marking method and the hedging of derivatives. Take the simple case of a vanilla equity option bought because it is considered cheap.
There are potentially three different volatilities here: By marking to market, or using a model-based marking that is as close to this as possible, your losses will be plain to see. You may be forced to close your position if the loss gets to be too large. Of course, you may have been right in the end, just a bit out in the timing. The loss could have reversed, but if you have closed out your position previously then tough. Having said that, human nature is such that people tend to hold onto losing positions too long on the assumption that they will recover, yet close out winning positions too early.
Marking to market will therefore put some rationality back into your trading. Prior to that it had been assumed that excess returns could be made by careful choice of investments.
These are weak form, semi-strong form and strong form. A trading strategy incorporating historical data, such as price and volume information, will not 88 Frequently Asked Questions in Quantitative Finance systematically outperform a buy-and-hold strategy. It is often said that current prices accurately incorporate all historical information, and that current prices are the best estimate of the value of the investment.
Prices will respond to news, but if this news is random then price changes will also be random. Share prices adjust instantaneously to publicly available new information, and no excess returns can be earned by using that information.
The same is true of previously neglected sources of convexity and therefore value. Essentially the job has been done for everyone. This is seen when one calibrates a model to market prices of derivatives, without ever studying the statistics of the underlying process. The validity of the EMH, whichever form, is of great importance because it determines whether anyone can outperform the market, or whether successful investing is all about luck.
Market bubbles, for example, do not invalidate EMH provided they cannot be exploited. There have been many studies of the EMH, and the validity of its different forms. Many early studies concluded in favour of the weak form. Short answer Performance measures are used to quantify the results of a trading strategy. They are usually adjusted for risk. The most popular is the Sharpe ratio.
Long answer Performance measures are used to determine how successful an investment strategy has been. The more sensible measures make allowance for the risk that has been taken, since a high return with low risk is much better than a high return with a lot of risk.
Sharpe ratio The Sharpe ratio is probably the most important non-trivial risk-adjusted performance measure. The Sharpe ratio will be quoted in annualized terms. A high Sharpe ratio is intended to be a sign of a good strategy. If returns are normally distributed then the Sharpe ratio is related to the probability of making a return in excess of Chapter 2: FAQs 91 the risk-free rate.
In the expected return versus risk diagram of Modern Portfolio Theory the Sharpe ratio is the slope of the line joining each investment to the risk-free investment. Choosing the portfolio that maximizes the Sharpe ratio will give you the Market Portfolio. We also know from the Central Limit Theorem that if you have many different investments all that matters is the mean and the standard deviation. So as long as the CLT is valid the Sharpe ratio makes sense.
This may be important if returns are very skewed. Modigliani—Modigliani measure The Modigliani—Modigliani or M2 measure is a simple linear transformation of the Sharpe ratio: This is easily interpreted as the return you would expect from your portfolio is it were de leveraged to have the same volatility as the benchmark.
Sortino ratio The Sortino ratio is calculated in the same way as the Sharpe ratio except that it uses the square root of the semi-variance as the denominator measuring risk. The semi-variance is measured in the same way as the variance except that all data points with positive return are replaced with zero, or with some target value. However, if returns are expected to be normally distributed the semi-variance will be statistically noisier than the variance because fewer data points are used in its calculation.
Information ratio The Information ratio is a different type of performance measure in that it uses the idea of tracking error. The numerator is the return in excess of a benchmark again, but the denominator is the standard deviation of the differences between the portfolio returns and the benchmark returns, the tracking error.
Tracking error This ratio gives a measure of the value added by a manager relative to their benchmark. Harvard Business Review 43 63—75 Chapter 2: Example You own a valuable work of art; you are going to put it up for auction.
Someone then offers you a guaranteed amount provided you withdraw the painting from the auction. Should you take the offer or take your chances? Utility theory can help you make that decision. When a meaningful numerical value is used to represent utility this is called cardinal utility. One can then talk about one thing having three times the utility of another, and one can compare utility from person to person.
If the ordering of utility is all that matters so that one is only concerned with ranking of preferences, not the numerical value then this is called ordinal utility. If it is a strict inequality then satiation is not possible, the investor will always prefer more than he has.
This slope measures the marginal improvement in utility with changes in wealth. FAQs 95 When the wealth is random, and all outcomes can be assigned a probability, one can ask what amount of certain wealth has the same utility as the expected utility of the unknown outcomes.
The quantity of wealth Wc that solves this equation is called the certainty equivalent wealth. One is therefore indifferent between the average of the utilities of the random outcomes and the guaranteed amount Wc. So we would pay this amount or less to play Certainty equivalent 80 60 40 20 0 0.
Certainty equivalent as a function of the risk-aversion parameter for example in the text. Observe how this decreases the greater the risk aversion. Example Actuaries work more than quants with historical data and that data tends to be more stable. Think of mortality statistics. Quants often project forward using information contained in a snapshot of option prices. Long answer Note: The following was published in The Actuary in September Quants are the relative newcomers, with all their fancy stochastic mathematics.
Since and the publication of the famous papers, all that has been made redundant. Quants are making their models increasingly complicated, in the belief that they are making improvements. This is not the case.
If this were a proper hard science then there would be a reason for trying to perfect models. Finance, thanks to it being underpinned by human beings and their wonderfully irrational behaviour, is forever changing.
It is therefore much better to focus your attention on making the models robust and transparent rather than ever more intricate. As I mentioned in a recent wilmott. They use their complex models, with sophisticated numerical solutions, to come up with the perfect value.
Having gone to all that effort for that contract they then throw it into the same pot as all the others and risk manage en masse. FAQs 99 bonus is tied to it. But each contract? Money is made by portfolios, not by individual contracts. We need models which people can understand and a greater respect for risk. Actuaries and quants have complementary skill sets. References and Further Reading Visit www.
Short answer The Wiener process or Brownian motion is a stochastic process with stationary independent normally distributed increments and which also has continuous sample paths. Example Pollen in water, smoke in a room, pollution in a river, are all examples of Brownian motion. And this is the common model for stock prices as well. The mathematics of this process were formalized by Bachelier, in an option-pricing context, and by Einstein.
The mathematics of BM is also that of heat conduction and diffusion. Mathematically, BM is a continuous, stationary, stochastic process with independent normally distributed increments. The important properties of BM are as follows: However, the path is fractal, and not differentiable anywhere.
It might also appear as dX or dB, different authors using different letters, and sometimes with a time subscript. But these are all the same thing! Its simplicity allows calculations and analysis that would not be possible with other processes. It can be used as a building block for random walks with characteristics beyond those of BM itself. For example, it is used in the modelling of interest rates via mean-reverting random walks.
Higher-dimensional versions of BM can be used to represent multi-factor random walks, such as stock prices under stochastic volatility. In practice, asset returns have tails that are much fatter than those given by the normal distribution of BM.
London Stachel, J ed. Journal of Mathematics and Physics 58 — Chapter 2: Example You roll a die, square the number of spots you get, you win that many dollars. For this exercise f x is x2 a convex function. Graphically this means that the line joining the points x, f x and y, f y is nowhere lower than the curve. Suppose that a stock price S is random and we want to consider the value of an option with payoff P S.
We could calculate the expected stock price at expiration as E[ST ], and 1 This is the probabilistic interpretation of the inequality. Alternatively, we could look at the various option payoffs and then calculate the expected payoff as E[P ST ]. The latter makes more sense, and is indeed the correct way to value options, provided the expectation is with respect to the risk-neutral stock price.
We can get an idea of how much greater the left-hand side is than the right-hand side by using a Taylor series approximation around the mean of S. The convexity of an option. As a rule this adds value to an option. Randomness in the underlying, and its variance.
Modelling randomness is the key to modelling options. The lesson to learn from this is that whenever a contract has convexity in a variable or parameter, and that variable or parameter is random, then allowance must be made for Chapter 2: FAQs this in the pricing.
To do this correctly requires a knowledge of the amount of convexity and the amount of randomness. It tells you that if you have a random walk, in y, say, and a function of that randomly walking variable, call it f y, t , then you can easily write an expression for the random walk in f.
A function of a random variable is itself random in general. What is the stochastic differential equation for the logarithm of S, ln S? You can think of it as a way of expanding functions in a series in dt, just like Taylor series. If it helps to Chapter 2: FAQs think of it this way then you must remember the simple rules of thumb as follows: Whenever you get dX 2 in a Taylor series expansion of a stochastic variable you must replace it with dt.
This means that dt 2 , dX 3 , dt dX, etc. This is important both in the theory of derivatives pricing and in the practical management of market risk. Sometimes we have a function of more than one stochastic quantity. Suppose that we have a function f y1 , y2 ,. These are usually modelled by a Poisson process.
Short answer Risk-neutral valuation means that you can value options in terms of their expected payoffs, discounted from expiration to the present, assuming that they grow on average at the risk-free rate. By eliminating risk in this way we also remove any dependence on the value of risk.
End result is that we may as well imagine we are in a world in which no one values risk at all, and all tradeable assets grow at the risk-free rate on average.
For any derivative product, as long as we can hedge it dynamically and perfectly supposing we can as in the case of known, deterministic volatility and no defaults the hedged portfolio loses its randomness and behaves like a bond.
What can you do? Then calculate the call payoff for each of these paths. Present value each of these back to today, and calculate the average over all paths. Long answer Risk-neutral valuation of derivatives exploits the perfect correlation between the changes in the value of an option and its underlying asset.
As long as the underlying is the only random factor then this correlation should be perfect. The resulting portfolio is risk free. Of course, you need to know the correct number of the stock to sell short. There are several such imperfections with risk-neutral valuation.
First, it requires continuous rebalancing of the hedge. FAQs Delta is constantly changing so you must always be buying or selling stock to maintain a risk-free position. Obviously, this is not possible in practice. Second, it hinges on the accuracy of the model.
The underlying has to be consistent with certain assumptions, such as being Brownian motion without any jumps, and with known volatility. One of the most important side effects of risk-neutral pricing is that we can value derivatives by doing simulations of the risk-neutral path of underlyings, to calculate payoffs for the derivatives.
Here are some further explanations of risk-neutral pricing. Explanation 1: If you hedge correctly in a Black—Scholes world then all risk is eliminated. If there is no risk then we should not expect any compensation for risk. We can therefore work under a measure in which everything grows at the risk-free interest rate. Explanation 2: Explanation 3: Two measures are equivalent if they have the same sets of zero probability. Therefore a price is non-arbitrageable in the real world if and only if it is non-arbitrageable in the risk-neutral world.
The risk-neutral price is always non-arbitrageable. If everything has a discounted asset price process which is a martingale then there can be no arbitrage.
Therefore there is no arbitrage in the real world. Explanation 4: When such a static replication is possible then it is model independent, we can price complex derivatives in terms of vanillas. Of course, the continuous distribution requirement does spoil this argument to some extent.
It should be noted that risk-neutral pricing only works under assumptions of continuous hedging, zero transaction costs, continuous asset paths, etc.
Academic Press Chapter 2: We can change from a Brownian motion with one drift to a Brownian motion with another. Long answer First a statement of the theorem. It will be helpful if we explain some of the more technical terms in this theorem. All possible future states or outcomes. A previsible process is one that only depends on the previous history. The two measures can have different probabilities for each outcome but must agree on what is possible. This implies that in the Black—Scholes world there is only the one equivalent risk-neutral measure.
If this were not the case then there would be multiple arbitrage-free prices. This is often the case in the world of equity derivatives. Straightforward Black—Scholes does not require any understanding of Girsanov. Once you go beyond basic Black—Scholes it becomes more useful. For example, suppose you want to derive the valuation partial differential equations for options under stochastic volatility. Using Girsanov you can get the governing equation in three steps: FAQs 1. This is the reason for the popularity of the BGM model and its ilk.
These two examples are called greek because they are members of the Greek alphabet. It is the rate of change of value with respect to the asset: By doing this they eliminate market risk. FAQs Typically the delta changes as stock price and time change, so to maintain a delta-neutral position the number of assets held requires continual readjustment by purchase or sale of the stock.
This is called rehedging or rebalancing the portfolio, and is an example of dynamic hedging. You then sell what you have borrowed, buying it back later. This can be costly, you may have to pay a repo rate, the equivalent of an interest rate, on the amount borrowed. If there are costs associated with buying or selling stock, the bid—offer spread, for example, then the larger the gamma the larger the cost or friction caused by dynamic hedging. Because costs can be large and because one wants to reduce exposure to model error it is natural to try to minimize the need to rebalance the portfolio too frequently.
This means buying or selling more options, not just the underlying. Speed The speed of an option is the rate of change of the gamma with respect to the stock price. The delta may change by more or less than this, especially if the stock moves by a larger amount, or the option is close to the strike and expiration.
Hence the use of speed in a higher-order Taylor series expansion. It is the sensitivity of the option price to volatility. This can be important. It is perfectly acceptable to consider sensitivity to a variable, which does vary, after all. However, it can be dangerous to measure sensitivity to something, such as volatility, which is a parameter and may, for example, have been assumed to be constant.
That would be internally inconsistent. See bastard greeks. This is a major step towards eliminating some model risk, since it reduces dependence on a quantity that is not known very accurately. There is a downside to the measurement of vega. It is only really meaningful for options having single-signed gamma Chapter 2: FAQs everywhere. For example, it makes sense to measure vega for calls and puts but not binary calls and binary puts.
The reason for this is that call and put values and options with single-signed gamma have values that are monotonic in the volatility: Contracts with a gamma that changes sign may have a vega measured at zero because as we increase the volatility the price may rise somewhere and fall somewhere else. Such a contract is very exposed to volatility risk but that risk is not measured by the vega. Rho would then be the sensitivity to the level of the rates assuming a parallel shift in rates at all times.
But see bastard greeks again. Rho can also be sensitivity to dividend yield, or foreign interest rate in a foreign exchange option. Charm The charm is the sensitivity of delta to time. This can be important near expiration. Colour The colour is the rate of change of gamma with time. It can be misleading at places where gamma is small. Vomma or Volga The Vomma or Volga is the second derivative of the option value with respect to volatility. For example, a fall in the stock price might be accompanied by an increase in volatility.
So one can measure sensitivity as both the underlying and volatility move together. Short answer Because they are fast to compute and easy to understand. Long answer There are various pressures on a quant when it comes to choosing a model.
Fast may also mean easy to calibrate, but not necessarily. Accurate and robust might be similar, but again, not always. The least important is speed. To the scientist the question of calibration becomes one concerning the existence Chapter 2: FAQs of arbitrage. To the practitioner he needs to be able to price quickly to get the deal done and to manage the risk. If he is in the business of selling exotic contracts then he will invariably be calibrating, so that he can say that his prices are consistent with vanillas.
So to the practitioner speed and ability to calibrate to the market are the most important. And the practitioner usually wins. And what could be faster than a closed-form solution?
This is why practitioners tend to favour closed forms. They also tend to be easier to understand intuitively than a numerical solution. Suppose you want to price certain Asian options based on an arithmetic average. To do this properly in the Black—Scholes world you would do this by solving a three-dimensional partial differential equation or by Monte Carlo simulation.
But if you pretend that the averaging is geometric and not arithmetic then often there are simple closed-form solutions. So use those, even though they must be wrong. The point is that they will probably be less wrong than other assumptions you are making, such as what future volatility will be.
Some Asian options can be priced that way. Or what about a closed form involving a subtle integration in the complex plane that must ultimately be done numerically? That is the Heston stochastic volatility model. If closed form is so appreciated, is it worth spending much time seeking them out? Probably not.
There are always new products being invented and new pricing models being devised, but they are unlikely to be of the simple type that can be solved explicitly. Chances are that you will either have to solve these numerically, or approximate them by something not too dissimilar. Finance Press Chapter 2: Short answer Forward and backward equations usually refer to the differential equations governing the transition probability density function for a stochastic process.
They are diffusion equations and must therefore be solved in the appropriate direction in time, hence the names. Example An exchange rate is currently 1. What is the probability that it will be over 2 by this time next year? If you have a stochastic differential equation model for this exchange rate then this question can be answered using the equations for the transition probability density function. Here A and B are both arbitrary functions of y and t. Many common models can be written in this form, including the lognormal asset random walk, and common spot interest rate models.
These two equations are parabolic partial differential equations not dissimilar to the Black—Scholes equation. Example An important example is that of the distribution of equity prices in the future. FAQs 0. The probability density function for the lognormal random walk evolving through time.
We can conclude Frequently Asked Questions in Quantitative Finance that the fair value of an option is the present value of the expected payoff at expiration under a risk-neutral random walk for the underlying.
Short answer The Black—Scholes equation is a differential equation for the value of an option as a function of the underlying asset and time. Long answer Facts about the Black—Scholes equation: If they were of opposite signs then it would be a forward equation.
The equation is an example of a diffusion equation or heat equation. Such equations have been around for nearly two hundred years and have been used to model all sorts of physical phenomena. But the volatility is another matter, rather harder to forecast accurately. Because the main uncertainty in the equation is the volatility one sometimes thinks of the equation less as a valuation tool and more as a way of understanding the relationship between options and volatility.
The equation can be generalized to allow for dividends, other payoffs, stochastic volatility, jumping stock prices, etc. The equation contains four terms: Monte Carlo is great for complex path dependency and high dimensionality, and for problems which cannot easily be written in differential equation form.
Finite difference is best for low dimensions and contracts with decision features such as early exercise, ones which have a differential equation formulation. Numerical quadrature is for when you can write the option value as a multiple integral. Which numerical method should you use? BGM is geared up for solution by simulation, so you would use a Monte Carlo simulation. You want to price a European, non-path-dependent contract on a basket of equities. This may be recast as a multiple integral and so you would use a quadrature method.
Since we work with Chapter 2: The only real difference between the partial differential equations are the following: Number of dimensions Is the contract an option on a single underlying or many? Is there any strong path dependence in the payoff? Answers to these questions will determine the number of dimensions in the problem. At the very least we will have two dimensions: Finite-difference methods cope extremely well with smaller number of dimensions, up to four, say.
Above that they get rather time consuming. Does this matter? Boundary conditions are where we tell the scheme about things like knock-out barriers. Some more modern models are nonlinear. They are excellent for non-linear differential equations. Note that we may need one piece of code per option, hence M in the above.
Not much harder is the application of the explicit method to American options. This is harder to program, but you will get a better accuracy. Start with an Asian option with discrete sampling, and then try a continuously-sampled Asian.
Finally, try your hand at lookbacks. Interest rate products: Repeat the above programme for non-path-dependent and then path-dependent interest rate products.
Two-factor explicit: To get started on two-factor problems price a convertible bond using an explicit method, with both the stock and the spot interest rate being stochastic. Two-factor implicit: So, in a sense they get right to the heart of the problem. When implementing a Monte Carlo method look out for the following: Number of dimensions For each random factor you will have to simulate a time series. Decision features When you have a contract with embedded decisions the Monte Carlo method becomes cumbersome.
This is easily the main drawback for simulation methods. But to correctly price an American option, say, we need to know what the option value would be at every point in stock price-time Chapter 2: FAQs space. Linear or non-linear Simulation methods also cope poorly with non-linear models.
It will take longer to price the greeks, but, on the positive side, we can price many options at the same time for almost no extra time cost. Programme of study Here is a programme of study for the Monte Carlo path-simulation methods. Path-dependent option on a single equity: