# Back of the Envelope

Observations on the Theory and Empirics of Mathematical Finance

## BR: Ito Calculus – I

Time to get down to business. Having build up the necessary intuition that GBM seems to be the way to go, the next step is build up the toolkit to work with BM. We need to make sure that we can take derivatives, do integration and the like with in our Brownian world.

Well, the fact that it needs mention itself suggests that probably the rules of ordinary calculus won’t work. That’s correct. When dealing with BMs, we need to be extra careful when doing calculus. BR next develop the intuition of why ordinary calculus is not useful when dealing with BM.

BR take the examples of some simplest Newtonian differentials and remind “…that though ODEs are powerful construction tools, they are also dangerous ones. There are plenty of bad ODE which we haven’t a clue how to explore”.

In fact, I very much liked their example where they zoom in on a general curve, and ‘show’ that differentiable functions “…are at heart built from straight line segments”.

Their further discussion is fine, and helps ease one into thinking about ODEs – but from that point on, I think it’s sufficient to argue that because BMs are nowhere differentiable, this makes it even more difficult to rely on Newtonian calculus – zooming in on a BM doesn’t produce a straight line! It’s self similar.

But self similarity, BR note, in itself is not a curse. In fact:

…this self similarity is ideal for a building block – we could build global Brownian motion out of lots of Brownian motion segments. And we could build general random processes from small segments of Brownian motion (suitably scaled). If we built using straight line segments  (suitably scaled) too, we could include Newtonian functions itself.

From there they take the jump to introducing the first Stochastic Differential Equation (SDE) [till this point they are talking the language of ODEs, and it’s not too abrupt]:

$dX_t = \sigma_t dW_t + \mu_t dt$

The “noisiness” $\sigma_t$ is called the volatility, and $\mu_t$ the drift of the process $X_t$. For processes such as above to be well-defined, it’s required that the variables of $X_t, \sigma_t, \mu_t$ depend only on the history/filtration $F_t$, but not the future. In that case, the variables are called $F_t - adapted$.

Written by Vineet

October 1, 2010 at 6:34 pm

## BR: Brownian Motion – II

It’s obvious that a simple BM cannot be used as a model for evolution of asset returns. But, as mentiond earlier, it can serve as a useful ‘basis’ to construct more sophisticated stochastic processes. BR first make the reader consider the following scaled BM:

$S_t = \sigma W_t + \mu t$

Let’s call this model the Normal BM. They show/argue that while this scaling (by $\mu$) and shifting (by $\mu t$) allows one to ‘map’ real data better, the simulated sample path have a distrubing property, that the asset values can go negative under this ‘model’. In fact, for any time $T > 0$, there is a positive probability that $S_T < 0$. This is a problem – as asset prices cannot go negative.

The next transformation they consider is:

$S_t = exp(\sigma W_t + \mu t)$

Being an exponential transformation, this model obviates negative asset prices – and turns out is also more realistic in the paths it generates. It happens to be a well known stochastic process and is called the Geometric BM (GBM). Prima facie, GBM does look like a reasonable choice.

It’s good that just a simple exponential transformation of the BM allows us to characterize the asset price process – at least for now. As BR say before moving on to developing the calculus of BMs (stochastic/Ito calculus), “Brownian motion can prove an effective building block”.

Until we have further evidence to the contrary, we will stick to GBM as our model of choice (in general, this is always a good way of building models – start with a simple one that can characterize the most basic features of the data well, and then if it fails (say, we have more information at some point) consider more complex ones. It’s good to remember that models are at best idealizations/abstractions – and (can) never tell the whole story)

Written by Vineet

September 30, 2010 at 6:48 pm

## BR: Brownian Motion – I

BR’s third chapter forms the core of the book. Here they take the intuition developed in ‘Discrete Processes’ to a more formal continuous setting. The chapter can be seen as divided into three parts:

• Brownian Motion
• Ito Calculus
• Change of Measure

In this post we briefly take a look at BR’s treatment of the construction of Brownian Motion (BM).

Looking at a real time series of asset returns, BR argue that BM, prima facie, offers a good way of characterizing it. Not only is BM “…sophisticated enough to produce interesting models, but is simple enough to be tractable” mathematically. Even though a real financial time series is ‘noisier’ than the paths generated by BM, it offers a ‘basis’ to build more sophisticated continuous processes with.

Like in most other introductory textbooks, BR also develop BM from the simple binomial setting. Their treatment is quite concise, and not the most formal (i.e. starting with definitions etc.), but is clear enough to be understandable.

The Binomial Random Walk

A binomial random walk $W_n(t)$ is defined as:

$W_n(\frac{i}{n}) = W_{n}(\frac{i - 1}{n}) + \displaystyle \frac{X_i}{\sqrt{n}}$

where $X_i$ are a sequence of independent binomial random variables taking values $+1$ or $-1$ with equal probability – i.e. after each time step the change in the value of the random walk is either $\frac{1}{\sqrt{n}}$ or $-\frac{1}{\sqrt{n}}$.

Given that the jump sizes are scaled by the inverse of $\sqrt{n}$, BR show that this “…seems to force some kind of convergence” – in that the binomial random walk $W_n(t)$ doesn’t “blow up”. To see this, write:

\begin{aligned} W_n(\frac{nt}{n}) &= W_{n}(\frac{nt - 1}{n}) + \displaystyle \frac{X_{nt}}{\sqrt{n}} \\&= W_{n}(\frac{nt - 2}{n}) + \displaystyle \frac{X_{nt-1}}{\sqrt{n}} + \displaystyle \frac{X_{nt}}{\sqrt{n}}\end{aligned}

Recursively expanding $W_n(\frac{nt - i}{n})$ and using $W_n(0) = 0$ gives:

$W_n(t) = \sqrt{t} \Big(\frac{\displaystyle\sum_{i = 1}^{nt} X_i}{\displaystyle\sqrt{nt}}\Big)$

where we have multiplied both the numerator and denominator by $\sqrt{t}$.

From the central limit theorem, for $n$ large, the term in parentheses above tends to a Gaussian $N(0, 1)$ distribution, implying that the distribution of $W_n(t)$ tends to a Gaussian $N(0, t)$ distribution.

The moral of the story is that at the end of any time $nt$, the distribution of the sum $W_n(t)$ tends to Gaussian – i.e. at the end of any time the distribution of possible values of the random walk is Gaussian. This completes the characterization of the BM in the binomial setting. Let $n \rightarrow \infty$ and we have constructed our BM/Wiener Process in continuous time.

They go and define the continuous BM formally. It’s standard stuff and there are no surprises here. But I like their concise description of the construction of BM – it assumes certain comfort level with probability theory, but is quite succint and to-the-point.

Next: From BM as a ‘basis’ to a stochastic process for asset returns.

Written by Vineet

September 30, 2010 at 3:29 pm

## BR: The Binomial Representation Theorem – III

BRP tells us that a previsible $\phi_k$ exists. Now the trick to use BRP to get a price for our claim $X$ is to do the following:

Let’s start at time $0$. Construct a portfolio $\Pi_0$:

\displaystyle \begin{aligned} \Pi_0 &= \phi_1 S_0 + \psi_1 B_0\\&=B_0(\phi_1 S_0 B^{-1}_0 + \psi_1)\\&=B_0(\phi_1 Z_0 + Y_0 - \phi_1 Z_0)\\&=B_0Y_0\\ &= Y_0 \\&= E^Q[B^{-1}_T | X]\end{aligned}

$; \psi_1 = Y_0 - \phi_1 B^{-1}_0 S_0 = Y_0 - \phi Z_0$

The value of this portfolio at time $1$, then, becomes:

$\Pi_1 = \phi_1 S_1 + \psi_1 B_1$

Substituting $\psi_1$ as above and rearranging gives:

\displaystyle \begin{aligned} \Pi_1 &= \phi_1 S_1 + B_1 Y_0 - B_1 \phi Z_0\\&=B_1(Y_o + \phi_1 Z_1 - \phi_1 Z_0)\\&=B_1(Y_0 + \phi_1 \Delta Z_1) \end{aligned}

Then, here we can exploit BRP to write $Y_1 = Y_0 + \phi_1 \Delta Z_1$, and this gives us:

$\Pi_1 = B_1(Y_0 + \phi_1 \Delta Z_1) = B_1Y_1$

Next, our strategy requires that we construct a porfolio $\Pi_1$:

\displaystyle \begin{aligned} \Pi_1 &= \phi_2 S_1 + \psi_2 B_1\\&=B_1(\phi_1 S_1 B^{-1}_0 + \psi_2)\\&=B_1(\phi_2 Z_1 + Y_1 - \phi_2 Z_1)\\&=B_1Y_1\end{aligned}

That is, it costs exactly the same as the value of the portfolio we ended up with at time $1$ when we created $\Pi_0$ at time $0$.

This can be carried on recursively to show that our original portfolio $\Pi_0$ is self-financing. At ‘maturity’ $T$, we end up with $B_T Y_T = B_T E^Q[B^{-1}_T X_T | F_T] = B_T B^{-1}_T X_T = X_T$, i.e. our original claim. That is, the price of claim $X_T$ should be exactly equal to the cost of the portfolio created originally (no money has come in or left the system) $E^Q[B^{-1}_T X] = Y_0 = \phi_1 S_0 + \psi_1 B_0$.

The moral of the story is that within a binomial two-asset setting there exists a self-financing strategy $(\phi_i, \psi_i)$ that duplicates the claim that we want to price. That is, the claim can be priced no matter what happens to the path of $S$ / filtration $F_i$ (ironically, then, the claim $X$ in the binomial world is redundant – as it can be reconstructed from the already existing $S$ and $B$).

I think it’s a good way to introduce the martingale representation theorem – but I personally would have preferred that it be done using the language of delta-hedging / risk-neutralily / no-arbitrage. In a way, I think it hampers BR in formally introducing the first fundamental theorem of asset pricing (they do point towards it “…as an afterthought”). Let’s conclude this post with the formal statement then:

First Fundamental Theorem of Asset Pricing

No-arbitrage implies that there exists a probability measure $Q$ under which the discounted price process $B^{-1}_i S_i$ is a martingale, and vice-versa. The measure $Q$, then, is called an equivalent martingale measure.

Written by Vineet

September 28, 2010 at 7:24 pm

## BR: The Binomial Representation Theorem – II

Having laid down the building blocks, now we are ready to define the Binomial Representation Theorem (BRP).

The Binomial Representation Theorem

Given a binomial price process $S$ which is a $Q$ martingale, if there exist another process $N$ which is also a $Q$ martingale, then there exists a previsible process $\phi$ such that:

$N_i = N_0 + \displaystyle\sum\limits_{k = 1}^n \phi_k \Delta S_k$

The basic idea is that if there are two martingale processes under the same measure, then we can find the value of one given the other in a ‘previsible’ (read deterministic) way. As long as we know the possible states of the world in both measures,  we just need to “match the widths” (how much the two processes vary – variance if you will) and “match the offsets” (how the mean of the two processes differ). Stated simply, it’s just like change of cordinates in geometry. In geometry we require that the change of co-ordinates be in the same X-Y plane, and here we require that both processes be martingale under the same measure $Q$.

Next step is to see what to exploit this useful change of co-ordinate trick in finance.

The first thing to note is that we have the claim $X$, which is a random variable and not a martingale. However, we know from the Tower Property of expectations discussed earlier that the conditional process of a claim is always a martingale under any arbitrary measure. This immediately tells us that we can, in principle, given the BRP, we should be able to find a previsible process $\phi_k$ that allows us to go from $S$ to $E[X|F_i]$

But before we do it systematically, we need to set up a little bit more machinery:

9. Bond process: Don’t think it needs any explanation. Process for bond price is a previsible process. We call $B_i$ (with $B_0 = 1$) the bond process

10. Discount process: Since $B_i$ is previsible, so is its inverse. We call $B^{-1}_i$ as the discount process

11. Discounted stock process: Given the discount process $B^{-1}_i$, $Z_i = B^{-1}_i S_i$ is a process that can be observed on the same binomial tree as $S$. We call $Z_i$ the discounted stock process

12. Discounted Claim: The process $B^{-1}_T X$ is called a discounted claim

Given that the process $Y_i = E^Q[B^{-1}_T X | F_i]$ is a martingale (from the Tower property implication), and so is $Z$, the BRP implies there exist a $\phi$:

$Y_i = Y_0 + \displaystyle\sum\limits_{k = 1}^n \phi_k \Delta Z_k$

What we now need to do is to show that  the previsible process $\phi$ is something that has a physical counterpart in the the market BRP allows us price the claim $X$ – which was our objective. That we leave for the next post.

Written by Vineet

September 27, 2010 at 3:32 pm

## BR: The Binomial Representation Theorem – I

The second half of the second chapter of BR’s book uses the binomial tree model discussed so far to introduce some of the basic probabilistic concepts in the theory of mathematical finance (in particular, the ones they need to build the theory in continuous time)

1. Process: The set of of possible values the underlying can take. The random variable $S_i$, then, denotes the value of the process at time $i$.

2. Measure: How probabilities evolve over the tree

3. Filtration: The history of the asset price $S_i$ up until the time $\displaystyle i$ on the tree: $F_i$. It corresponds to the history of $S_i$ on a particular node as of time $i$. The binomial structure ensures that there is only history corresponding to any node. Given a node and a point in time filtration fixes the history “so far”. It is a useful construct to talk about where we are at a point in time and how we reached there.

4. Claim: The derivative in question being priced, which gives a payoff at “maturity”: $X$ – it’s value is a function of the filtration $F_T$.

5. Conditional Expectation of a Claim: Given a claim $X$, we can talk about it’s expectation given the history “so far”, i.e. a filtration $F_i$ – or in other words, the expectation $E^Q[X|F_i]$. Note that this makes this conditional expectation also a random variable, as it depends on the filtration – i.e. where we are on the tree.

For each node/point in time, given the probability measure $Q$, $E^Q[X|F_i]$ denotes the expectation of the claim $X$ if we have observed a filtration $F_i$. It is clear that the unconditional expectation is the same as $\displaystyle E^Q[X|F_0]$

6. Previsible process: It’s a process $\phi_i$ whose value at any time is dependent only the history upto one time earlier, i.e. $F_{i - 1}$ (e.g. $\phi_i = S_{i - 1}$)

7. Martingale: A process $S$ is a martingale with respect to a measure $Q$ and a filtration $F$ if $\forall j\leq i$:

$E^Q[S_i | F_j] = S_j$

That is, the process $S$, if it is a martingale does not move up or down systematically – i.e. it has no drift.

8. Tower Property: $\forall k\leq j\leq i$

$E^Q[E^Q[S_i | F_j]|F_k] = E^Q[S_i|F_k]$

It’s a useful mathematical property of expectations not only in mathematical finance but also in time series analysis. Another way to interpret the Tower property is to say that the conditional expectation process of a claim is always a martingale (obviously, this is just a consequence of the mathematical property of conditional expectations).

Please note that previsibility is closely related to the notion of predictability. It essentially says that a process whose value is “known” at any time given the “history so far” is previsible. Technically, a process $X_t$is previsible, if given a filtration $F_t$ over a probability space $(\Omega, P, F)$, $X_t$ is adapted to $F_t$ or, alternatively, is $F_t$ – measurable. On the other hand, the martingale property is a probabilistic concept relating best forecast to conditional expectation of the random variable (in finance with a clear physical interpretation of “no drift”).

Written by Vineet

August 27, 2010 at 4:26 pm

## BR: The Binomial Model

BR’s second chapter is devoted to introducing the main probabilistic concepts in mathematical finance using the binomial model (they call it the “branch model”).

They start off with the standard one-period “branch model”, and then take forward the discussion over a binomial tree. This is mostly standard stuff. The examples and the general description follows on familiar lines.

They conclude their discussion about the “branch model” by pointing out that arbitrage ensures that there is only one “fair” price of a derivative. Also, that one can think of the price as a discounted value of “local expectation” with probabilities measured differently.

There is very little one can do new when discussing binomial model – so not much to comment here. Still a few things stand out in their approach:

1. First thing that strikes as a little odd is that they go through the entire sub-section on the binomial model without ever talking about hedging or even defining arbitrage. One could argue that the warning in chapter 1 is sufficient enough, but they obviously expect more than a bit from their readers.
2. They again get themselves stuck in the “muddy waters” they created for themselves by repeatedly having to warn the readers that “expectation pricing” doesn’t work. And then they have to clarify that while “expectation pricing” doesn’t work, one could still think of pricing a derivative in the binomial world as an expectation but with different probabilities – totally unnecessary, but inevitable given the way they motivated no-arbitrage pricing.
3. At this stage, I think, they also miss out on utilizing the opportunity of introducing the idea of a risk-neutral world/measure. In fact, they have no discussion on risk at all till this point in the book (and am not sure they will get such an opportunity later)
4. It’s a good sign, however, that they clearly mention that the “construction portfolios $(\phi_i, \psi_i)$  are also random” – a fact which is often not made explicit in introductory mathematical finance texts.

As an aside – their choice of words is a little odd (non-standard use of “expectation pricing”, “construction portfolio” instead of hedging portfolio etc). Not sure a guy who studied finance at a business school would enjoy it too much. But given that the authors are ex-mathematicians and heading a quant-desk may have something to do with it. In fact, their terminology may even be preferred by quants – who themselves have a physics/math background – working in banks.

Their next section is devoted to “binomial representation theorem”. From the name it seems it’s going to be a discussion of the martingale representation theorem in the binomial setting. Not sure I have come across this elsewhere before, and I am quite looking forward to reading it.