Lecture Note 4.2: Multiple Sources of Risk

Introduction:

This note generalizes the lessons we learned in developing models with

stochastic volatilty. The techniques we used there can be used to value

derivatives with any number of underlying sources of randomness.

Once we have articulated the problems, we also have to learn how to

solve them. We will talk about some issues of implementation. Most

crucially, we will discuss how the idea of risk-neutral pricing extends

to the multidimensional case. This will give us an extremely powerful

tool, that we can begin to use right away.

We will consider a range of examples. We will also discuss direct

numerical solutions of PDEs.

Outline:

I. Multidimensional topics

II. Implementability issues

III. Probabilistic solutions to general models

IV. Applications

V. Is the Market Complete?

VI. Beyond Monte Carlo

VII. Summary

I. Some new classes of problems.

� Previously we put a lot of effort into figuring out how to cope

with a second source of risk in derivatives pricing. Now let’s

step back and recognize that how useful that was.

There are many other situations in which we will want to

analyze multiple sources of risk.

� Here are a few.

(A) Other random parameters.

� The motivation for our investigation of stochastic volatility

was to address the sensitivity of the B-S formula to the con-

stant volatility assumption – within a consistent framework.

� We could now proceed the same way to examine the other

parameter assumptions, like constant interest rates.

� As before, our method will be to specify new stochastic pro-

cesses for the thing we previously assumed was constant:

dS = µ S dt + σ S dW S

dr = m(·) dt + b(·) dW r.

(We’re keeping σ constant for simplicity.)

� Now, we have our option, C, and it varies with both S and

r. So just as we did before, we use the two-dimensional Itô

rule to write down how it changes. Then we get rid of the

two types of risk, just like we did with stochastic volatility.

Then ….

� Hey, the whole argument is the exact same!

� So we already know what the PDE will turn out to be:

1

2b2Crr + ρbσSCrS +

1

2σ2S2CSS + rSCS − rC + Ct + [m − λb]Cr = 0.

� This is the exact same equation we got last week but with

r substituted for σ in the partial derivatives.

· And now ρ is the correlation between dW S and dW r;

· and λ = λr is the market price of interest-rate risk.

� The point is, when we derived the stochastic volatility PDE,

our argument never really made reference to what σ was.

� So now we know how to handle any type of random

parameter.

� Recall how the whole thing worked:

· The key was there had to be a second security out therethat we could use to (continuously) hedge the new risk.

· And we have to know the new risk’s market price, λ.

(B) Derivatives on Non-traded Underlyings

� We have been discussing the effect of random parameters in

valuing derivatives on some asset S.

� But our methods point us to a further generalization:

· As we saw last time, we can price derivatives whose payoffdepends on the random parameter.

· The no-arbitrage arguments still works even if there is notraded underlying security – no S – at all.

� Interest rate derivatives are one example.

· If r is the ONLY random thing affecting the payoffs ofsome claim, then our PDE is even simpler:

1

2b2Crr − rC + Ct + [m − λb]Cr = 0.

Note: all the S terms are gone.

� Our approach can value derivatives on any quantity.

� The underlying can be anything. It doesn’t even have to

have any connection to finance. It could be:

· A weather-related quantity.

· A catastrophe-loss index.

· An economic statistic.

· Whatever you can imagine…

(C) Higher Dimensional Models

� Another direction in which we can exploit the work we have

done is to add still more types of risk.

� We have shown how it works for N = 2 variables, but noth-

ing stops us from building general N−dimensional modelswith N = 3, 4, . . .

� For example, we could handle:

· Currency options with stochastic volatility and domesticand foreign interest rates (N = 4).

· Multifactor term-structure models, e.g. bond prices thatdepend on both r and inflation.

· Options on portfolios with N correlated assets.

� Solving the PDE takes more computer power. But it can be

done.

� One goup of researchers has solved a pricing problem in 360

dimensions!

· The product was a collaterallized mortgage obligation (CMO).

· This is a security that pools together cash-flows frommany underlying loans and re-sells them in packages.

· They wanted to let the re-payment rates on each of theunderlying mortgages be its own random process.

� We can summarize what happens to the no-arbitrage PDE

when we add a new factor, X, as follows:

1. Add a second derivative term 12X2CXX times the squared

volatility of X.

2. For each other factor, Y , add a cross-derivative termXY CXY times the covariance between X and Y .

3. Add a first derivative term CX, whose coefficient is

(i) X times the risk-free rate minus the payout rate if Xis a traded factor;

(ii) the drift of X minus the product of the market priceof X risk with the diffusion coefficient of X, if X is anon-traded factor;

� The derivation of these rules just follows the lines of the

argument we went through for the stochastic volatility case.

· Recall that “X is a traded factor” means that there is asecurity that we can buy and sell whose price is X. Any

other risk factor is “non-traded”.

� It is starting to look like we can handle almost unlimited

complexity.

II. Implementability Issues

� Before getting carried away, let’s just keep track of the prac-

ticalities of employing an N-dimensional model on, perhaps,

arbitrary quantities.

1. You have to write down a stochastic model for each randomquantity that you believe.

2. You have to have estimates of the unknown parameters ofthat model that you trust.

3. There has to be at least one other security in the market thatyou can buy and sell to hedge the risk of each new quantity.

4. The risk of non-hedgeable jumps in the new quantity mustbe negligible.

5. You have to have some way of finding the market price ofrisk for each non-traded quantity.

� Let’s think about each of these for a moment.

� Issues 1 and 2 are really about econometrics.

· For the models to be improvements over simpler models, wehave to be confident that:

* Our specification of the model is a better description of

the true underlying process that we are trying to describe

– as it will evolve in the future; and

* We can estimate (and forecast) the new parameters re-

quired with a high degree of confidence; and

* Our uncertainty about them has smaller effects on pricing

and hedging than a simpler model with fewer parameters.

· This will not always be the case. In fact there are dangersfrom two directions.

i. We can’t estimate the new model’s parameters with anyaccuracy at all.

ii. We estimate them “too well” and end up overfitting.

· When we have N sources of risk, and N models for them,we could have around N2/2 new parameters to estimate.

* For each new factor we include, we have to specify its

correlation with each old one. Then we have to estimate

means, variances, and covariances.

· The task of model validation is an important part of riskmanagement in any financial engineering exercise.

* Firms that use proprietary models typically impose formal

procedures for assessing both estimation risk and

specification risk.

� Issues 3 and 4 on my list above concern the realism of the

continuous trading/hedging assumption.

· Economists call this the market completeness condition.

· It is unrealistic if

* There is not a way to create a pure hedge for each source

of risk.

– For example, an individual homeowner’s mortgate repay-ment decision.

* The hedge cannot be undertaken from both sides: long

and short.

* Market liquidity is not sufficient to allow frequent re-hedging

(without affecting prices).

We’ll return to market completeness later in the lecture.

� This brings us to issue 5. Recall our earlier discussion of λs.

· Once our model involves them, we are no longer in the (nice)position of being able to ignore economics for valuing deriva-

tives.

· We need to know something about people’s risk/reward pref-erences for each type of risk.

· There is no reason why people’s preferences can’t changeover time!

* Our PDE allows us to incorporate changing market prices

of risk, by making them functions of other state variables.

* Sometimes economic theories can provide guidance.

* Example: What does the CAPM assert about the marketprice of risk for any factor X?

– CAPM: πX = βX,M πM.

– Definition: βX,M = ρX,MσX/σM

– Hence: λX = ρX,M λM

· I’m not saying this is true! But it is an alternative to com-puting historical or implied lambdas.

� To summarize the implementability issues, multidimensional

models are exciting, but not always feasible in practice.

� Hence, more sophisticated models require more sophisticated

scrutiny.

� More complicated isn’t necessarily better.

III. Probabalistic Solutions to General Models

� I now want to show you one solution technique for the general

multi-dimensional PDE we have developed.

� The idea is to extend to the general case the correspondence

between solutions and discounted expectations using suitably

adjusted probabilities.

· Often, it is possible to find these expectations directly.

· Even when it’s not, the intuition is important.

� How do we go about “risk-neutralizing” these more complex

models?

� To review, when we considered the Black Scholes PDE, we

learned that one way to find the solution was to average all

possible terminal payoffs (discounted at the riskless rate), after

having adjusted the drift on the underlying process.

� Specifically, we set the expected percentage change to the risk-

free rate (or that rate minus the payout rate).

· We interpreted that adjustment as switching to an alterna-tive model in which the stock price was determined by risk-

neutrality. Hence we called the pricing approach “taking the

discounted risk-neutral expected payoff.”

� But recall the subtle underpinnings of the logic that allowed us

to reach this conclusion.

� Then, the justification for this result was that the equation

we had to solve made no reference to people’s preferences for

risk, so that risk neutral people (who value all cashflows as

their expected present value) must reach the same solution as

anybody else. So we could pretend we were risk-neutral.

� Now, the more general equation we have to solve does involve

people’s preferences. That’s what that λ was.

· So it’s not clear whether we should expect to get away witha similar trick.

� Well, luckily for us, there is a theorem in mathematics that tells

us precisely what to do.

� I’ll describe it in terms of the stochastic volatility model we

have been using. Recall the two processes we specified were

dS = µS dt + σS dW S

dσ = κ(σ0 − σ) dt + s0σ dW σ.

and the equation we want to solve is:

1

2s20σ

2Cσσ+ρs0σ2SCσS+

1

2σ2S2CSS+rSCS−rC+Ct+[κ(σ0−σ)−λs0σ]Cσ = 0.

� The result we need is a version of what is called the

Feynman-Kač Theorem.

� This theorem tells us how to “risk-neutralize” any risk factor to

get an adjusted model that we can simulate in order to compute

prices.

� The message of FK is:

The value of any security that satisfies the PDE is the ex-

pectation of its payoffs (discounted at the riskless rate),

after each of the underlying processes has had its drift

changed to whatever multiplies the first partial deriva-tive term with respect to it in the PDE.

� In other words, here

(A) Set the drift of the stock to r · S, as before(which is the same as setting its expected return to r).

(B) Set the drift of σ to m() − λb().

(C) Don’t change anything else. Most notably, this says that thevolatilities and correlations don’t get changed.

� Then all you have to do is simulate to figure out the average

payoff.

� The theorem tells us, once we have shown that our derivative

satisfies the PDE, we can conclude that that adjustedaverage payoff is the solution.

� Before looking at examples, I want to call your attention to

one subtle point about the recipe. If the interest rate itself is

one of the random variables, then there is a difference between

the expected payoff, discounted and the expected discounted

payoff. The theorem tells us we are supposed to use the latter:

· The discount factor goes inside the expectation.

� To put it in terms of equations, let X stand for the vector of

all our state variables.

Then, if the derivative has terminal payoff C(XT), and E⋆ de-

notes averaging over future outcomes with the adjusted drifts,

then the result says its price at t is

E⋆[DFt,T · C(XT)]

where

DFt,T ≡ e−∫ Tt ru du

is the integrated discount factor.

� Putting it inside the expectation takes into account the possi-

bility that the evolution of X may be correlated with that of

r.

� If there are cash-flows Γs(Xs) at times t < s < T , then the

formula is

E⋆[

∫ Tt

DFt,s · Γs ds + DFt,T · C(XT)]

· Recall that these cash-flows (such as interest payments) justtack on the additional term Γ to the PDE.

· If we were simulating the process over intervals ∆t then thediscount factor would be the same as(

e−rt0∆t × e−rt1∆t × e−rt2∆t × . . . e−rtN ∆t)

which is approximately equal to(1

(1 + rt0∆t)×

1

(1 + rt1∆t)×

1

(1 + rt2∆t)× . . .

1

(1 + rtN ∆t)

)i.e the product of all the little one-period discount factors

that will apply between now and T .

· Or, in words, when interest rates are random, thediscount factors between t and T is the product ofall the future one-period discount factors multipliedtogether – after adjusting the drift of r.

· Of course, if we are using a model with constant r, that isjust e−r(T−t).

· Also notice that, if rates are random – but they are indepen-dent of the state variables X – then we can observe that

E⋆[DFt,T · C(XT)] = E⋆[DFt,T] · E⋆[C(XT)]= Bt,T · E⋆[C(XT).]

So we don’t have to model the bond prices if they don’t

interact with the variables that determine our cashflows.

� Let’s continue with the stochastic volatility example, so that

you get the idea of how to use the Feynman-Kač result.

� Our general formula (for a call) requires us to compute

Bt,T · E⋆[max(ST − K, 0)]But it is not clear how to compute that expectation mathemat-

ically.

� So just simulate! Here’s how it goes.

(A) Adjust the stock and volatility processes according to therule.

dSt = rSt dt + σtSt dWS

dσt = [κ(σ0 − σt) − λs0σt] dt + s0σt dW σ.

(B) Approximate the continuous time model by a discrete timeversion, i.e. replace dt with ∆t and dW σ and dW S with

two normal random variables with mean zero, variance ∆t

and correlation ρ.

(C) Draw one realization of the random sequences.

(D) Compute the terminal payoff, max(ST − K, 0), for this oneoutcome.

(E) Multiply by the discount factor.

(F) Average the result over a whole lot of draws.

� This is the Monte Carlo method of evaluating expectations.

IV. Applications.

� Sometimes we don’t need to simulate. For some stochastic

processes we can compute the adjusted expectations directly.

(A) Forwards.

� Go back to the simple, one-variable model. One fact you

need to know is that, if we have any process described by

the modeldX

X= c0 dt + c1 dW

where c0 and c1 are constant, then, if we are at time t, for

any horizon T , the expected value of XT is

E[XT] = Xtec0(T−t) or

1

(T − t)log

E[XT]

Xt= c0.

· Example: A stock pays no dividends and has a constant9% expected return and constant volatility. Today it is at

100. What is its expected price in 10 years?

* The model here is

dS

S= 0.09 dt + σ dW

* So the formula says

E[ST] = 100 · e0.09·10 = 246

� We can use this recipe for expectations to immediately derive

some derivatives prices.

� Suppose we want to price an outstanding forward contract

with forward price F0 on a non-dividend paying stock that

follows that same type of model.

· Assume r is constant.

· Then using our recipe, we first change the expected returnto r, and then compute the expected discounted payoff

VF = E⋆[e−r(T−t)(ST − F0)]

= e−r(T−t)(E⋆[ST] − F0)where the E⋆ notation is to remind us that this isn’t our

true expectation.

· Now use the result we just learned to evaluate E⋆:

VF = e−r(T−t)(Ste

r(T−t) − F0) = Bt,T(Ft,T − F0)

which is a formula we saw in week one.

· Also, if we solve for the forward rate that makes VF zero,it’s given by F0 = Ste

r(T−t), which we also knew.

� Old stuff there. But what about a forward on, say, the con-

sumer price index? We couldn’t do that in week one. But

now we at least have a framework for approaching it.

Let πt denote the price level index.

· First, we are going to suppose π follows

dπ = ιπ dt + σπ dW.

That says, essentially that ι is the expected rate of in-

crease, which is the same as the inflation rate. And now

we are supposing ι and σ are constant.

· How do we risk-neutralize this thing? What’s our recipe?

· Since the index itself isn’t traded, our generic formula sayjust change the drift to m − λb, or, here,

πι − λπσ

· So this makes the adjusted price level process

dπ

π= (ι − λσ) dt + σ dW

· And this is still just a constant-coefficient lognormal pro-cess assuming λ is a constant.

· Suppose it is. What is the forward worth?

· Do the same steps as for the stock, replacing S by π:

V (πt, t) = E⋆[e−r(T−t)(πT − F0)]

= e−r(T−t)(E⋆[πT] − F0)= e−r(T−t)(πte

(ι−λσ)(T−t) − F0)which is a new result.

· Setting this to zero to solve for the current forward price

F = πte(ι−λσ)(T−t).

� Notice some interesting things about this formula:

1. If we know the true process (ι and σ), the forwardprice would give us λ. This is an instances of theobservation we made earlier in the term that we may not

need to estimate market prices of risk if we take the prices

of other derivatives as given.

2. The forward price is the expected future level of πif and only if λ = 0. The formula says the F = E⋆[πT],and, in this example, E[πT] = πte

ι(T−t). The two expres-

sions are the same if people really are neutral with respect

to π risk.

This conclusion turns out to hold much more generally,

i.e., whether or not the underlying process is a geometric

Brownian motion

(B) Treasury Bonds.

� Consider pricing a zero-coupon riskless bond in a one state-

variable model.

� The simplest model of interest rates people use is the

Vasicek model

dr = κ(r̄ − r) dt + b dW

� This is an Ornstein-Uhlenbeck model. It has several conve-

nient features, including the fact that, given today’s rate rt,

the distribution of the rate at every future date is normal:

rT ∼ N(rt e−κ(T−t) + r̄(1 − e−κ(T−t)),b2

2κ(1 − e−2κ(T−t)) ).

� The risk-neutralized version of the Vasicek process is

dr = [κ(r̄ − r) − λb] dt + b dW

= κ(r⋆ − r) dt + b dWwhere r⋆ ≡ r̄−λb/κ is the new “long run mean” that resultsfrom our risk-neutralization.

� The payoff function for our bond is always one. So, according

to our formula, its price must be

Bt,T(rt) = E⋆[DFt,T · 1]

where DFt,T is the riskless discount factor to time T .

� Now that rates are random, this discount factor will be dif-

ferent along each path that r takes.

· So we need to use

DFt,T = e−∫ Tt ru du

� So our pricing law just tells us that zero coupon bond prices

are the risk-neutralized expectation of this discount factor.

� It happens that this expectation is available in closed-form

for this particular model. The solution is

Bt,T(rt) = eG0(t,T)−rt G1(t,T)

where

G1(t, T) =1

κ(1 − e−κ(T−t))

G0(t, T) =1

κ2(G1(t, T) − T + t)(r⋆κ2 − b2/2) −

(bG1(t, T))2

4κ

which describes a complete term-structure model in terms

of the spot rate and four parameters.

� Do you see how you could now use this to try to extract λ

from the government bond yield curve?

� For more complicated payoffs (like interest-rate options) there

aren’t closed-form formulas.

� But even without formulas, it is still really simple to just

program a routine to simulate future paths of r under the

adjusted model, compute the discount factor along each fu-

ture path, and take the average payoff.

· In Vasicek, all we do is change the long-run spot ratebefore simulating.

The risk adjustment is easy. So pricing bonds is easy!

� If you want to keep the r process from going negative, you

could use the Cox, Ingersoll, Ross (CIR) specification:

dr = κ⋆(r⋆ − r) dt + s√r dW

which is another version of the CEV model.

� Assuming that this is the risk-neutralized version, the zero-

coupon bond prices again have closed form

Bt,T(rt) = eH0(t,T) −rt H1(t,T)

where

H1(t, T) =2 (exp(h(T − t)) − 1)

2h + (κ⋆ + h)(exp(h(T − t)) − 1)

eH0(t,T) =

[2h exp(1

2(κ⋆ + h)(T − t))

2h + (κ⋆ + h)(exp(h(T − t)) − 1)

]Xwith X = 2κ⋆r⋆/s2 and h =

√(κ⋆)2 + 2s2.

(C) Options on Strategies

� Consider an option on a portfolio of stocks whose weights

change over time.

� Suppose you are a fund manager and you want to insure

a portfolio whose holdings (shares of each stock) is held

fixed. Then your percentage allocations vary with their per-

formances.

� For example, suppose there are just two stocks A and B

with prices SA and SB, and you hold NA shares of A and

NB shares of B.

� Then, even if they are uncorrelated, the volatility of your

portfolio is √N2AS

2Aσ

2A + N

2BS

2Bσ

2B

NASA + NBSB

which is a complicated, time varying function of SA, and SB.

� But we know how to price options on it! Just set the ex-

pected returns of both stocks to r (no prices of risk neces-

sary) and start simulating.

� You can extend this idea to price derivatives on arbitrary

time-varying strategies, as long as you can specify the port-

folio weights as functions of the state variables.

(D) Structured Products

� Structured notes are bonds with exotic features determining

the “interest” payment and/or the principal repayment.

� As you may know, these products are often invented to be

sold to unsophisticated investors.

� So having a good model is really important in order not to

get fooled into over-valuing them.

� Some examples:

Equity appreciation notes. These notes determine theirquarterly coupon based upon the appreciation on some

basket of stocks over the quarter. The payment might

look like

Γti = min(10%, max(0, (Sti/Sti−1 − 1)))which also features a cap (ten percent) and a floor (zero).

Each coupon thus has the features of an option (or op-

tion spread) whose strike price is determined in the future.

Options like this are called cliquets.

Rainbow notes. These pay off in one of several differentcurrencies, depending on which has appreciated most.

F(T) = maxn=1:N

(S(n)T /S

(n)0 ).

One recent issue specified that it would pay off in the

second most appreciated currency, but only if at least one

other currency had depreciated!

Range accrual notes. The coupon payment on this prod-uct ceases or gets reduced for every day that some under-

lying index goes out of a pre-defined range.

A currently very popular version pays a large fixed coupon

unless the 10-year swap rate goes below the 2-year swap

rate. So investors are short a series of binary options on

the yield curve slope, with strike price zero.

� Even with the complex path-dependency in these products,

simulation methods easily give their payoffs.

� As you simulate each path forward, you just compute each

contingent payoff Γti at the coupon dates ti, multiply them

by the discount factor to that date DFti and add this amount

to the running total for that path.

V. The Boundaries of Market Completeness

� Earlier, I called your attention to the facts that (a) our tech-

niques can be applied to derivatives on arbitrary underly state

variables, but (b) the techniques only should be applied when

the state variables can be (perfectly) hedged.

� Sometimes it is not immediately obvious if the second condition

is or is not met.

� Let’s look at some examples.

(A) Economic statistics.

� We did an inflation example already, and I’ve told you that

there is an active market for inflation swaps in the major

economies. So inflation risk can be hedged.

� What about GDP or unemployment? If you worked for a

bank and you sold an option on one of these variables, what

would you say when your risk manager asks you how you

intend to hedge it?

� In practice, you could look for a spanning portfolio oftraded assets whose returns are highly correlated with the

statistic. I don’t know how well this would work.

(B) Weather/climate risks.

� It’s conceivable that you could construct spanning portfolios

for broad-based weather variables, like the global average

temperature.

� For local variables (temperature or rainfall, say, in a specific

location) it is not generally plausible.

� The same would be true of most catastrophe-related pay-

outs, but there is an interesting twist. There are sometimes

companies – local insurance underwriters perhaps – who have

enough exposure to the events, then you can short them.

(C) Company results.

� Some banks will sell companies options on their own oper-

ating performance. Revenue puts, for example, give thecompany protection against shortfalls in sales.

� Obviously this is not a hedgeable risk. But sometimes the

banks convince the companies to instead tie the payouts –

not to revenue itself – but to a model of what their revenue

should be, given some market related variables.

� This is called parametric insurance. For a power com-pany, the model could be based on the price of inputs (e.g.,

natural gas) and outputs (electricity). The whole point of

the product is that now the risk may be hedged if there are

forwards on those quantities.

� Other company-specific variables are definitely not spanned

risks. This year saw a one company do a swap where its

payments were tied to improvements in its ESG score!

(D) Path-dependent functionals.

� Some exotic options have payoffs tied to a traded-asset, but

determined by the whole history of prices. Are these statistics

spanned?

� For example, a lookback option could give you the rightto buy at the low. So if the option is written at date t, the

payoff at T is max[ST − LT, 0] where Lt′ = mint≤u≤t′ Su.Note that Lt′ is not itself the price of a traded asset.

� The trick here is that we can also view the payout as a

function of Vt′ = St′ − Lt′, the current distance from thelow. If S is a Brownian motion, V is a reflected Brownian

motion.

� This variable is perfectly correlated with S. So yes V -risk is

spanned.

� Lookback options are really easy to price via Monte Carlo

because your simulations are building the price paths for you.

All you have to do is to update the path statistic at each step.

VI. Beyond Monte Carlo

� Before we conclude that any derivative problem can be tackled

by the probabalistic approach wechave learned today, we do

have to acknowledge some practical drawbacks of Monte Carlo.

� Roughly, these fall into two categories: computational difficul-

ties, and inherent limitations.

� The computational difficulties arise because the averages you

calculate are, by definition, approximations. That means there

is always approximation error. That means you have to (a) keep

track of the error; and (b) make it small.

� In fact there are two sources of approximation error:

· The error that comes from only being able to approximatean expectation with an average of a finite number of draws;

and

· The error that comes from approximating a continuous-timeprocess by a discrete-time one.

� In principle, we can deal with both issues with enough computer

power. Just scale up the number of paths, and scale down the

time interval ∆t.

� And, in fact, the second type of error declines quickly as you

shrink the time interval. It is essentially zero when you use

∆t = one day.

� But, an unfortunate mathematical law tells us that the first

type of error goes down at a slow rate. Specifically, the error

in the average for a payoff function g(ST) with M simulations

of S is approximately √Var[g(ST)]

M.

The numerator is the variance of the payoffs experienced over

the M draws. It doesn’t change (much) with M. The problem

is the denominator only gets large very slowly.

� For instance, if you start with 1000 paths and discover that

your error is 10%, and your boss tells you to get it down to 1%,

you have to draw

old error

new error=

.10

0.01= 10 =

√new M

old M

which means new M = 100× old M or 100,000 draws!

� As you know from your other courses, there are various numer-

ical tricks for making things better.

� Even with state-of-the-art technology, though, Monte Carlo

techniques may still be too slow for:

(A) Valuing lots of derivatives in real-time; and

(B) Valuing securities whose payoff variance is high, such as AAAcoporate bonds or deep out-of-the-money options.

� In addition, the nature of derivatives problems also imposes

some inherent limitations on the use of Monte Carlo.

� There are two things you need to be aware of:

1. It’s very hard to value American options with MonteCarlo. This seems strange, given all the other exotics we cando. But Americans are different, because the early exercise

floor imposes a boundary condition that depends on the value

of the derivative at the time. If you think about a grid,

there we valued by going backwards in time. So at each

date we knew what the option’s value was. But now we

are trying to value by simulating forward in time. And, at

intermediate time-steps, we don’t have any way of knowing

what the derivative will turn out to be worth.

2. Monte Carlo only delivers the price of the derivative.For most applications, we need to know more than the price.

We need all the hedge ratios to tell us how to replicate it,

or take away our risk. Monte Carlo tells you what the thing

is worth starting from the world right now. It can’t tell you

what it would be worth if things changed a little bit (like a

small move in the underlying). To get those, we just have to

change the starting conditions a little bit and run the whole

thing again. If we want to know hedge ratios with respect

to 5 quantities (perhaps volatility, interest rates, time, and

two underlyings), we have to repeat the process 5 times.

� So sometimes we have to revert to other PDE solution

techniques.

� Let me briefly sketch one alternative approach:

finite difference methods.

� Suppose we have an asset, C, whose payoff we know at time-T ,

and which obeys:

−Ct =1

2s2σ2Cσσ +

1

2σ2S2CSS + rSCS − rC + κ̃(σ̃ − σ)Cσ.

� If we approximate dt as ∆t, then we can view Ct =∂C∂t

as

[C(t)−C(t−∆t)]/∆t and view the PDE as specifying a recipefor moving backwards in time.

· Conceptually, if we know the solution at T we could evaluateall the partial derivatives using that solution.

· Then we could just iterate back:

C(t−∆t) = C(t)+∆t (all the other terms evaluated at T) .

· Then, just like in the binomial model, we’d repeat the processagain for time t − 2∆t, and continue back to today.

� Finite difference methods are ways of making this practical by

choosing a discrete grid of [S, σ] points and then, at any point

in the grid, say (Si, σj), approximating, e.g.,

∂2C

∂S2≈

C(Si+1, σj) − 2C(Si, σj) + C(Si−1, σj)(∆S)2

and∂C

∂σ≈

C(Si, σj+1) − C(Si, σj−1)2∆σ

.

And so on.

� While this is all fine conceptually (as long as the grid spacing

is small enough), it turns out the simple procedure I outlined

above is not very good numerically.

· It doesn’t keep the solutions smooth enough.

· And little errors can cause it to blow up.

� To ensure that the solutions are stable, one usually has to eval-

uate some of the approximate partial derivatives using the grid

values at the earlier time step.

· These are called implicit methods, because you are solvingfor those earlier time step values at the same time that you

are using them implicitly to approximate the derivatives.

� To illustrate, suppose we are at time tn−1 and we are solving

for the values along one column of our grid with σ = σj fixed.

· Then we can write a linear system of equations that solvesfor C for all the Si values at once, as follows.

· First evaluate ∂C∂σ

and ∂2C∂σ2

using the simple explicit replace-

ments using the C function at tn. Call the sum of these

terms W 0j (a column vector).

· Then bring all the other terms over to the other side:

Ci,j−∆t(1

2σ2jS

2i

Ci+1,j − 2Ci,j + Ci−1,j(∆S)2

+ rSiCi+1,j − Ci−1,j

2∆S− rCi,j

)= Ci,j(t) + ∆t W

0i,j.

where all the Cs on the left are the values at tn−1 that we

have to solve for.

· This is a linear equation involving Ci1,j, Ci,j, and Ci−1,j.

* And we have one such equation for each i.

* So if we stack all Ni equations, we have a complete system

Wj = Kj Cj that looks like:



…

…

Wi,j……

 =



…

0 k+i+1,j k0i+1,j k

−i+1,j 0 · · ·

· · · 0 k+i,j k0i,j k

−i,j 0 · · ·

· · · 0 k+i−1,j k0i−1,j k

−i−1,j 0…

·…

Ci+1,jCi,jCi−1,j…



where Wi,j = Ci,j(t) + ∆t W0i,j and

k+i,j = −σ2jS

2i

∆t2∆S2

− rSi ∆t2∆S andk−i,j = −σ

2jS

2i

∆t2∆S2

+ rSi∆t2∆S

and

k0i,j = 1 + r∆t + σ2jS

2i

∆t∆S2

.

* In fact we don’t even have to invert the matrix Kj here

because it is a tri-diagonal form that allows the system to

be solved quickly by back-substitution. (Matlab does

this automatically via their backslash matrix division op-

erator.)

· Using this approach we can solve the linear system for eachvalue j of our grid and thus fill in the time tn−1 grid.

· I skipped a step here however: we actually can’t build thesystem as I described at the very top and very bottom points

of our i-vector.

* The i = 1 and i = Ni lines will involve C values that are

off the grid.

* So the PDE only actually yields Ni − 2 equations.

* We have to supply two more equations to close the system.

* If we know the value exactly on one of the boundaries,

that provides an additional equation.

* Otherwise, another possibility is to make the grid large

enough that we feel confident that the curvature is going

to zero near the boundary. (Of course we need to test this

assumption.)

– Then we would have an additional equation that said,for example,

[1, −2, 1] CNi,jCNi−1,jCNi−2,j

 = 0.

� Finally, the routine I just described probably struck you as

odd becuase it treated the σ derivatives differently from the

S derivatives.

� In fact, the alternating direction implicit algorithm (ADI) is a

2-step procedure:

1. To go from tn to tn−1, do the steps above.

2. Then, to go from tn−1 to tn−2, switch i and j. That is, eval-uate the S partials explicitly, and solve the implicit system

along line of fixed S, evaluating the σ partials implicitly.

� Finite difference methods restore many of the benefits of bino-

mial trees, because, by working backwards in time, it is easy to

evaluate early-exercise type decisions.

VII. Summary

� We learned that, in principle, we can handle derivative problems

involving any number of sources of risk. We discussed several

types of applications:

� In practice, these models are tricky to fit and require strong

assumptions. Use the simplest model you can get awaywith.

� The method of pricing derivatives by taking “risk-neutral” dis-

counted expectations applies to all problems which can be de-

scribed in terms of our fundamental PDE

� The general transformation is to change the expected returns

of all traded processes to r (or r − y), and to change the driftsof all non-traded processes to m − λb. Also:· Volatilities and correlations don’t get changed.

· The discount factor goes inside the expectation.

� When we can’t figure out the exact expectation, we can often

approximate it accurately by Monte Carlo simulation.

� When Monte Carlo isn’t practical, we may need finite differ-

ences or other approaches.

Lecture Note 4.2: Summary of Notation

Symbol Page Meaning

dr p2 change in instantaneous risk-free rate over dt

m(), b() p2 drift and diffusion functions of dr

ρ, λ p3 correlation between r and S, and market price of r risk

X p6 an arbitrary risk factor obeying a diffusion process

CX, etc p6 sensitivity (∂C/∂X) of security C to X

πX p10 risk premium for asset exposed to X risk

πM p10 stock market risk premium

ρX,M p10 correlation of stock returns with changes in X

DFt,T p14 discount factor to apply to random payouts at time T

when future interest rates are unknown

E⋆[ ] p45 expectation using risk-neutralized processes

C(), Γ() p14 payoff and cash-flow as a function of state-variables X

c0, c1 p17 expected growth and volatility of generic process dX

VF p18 value of an old forward contract at forward price F0when the current spot price is St at t

π, ι p18 consumer price index and inflation rate

σ, λ p19 volatility of price index and market price of π risk

κ, r0 p21 mean-reversion rate and long-run level of riskless rate r

b, λ p21 diffusion coefficient of riskless rate and market price of r risk

r⋆ p21 risk-neutralized long-run level of riskless rate∫ Ttru du p22 sum of future instantaneous rates between t and T

G0, G1 p22 coefficient functions in Vasicek bond solution

H0, H1 p23 coefficient functions in CIR bond solution

Var[g(ST)] p31 variance of realized payoffs g(ST) in a sample of M

random paths of S

Wi,j p35 explicit terms in implicit finite-difference scheme