Financial Derivatives and
Partial Differential Equations∗
Robert Almgren
July, 2001
1. ASSETS AND DERIVATIVES. Assets of all sorts are traded in
financial markets: stocks and stock indices, foreign currencies, loan contracts
with various interest rates, energy in many forms, agricultural products,
precious metals, etc. The prices of these assets fluctuate, sometimes wildly.
As an example, Figure 1 shows the price of IBM stock within a single day.
The picture would look more or less the same across a month, a year, or a
decade, though the axis scales would be different.
If you could anticipate the price fluctutations to any significant extent,
then you could clearly make a great amount of money very quickly. The
fact that many people are trying to do exactly that makes the fluctuations
essentially unpredictable for practical purposes. A fundamental principle
of finance, the efficient market hypothesis [9] asserts that all information
available to anyone anywhere is instantly expressed in the current price,
as market participants race to be the first to profit from new information.
Thus successive price changes may be considered to be uncorrelated random
variables, since they depend on as-yet unrevealed information. This principle
is the subject of intensive analytical testing and some controversy [7], but is
an excellent approximation for our purposes.
Although the directions of the price motions are completely unpredictable,
statistics can tell us a lot about their expected size. Figure 2 shows the distribution of percentage changes in IBM stock price across half hour time
intervals. We can identify a typical size of the fluctuations, about half of
one percent in this example. Since the fluctuations are uncorrelated and
have mean near zero, this typical size is the single most important statistical quantity that we can extract from the price history. We may additionally
ask about the form of this distribution, for example, whether or not it is a
Gaussian. Again, this is the subject of active research [10].
∗
To appear in American Mathematical Monthly
1
Robert Almgren/July 2001
Financial Derivatives and PDEs
2
95
94.5
94
93.5
93
10
12
14
16
Figure 1: Price of one share of IBM stock, on November 16, 1999; the xaxis is time of day. These are prices at which trades actually occured; this
picture contains 5400 data points. The fastest oscillations, on scales of a few
seconds, represent “bounce” between bid and ask prices. But complicated
structure is clear on all higher scales, and continues across decades.
Number of events
300
250
200
150
100
50
0
−0.02
−0.01
0
0.01
Fractional price change
0.02
Figure 2: Fractional price change in IBM stock price across half-hour time
intervals, for 1999 (about 3000 data values). Although the direction of the
changes is unpredictable, we can still identify a characteristic size of the
changes, about half a percent in this example.
Robert Almgren/July 2001
Financial Derivatives and PDEs
3
A derivative is an asset whose value depends on the price of some other
asset, the underlying. Derivatives permit investors to customize their exposure to the market: they can speculate, hoping to win a large amount with
a small initial investment, or hedge, investing in the new asset in order to
offset risks they already have. Derivatives provide a rich means for market participants to exchange risk and thereby achieve their preferred profile
of exposure to marketplace fluctations. Many books explain the basics of
derivatives [5] and the mathematical models for valuing them [13].
The simplest derivative is a forward or futures contract: a committment
to buy a particular asset on a specified future expiration date T , for a specified strike price K. If at expiration the asset is trading in the marketplace
for price S, then the holder of the contract makes a profit or loss of S − K.
Futures are widely used to take positions in assets that are not themselves
easily traded, such as agricultural commodities. For example, a farmer may
sell her crop as she plants it, thus guaranteeing her price before the objects
being sold even exist. By selling the futures contract, the farmer acquires
short exposure to the crop price, which hedges the long exposure she already
has by her ownership of the crop. The speculator at the Chicago Mercantile
Exchange who buys the contract hopes to make money if crop prices rise.
An option gives the contract holder the decision whether or not to execute the final transaction. A call option gives the holder the right to purchase
the asset (from the counterparty) on date T for price K; a put option gives
her the right to sell the asset on date T for price K. For a call option, the
payoff is thus Λ(S) = max{S − K, 0}, since the option holder will choose
to exercise only if the market price S is greater than K; she can then turn
around and sell the asset, pocketing a profit S − K. Similarly, the payoff
of a put option is Λ(S) = max{K − S, 0}. In both cases, the option holder
has the possibility of profit with no chance of loss; she must therefore pay
something to acquire the contract.
If you believe an asset will rise in price, then you may buy a call option
to capture a very large potential gain with a small investment; however, if
your belief is wrong then you may very easily lose your entire option investment. Options may also be used for hedging. For example, the Canadian
Imperial Bank of Commerce offers an “Index-Linked GIC” that “allows you
to take advantage of increases in the value of the S&P/TSE 60 Index, but
with no risk to your principal if the index declines.” In effect, this is an
investment in the index, combined with a put option to sell at the original value. Mortgages in the United States commonly combine a fixed rate
with a prepayment feature, which permits the borrower to take advantage
of decreases in interest rates, while being protected against increases.
Robert Almgren/July 2001
Financial Derivatives and PDEs
4
Options trade in marketplaces, just like the assets on which they are
based. Figure 3 shows market prices of 117 call and put options on IBM
stock on the same day as in Figure 1. The expiration dates range from a
few weeks to six months into the future (longer-term options also exist), and
the strike prices of the options range over about a factor of two above and
below the current stock price. The prices shown are daily closing values;
throughout the day the options fluctuate just as actively as the stock itself.
The first attempt to explain option prices was by Poincaré’s student
Louis Bachelier in 1900 [1], [3]. He proposed that the “correct” value of an
option was the expected value of its payoffs, and by introducing a specific
probabilistic model for the underlying price motion, he was able to calculate
this expectation and compare his results with market prices. In his formulation, the option holder and the writer both took on risk associated with
fluctuations about this mean value.
The surprising fact is that, under suitable assumptions about the statistics of the asset price motion, the risk can be eliminated by following a
suitable hedging strategy. The value of the option is then uniquely determined. Again, the value is obtained by computing an expectation, which
can be carried out by solving a partial differential equation, though the interpretation is quite different than in Bachelier’s model. This observation in
1973 by Black, Scholes, and Merton [2], [11] led to the development of large
options exchanges—and to the 1997 Nobel Memorial Prize for Merton and
Scholes.
2. EUROPEAN OPTION VALUATION. Let us consider an option,
written on an underlying asset—a stock, say—that trades in the market at
price S(t). Some payoff function Λ(S) has been specified, which determines
the value of the option at expiration t = T . For t < T , the option value V
should depend on the underlying price S and on time t; in the hope (to be
justified later) that these are the only relevant parameters, we denote the
price as V (S, t). So far, we know only that V (S, T ) = Λ(S).
We assume that the option is European, meaning that it can be exercised
only on the future date T ; thus any value it has for t < T comes from
passively waiting to receive this possible payout. An American option can
be exercised at any time up to and including T ; we consider options of this
kind in Section 3.
As time progresses, the value of the option changes, both because the
expiration date approaches and because the stock price changes. Across a
Robert Almgren/July 2001
Financial Derivatives and PDEs
5
60
40
Calls
20
Apr 2000
0
60
80 100
120 140
160
K
Jan 2000
Dec 1999
Nov 1999
T
60
40
Puts
20
0
Apr 2000
Jan 2000
Dec 1999
Nov 1999
T
60
80
140
100 120
K
160
Figure 3: Options on IBM stock, closing prices on Nov. 16, 1999. The x-axis
is the strike price K, at which the option holder may eventually buy or sell
the stock; the dashed line at about 95 shows the closing price of the stock
itself. The y-axis is the expiration date T . The z-axis is the price at which
the option defined by parameters (K, T ) could be either bought or sold at
the Chicago Board of Options Exchange. Call options are more valuable
for lower strike prices, and conversely for puts. The Black-Scholes theory
largely explains the structure of these prices.
Robert Almgren/July 2001
Financial Derivatives and PDEs
6
short time interval δt, we may write
δV = Vt δt + VS δS +
1
2 VSS
δS 2 + · · ·
(1)
in which the neglected terms are of size O( δt2 , δS δt, δS 3 ). It will be made
clear later why we need to include the terms of size δS 2 .
To determine the option price, we construct a replicating portfolio. This
will be a specific investment strategy, involving only the stock and a cash
account, that will yield exactly the same eventual payoff as the option in
all possible future scenarios. Its present value must therefore be the same
as the present value of the option, and if we can determine one we have the
other. We thus define a portfolio Π, consisting of D(S, t) shares of stock and
a cash account with balance C. Both D and C can be positive or negative,
corresponding to long or short positions. The value of this portfolio is
Π(S, t) = D(S, t) S + C(S, t).
The stock holdings D(S, t) are chosen by a specific rule (determined in what
follows) depending on the underlying price and on time. The investor holding this portfolio is also exposed to risk as long as she holds a nonzero
amount of stock. She could eliminate this risk by taking D = 0, but instead
we are going to choose D so that the risks of the portfolio exactly match the
risks of the option. We will then say that Π replicates V .
The change in value of Π across time δt is not determined by a passive
expansion as in (1); rather, we follow the investing actions in detail. During
the time interval δt, the stock holding is held constant at D, and the value of
Π changes only due to fluctuations in the stock price S and due to interest
paid on the cash balance:
δΠ = D δS + r C δt.
(2)
The interest payment rC δt is approximate for small δt, depending on whether
interest is paid continuously or discretely, but the D δS term is exact; there
is no term δS 2 .
At the end of the time interval, before the start of the next interval,
D is changed to respond to the new price S(t + δt). The amount of money
required for or generated by this rebalancing is withdrawn from or deposited
into the cash account. Thus the total value Π does not change in this step,
and (2) represents the entire change across the time δt. We say that the
portfolio Π is self-financing, since no money is put in or taken out.
Thus the difference in value between the two portfolios changes by
¡
¢
¡
¢
¡
¢
δ V − Π = Vt − rC δt + VS − D δS + 21 VSS δS 2 + · · · ,
Robert Almgren/July 2001
Financial Derivatives and PDEs
7
which depends on the unknown change δS. But we can eliminate the firstorder dependence by choosing
D(S, t) = VS (S, t).
That is, if the investor is able to compute the function V (S, t), then she can
compute its derivative with respect to S and artificially implement a trading
strategy that at first order tracks the same risks.
Assuming this strategy has been implemented, we then have
¡
¢
¡
¢
δ V − Π = Vt − rC δt + 12 VSS δS 2 + · · · ,
(3)
where the higher-order terms in δt and δS are small if the time interval and
the corresponding price changes are small. This change is still an uncertain
quantity, since we do not know δS 2 .
However, we argued in the introduction that we know much more about
the size of the changes δS than about their direction. As a consequence,
it may be that δS 2 is effectively deterministic, as long as we average over
enough small steps. Let us consider a time interval ∆t that is small compared
with the overall lifetime of the option, yet large compared with the time
intervals δt at which we are able to trade. Let us take ∆t = N δt, and
denote the small price changes by δSj for j = 1, . . . , N .
Now, for the first time, we write down a specific probabilistic model for
the price changes. We suppose that
√
(4)
δSj = a δt + b δt ξj ,
where a is the expected rate of change and b is an “absolute volatility”
measuring the expected size of the motions. At each step, ξj is a random
variable of mean zero and unit variance, and these variables are independent
across successive steps. In general, a and b may depend on S and t, but for
now they can be considered constant.
When δt is small, the stochastic process S(t) corresponding to the model
(4) becomes a Brownian motion or Wiener process, although it was first introduced by Bachelier for the purpose of option pricing. The Brownian
motion model is extremely popular, not primarily because of statistical evidence, but because it is only with this model that we can determine option
prices exactly.
The accumulated change across the time interval ∆t is
∆S =
N
X
j=1
√
δSj = a ∆t + b ∆t X,
N
1 X
ξj .
X=√
N j=1
(5)
Robert Almgren/July 2001
Financial Derivatives and PDEs
8
By standard probability theory, since the ξj are independent, the random
variable X has mean zero and unit variance; if N is large, then X has a
Gaussian distribution. The resemblance between (4) and (5) is the reason
for choosing the specific time scalings in (4): they are stable across different
time scales. If the ξj have a Gaussian distribution, then the law is precisely
the same on all time scales.
As suggested above, the sum of the squares of price changes is much less
random than the changes themselves. We readily calculate
(δSj )2 = b2 δt ξj2 + 2ab δt3/2 ξj + a2 δt2 ,
and find that
N
N
N
X
1 X 2
1 X
1
3/2
2
2
ξj + 2ab(∆t)
ξj + a2 (∆t)2 2
(δSj ) = b ∆t
3/2
N
N
N
j=1
j=1
j=1
→ b2 ∆t
as N → ∞. The latter limit is a consequence of our assumption that the
ξj are independent with unit variance. Thus our intuition was correct: although the square of change in price is random on any one step, when we
average across a large number of steps, it becomes certain.
Since we have eliminated the first-order dependence on S by taking D =
VS , the total accumulated change from (3) is now
N
X
¡
¢
¡
¢
∆ V −Π =
δ V −Π j
j=1
¡
¢
= Vt − rC ∆t +
=
¡
Vt − rC +
1
2
VSS
1 2
2 b VSS
¢
N
X
(δSj )2
j=1
∆t.
This expression has no randomness anywhere in it. By adjusting the holdings
D(S, t) at every microscopic time interval to follow changes in price, we have
replicated V . In this expression we have neglected the changes in VSS caused
by changes in S from step 1 to N . It is not difficult to see that these are of
higher order in ∆S. In effect, we have just given a heuristic derivation of a
result known as Itô’s Lemma.
Since the difference portfolio V − Π is nonrisky, it must grow in value at
exactly the same rate as as any risk-free bank account:
¡
¢
¡
¢
∆ V − Π = r V − Π ∆t.
(6)
Robert Almgren/July 2001
Financial Derivatives and PDEs
9
The reasoning is a classic application of lack of arbitrage: lack of possibility
to make a profit without risk.
If ∆(V − Π) > r(V − Π)∆t, then anyone could borrow money at rate r
to acquire the portfolio V − Π. “Acquiring the portfolio” means buying the
option in the marketplace for V , shorting D shares of stock, and loaning out
cash C. By holding the option for a time ∆t, following the above hedging
strategy, and selling at price V (t + ∆t), the growth in value of the difference
portfolio is certain to more than cover the interest costs on the loan.
Conversely, if ∆(V − Π) < r(V − Π)∆t, then the reverse strategy yields a
risk-free profit. This strategy consists of selling the option in the marketplace
for V , covering the risk by purchasing D shares of stock, and loaning out
the cash left over at rate r.
Recalling that V − Π = V − DS − C and that D = VS , we obtain the
general version of the Black-Scholes equation:
Vt + 21 b(S, t)2 VSS + rSVS − rV = 0.
(7)
Note that the “drift coefficient” a has disappeared, hence does not affect
the value of the option. The only coefficients that appear are b and r, the
size of the motions and the risk-free interest rate, respectively. This partial
differential equation must be satisfied by the value of any derivative security
depending on the asset S.
The PDE (7) is linear: two options are worth twice as much as one
option, and a portfolio consisting of two different options has value equal to
the sum of the individual options.
The PDE is backwards parabolic. Thus, terminal values V (S, T ) must be
specified at some future time T , from which values V (S, t) can be determined
for t < T . Typically, the value of an option is known at expiration, and this
equation is solved to determine its values for earlier times. In addition,
boundary conditions may arise from features of the option specification.
The option price may also be calculated by probabilistic methods [6]. In
this equivalent formulation, the discounted price process e−rt S(t) is shifted
into the “risk-free measure” using the Girsanov theorem, so that it becomes
a martingale. The option price V (S, t) is then the discounted expected value
of the payoff Λ(S) in this measure, and the PDE (7) is obtained as the backwards evolution equation for the expectation. Our derivation has followed
the original reasoning of Black and Scholes, although the probabilistic view
is more modern and can more easily be extended to general market models.
We have achieved the remarkable conclusion that the market value of
the option V (S, t) is uniquely determined by its boundary conditions and
Robert Almgren/July 2001
Financial Derivatives and PDEs
10
by the parameters of the probabilistic model for the underlying asset price
motion. Let us summarize the assumptions that went into this model:
• The asset S can be bought and sold. This is essential for us to construct
a suitable hedging portfolio. Thus the model cannot be applied to
risk factors that don’t exist in a market. For example, home heating
costs depend on outside temperature. You are at the mercy of this
risk, unless you can find a way to buy and sell temperature (this is
now possible, precisely by using hedging portfolios composed of energy
futures).
• Assets can be bought and sold with no transaction costs. In practice,
trading costs are substantial, including both the bid-ask spread on
small amounts and liquidity limits on large amounts. This constrains
the frequency with which new values of D can be taken and means
that risk cannot be eliminated completely.
• The market parameters r and b are known. The interest rate r is different for different investors (usually the rate taken is the “overnight”
rate for short-term cash deposits between major banks), but does not
have too large an effect on the result. However, the computed value
V is very sensitive to the input value of b, and this value is very difficult to estimate empirically. In practice, option prices quoted on the
exchanges are usually used to determine appropriate “implied” values
for the parameters of the price process.
• The asset price follows a Brownian motion. This model follows from
the assumptions that (a) price is a continuous function of time, (b) its
increments are independent random variables, even when viewed on
arbitrarily short time intervals, and (c) variance is finite and constant.
As a consequence, the price changes ∆S across intermediate time intervals have a Gaussian distribution. Although these assumptions are
very plausible, the results are not consistent with empirical observations either of the asset prices themselves [10] or of option prices.
Constructing improved models for asset price motion and for option
pricing is a subject of active research.
An especially popular choice is the lognormal model b(S, t) = σS, so
δSj = a(S, t) δt + σ S ξj .
That is, the percentage size of the random changes in S, rather than the
absolute sizes, are assumed to be constant as S varies. The parameter σ is
Robert Almgren/July 2001
Financial Derivatives and PDEs
11
√
called the volatility, and σ ∆T is the expected size of changes across a time
interval ∆T . For this model, the Black-Scholes equation is
Vt + 21 σ 2 S 2 VSS + rSVS − rV = 0.
This PDE has nonconstant coefficients, depending on the independent variable S. At the edge S = 0 of the domain of solution, the coefficients have
a strange singularity. However, by changing the independent variable to
x = log S, these difficulties disappear and, with the aid of Green’s functions, it is easy to construct exact solutions. These deliver the famous
Black-Scholes formula for the price of a European call option:
Ã
!
¡
¢
log(S/K) + r + 12 σ 2 (T − t)
√
V (S, t; K, T ) = S N
σ T −t
Ã
!
¡
¢
1 2
log(S/K)
+
r
−
σ
(T
−
t)
2
√
− Ke−r(T −t) N
,
σ T −t
in which N (·) is the cumulative normal distribution
Z x
1
2
e−y /2 dy.
N (x) = √
2π −∞
(Prices for put options can be determined in the same way, or by using the
symmetry principle of “put-call parity.”)
In Figure 4, the Black-Scholes solution is compared with the market
prices of call options for a single expiration date T . The value of σ has
been chosen to give a reasonable fit over the whole range of strike prices
K. A more detailed examination shows that the lognormal model with a
constant value of σ simultaneously overprices “at-the-money” options—that
is, with K near S—and underprices options at the ends—either deep “in
the money” or deep “out of the money.” This is an indication that the
price process has “fat tails”: large changes are more frequent than would
be predicted by extrapolation from the statistics of small changes. In fact,
market practioners quote prices in terms of “implied volatility:” for each
strike, the value of σ that gives the market option price. Nonetheless, this
simple model does a remarkably good job of reproducing actual market
behavior.
3. AMERICAN AND EXOTIC OPTIONS. The foregoing model
may be extended in various ways, giving rise to a variety of interesting
mathematical problems.
Robert Almgren/July 2001
Financial Derivatives and PDEs
12
Call option price
40
30
20
S
10
0
60
80
100
120
Strike price K
140
160
Figure 4: Comparison of Black-Scholes formula with IBM call option prices
as in Figure 3 (Jan. 2000). Parameters are σ = 35% and r = 5% per year.
Many real options have “American” features: at any time t before T ,
the option holder may choose to exercise the option and receive the payout Λ(S, t) at that time. Mathematically, this adds a free boundary to the
problem, corresponding to the optimal exercise strategy, and makes explicit
solution impossible.
In fact, equity options are generally American rather than European,
so our description in Section 2 was slightly over-simplified. However, for a
simple call option, it is never optimal to exercise early (you would incur interest costs by paying the strike price, unless the asset itself pays dividends),
so our analysis was correct in that case. For put options, early exercise is
important.
Many options are complicated custom contracts traded privately “over
the counter” between professional market participants; an immense variety of possible structures exists. Black-Scholes theory is used to determine
model parameters from prices of exchange-traded “vanilla” options, then
solved to quote prices for arbitrary “exotics,” and to determine optimal
hedging strategies for reducing risk after the option contract has been written.
Robert Almgren/July 2001
Financial Derivatives and PDEs
13
American options. Let us again suppose that the option price V (S, t)
is a smooth function, and denote by δV the change in value across a short
time interval, δV = V (t + δt) − V (t). Earlier, we argued that δV was given
by the expression on the right-hand side of (1).
Now, however, the reasoning needs to be modified. We argued previously that ∆(V − Π) = r(V − Π)∆t, for if the equality were violated in
either direction, there would be a strategy available to the option holder
that would guarantee a profit. One side of that strategy depended on shorting the option. But for an American option, shorting it means giving the
counterparty the decision whether to exercise, and it is unlikely that he will
do it in a way that is optimal for the holder of the option. Thus (6) needs
to be modified to
¡
¢
¡
¢
∆ V − Π ≤ r V − Π ∆t,
and the PDE (7) becomes
Vt + 21 b2 VSS + rSVS − rV ≤ 0.
(8)
Of course, a partial differential inequality is not sufficient to determine
V (S, t). The second piece of information needed is that
V (S, t) ≥ Λ(S, t),
(9)
precisely because the option may be exercised at any time: if it ever happened that V < Λ, then a risk-free profit could be made by purchasing the
option for V and immediately exercising it to collect Λ.
One further statement is required to determine V uniquely. At each
moment, the option value arises purely from the possibility either to exercise
immediately or to hold and get the exercise value at a later time. This yields
the third constraint:
At each (S, t), at least one of (8) and (9) is an equality.
(10)
The three pieces (8)–(10) constitute a linear complementarity or obstacle
problem [12]. The (S, t)-plane may be divided into two pieces, an exercise
region in which (9) is an equality and a hold region in which (8) is an equality. Assuming everything is regular, the boundary between these regions is a
curve S∗ (t), the optimal exercise boundary. The option holder should exercise when the price S(t) crosses this level. The motion of the boundary can
be modeled as a free boundary problem, mathematically similar to the Stefan problem of freezing/melting. Solutions may be obtained numerically by
adding obstacle features to a finite-difference code or by solving an integral
equation for the motion of the boundary.
Robert Almgren/July 2001
Financial Derivatives and PDEs
14
Other exotics. Almost any contract between consenting parties that could
possibly be written is probably being traded somewhere right now. Here is
a brief list of possible extensions, any one of which may include American
features as well.
• The payoff function may have a more complicated form. For a digital
option, Λ(S) is a step function: it pays a fixed amount if S > K
at expiration, zero if S < K. By putting together piecewise linear
payoffs, one obtains spreads, straddles, collars, etc. These are valued
in the same way as the vanilla assets, namely, by solving the PDE with
appropriate terminal data.
• A barrier option loses its value if the underlying asset price crosses a
specified level (which may be either above or below the current price)
at any time before expiration. It is valued by adding a homogeneous
Dirichlet boundary condition to the diffusion equation (7).
• Asian options depend in some way on a time average of the price. For
example, one may have the right to acquire the asset for its average
price over a specified time period. These are valued by adding an
additional state variable. The option price V (S, I, t) also depends on
the value of the running average I, and the Black-Scholes equation (7)
becomes a degenerate diffusion equation in two variables [13]. Lookback
options depend on the maximum or minimum asset price in a time
window and are priced by similar means.
• Multi-asset or rainbow options may give, for example, the right to
acquire any one of a set of assets for a given price. They are valued by
solving a diffusion equation in as many variables as there are assets,
with a diffusivity matrix representing correlations among the price
motions as well as their volatilities.
In addition to changing the definition of the option, one may make the
market model more realistic. For example, transaction costs are important
in real life, preventing the hedging strategy of Section 2 from being implemented in continuous time.
Finally, a more realistic statistical model may be used for the price process itself. Two prominent models for incorporating realistic statistics are
stochastic volatility [4] and jump models such as variance gamma [8]. Both
of these incorporate fat tails, hence are capable of correcting the systematic
mispricings mentioned at the end of Section 2.
Robert Almgren/July 2001
Financial Derivatives and PDEs
15
In this note, we have reviewed the basic theory of pricing and hedging
options, and a few of its extensions. Besides the practical importance of the
calculations, the development of such quantitative models represents a real
advance in our ability to think rationally about risk and hedging. Although
the classical framework of Brownian motion and Black-Scholes hedging has
been developed to a high degree of refinement, there still remains a lot to do
in the area of more realistic stochastic models, not to mention the connection
of these ideas with traditionally less quantitative areas such as insurance and
risk management.
Acknowledgement This article is based on an MAA invited lecture at
the Joint Mathematics Meetings in New Orleans, January 2001. The author
would like to thank the MAA both for their invitation and for their support
in completing the article.
Robert Almgren is Associate Professor of Mathematics and Computer
Science at the University of Toronto, where he teaches numerical methods for PDEs in the Mathematical Finance Program. His Ph.D. thesis in
Applied and Computational Mathematics at Princeton University was on
high-frequency asymptotics in reacting gas dynamics. He has since worked
on free boundary problems in materials science and in fluid dynamics, and
on optimal trading strategies in finance. In good weather he soars crosscountry in his Pegasus sailplane.
University of Toronto, Department of Mathematics, 100 St. George St.,
Toronto ON M5S 3G3, Canada.
almgren@math.toronto.edu
Robert Almgren/July 2001
Financial Derivatives and PDEs
16
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