Lecture 11: A Brief Introduction to Continuous Time Option Pricing
- Readings:
- Ingersoll Chapters 14 – 17
- Cochrane Chapter 17
- Shimko – Finance in Continuous Time: A Primer (from which these notes are largely drawn)
We will spend some time here building up the tools we need to develop the Black-Scholes Partial Differential Equation. This will be done in a relatively informal way and you should consult other texts if you wish to pursue these issues in more depth.
First we need to introduce an “Ito Process.” I’ll build this idea up slowly so bear with me if you are already familiar with the concept.
Definition: A stochastic process, defined by
B(0) = 0 (more generally B(0) = B0 a fixed starting point) and
B(t + 1) = B(t) + (t + 1) t {0, 1, 2, …}
where the innovations in B are independent standard normal random variables:
(t + 1) ~ iid N(0,1) t, is a special version of a random walk – special in that it has normally distributed increments.
This is a simple example of a discrete time stochastic process where we see a new realization of the process B(t) at each point in time, i.e. at each time t.
The realization at any time t of the process can be arbitrarily high or low. At each time t the innovations in the process B are unpredictable (and normally distributed). In other words, as with all random walks, the expected value of a future realization of the process as of date t is simply B(t). The expected change in the process is always zero and the variance of the change depends on how far into the future you are trying to forecast. Over one period the variance is 1. Over five periods (you know B(11) and are forecasting B(16)) the variance is 5 (the expectation is still: E11(B(16)) = B(11)).
Now suppose we “observe” the process more frequently than at each fixed time interval. Let = 1/n for some arbitrary integer n > 1. We want to describe a process with the same characteristics as the random walk described above but observed more frequently:
B(t + ) = B(t) + (t + ), with B(0) = 0 andB = B(t + ) – B(t) =(t + ) ~ iid N(0,)
Over n periods of length Δ this new process has the same expected change(or “drift”– in this example there is none) and the same variance as the original has over one fixed time “interval” or period.
Finally let dt, a very small increment of time (so n is very large). Define “small” heuristically by letting dt be the smallest positive real number such that dt = 0 whenever > 1.
Then:
B(0) = 0
B(t + dt) = B(t) + (t + dt), t [0, T]
where (t + dt) ~ iid N(0, dt)
Define dB(t) = B(t + dt) – B(t) = (t + dt), as the increments in the process B(t).
dB(t) may be thought of as a normally distributed random variable with mean 0 and variance dt. It is often referred to as white noise. The process B(t) is a standard Wiener process.
Some properties of dB(t) that follow by construction are:
(1) E[dB(t)] = 0 the expected change in B(t) is zero
E[dB(t)] = E[B(t + dt) – B(t)] = E[(t + dt)] = 0 since (t + dt) ~ N(0, dt)
(2) E[dB(t) dt] = 0
E[dB(t) dt] = E[dB(t)]dt = 0 since dt is a constant.
(3) E[(dB(t) dt)2] = 0
E[(dB(t) dt)2] = dt2 E[dB(t)2] = dt2 Var(dB(t)) = dt2(dt) = 0 since dt = 0 for all > 1.
(4) Var[dB(t) dt] = 0
Var[dB(t) dt] = dt2 Var[dB(t)] = dt2 dt = 0 as above.
Note: (2) and (3) or (4) imply dB(t)dt = 0 since its expectation and variance are both zero.
(5) E[dB(t)2] = dt
E[dB(t)2] = Var[dB(t)] = dt since dB(t) = ((t + dt) and (t + dt) ~ N(0, dt) and since
E[(t + dt)] = 0 implies that E[(t + dt)2] = Var[(t + dt)]
(6) Var[dB(t)2] = 0
Var[dB(t)2] = E[dB(t)4] – (E[dB(t)2])2 = E[(t + dt)4] – dt2 = 3dt2 – dt2 = 0
Follows since if (t + dt) ~ N(0, 2), then E[(t + dt)4] = 34.
Similar to the note above, (5) and (6) imply that dB(t)2 = dt, a constant.
These properties are important in that they demonstrate that the variance of a function of a random variable can vanish. When this is true, the expectation sign is redundant: E[f(dB(t))] = f(dB(t)) if Var[f(dB(t))] = 0.
These properties lead to the following “multiplication rules” that will come in handy:
dt2 = 0 – this essentially says dt is small
dB(t) dt = 0 and dB(t)2 = dt – these just eliminate the redundant expectations operator since the variances of these functions of the random variable dB( ) are zero.
Standard Brownian Motion or Standard Wiener Process
Definition: A standard Brownian motion, denoted B(t), is a stochastic process defined by:
B(0) = B0(= 0) a.s. (with probability 1)
has increments Bs – Bt ~ N(0, s – t) s,t with s > t
are independently distributed for any
0 < t0 < t1 < t2 < …< tn-1 < tn T. So the increments are independent normal
random variables. (Or simply, dB(t) - a standard Wiener process - is the differential representation.) B is continuous in each sample path. “Continuous means you can draw the sample path without lifting your pen from the paper.” This is true because while dB(t) is a random variable it is of infinitesimal magnitude (no jumps).
Alternative representation for B(t): Integral form:
Notes:
(a) B is nowhere differentiable. The intuition for this is that for any point in a sample path, the change to the right and to the left are independent random variables.
(b) Et[Bs] = Et[Bt + (Bs – Bt)] = Bt + E[Bs – Bt] = Bt – The forecast of Bs made at time t is always Bt.
(c) Vart[Bs] = Vart[Bt + Bs – Bt] = s – t - since Bt is known at time t. This tells us about the volatility of the realization around the guess made in note (b).
(d) Vart[Bs] as s
Despite all our time developing the idea, the standard Brownian motion is not a good model for stock price movements. We want a process that allows for a drift in prices, i.e. a generalization of a standard Brownian motion for which the expected change, over any future interval of the process is non-zero (we would like to have an expected change in price, an expected return, given the observed behavior of prices).
Ito Process
Consider two processes
(1) a standard Brownian motion – volatility, but zero expected change (no drift)
and
(2) a process that is a constant change for each increment of time f(t) = t for some constant α – a drift but no volatility.
Now add the two together – this is a simple version of an Ito process.
Definition: A stochastic process X defined by:
X(0) = X0
dXt = (Xt, t)dt + (Xt, t)dBt where dBt is the instantaneous increment of a
standard Brownian motion
is an Ito process. Note for notational simplicity I am writing Xt and Bt rather than X(t) and B(t).
: [0, T] is the drift of the process
: [0, T] is the diffusion
In the simple example given above(Xt, t) = and (Xt, t) = 1 Xt and t. In general it need not be this simple (with and being constants) but it is always true (since both Xt and t are known at each time t) that the values(Xt, t) and (Xt, t),which determinethe drift and diffusion of the process over the next instant in time, are known at time t.
Some examples:
(1) if (Xt, t) = 0 and (Xt, t) = 1 then we have a standard Brownian motion
(2) if (Xt, t) = (a constant) and (Xt, t) = (a constant) then gives the drift or expected change for each increment of time. can enhance or diminish (depending on whether > 1 or < 1) the changes in the Brownian motion.
This is an “arithmetic Brownian motion,” it allows for negative realizations and the expected growth is linear (constant absolute growth), not the limited liability and the exponential expected growththat stock prices exhibit. For the arithmetic Brownian motion:
P(0) = P0
P(1) = P0 + ε(1)
P(2) = P1 + ε(2) = P0 + ε(1) + ε(2)
Linear growth: example α = 1
Stock prices on the other hand exhibit exponential expected growth:
P(0) = P0
P(1) = P0(1 + r)
P(2) = P1(1 + r) = P0(1 + r)2
Notes on the arithmetic Brownian motion
(a) X can be positive or negative and arbitrarily large in either direction
(b) Xs – Xt ~ N((s – t), 2(s – t))
(c) Vart(Xs) as s
(3) Geometric Brownian motion
so that
a process with a constant expected return over time and a constant variance of return.
This is a simplified but more natural model of stock prices:
Notes:
(a) if X starts positive it remains positive.
(b) X has an absorbing barrier at 0.
(c) the conditional distribution of Xs given Xt is lognormal. Ln(Xs) is normally distributed and the conditional mean of Ln(Xs) for s > t is Ln(Xt) + (s – t) – ½ 2(s – t) and the conditional standard deviation of Ln(Xs) is . The conditional expected value of Xs is , to find expected future price inflate current price by the continuously compounded expected rate of return.
(d) The variance of a forecast of Xs tends to as s tends to .
(4) Mean Reverting Process
where k, , and are all greater than zero.
If = 1 this is called an Ornstein – Uhlenbeck process.
For Xt the drift is negative and for Xt the drift is positive. This type of process is used a lot in the modeling of interest rates.
Notes:
(a) X is always positive if it starts positive
(b) as X approaches 0 the drift is positive and the volatility is zero
(c) as s the variance of a forecast of Xs is finite
(d) if = ½ (as pictured above) the distribution of Xs given Xt (s > t) is non central 2 with mean
and variance
We now turn to Ito’s lemma:
If we have Xt an Ito process and Yt = f(Xt) what does dYt look like? In other words, we know dXt has a drift and a diffusion – what are they for dYt?
Clearly this is an important question for derivative pricing if we think of Xt as the price of the underlying asset then for the right f( ), Yt = f(Xt) is the price of the derivative.
Ito’s Lemma (Univariate Case)
Let X be an Ito Process defined by
where the dependence of and on (Xt, t) is suppressed for notational convenience.
Let f: [0, T] . Then Yt = f(Xt, t) is an Ito Process defined by
Typically it is assumed that the function f(.) is twice continuously differentiable in both Xt and t, however, we only require thatfx, fxx, and ft exist and are continuous.
Intuition: Take a 2nd order Taylor series expansion of f(Xt + dt, t + dt) around (Xt, t) then dYt = f(Xt +dt, t + dt) – f(Xt, t).
Now the famous line, “it can be shown that” the residual R 0 as dt 0.
Consider the different terms and use the multiplication rules. We know what dXt, dt, and dt2 are but what are (dXt)2 and dXtdt?
Then collecting terms from the Taylor’s series expansion
Note that the residual includes only higher order terms and the multiplication rules therefore imply that it indeed vanishes. This finishes a heuristic proof of the lemma.
Note that in ordinary calculus dX is small enough so that dX2 vanishes. In stochastic calculus dX is a random variable so dX2 does not vanish (instead it converges to ) but terms of higher order, dX3 or dXdt do vanish.
Examples:
(1) Consider an Ito process Yt = Bt2 for t 0, where Bt is a standard Brownian motion. Find dYt. We first identify Xt then f() and finally compute fx, fxx, and ft in order to use the lemma.
then
and we arrive at
So Yt is an Ito process with a drift () of 1 and a diffusion () of 2Bt.
(2)
(3) an arithmetic Brownian motion, can also be written
(since is a constant). Consider a process defined by with S0 > 0.
St = f(Xt, t) where the function f:[0, T] is defined by f(Xt, t) = S0exp(Xt).
and
and we see that S is a geometric Brownian motion.
Xt is normally distributed so Ln(St) = Ln(S0)+Xt is normal with mean Ln(S0) + tand variance 2t. The log of S has normal increments so S has lognormal increments.
Solving a simple problem:
Start with a deterministic example. Suppose that a security with current value V guarantees a flow of cash payments at the rate $1dt every instant of time forever. This is a continuous-time equivalent of a risk-free perpetuity paying $1 each discrete period. If theinstantaneous risk-free rate is a constant r, what is the current value of the security?
First write the law of motion for V using Ito’s lemma. Here it is simple, because V does not depend on any stochastic variable: V = V(t). Thus dV = Vtdt.
Calculate the expected capital gain from owning the security: E[dV] = Vtdt.
Calculate the expected cash flow to the security: $1dt.
Thus the total expected return on the security is: Vtdt + 1dt = [Vt + 1]dt.
Now, set the total expected return equal to the risk free dollar return: rVdt = [Vt + 1]dt.
Divide both sides by dt: rV = Vt + 1
and we get a differential equation whose value V must satisfy. Then we just have to solve the equation for V some how. Here just guess that V cannot depend on t since it is the value of a constant, risk-free perpetuity. Thus Vt = 0 and the solution is V = 1/r. This is equivalent to the present value of a risk-free perpetuity in discrete time but here r is the instantaneous risk free rate.
Now, suppose X follows a geometric Brownian motion: . A security with value V collects Xdt continuously forever. V represents the current value of a perpetuity whose continuous cash flowXdt grows at the average exponential rate . Suppose further that the risk of the cash flow variation is considered diversifiable risk. The economy is risk neutral and the instantaneous risk free rate is a constant, r. What is the value of this security?
V = V(Xt, t) – however, just as above, since it’s perpetual (and so looks the same today as it does in 2 years) V can’t actually depend on t only on X, so V = V(Xt). What is V(X)?
Using Ito’s lemma:
The expected capital gain on this security is E[dV]
The expected cash flow is Xdt by the construction of the security.
The total expected return on the security is the sum of these components:
Since the risk of the security is diversifiable, to avoid arbitrage it must be that this total return equals rVdt (a riskless return on the security’s value). Setting them equal we obtain a differential equation for V:
To solve the differential equation, guess that doubling the current level of X (current cash flow and so the expected future cash flow) will double V. Then V = X for some constant . This implies that Vx = , and Vxx = 0. Substituting these into the differential equation we get:
and so
This is the equation for the present value of a perpetuity growing at a constant rate of when the riskless rate is r.
The Multi-Variate Case:
Now introduce a second Ito process Y to the system so we have:
Define E[dB1dB2] = dt as the correlation between B1 and B2. We can also show that E[(dB1dB2)2] = 0, which says the variance of dB1dB2 is zero. Therefore dB1dB2 = E(dB1dB2) = dt.
Now suppose that Wt = f(Xt, Yt, t). The multivariate extension of Ito’s lemma has the following Ito differential:
terms of higher order than dX2 or dY2 vanish as before.
Now we have to expand our list of multiplication rules:
Then
Making these substitutions into the Ito differential we have:
Again if B1 and B2 were deterministic, terms of higher order than dB1 and dB2 would vanish. Since B1 and B2 are stochastic variables terms of higher order than (dB1)2 and (dB2)2 vanish.
Application:
Let X and Y be two real valued Ito processes with:
Show that Z = XY is an Ito process, that is show that it can be written:
First note so .
Using Ito’s lemma
And the result follows by simple substitution.
Stochastic Integrals and Trading Gains
Trading Strategies:
A trading strategy is a stochastic process that specifies at each state and each date the holdings of each of the available securities.
Formally: t : N is a measurable function with respect to Ft (the information set, a subset of , that is available at time t). then is : [0, T] N , the specification of t for each t.
Trading Gains for Brownian Prices:
Let B be a Brownian motion (representing the price processes for the assets) and let be a trading strategy.
The gains of trading strategy are given by the stochastic integral of with respect to B:
where B may be an N dimensional Brownian motion
This integral is stochastic since B is stochastic.
Let’s consider some simple examples:
(1) Suppose that where is a constant for all t, then:
(2) Consider a piecewise constant strategy on [0, T] so for some 0 = t0 < t1 < … < tk = T t = (tm-1) t [tm-1, tm) m = 1, …, k
Then is called a “simple” trading strategy and
(3) The stochastic integral is also defined for trading strategies that are not simple – the idea is to find a simple trading strategy m that approaches the strategy in the sense that
The stochastic integral is then defined by
Informally we can think of this as
Trading Gains for Ito Prices
Let S be an Ito process (security price process) defined by:
Then the gains of trading strategy are given by the stochastic integral of with respect to S:
an integral with one nonstochastic and one stochastic term.
The Black-Scholes Model
Review:
Binomial option pricing formula – single period
2 states of nature
3 securities
1 – risky stock
2 – riskless bond
3 – call option on the stock
the idea is that you replicate the call option using the stock and the bond
a simple portfolio of the 2 securities spans the 2 states of nature
Binomial option pricing formula – multi (T) period
2T states of nature
3 securities
1 – risky stock
2 – riskless bond
3 – call option on the stock
the idea is that you replicate the option with the stock and the bond at each date
2 long-lived securities and a dynamic trading strategy span the 2T states of nature
Black-Scholes Model (continuous time)
Infinite number of states of nature
3 securities
1 – risky stock
2 – riskless bond
3 – call option on the stock
the idea is that you replicate the option with the stock and the bond using a continuous trading strategy
2 long-lived securities and a continuous trading strategy span an infinite number of states of nature
The Available Securities
Stock – the price of the risky stock is assumed to follow a geometric Brownian motion
Bond – the price of the riskless bond follows the process
where r is the (assumed constant) instantaneous riskless rate of return.
We can derive this price process as follows. By definition, for any t, . Differentiate this with respect to t:
(constant exponential growth)then
Option – a European call option that matures at time T and has an exercise price equal to k. The price process of the European call option is denoted C, where .
A derivation of the Black-Scholes PDE
1st – a trading strategy is self financing if it generates no dividends (positive or negative) for any time t in 0 < t < T. Let at be the holdings of the stock at time t and bt be the holdings of the riskless bond. Then the trading strategy (a, b) is self financing if
for any time t in 0 < t < T.
Assume that C(St, t), the function relating the price of the call option to the stock price (or the call price process), is twice continuously differentiable. Then, using Ito’s lemma
Claim
Suppose that there exists a self financing strategy (a, b) with
(so the time T payoffs on the strategy and the call are the same)
then in the absence of arbitrage opportunities the current (t = 0) price of the European call option on the stock is given by .
Proof
Assume for contradiction that
.
Then the trading strategy (a, b, -1) (long the strategy and short the call option) has a t = 0 payoff (cost) of and the time T payoff of this position is zero hence it is an arbitrage opportunity.
If we instead assume that
.
Then the trading strategy (-a, -b, 1) (short the strategy and long the call option) is an arbitrage. So the replicating portfolio’s (trading strategy’s) initial cost must be the initial option price if the strategy is self financing.
The same argument applied at each date tin the interval [0, T] implies that
It follows that