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Lecture 11 Continuous Time Option Pricing

By Mike Weaver,2014-11-26 16:55
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Lecture 11 Continuous Time Option Pricing

Lecture 11: A Brief Introduction to Continuous Time Option Pricing

o Ingersoll Chapters 14 17

o Cochrane Chapter 17

o Shimko Finance in Continuous Time: A Primer (from which these notes

are largely drawn)

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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

a fixed starting point) and B(0) = 0 (more generally B(0) = B0

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: E(B(16)) = B(11)). 11

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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 and B = 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” = 0 whenever heuristically by letting dt be the smallest positive real number such that dt > 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.

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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.

2] = 0 (3) E[(dB(t) dt)22222 E[(dB(t) dt)] = dt E[dB(t)] = dt Var(dB(t)) = dt(dt) = 0 since dt = 0 for all > 1.

(4) Var[dB(t) dt] = 0 22 Var[dB(t) dt] = dt Var[dB(t)] = dt dt = 0 as above.

Note: (2) and (3) or (4) imply dB(t)dt = 0 since its expectation and variance are both zero.

2(5) E[dB(t)] = dt 2 E[dB(t)] = Var[dB(t)] = dt since dB(t) = ?((t + dt) and ?(t + dt) ~ N(0, dt) and since 2 E[?(t + dt)] = 0 implies that E[?(t + dt)] = Var[?(t + dt)]

2(6) Var[dB(t)] = 0 24224222 Var[dB(t)] = E[dB(t)] (E[dB(t)]) = E[?(t + dt)] dt = 3dt dt = 0 244 Follows since if ?(t + dt) ~ N(0, ?), then E[?(t + dt)] = 3?.

2Similar to the note above, (5) and (6) imply that dB(t) = dt, a constant.

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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: 2 = 0 this essentially says dt is small dt2dB(t) dt = 0 and dB(t) = 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) = B (= 0) a.s. (with probability 1) 0

has increments B B ~ N(0, s t) ; s,t with s > t st

BB,BB,BB,...,BB are independently distributed for any t0tttttt01021nn1

0 < t < t < t < …< t < t ? T. So the increments are independent normal 012n-1n

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).

t

Alternative representation for B(t): Integral form: B(t)BdB()0?0

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) E[B] = E[B + (B B)] = B + E[B B] = B The forecast of B made at time t is tsttsttstts

always B. t

(c) Var[B] = Var[B + B B] = s t - since B is known at time t. This tells us about tsttstt

the volatility of the realization around the guess made in note (b).

(d) Var[B] as s ts

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).

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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

dX = (X, t)dt + ?(X, t)dB where dB is the instantaneous increment of a ttttt

standard Brownian motion

is an Ito process. Note for notational simplicity I am writing X and B rather than X(t) tt

and B(t).

: ? ? [0, T] ? is the drift of the process

?: ? ? [0, T] ? is the diffusion

In the simple example given above (X, t) = and ?(X, t) = 1 ; X and t. In general it ttt

need not be this simple (with and ? being constants) but it is always true (since both X t

and t are known at each time t) that the values (X, t) and ?(X, t), which determine the tt

drift and diffusion of the process over the next instant in time, are known at time t.

Some examples:

(1) if (X, t) = 0 and ?(X, t) = 1 then we have a standard Brownian motion tt

(2) if (X, t) = (a constant) and ?(X, t) = ? (a constant) then gives the drift or tt

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 growth that stock prices exhibit. For the arithmetic Brownian

motion:

P(0) = P 0

P(1) = P + ε(1) 0

P(2) = P + ε(2) = P + ε(1) + ε(2) 10

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Linear growth: example α = 1

f(t) = t 101 1% 100 Growth

2 100% 1 Growth

99 100 1 2

Stock prices on the other hand exhibit exponential expected growth:

P(0) = P0

P(1) = P(1 + r) 02 P(2) = P(1 + r) = P(1 + r) 10

Notes on the arithmetic Brownian motion (a) X can be positive or negative and arbitrarily large in either direction 2(b) X X ~ N(：?(s t), ??(s t)) st

(c) Var(X) as s ts

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(3) Geometric Brownian motion

(X,t)Xand?(X,t)?X tttt

so that

dXtdXXdt?XdBordt?dB tttttXt

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 X given X is lognormal. Ln(X) is normally sts2distributed and the conditional mean of Ln(X) for s > t is Ln(X) + (s t) ? ?(s t) st

?stand the conditional standard deviation of Ln(X) is . The conditional expected s

(st)Xevalue of X is , to find expected future price inflate current price by the st

continuously compounded expected rate of return. (d) The variance of a forecast of X tends to as s tends to . s

(4) Mean Reverting Process

?(X,t)k(?X)and?(X,t)?X tttt

where k, ?, and ? are all greater than zero.

If ? = 1 this is called an Ornstein Uhlenbeck process.

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> ? the drift is negative and for X < ? the drift is positive. This type of process is For Xtt

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 X is finite s2(d) if ? = ? (as pictured above) the distribution of X given X (s > t) is non central ~ stk(st)(X?)e?with mean t22k(st)2k(st)k(st)2??X()(ee)?()(1e)and variance k2kt

We now turn to Ito’s lemma:

If we have X an Ito process and Y = f(X) what does dY look like? In other words, we tttt

know dX has a drift and a diffusion what are they for dY? tt

Clearly this is an important question for derivative pricing if we think of X as the price of t

the underlying asset then for the right f( ), Y = f(X) is the price of the derivative. tt

Ito’s Lemma (Univariate Case)

dXdt?dBLet X be an Ito Process defined by tttt

where the dependence of and ? on (X, t) is suppressed for notational convenience. t

Let f: ??[0, T] ?. Then Y = f(X, t) is an Ito Process defined by tt

21dY[f(X,t)f(X,t)f(X,t)?]dtf(X,t)?dB 2txttttxxttxttt

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Typically it is assumed that the function f(.) is twice continuously differentiable in both

and t, however, we only require that f, f, and f exist and are continuous. Xtxxxt

ndIntuition: Take a 2 order Taylor series expansion of f(X, t + dt) around (X, t) then t + dttdY = f(X, t + dt) f(X, t). tt +dtt

f(X,tdt)f(X,t)f(X,t)dXf(X,t)dttdttxtttt

2211?f(X,t)(dX)?f(X,t)(dt) 22xxttttt

12??f(X,t)dXdtR(residual)2xttt

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 dX, dt, and t22dt are but what are (dX) and dXdt? tt22?(dX)(dtdB)tttt

2222??dtdB2dBdt tttttt

220?dt0?dttt

dXdt(dt?dB)dt000 tttt

Then collecting terms from the Taylor’s series expansion

dYf(X,tdt)f(X,t)f(XdX,tdt)f(X,t)ttdttttt 21[f(X,t)f(X,t)?f(X,t)?]dtf(X,t)?dB2xttttxxttxttt

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.

2Note that in ordinary calculus dX is small enough so that dX vanishes. In stochastic

22?dtcalculus dX is a random variable so dX does not vanish (instead it converges to ) t3but terms of higher order, dX or dXdt do vanish.

Examples:

2(1) Consider an Ito process Y = B for t ? 0, where B is a standard Brownian motion. ttt

Find dY. We first identify X then f() and finally compute f, f, and f in order to use the ttxxxt

lemma.

01andXBsodXdB?tttttt 2Yf(X,t)Xandf:??[0,T]?;X??tttt

then

f(X,t)2Xf(X,t)2andf(X,t)0 xttxxttt

and we arrive at

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