STOCHASTIC VOLATILITY MODELS

9.6 STOCHASTIC VOLATILITY MODELS

Stochastic volatility models presume that a process’s volatility (variance) varies over time following some stochastic process, usually well specified. As a result, it is presumed that volatility growth increases market unpredictability, thereby rendering the application of the rational expectations hypothesis, at best, a tenuous one. Modelling volatility models might require then a broad number of approaches not falling under the ‘random walk hypothesis’. Techniques such as ARCH and GARCH, we referred to, might be used to estimate empirically the volatility in such cases. Below, we consider a number of problems and issues associated with stochastic volatility in the valuation of financial assets.

Stochastic volatility introduces another ‘source of risk’, a volatility risk, when we model an asset’s price (or returns). This leads to incompleteness and thus to non-unique asset prices. Risk-neutral pricing is no longer applicable since the probabilities calculated by the application of rational expectations (i.e. hedging to eliminate all sources of risk and using the risk-free rate as a mechanism to replicate assets) do not lead to risk-neutral valuation. For this reason, unless some other asset can be used to ‘enrich’ a hedging portfolio (for the volatility risk as well), we are limited to using approximations based on an economic rationale or on some other principles so that our process can be constructed (and on the basis of which risk-neutral pricing can be applied). A number of approaches can be applied including:

r time contraction, r approximate replication, r approximate risk–neutral pricing valuation, r bounding.

These approaches and related ones are the subject of much ongoing research. Again, we shall consider some simple cases and, in some cases, define only a quantitative framework of the problem at hand.

9.6.1 Stochastic volatility binomial models ∗ Stochastic volatility has an important effect on the process underlying uncertainty,

altering the basic assumption of ‘normal’ or ‘binomial’ driving disturbances. To see these effects we consider the simple binomial model we have used repeatedly

STOCHASTIC VOLATILITY MODELS

and given below:

x t +1 −x t = αε t ;ε t = Lq ; x t

Here α denotes the process constant volatility. Now assume a mild stochastic volatility. Namely, we let the volatility α assume a value of 1 and zero with probabilities ( p α , q α ) leading to:

x t +1 −x t = ˜α α

Lq ; ˜α t x = 0q α In this case, the random (binomial) volatility is reduced to a trinomial model (see

t ε t ;ε t =

Figure 9.2) where p is the probability of the constant volatility model and p α is the probability that volatility equals one and q α is the probability that there is no volatility. For this simple case, already it is not possible to construct a perfect hedge for, say, an option as we have done earlier. This is because there are two sources of risk – one associated with the price and the other with volatility. Assuming one asset only, the number of risk sources is larger than the number of assets and therefore we have an incomplete market situation where prices need not be unique.

When volatility is constant, note that α t ε t is a random walk, but when volatility

is a random variable, the process ˜α t ε t is no longer a random walk. Let z t = ˜α t ε t have a density function F z t (.) and assume that the random walk and the volatility are statistically independent (which is a strong assumption). Using elementary probability calculations, we have:

F α (α)F ε

F α (α)F ε

For example, if the random walk ε t = (+1, −1) is biased and with probabilities ( p, ¯p), while volatility assumes two values ˜α = (a, b) with probabilities (q, ¯q),

the following quadrinomial process results:

In other words, stochastic volatility has generated incompleteness in the form of

INCOMPLETE MARKETS AND STOCHASTIC VOLATILITY

a quadrinomial process. By enriching the potential states volatility may assume we are augmenting the ‘volatility stochasticity’. Note, that it is not the size of volatility that induces incompleteness but its uncertainty. We shall see below, using a simple example, that the option of a ‘mild volatility’ process is larger than a larger (constant) volatility process – hence, emphasizing the effects of incompleteness (stochastic volatility) on option prices which in some cases can

be more important to greater (but constant) volatility.

Time contraction

The underlying rationale of ‘time contraction’ is a reverse discretization. In other words, assuming that at the continuous-time limit, the underlying price process can be represented by a stochastic differential equation of the Ito type, it is then reasonable to assume that there is some binomial process that approximates the underlying process. Of course, there may be more than one way to do so (thus leading, potentially, to multiple prices) and therefore, this approach has to be applied carefully to secure that the limit makes economic sense as well. In this approach, a multinomial process is replaced by a binomial tree, consisting of as many stages (discretized time) as are needed to replicate the underlying model. For example, the trinomial process considered earlier can be reduced to a two-

stage tree as shown in Figure 9.3, where ( p 1 , p 2 , p 3 ) are assumed to be risk-neutral probabilities, appropriately selected by replication. Note that we have necessarily:

p 1 p 2 = pp α p 1 ¯p 2 + ¯p 1 p 3 =1−p α ¯p 1 ¯p 3 = q ¯p α

Since there are only two independent equations, we have in fact a system of three variables in two equations that can be solved in a large number of ways (for

example as a function of ¯p 3 ). This means also that there is no unique price. In this case,

p 2 = [ pp α / ( p ¯p α / ¯p 3 )] p 1 = 1 − ( p ¯p α / ¯p 3 )

Hx

Hx

Lx

Lx

Figure 9.3

285 However, if we assume rational expectations, then:

STOCHASTIC VOLATILITY MODELS

A=

[p 2 (1 + H)x + ¯p 2 x ] and B =

[p 3 x + ¯p 3 (1 + L)x]

which provides a third equation in ( p 1 , p 2 , p 3 ). Of course, for p α = 1 we have nonstochastic volatility and, therefore, we can calculate the approximate risk- neutral probabilities as a function of 0 < p α <

1. For simplicity, set p 1 =p 2 = p 3 , then we have a quadratic equation:

f −1−L 0= p −2

p−

(H + L) whose solution is given by: L 2 L

(H + L)

f ) (1 + R 2 −1−L p=

(H + L) If we use the following parameters as an example, 1 + H = 1.4, 1 + L = 0.8,

(H + L)

H+L

R f = 0.04, the only feasible solution is:

p= 1+ − 1 or p = 0.55177

Inserting in our equations:

A/x = 0.9615 [(0.55177)(1.4) + 0.44823] = 1.1737 B/x = 0.9615 [(0.55177) + 0.44823(0.8)] = 0.8753

In this particular case, the option price is given by:

1 C=

2 (H − K )x] = 0.9245[0.3(0.55177) x ] = 0.084439x (1 + r)

[p ∗2

We consider next the problem with mild volatility (see Figure 9.4) and set: p ′ =p ∗2 / p α = 0.3044/ p α . If p α = 0.6, p ′ = 0.50733. If we assume no stochastic volatility but α =1 with probability 1 (that is a process more volatile than the previous one), then the value of an option is calculated by p ′ =p ∗2 / p α = 0.3044/ p α . In addition, since p α = 1 we have p ′ = 0.3044. As a result, the option price is:

1 C=

[p ′ (H − K )x] = 0.9245 [0.3044(1.4 − 1.1)x] = 0.08442x 1+R

compared with an option price with mild volatility given by C = 0.084439x.

INCOMPLETE MARKETS AND STOCHASTIC VOLATILITY

Max (0, Hx − K )

pp ' α

(1 − pp ') α

Figure 9.4

Thus, the difference due to the stochastic volatility growth is equal to 0.08442x − 0.084439x = 0.00003x. In other words the value of an option increases both with volatility and with stochastic volatility.

We can generalize this approach further. For example, if the volatility can assume a number of potential values, say, ˜α = (0, 1, 2, 3, 4, 5), then it is possible

to reduce the ten-nomial process to a ten-stage binomial process as shown below. Mathematically, this is given by:

An analysis similar to the previous one provides the mechanism to calculate the approximate binomial risk-neutral probability. Of course, in this model, the single stage ten-nomial process is transformed into a nine-stage binomial process (see Figure 9.5). The number of ways to do so might be very large, however. Additional information and assumptions might then be needed to reduce the number of possibilities and thereby constrain the set of prices the financial asset can assume.

Ten-- nomial

Nine-- stage binomial

Figure 9.5

STOCHASTIC VOLATILITY MODELS

Problem

Consider the following stochastic volatility process: q

x t +1 =x t + ˜αε t ; ε t = −1 1/2 ; ˜α = 31−q Construct the equivalent binomial process and find the risk-neutral probabilities

coherent with such a process.

 +3 w. p.  

(0.5)(1 − q)

(0.5)q x α

+2 w. p.

t +1 =x t +z t ;z t =   −2 w. p. (0.5)q

  −3 w. p. (0.5)(1 − q) This approach can be extended to continuous-time models. For example, assume

a mean reversion interest-rate model:

dS = µ(α − S) dt + V dW or S (t) = α + [S(0) − α ± V ε(t)] e −µt where ε(t) is a standard random walk and V is a stochastic volatility given by: dV = θ(β − V ) dt + κ dW or V (t) = β + [V (0) − β ± κη(t)] e −θt

We assume that V ≥ 0 for simplicity. Note that the interest-rate process combines two sources of risk given by (η(t), ε(t)) and therefore:

S (t) = S(0) e −µt + α(1 − e −µt ) ±

−(θ+µ)t + β(e −µt −e −(θ+µ)t ) −(θ+µ)t η (t)ε(t) The mean rate is therefore:

ˆS(t) = S(0) e −µt + α(1 − e −µt )+κe −(θ+µ)t E [η(t)ε(t)] Since [η(t), ε(t)] are standard random walks their covariation E[η(t)ε(t)] is equal

to 1/4, thus:

ˆS(t) = α + e −θt [S(0) − α + κ e / 4]

−µt

Since the resulting process is given explicitly by a quadrinomial process: S

(t) = S(0) e −µt + α(1 − e −µt )± +e θ −(µ+θ)t t [V (0) + β(e − 1) + κ] w. p. 1/4

+e −(µ+θ)t

θ [V (0) + β(e t − 1) − κ] w. p. 1/4 −e −(µ+θ)t

θ [V (0) + β(e t − 1) − κ] w. p. 1/4 −e −(µ+θ)t

θ [V (0) + β(e t − 1) + κ] w. p. 1/4 The interest-rate variance can be calculated easily since:

−2(θ+µ)t

2 Var(S(t)) = e 2 [V (0) + β(e − 1) − (κ) /4] +κ e −2(µ+θ)t

INCOMPLETE MARKETS AND STOCHASTIC VOLATILITY

∼ W (0) ⇐W ⇔ (1)

C (0)

C (1)

cash flow

W(0) C(0)

Figure 9.6

This process can be replaced by an approximate risk-neutral pricing process by time contraction, by mean–variance hedging or by applying the principle of least divergence as we shall see below.

(2) Mean variance replication hedging

This approach consists in the construction of a hedging portfolio in an incomplete (stochastic volatility) market by equating ‘as much as possible’ the cash flows resulting from the hedging portfolio and the option’s contract. We seek to do so, while respecting the basic rules of rational expectations and risk-neutral pricing. In particular, say that we consider a two-stage model (Figure 9.6; see also Chapter 6). Thus, assuming that a portfolio and an option have the same cash flow, (i.e.

W ˜ (1) ≡ ˜ C (1)), their current price are necessarily equal, implying that: W (0) =

C (0). Since, by risk-neutral pricing,

C (0) =

E˜ C (1); W (0) =

E˜ W (1) and C(0) = W (0)

1+R f 1+R f

or equivalently: E˜ W (1) = ˜ EC (1) and further, E˜ W 2 (1) = ˜ EC 2 (1) These equations provide only four equations while the number of parameters

might be large. However, since a hedging portfolio can involve a far greater number of parameters, it might be necessary to select an objective to minimize.

A number of possibilities are available.

A simple quadratic optimization problem consisting in the minimization of the squared difference of probabilities associated to the binomial tree might be used. Alternatively, the minimization of a hedging portfolio and the option ex-post values of some option contract leads to the following:

Min Φ

p 1 ,..., p n = E( ˜ (1) − ˜

C (1)) 2

289 subject to: W (0) = C(0) or

STOCHASTIC VOLATILITY MODELS

[E ˜ W [E ˜ C (1)] and E ˜ W (1)] = 2 (1) = E ˜ C 2 (1)

1+R f 1+R f

Of course the minimization objective can be simplified further to:

Min

p (C 2 p 1 ,..., Φ p n =E˜ (1) − E ˜

C 2 W (1) ˜ C (1) or Mi n

p 1 ,..., Φ p

n = 1i −W 1i C 1i )

i =1

where W 1i , C 1i are the hedging portfolio and option outcomes associated with each of the events i that occurs with probability p i , i = 1, 2, . . . , n. For example, consider the four-states model given above and assume a portfolio consisting of stocks and bonds a S + B. Let K be the strike price with S i − K > 0, i = 1, 2 and S i − K < 0, i = 3, 4 then the cash flows at time ‘1’ for the portfolio and the op-

tion are respectively a S i + (1 + R f )B, i = 1, 2, 3, 4 and (S 1 −K,S 2 − K , 0, 0) as shown in Figure 9.7. The following and explicit nonlinear optimization problem results:

Min

p 1 ,..., Φ p 4 =

i (S i − K )[(1 − a)S i − K − (1 + R f )B)]

i =1

Subject to:

Explicitly, say that we have the following parameters: S 1 = 110, S 2 = 100, S 3 = 90, S 4 = 80, S = 90, K = 95, R f = 0.12

aS

1 (1 rB )

Option cash f low

p 3 0 C(0)

aS+(1+r)B

Hedging portfolio

aS 4 (1 rB )

Figure 9.7

INCOMPLETE MARKETS AND STOCHASTIC VOLATILITY

Then, our problem is reduced to a nonlinear optimization problem: Min

p 1 ,..., Φ p 4 =3p 1 (110a − 15 − 1.12B)) + p 2 (100a − 5 − 1.12B)) Subject to: 4ap 1 + 2ap 2 − 2ap 4 + 0.024B = 3 p 1 +p 2

1.09126 p 1 + 0.99206 p 2 + 0.8928 p 3 + 0.79365 p 4 =1

p 1 +p 2 +p 3 +p 4 = 1; p i ≥0

The problem can of course be solved easily by standard nonlinear optimization software.

Example

Let S j , j = 1, 2, . . . , n be the n states a stock can assume at the time an op- tion can be exercised. We set, S 0 < S 1 < S 2 <···<S n and define the buy- ing and selling prices of the stock by: S a , S b respectively. By the same token,

we define the corresponding observed call option prices C a , C b . Let p be the probability of a price increase. Of course, if the ex-post price is, S n , this will correspond to the stock increasing each time period with probability given by:

=p n . By the same token, the probability of the stock having a price S j corresponding to the stock increasing j times and decreasing

(1 − p) n−n

n − j times is given by the binomial probability:

= j p (1 − p) n− j

As a result, we have, under risk-neutral pricing:

S= P j S j =

p j j (1 − p) n− j ;S a ≤S≤S b 1+R f 1+R f j =0 j j =0

and the call option price is:

C=

(1 − p) Max(S j − K , 0)

subject to an appropriate constraint on the call options values, C a ≤C≤C b . Note that S and C, as well as p, are the only unknown values so far. While the buy and sell values for stock and option, the strike time n and its price K , as well as the risk-free discount rate and future prices S j , are given. Our problem at present is to select an objective to optimize and calculate the risk-neutral probabilities. We can do so by minimizing the quadratic distance between a portfolio of a unit

of stock and a bond B. At time n, the portfolio is equal a S j + (1 + r) n B if the

291 price is, S j . Of course, initially, the portfolio equals:

STOCHASTIC VOLATILITY MODELS

As a result, the least squares replicating portfolio is given by:

Φ= 2 P

j (a S j + (1 + R f )

B − Max(S j − K , 0))

j =1

leading to the following optimization problem:

n 2 1≥ p≥0,C,S

B − Max(S j − K , 0)]

j =1

Subject to:

≤S≤S b

(1 − p) n− j Max(S

j − K , 0); C ≤C≤C

(1 + r) n

j =0

aS+B=C This is again a tractable nonlinear optimization problem.

(3) Divergence and entropy

Divergence is defined in terms of directional discrimination information (Kullback, 1959) which can be measured in discrete states by two probability distributions say, ( p i , q i ), i = 1, 2, . . . , m, as follows.

p i while the divergence is:

p i and finally,

This expression measures the ‘distance’ between these two probability distribu- tions. When they are the same their value is null and therefore, given a distribution ‘p’, a distribution ‘q’ can be selected by the minimization of the divergence, subject to a number of conditions imposed on both distributions ‘p’ and ‘q’ (such as expec- tations, second moments, risk-neutral pricing etc.). For example, one distribution

INCOMPLETE MARKETS AND STOCHASTIC VOLATILITY

may be an empirical distribution while the other may be theoretical, providing the parameters needed for the application of approximate risk-neutral pricing.

This approach can be generalized further to a bivariate state, involving time as well as states. In this case, we have for each time period:

where the following constraints must be satisfied:

Moments conditions as well as other constraints may be imposed as well, pro- viding a ‘least divergence’ risk-neutral pricing approximation to the empirical (incomplete) distribution.

Example

Consider the following random volatility process

  +3 w.p. (0.5)(1 − q) x α

   +1 w.p. (0.5)q t +1 =x t +z t ;z t =

−1 w.p. (0.5)q −3 w.p. (0.5)(1 − q)

A three-stage standard binomial process leads to:

  +3 w.p.

(0.5)(1 − q) ↔ π

x (0.5)q ↔ 3π (1 − π)

 +1 w.p.

t +1 =x t +z t ;z t =

  −1 w.p. (0.5)q ↔ 3π(1 − π)  

−3 3 w.p. (0.5)(1 − q) ↔ (1 − π) As a result, we can calculate the probability π by minimizing the divergence J

which is given by an appropriate choice of π :

J = [π − (0.5)(1 − q)] log + [3π (1 − π) − (0.5)q]

1−q

(1 − π) 2 × log 2 q

+ [3π(1 − π) − (0.5)q] log q

+ [(1 − π) 3 − (0.5)(1 − q)] log

1−q

Of course other constraints may be imposed as well. Namely, if the current price is $1, we have by risk-neutral pricing the constraint:

[3π 3 2 2 +π 3 (1 − π) − π(1 − π) − 3(1 − π) ]; 0 ≤ π ≤ 1 1+R f

293 As a result, the least divergence parameter π is found by solving the optimization

EQUILIBRIUM , SDF AND THE EULER EQUATIONS

problem above.