SELECTED TOPICS ∗

9.8 SELECTED TOPICS ∗

When a process has more sources of risk than assets, we are, as stated earlier, in an incomplete market situation. In such cases it is possible to proceed in two ways. Either find additional assets to use (for example, another option with different maturity and strike price) or approximate the stochastic volatility process by another risk-reduction process. There are two problems we shall consider in detail, including (stochastic) jumps and stochastic volatility continuous type models.

Problems are of three types: first, how to construct a process describing reli- ably the evolution of the variance; second, what are the sources of uncertainty of volatility; and, third, how to represent the stochastic relationship between the

INCOMPLETE MARKETS AND STOCHASTIC VOLATILITY

underlying process and its volatility. These equations are difficult to justify ana- lytically and therefore we shall be satisfied with any model that practically can

be used and can replicate historical statistical data. There are, however, a num- ber of continuous-time models for stocks, returns, interest rates and other prices and their volatility that are often used. We shall consider such a model in the appendix to this chapter in detail to highlight some of the technical problems we must resolve in order to deal with such problems.

9.8.1 The Hull and White model and stochastic volatility

Hull and White (1987) have suggested a stochastic volatility model in which volatility is a geometric Brownian motion. This is written as follows:

√ dS/S = α dt + V dw, S(0) = S 0 ; dV / V = µ dt + ξ dz, V (0) = v 0 ;

E (dw dz) = ρdt

where V is the volatility while dw and dz are two Wiener processes, with cor- relation ρ. A call option would in this case be a function of both S and V , or

C (t, S, V ). Since there are two sources of risk, the hedging (replicating) portfolio must reflect this multiplicity of risks. Hull and White assume that the volatility risk is perfectly diversifiable, consequently the volatility risk premium is null. Using a Taylor series development of the option’s price allows the calculation of the value of a call option as a function of small perturbations in volatility. The resulting solution turns out to be (see the Appendix for a mathematical development):

2 [dS] + [dV ] dt

After some additional manipulations, we obtain a partial differential equation in two variables:

(S, V , T ) = Max (S(T ) − K, 0) where K is the strike price, λ V = (µ − R f )Vβ V and while β V is the beta of the

2 C∂ V 2

C∂S∂V

volatility. The analytical treatment of such problems is clearly difficult. In 1976, Cox introduced a model represented generally by:

dS = µ(S, t) dt + V (S, t) dW

297 Additionally a volatility state that V (S, t) = σ S δ with δ a real number between

SELECTED TOPICS

0 and 1. Application of Ito’s Lemma, as seen in Chapter 4, leads to:

dV = 2 ∂ S

(S, t) + ∂ S 2 V (S, t) dt + V (S, t) dW

and in this special case, we have:

∂ S 2 = δ(δ − 1)σ S which we insert in the equation above to obtain a stochastic volatility model:

A broad number of other models can be constructed. In particular, for interest rate models we saw in Chapter 8, mean reversion models. For example, Ornstein– Uhlenbeck models of stochastic volatility are used with both additive and geo- metric models for the volatility equation. The additive model is given by:

while the geometric model is: dS

dV

= µ dt + V dW t S and V = α(θ − V ) dt + ξ dW ′ In both cases the process is mean-reverting where θ corresponds to a volatility,

a deviation from which induces a volatility movement. It can thus be interpreted as the long-run volatility. α is the mean reversion driving force while ξ is the stochastic effect on volatility. The study of these models is in general difficult, however.

9.8.2 Options and jump processes (Merton, 1976)

We shall consider next another ‘incomplete’ model with two sources of risk where one of the sources is a jump. We treat this model in detail to highlight as well the treatment of models with jumps. Merton considered such a problem for the following price process:

dS = α dt + σ dw + K dQ S

Q infinitesimal time intervals:

w.p. q dt dQ = 0

w.p. (1 − q) dt

INCOMPLETE MARKETS AND STOCHASTIC VOLATILITY

Let F = F(S, t) be the option price. When a jump occurs, the new option price is F[S(1 + K )]. As a result,

dF = [F(S(1 + K )) − F] dQ

When no jump occurs, we have a process evolving according to the diffusion process:

(dS) dF = 2

∂ t dt + ∂ S dS + 2 ∂ S 2

Letting τ = T − t be the remaining time to the exercise date, we have: ∂ F ∂ F 1 ∂ 2 F ∂ F

Combining these two equations, we obtain:

dF = a dt + b dw + c dQ

2 ∂ S 2 ; b = Sσ ∂ S ; c = F[S(1 + K )] − F with

a= − + αS

∂τ ∂ S

E (dF) = [a + qc] dt since E(dQ) = q dt

To eliminate the stochastic elements (and thereby the risks implied in the price process) in this equation, we construct a portfolio consisting of the option and a stock. To eliminate the ‘Wiener risk’, i.e. the effect of ‘dw’, we let the portfolio

Z consist of a future contract whose price is S for which a proportion v of stock options is sold (which will be calculated such that this risk disappears). In this case, the value of the portfolio is:

dZ = Sα dt + Sσ dw + SK dQ − [va dt + vb dw + vc dQ] If we set v = Sσ/b and insert in the equation above (as done by Black–Scholes),

then we will eliminate the ‘Wiener risk’ since:

dZ = S(α − σ a/b) dt + (Sσ − vb) dw + S(K − σ c/b) dQ

or

dZ = S(α − σ a/b) dt + S(K − σ c/b) dQ

In this case, if there is no jump, the evolution of the portfolio follows the differ- ential equation:

dZ = S(α − σ a/b) dt

However, if there is a jump, then the portfolio evolution is:

dZ = S(α − σ a/b) dt + S(K − σ c/b) dQ

299 Since the jump probability equals, q dt,we obviously have:

THE RANGE PROCESS AND VOLATILITY

E (dZ ) = S(α − σ a/b) + Sq(K − σ c/b)

dt There remains a risk in the portfolio due to the jump. To eliminate it we can

construct another portfolio using an option F ′ (with exercise price E ′ ) and a future contract such that the terms in dQ are eliminated as well. Then, constructing a combination of the first (Z ) portfolio and the second portfolio (Z ′ ), both sources of uncertainty will be eliminated. Applying an arbitrage argument (stating that there cannot be a return to a riskless portfolio which is greater than the riskless rate of return) we obtain the proper proportions of the riskless portfolio.

Alternatively, finance theory (and in particular, application of the CAPM (cap- ital asset pricing model) state that any risky portfolio has a rate of return in a small time interval dt which is equal the riskless rate plus a premium for the risk assumed. Thus, using the CAPM we can write:

dZ

E (K − σ c/b)

Z dt =R f +λ

where λ is assumed to be a constant and expresses the ‘market price’ for the risk associated with a jump. This equation can be analysed further, leading to the following partial differential equation which remains to be solved (once the boundary conditions are specified):

∂ F ∂ F 1 ∂ 2 F SK

2 − 2 + (λ − q) − (F[S(1 + K ) − F) + S σ −R

f F=0 ∂τ

2 ∂ S 2 with boundary condition:

F (T ) = Max [0, S(T ) − E]

Of course, for an American option, it is necessary to specify the right to exercise the option prior to its final exercise date, or

F (t) = Max[F ∗ (t), S(t) − E]

where F ∗ (t) is the value of the option which is not exercised at time t and given by the solution of the equation above. The solution of this equation is of course much more difficult than the Black–Scholes partial differential equation. Specific cases have been solved analytically, while numerical techniques can be applied to obtain numerical solutions.