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Introduction to Implied, Local and Stochastic Volatility >
Coupled SDEs for Stochastic Volatility

We begin by writing down the usual Geometric Brownian Motion SDE where the volatility σ is written as the square root of a variance ν:

dS = S(r − D)dt + S√νdW_{1} (17)

and (r − D) is the deterministic instantaneous drift of stock price returns. The variance ν is constant in the original Black-Scholes model (1973). Now it is assumed to follow its own SDE in the form:

dν = ((ω − ζν) − Λ) dt + ξν^{γ} dW_{2 } (18)

where ξ is the volatility of volatility and ρ is the correlation between dW_{1}
and dW_{2}. This representation models mean-reversion in the volatility "σ" or
variance "ν". Conventionally (ω − ζν) is called the real world drift. Λ(S, ν, t)
is the market price of volatility risk, and it tells us how much of the expected
return of V is explained by the risk (standard deviation) of ν in the Capital Asset
Pricing Model framework.

Various economic arguments can be made (see reference [11] for example) that the market price of volatility risk Λ should be proportional to the variance ν. Then let Λ = λν for some constant or function λ. Furthermore, if the real world drift is re-parametrized in the form:

(ω − ζν) = &k(θ − ν)

we yield to:

dν = (k(θ − ν) − λν) dt + ξν^{γ} dW_{2 }(19)

where κ is the mean-reverting speed, θ is the long-run mean, λ is the market
price of risk function, ξ is the volatility of volatility, and dW_{1 }(19) and dW_{2 }(19) are two
Wiener processes (Brownian motion) with correlation coefficient, ρ. We use γ
to generalize {19}. The six parameters κ, θ, λ, ξ, γ and ρ are assumed to be
constant.

For our purpose, we will use the stochastic model {19} because it is both a general representation for all stochastic models that we outline in {13, 14, 15, 16}, and it uses the Heston parameters that are well known in the financial world.

**Applying Ito to the Hedging Portfolio**

We can not hold or "short" volatility as is, but we can hold a position in a second option to do hedging. So if we consider the valuation of the "volatility dependent instrument V ", we shall assume that we can take long or short positions in a second instrument U as well as in the underlying S. Now our candidate for an instantaneously risk-neutral portfolio Π is:

Π = V − φ_{1}S − φ_{2}U

The jump in the value of this portfolio in one time step is:

dΠ = dV − φ_{1}dS − φ_{2}dU − φ_{1}DSdt

where D is the dividend yield on the asset S. As is by now standard, the change in this portfolio in a time dt is given by:

dΠ = adS + bdν + cdt (20)

where:

Prof. Klaus Schmitz

**Next: **Risk-Neutralization and No-Arbitrage

**Summary: **Index