An Hilbert space approach for a class of arbitrage free implied volatilities models

In other words the results we give allow to identify a class of (non-trivial, arbitrage free) evolutions of the implied volatility starting from some the initial (market-given) surface.

Many aspects of implied volatility models have been diffusely studied and the reader is referred to [10], Chapter 7 for a review.

The setting of the model and some results from [3] Consider W (i) t , for i ∈ {1, .., m} and t ≥ 0, m independent real Brownian Motions on the probability space (Ω , F , P). We call F t the induced filtration. We consider a fixed T * > 0 and a market in which a bond (with interest rate equal to zero), a stock S t and a family of European call options O t (K, T ) for t ≥ 0, T ∈ (t,t + T * ], and K > 0 are liquidly traded. So at every time t we consider the call options expiring in the interval (t,t + T * ] for a fixed T * . Without losing in generality (changing if necessary the Brownian motions and the measure P) we can assume that the price of the stock S t depends only on the first BM, that S t is martingale and evolves following the SDE dS t = S t θ t dW

(1) t (1) for some one-dimensional process θ t . The Black and Scholes price for O t (T, K) is of course

where N is the cumulative distribution of the normal distribution and,

The implied volatility paradigm consists, as well known, in inverting ( 2) obtaining (and defining) the “(Black-Scholes) implied volatility” σt (T, K) as a function of C t (and K, T , S t ). So, once we have modeled the evolution of the implied volatility, thanks to its definition, we can use (2) to find the evolution (varying the time t) of the prices of the options O t (T, K) and we can wonder if the evolution of the market so obtained is arbitrage free namely, if the processes C t (T, K) := S t N(d 1 (S t , σt , K, T )) -KN(d 2 (S t , σt , K, T )) and S t have an equivalent common (varying T and K) local martingale measure.

In [3] the authors prove that, if we assume the implied volatility to follow a SDE of the form 1 d σt (T, K) = m t (T, K) dt + v t (T, K) * dW t , the arbitrage-free conditions for the market can be expressed (we do not write the dependence of σt , and u t := v t / σt on T and K in the second equation) as

(1) t dt + σt u * t dW t σ0 (T, K) initial condition σt (T, K) = θ t ℓ + u t ln K S t feedback condition .

(3) 1 Where m t and v t are respectively a one-dimensional and a m-dimensional process and they can depend explicitly, as we will assume when we give some sufficient conditions to prove the existence of the solution, on T , K, S t , σt and θ t . v t (T,K) * is the adjoint of the vector v t (T,K) so that v t (T,K)

where we called ℓ the vector of R m given by (1, 0, 0, …, 0), | • | is the norm in R m and the m-dimensional process u t = v t / σt . They also prove that such a system of SPDEs can be rewritten using the variable 2

feedback condition (4) where we used u

The feedback condition is obtained in [3] in order to avoid the phenomenon (already observed in [16], Section 3(a), see also ([1] and [2])) of the “bubble” of the drift for t → T . Such a condition, it will be clearer in the following, adds a certain number of difficulties in the study of the problem.

In [3] the author does not prove an existence result for equation ( 3) or (4) but they prove that such conditions are equivalent to the market being arbitrage-free. So, if we can find some sets of u (i) t and θ t of stochastic processes such that equations (4, 1) admit a positive solution (ξ t , S t ) (or, that is the same, (3, 1) admit a positive solution ( σt , S t )), the evolution of the market is arbitrage free.

In the present work we study a “reduced” problem: indeed we consider a fixed K and we study the existence and uniqueness for the system of SPDEs (4) varying T . We continue in the introduction to write the equations for the the general problem and we will fix a K in Section 1 (starting from equation (EQ)). For the general case we would need the “compatibility conditions” described in Section 5 to be satisfied.

We want to describe the system using the square root of the forward implied volatility introduced in [16]. We define X t , formally, as

The idea of use such a variable in the implied volatility models was introduced for the first time, as far as we know, in [14]. In the works [14,15] the authors use different techniques to deal with problems strictly related to the our. They use some results about strong solutions for functional SDE proven in [17] (see also [11]) to study the case of the family for a fixed

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