##
**Convergence with respect to the parameter of a series and the differentiability of barrier option prices with respect to the barrier.**
*(Ukrainian, English)*
Zbl 1224.91158

Teor. Jmovirn. Mat. Stat. 81, 102-113 (2009); translation in Theory Probab. Math. Stat. 81, 117-130 (2010).

The authors deal with a family of Black-Scholes models (denoted by \((A_n)\)), where every model is a continuous model of a financial market with a risky asset (a stock whose price at a moment \(t\) is \(S_n(t)\)) and a nonrisky asset (a bond whose price at a moment \(t\) is \(B_n(t)\)). The price \(B_n(t)\) is defined as a solution of the differential equation
\[
dB_n(t) = r_n(t)B_n(t) dt,
\]
where \(r_n = r_n(t)\), \(t\geq0\), is a nonnegative function that is Lebesgue integrable on an arbitrary interval. If \(B_n(0) = 1\), then \(B_n(t)=\exp\left\{\int_0^tr_n(s)\,ds\right\}\) for \(t\geq0\). The equation for the price \(S_n(t)\) of a risky asset can be written in the differential form
\[
dS_n(t) = S_n(t)\left( \mu_n(t)dt+\sigma_n(t)dW(t) \right),
\]
where \(\mu_n(t)\) and \(\sigma_n(t)\) are nonrandom functions and \(W(t)\) is a standard Wiener process with respect to the measure \(P\). A family \((A_n)\) of financial markets with continuous time is described by sequences of nonrandom functions \(\mu_n(t)\), \(\sigma_n(t)\), and \(r_n(t)\). Consider also numerical sequences \(H_n\) and \(K_n\), \(n\geq 0\), which define the payoff function for “up-and-out” barrier call options
\[
C_n=\begin{cases} (S_n(T)-K_n)^+,&\text{for \(\max_{0\leq t\leq T}S_n(t)<H_n\)},\\ 0,&\text{otherwise}, \end{cases}
\]
where \(H_n\) is a barrier and \(K_n\) is a strike price. Payoff functions are defined similarly for other types of barrier options. The fair price of a European up-and-out call option for the model has the form
\[
C_n = E_{P^*_n}\left( \exp\left\{ -\int_0^Tr_n(t)\,dt\right\} (S_n(T)-K_n)^+{\mathbb I}\{\max_{0\leq t\leq T}S_n(t)<H_n\}\right).
\]
The authors present conditions under which the price of a European up-and-out call option \(C_n\) in the prelimit model converges as \(n\to\infty\) to the corresponding price \(C_0\) in the case where \(H_n\to H_0\), \(K_n\to K_0\), \(\sigma_n(t)\to\sigma_0(t)\) in \(L_2[0,T]\) and \(r_n(t)\to r_0(t)\) in \(L_1[0,T]\). The explicit form of the price of the barrier option is not required. The result obtained allows one to prove the continuity of a solution of the corresponding boundary-value problem for the parabolic partial differential equation with respect to the parameter of a series. Applying Malliavin calculus, the existence of a bounded continuous density of the distribution of a Wiener integral with shift restricted to an arbitrary “positive” ray and differentiability of the fair price with respect to the barrier is proved (the differentiability with respect to other parameters is a classical result).

Reviewer: Mikhail P. Moklyachuk (Kyïv)

### MSC:

91G20 | Derivative securities (option pricing, hedging, etc.) |

91G80 | Financial applications of other theories |

60F17 | Functional limit theorems; invariance principles |

60H07 | Stochastic calculus of variations and the Malliavin calculus |