Limit value of price of European call option in binomial model. (Ukrainian, English) Zbl 1150.91363

Gorovyj, S. O.

Teor. Jmovirn. Mat. Stat. 74, 23-26 (2006); translation in Theory Probab. Math. Stat. 74, 25-28 (2007).

The author considers the binomial model for asset price \[ S(t+1)=\begin{cases} S(t)(1+b),&\text{with probability}\;p,\\ S(t)(1+a),&\text{with probability}\;q,\end{cases} \] where \(p+q=1,\;0<1+a<1+r<1+b.\) Let \((p^{*},q^{*})\) be a martingale measure \(p^{*}(1+b)+q^{*}(1+a)=1+r\), and let us denote by \(S_{n}\) the fair price of European call option after \(n\) periods. It is proved that \[ \lim_{n\to\infty}S_{n}=\begin{cases} S(0),& \lambda<\tilde p,\\ S(0)/2,& \lambda=\tilde p,\\ 0,& \lambda>\tilde p,\end{cases} \] where \(\lambda={-\ln(1+a)\over\ln(1+b)-\ln(1+a)},\;\tilde p={p^{*}(1+b)\over 1+r}\).

Reviewer: Aleksandr D. Borisenko (Kyïv)

MSC:

91B28

Finance etc. (MSC2000)

Keywords:

limit value; fair price; European call option; binomial model