Stochastic Processes and Financial Mathematics
(part two)
\(\newcommand{\footnotename}{footnote}\)
\(\def \LWRfootnote {1}\)
\(\newcommand {\footnote }[2][\LWRfootnote ]{{}^{\mathrm {#1}}}\)
\(\newcommand {\footnotemark }[1][\LWRfootnote ]{{}^{\mathrm {#1}}}\)
\(\let \LWRorighspace \hspace \)
\(\renewcommand {\hspace }{\ifstar \LWRorighspace \LWRorighspace }\)
\(\newcommand {\mathnormal }[1]{{#1}}\)
\(\newcommand \ensuremath [1]{#1}\)
\(\newcommand {\LWRframebox }[2][]{\fbox {#2}} \newcommand {\framebox }[1][]{\LWRframebox } \)
\(\newcommand {\setlength }[2]{}\)
\(\newcommand {\addtolength }[2]{}\)
\(\newcommand {\setcounter }[2]{}\)
\(\newcommand {\addtocounter }[2]{}\)
\(\newcommand {\arabic }[1]{}\)
\(\newcommand {\number }[1]{}\)
\(\newcommand {\noalign }[1]{\text {#1}\notag \\}\)
\(\newcommand {\cline }[1]{}\)
\(\newcommand {\directlua }[1]{\text {(directlua)}}\)
\(\newcommand {\luatexdirectlua }[1]{\text {(directlua)}}\)
\(\newcommand {\protect }{}\)
\(\def \LWRabsorbnumber #1 {}\)
\(\def \LWRabsorbquotenumber "#1 {}\)
\(\newcommand {\LWRabsorboption }[1][]{}\)
\(\newcommand {\LWRabsorbtwooptions }[1][]{\LWRabsorboption }\)
\(\def \mathchar {\ifnextchar "\LWRabsorbquotenumber \LWRabsorbnumber }\)
\(\def \mathcode #1={\mathchar }\)
\(\let \delcode \mathcode \)
\(\let \delimiter \mathchar \)
\(\def \oe {\unicode {x0153}}\)
\(\def \OE {\unicode {x0152}}\)
\(\def \ae {\unicode {x00E6}}\)
\(\def \AE {\unicode {x00C6}}\)
\(\def \aa {\unicode {x00E5}}\)
\(\def \AA {\unicode {x00C5}}\)
\(\def \o {\unicode {x00F8}}\)
\(\def \O {\unicode {x00D8}}\)
\(\def \l {\unicode {x0142}}\)
\(\def \L {\unicode {x0141}}\)
\(\def \ss {\unicode {x00DF}}\)
\(\def \SS {\unicode {x1E9E}}\)
\(\def \dag {\unicode {x2020}}\)
\(\def \ddag {\unicode {x2021}}\)
\(\def \P {\unicode {x00B6}}\)
\(\def \copyright {\unicode {x00A9}}\)
\(\def \pounds {\unicode {x00A3}}\)
\(\let \LWRref \ref \)
\(\renewcommand {\ref }{\ifstar \LWRref \LWRref }\)
\( \newcommand {\multicolumn }[3]{#3}\)
\(\require {textcomp}\)
\(\newcommand {\intertext }[1]{\text {#1}\notag \\}\)
\(\let \Hat \hat \)
\(\let \Check \check \)
\(\let \Tilde \tilde \)
\(\let \Acute \acute \)
\(\let \Grave \grave \)
\(\let \Dot \dot \)
\(\let \Ddot \ddot \)
\(\let \Breve \breve \)
\(\let \Bar \bar \)
\(\let \Vec \vec \)
\(\DeclareMathOperator {\var }{var}\)
\(\DeclareMathOperator {\cov }{cov}\)
\(\DeclareMathOperator {\indeg }{deg_{in}}\)
\(\DeclareMathOperator {\outdeg }{deg_{out}}\)
\(\newcommand {\nN }{n \in \mathbb {N}}\)
\(\newcommand {\Br }{{\cal B}(\R )}\)
\(\newcommand {\F }{{\cal F}}\)
\(\newcommand {\ds }{\displaystyle }\)
\(\newcommand {\st }{\stackrel {d}{=}}\)
\(\newcommand {\uc }{\stackrel {uc}{\rightarrow }}\)
\(\newcommand {\la }{\langle }\)
\(\newcommand {\ra }{\rangle }\)
\(\newcommand {\li }{\liminf _{n \rightarrow \infty }}\)
\(\newcommand {\ls }{\limsup _{n \rightarrow \infty }}\)
\(\newcommand {\limn }{\lim _{n \rightarrow \infty }}\)
\(\def \ra {\Rightarrow }\)
\(\def \to {\rightarrow }\)
\(\def \iff {\Leftrightarrow }\)
\(\def \sw {\subseteq }\)
\(\def \wt {\widetilde }\)
\(\def \mc {\mathcal }\)
\(\def \mb {\mathbb }\)
\(\def \sc {\setminus }\)
\(\def \v {\textbf }\)
\(\def \p {\partial }\)
\(\def \E {\mb {E}}\)
\(\def \P {\mb {P}}\)
\(\def \R {\mb {R}}\)
\(\def \C {\mb {C}}\)
\(\def \N {\mb {N}}\)
\(\def \Q {\mb {Q}}\)
\(\def \Z {\mb {Z}}\)
\(\def \B {\mb {B}}\)
\(\def \~{\sim }\)
\(\def \-{\,;\,}\)
\(\def \|{\,|\,}\)
\(\def \qed {$\blacksquare $}\)
\(\def \1{\unicode {x1D7D9}}\)
\(\def \cadlag {c\`{a}dl\`{a}g}\)
\(\def \p {\partial }\)
\(\def \l {\left }\)
\(\def \r {\right }\)
\(\def \F {\mc {F}}\)
\(\def \G {\mc {G}}\)
\(\def \H {\mc {H}}\)
\(\def \Om {\Omega }\)
\(\def \om {\omega }\)
\(\def \Vega {\mc {V}}\)
15.4 Martingales and ‘the risk-neutral world’
In this section we give a (brief) explanation of where the term ‘risk-neutral world’ comes from. Offering explanations for mathematical wording is something of a dangerous game – in practice terminology often
arises from accidents of history. There are many cases where “Someone’s Theorem” was not discovered by the same Someone whose name is generally quoted.
As we saw in Proposition 5.5.6, in the risk neutral world \(\Q \) the discounted stock price is a martingale. In continuous
time we have the precise equivalent:
Proof: Recall that \(C_t=e^{rt}\). Thus we are interested in the process \(X_t=e^{-rt}S_t\). In the
risk-neutral world \(\Q \) the process \(S_t\) satisfies \(dS_t=rS_t\,dt+\sigma S_t\,dB_t\), so by Ito’s formula we have
\(\seteqnumber{0}{15.}{17}\)
\begin{align*}
dX_t&=\l (-re^{-rt}S_t+rS_te^{-rt}+0\r )\,dt+\sigma S_te^{-rt}\,dB_t\\ &=\sigma S_te^{-rt}\,dB_t
\end{align*}
By Theorem 12.2.1, we have that \(X_t\) is a martingale (under \(\Q \)). ∎
In fact, more is true. Our next result says that, in the risk neutral world, once we have discounted for interest rates (i.e. divided by \(C_t\)), the price of any contingent claim is a martingale. As we
saw in Section 3.3, martingales model fair games that are (on average) neither advantageous or disadvantageous to their players. The term ‘risk-neutral’
captures the fact that, inside the risk-neutral world, if we forget about interest rates, buying/selling on the stock market would be a fair game.
It is important to remember that we do not believe that the risk-neutral world is the real world. We believe in the absence of arbitrage, and we only care about the risk neutral world because it is useful when
calculating arbitrage free prices.
Proof: Our strategy, which is based on the proof of Lemma 15.4.1, is to set \(X_t=\frac {\Pi _t}{C_t}=e^{-rt}\Pi _t\) and calculate \(dX_t\) using Itos formula. If \(X_t\) is to be a martingale,
it should have the form \(dX_t=(\ldots )\,dB_t\), in which case we can use Theorem 12.2.1 to finish the proof. We’ll carry out the whole proof in the
risk-neutral world \(\Q \).
To avoid arbitrage, \(\Pi _t\) is equal to the value of the replicating portfolio for \(\Phi (S_T)\) at time \(t\). Hence, by Theorem 15.3.1,
\(\seteqnumber{0}{15.}{17}\)
\begin{equation}
\label {eq:PiF} \Pi _t=F(t,S_t).
\end{equation}
Recall that, in the risk neutral world \(\Q \), we have \(dS_t=r S_t\,dt+\sigma S_t\,dB_t\). By Ito’s formula (applied in the risk neutral world!), we have
\[d\Pi _t=\l (\frac {\p F}{\p t}+rS_t\frac {\p F}{\p x}+\frac 12\sigma ^2S_t^2\frac {\p ^2 F}{\p x^2}\r )\,dt+\sigma S_t\frac {\p F}{\p x}\,dB_t\]
where we have suppressed the \((t,S_t)\) arguments of \(F\) and its partial derivatives. We know that \(F\) satisfies the Black-Scholes PDE (15.10). Hence,
\[d\Pi _t=rF\,dt+\sigma S_t\frac {\p F}{\p x}\,dB_t\]
and using (15.18) again we have
\[d\Pi _t=r\Pi _t\,dt+\sigma S_t\frac {\p F}{\p x}\,dB_t\]
which represents \(\Pi _t\) as an Ito process.
We have \(X_t=e^{-rt}\Pi _t\). Using Ito’s formula again, this gives us
\(\seteqnumber{0}{15.}{18}\)
\begin{align*}
dX_t&=\l (-re^{-rt}\Pi _t+r\Pi _te^{-rt}+0\r )\,dt+\sigma S_t\frac {\p F}{\p x}e^{-rt}\,dB_t\\ &=\sigma S_t\frac {\p F}{\p x}e^{-rt}\,dB_t.
\end{align*}
Hence, by Theorem 12.2.1, \(X_t\) is a martingale (under \(\Q \)). ∎