a***u
发帖数: 67 | 1 how to price a derivative which pays S_t*log S_t, S_t follows geometric
brownian motion |
m*********g
发帖数: 646 | 2 By Feynman-Kac ?
【在 a***u 的大作中提到】
: how to price a derivative which pays S_t*log S_t, S_t follows geometric
: brownian motion
|
c****o
发帖数: 1280 | 3 change of numeria will do the trick.
【在 a***u 的大作中提到】
: how to price a derivative which pays S_t*log S_t, S_t follows geometric
: brownian motion
|
w******i
发帖数: 503 | 4 it is an integration. would change of numeraire really simply? |
m*********g
发帖数: 646 | 5 Use s(t) as the numeriate, then the payoff under the new measure is just E [ log(sT) ] ?
【在 w******i 的大作中提到】
: it is an integration. would change of numeraire really simply?
|
q**********a
发帖数: 4 | 6
geometric
change measure, Girsanov theorem
E_tail(Y) = E(YZ)
Here Z(t) = EXP(integral(sigma*dt)-1/2*integral((sigma^2)*dt))=
((S(t)/S(0))*EXP(integral(rdt)),
Y(t) = log (S(t))
dY = rdt + sigma*dw (under risk neutral)
and under new measure
d_w_tail = dw - sigma*dt
E(YZ) can be written as
E(log(S)*S/S_0*EXP(integral(rdt)) = E_tail(Y)
E(log(S)*S) = E_tail(Y)*discount*S_0
under tail measure
dY = (r + sigma^2)dt + sigma*dw_tail
Y(t) = log(S_0) + (r +sigma^2)t + N(0, sigma^2*t)
E(log(S)*S) = S_0*discount*(log(S_0)+ (r + sigma^2)*t)
【在 a***u 的大作中提到】
: how to price a derivative which pays S_t*log S_t, S_t follows geometric
: brownian motion
|
M****i
发帖数: 58 | 7 Assume that dS(t)=rS(t)dt+\sigma S(t)dW(t) under risk neutral measure,
then the price process of your claim is
V(t)=S(t)(lnS(t)+(\sigma^2/2+r)(T-t)).
Key points:
1) Formula for conditional expectation under change of measure;
2) Girsanov theorem.
Proof: Let Z(t)=\sigma W(t)-(\sigma^2)t/2,
P'=Z(T)P on F(T), W' is P' Brownian motion.
By risk neutral pricing formula,
V(t)=E[exp(-r(T-t))S(T)lnS(T) | F(t)]
=S(0)exp(rt)E[Z(T)(\sigmaW(T)-(\sigma^2)T/2)+lnS(0)+rT | F(t)]
=S(t)E'[\sigmaW(T)-(\sigma^2)T/2)+lnS(0)+rT|F(t)]
=S(t)E'[\sigma(W'(T)+\sigma T)-(\sigma^2)T/2)+lnS(0)+rT | F(t)]
=S(t)(lnS(t)+(\sigma^2/2+r)(T-t)). CQFD |
k*******d
发帖数: 1340 | 8 还可以对exp(-rt)*S_t*log S_t用Ito's lemma |
L**********u
发帖数: 194 | 9 扯淡的人还真多。
这个问题就是直接积分，根本没有不需要用任何高深的东西。
under risk-neutral measure, the stock price satisfies
d(ln S)=rdt+\sigma dW,
therefore the pdf of y=ln S is
g(y)=1/\sqrt{2\pi \sigma^2 t}e^{\frac{-(y-\ln S_0-(r-\sigma^2/2)t)^2}{2\
sigma^2 t}}.
The price of the option is given by
e^{-rt}\int_{-\infty}^{+\infty}ye^yg(y)dy.
|
t**********a
发帖数: 166 | 10 Is this implying that you only know integration? learn more pls...
【在 L**********u 的大作中提到】
: 扯淡的人还真多。
: 这个问题就是直接积分，根本没有不需要用任何高深的东西。
: under risk-neutral measure, the stock price satisfies
: d(ln S)=rdt+\sigma dW,
: therefore the pdf of y=ln S is
: g(y)=1/\sqrt{2\pi \sigma^2 t}e^{\frac{-(y-\ln S_0-(r-\sigma^2/2)t)^2}{2\
: sigma^2 t}}.
: The price of the option is given by
: e^{-rt}\int_{-\infty}^{+\infty}ye^yg(y)dy.
:
|
