Quantitative Finance Stack Exchange is a question and answer site for finance professionals and academics. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

1) Suppose S is the stock price, how to hedge a derivative that pays $1/S_t$ at time $t$?

2) Suppose there will be a dividend of amount $d$ between $t$ and $T$, how to hedge a derivative that pays $100 $*$ S_T/S_t$ at time $T$?

The person who asked me the question said we don't need to assume the distribution of S here.


share|improve this question
You should clarify your questions: First, are you looking for a static or a dynamic hedge? Second, is the dividend paid continuously between t and T, or is it a lump-sum at a random time between t and T? – pbr142 Apr 1 '14 at 9:00
Hint: $\log(1/S_t) = -\log(S_t)$ – Brian B Apr 1 '14 at 12:42
The dividend is a lump sum of amount of $d$ at a fixed time, say $t < t_1 < T$ – benh Apr 2 '14 at 1:01
I am looking for static hedge. Thanks – benh Apr 2 '14 at 1:02
Is this under Black-Scholes world? – emcor Jul 30 '14 at 20:24

Note that, for a smooth function and constant a $$f(S_t) = f(a) + f'(a) (S_t-a) + \int_a^{\infty}(S_t-x)^+f^{''}(x)dx + \int_{0}^a(x - S_t)^+f^{''}(x)dx.$$ Then, the payoff $1/S_t$ can be approximately hedged by call and put options: $$\frac{1}{S_t} = \frac{1}{a} -\frac{1}{a^2}(S_t-a)+ 2\bigg[\int_a^{\infty}\frac{(S_t-x)^+}{x^3}dx + \int_{0}^a\frac{(x - S_t)^+}{x^3}dx \bigg], $$ where $a = E(S_t)$.

As for $S_T/S_t$, let $d$ be the dividend paid at $t_1$, where $t<t_1<T$. Note that $$E(S_T \mid \mathcal{F}_t) =S_t \exp\Big(\int_t^T r_s ds \Big) - d\exp\Big(\int_{t_1}^T r_s ds \Big). $$ We replicate the payoff $1/S_t$ at time $t$. Then we replicate by forwards and bonds.

share|improve this answer

We can explicitly value the Inverted Option under Black-Scholes Model as follows:

enter image description here

Then the delta-hedging ratio is given as:

enter image description here

share|improve this answer

For question 2): At time T, we need to pay 100*S_T/S_t (in domestic currency, say \$). To do this, we need to buy 100\$ worth of shares at time $t$: that gives us $N=100/S_t$ shares, with the desired final value of $N\,S_T = 100*S_T/S_t$ at expiry. Needless to say, today's PV of 100\$ at time $t$ is $100\,B(0,t)$.

However, then at time $t_1$, we hold $N$ shares, so we get a dividend of $d$ per share, so we receive $d\,N = 100\cdot d/S_t$. Being the nice investment bankers that we are, we charge the client correspondingly less, namely the PV of that, which is $100\,d$ times the answer to question 1.

Thus, final answer: $100\,(B(0,t) - d\cdot Q_1)$

share|improve this answer
What is $Q_1$ ? – emcor Aug 11 '14 at 23:28
Hi @emcor, $Q_1$ is the answer to the previous question 1), namely the price of a derivative that pays $1/S_t$ at time $t$... It's European, some sort of hyperbola, and replicable with calls and puts and what have you, and you and Gordon answered that question :-) – Fab Aug 12 '14 at 16:22

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.