How to hedge a derivative that pays the reciprocal of the stock price?

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.

Thanks!

• 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
• Is this under Black-Scholes world? – emcor Jul 30 '14 at 20:24
• If you do not know, this is the dynamic hedge for a variance swap. – Dom Oct 29 '16 at 17:07

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.
For question 2): At time $T$, we need to pay $100\cdot\frac{S_T}{S_t}$ (in domestic currency, say \$). To do this, we need to buy 100\$ worth of shares at time $t$: that gives us $N=100\cdot \frac{1}{S_t}$ shares, with the desired final value of $$N\cdot S_T = 100\cdot\frac{S_T}{S_t}$$ 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:$100\,(B(0,t) - d\cdot Q_1)$• 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