# Completeness and Hedging Question

A question in some private notes I'm struggling to work through (exam. prep.). (iii) is where I hit a wall with my understanding & I'm lost thereafter. Any help/clarification gratefully received.

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Consider a financial market with $d = 1$ risk security, whose price $S^1$ is determined by $$dS_t^1 = (1/{S_t^1})dt + dW_t^1$$

In addition, the risk-free rate is $r_f=0$, so the price of the money-market account is $S^0 = 1$.

i) Derive an expression for the market price of risk

ii) Derive a stochastic differential equation for the numeraire portfolio

iii) Use the real-world pricing forumla to derive an expression for the price of a zero-coupon bond (ZCB) with a face-value of $1 (Note: Notice the ZCB is an equity derivative in this model!) iv) Derive expression for the weights of the hedge portfolio for the ZCB v) Consider a portfolio consisting of a long position in the hedge portfolio above, and a short position in the money-market account, structured in such a way that its initial value is zero. What is the final payoff of this portfolio? What type of arbitrage is it? vi) Does the model in question satisfy NA$_+$? Justify. vii) Does the model in question satisfy NA? Justify. viii) Does the model in question satisfy NUPBR? Justify. ix) Does the model in question satisfy NFLVR? Justify. • I think you are missing another$S_t^1$term before$dW_t^1$in your dynamics? – emcor Jun 14 '15 at 8:01 • You only want to solve (iii) here? – emcor Jun 14 '15 at 8:03 • Hi emcor, thanks for the response. Nope, there's no additional St term preceding dWt. +24hrs after posting I'm now past i, ii & iii & am looking at iv, v, vi – Nigel M. Jun 14 '15 at 23:50 • Eckhard Platen's excellent research paper here has helped with i...iii (qfrc.uts.edu.au/research/research_papers/rp262.pdf) – Nigel M. Jun 15 '15 at 0:10 • What is the economic sense of this dynamics? Do you need to have a volatility term? In risk-neutral world, should the drift be zero as you assumed zero interest rate? – Gordon Nov 24 '15 at 14:03 ## 2 Answers$i,ii,iii)$Define$x_t$by$x_t=(S_t^1)^2. Ito's formula give us \begin{align} & dx_t=d(S_t^1)^2=2S_t^1dS_t+\frac{1}{2}(2)d[S_t^1,S_t^1](t)\\ & dx_t=d(S_t^1)^2=2S_t^1(\frac{1}{S_t^1}+dW_t)+dW_tdW_t=3dt+2S_t^1dW_t\\ \end{align} then \begin{align} & dx_t=3\ dt+2\sqrt x_t\ dW_t\\ \end{align} The following portfolio is constructed: we buy a bond of dollar valueV_1$with maturity$T_1$and sell another bond of dollar value$V_2$with maturity$T_2$. The portfolio value$\Piis given by \begin{align} \Pi=V_1-V_2 \end{align} where thetsubscripts are omitted for convenience. Assuming the portfolio is self-financing, the change in portfolio value is \begin{align} d\,\Pi=dV_1-dV_2 \end{align} The strategy is similar to that for the Black-Scholes case. I apply Ito’s lemma to obtain the processes forV_1$, which allows us to find the process for$\Pi$.To form the hedging portfolio, first apply Ito’s lemma to the value of the derivative,$V_1=V_1(x_t,t)$We must differentiate$V_1$with respect to the variables$t$and$x$,The result is that$dV_1follows the process \begin{align} dV_1=\frac{\partial V_1}{\partial t}dt+\frac{\partial V_1}{\partial x}dx+\frac{1}{2}\frac{\partial^2 V_1}{\partial x^2}d[x,x](t) \end{align} in other words \begin{align} dV_1=(\frac{\partial V_1}{\partial t}+3\frac{\partial V_1}{\partial x}+2x_t\frac{\partial^2 V_1}{\partial x^2})dt+2\sqrt x_t \frac{\partial V_1}{\partial x}dW_t \end{align} I have use the fact thatd[x,x](t)=(2\sqrt x_t\ dW_t)(2\sqrt x_t\ dW_t)=4x_t dt$,$dt\,dW_t=0$and$dt\,dt=0. Then portfolio value can be written \begin{align} &d\,\Pi=(\frac{\partial V_1}{\partial t}+3\frac{\partial V_1}{\partial x}+2x_t\frac{\partial^2 V_1}{\partial x^2})dt+2\sqrt x_t \frac{\partial V_1}{\partial x}dW_t\\ &\,\,\,\,\,\,\,\,\,\,-(\frac{\partial V_2}{\partial t}+3\frac{\partial V_2}{\partial x}+2x_t\frac{\partial^2 V_2}{\partial x^2})dt-2\sqrt x_t \frac{\partial V_2}{\partial x}dW_t\\ \end{align} In order for the portfolio to be hedged against movements Wiener process last two terms in this equation must be zero. This implies that the hedge parameters must be \begin{align} \frac{\partial V_1}{\partial x}=\frac{\partial V_2}{\partial x} \end{align} then \begin{align} &d\,\Pi=(\frac{\partial V_1}{\partial t}+2x_t\frac{\partial^2 V_1}{\partial x^2})dt\\ &\,\,\,\,\,\,\,\,\,\,-(\frac{\partial V_2}{\partial t}+2x_t\frac{\partial^2 V_2}{\partial x^2})dt\\ \end{align} The condition that the portfolio earn the risk-free rate,S^0=1$, implies that the change in portfolio value is$d\,\Pi=S^0\,\Pi dt\begin{align} d\,\Pi=S^0\,\Pi\,dt=\Pi\,dt=(V_1-V_2)dt \end{align} By combination these equations we have \begin{align} \frac{\partial V_1}{\partial t}+2x_t\frac{\partial^2 V_1}{\partial x^2}-V_1=\frac{\partial V_2}{\partial t}+2x_t\frac{\partial^2 V_2}{\partial x^2}-V2 \end{align} The above relation is valid for arbitrary maturity datesT_1$and$T_2$, so this equation should be independent of maturity$T.Then \begin{align} \frac{\partial V}{\partial t}+2x_t\frac{\partial^2 V}{\partial x^2}-V=\lambda \end{align} If moneymarket rate is always 0, then the bond is also priced at 1 by no arbitrage. However, if you have to go through all the measure change exercise... To price the zero-bond as a derivative: Let B(t,S) be the price of the bond as a function of S. We know thatB(T, S_T) =1$. To price any derivative we will use martingale pricing:$\frac{B(t_0)}{N(t_0)} = E^Q \frac{B(T)}{N(T)}$where N(t) is our numeraire. We will take the moneymarket account to be our numeraire. Note that since the riskfree rate is always zero in this example we have$dN = 0 dt = 0$. The dynamics of the discounted stock are given by:$d\frac{S}{N} = \frac{1}{N}dS - \frac{S}{N^2}dN = \frac{dS}{N} = dS$, since N(t) =1 for all t. We change the measure from P to Q in order to make$d\frac{S}{N}$a martingale. In this case it is equivalent to making$dS$a martingale. We apply girsanov and endup with$dS = dW^Q$.$B(t_0) = E^Q(B(T)) = \int 1 dW^Q = 1\$

(chapter 11 of Bjorks book is good reference for girsanov).