Change of numéraire for two risky assets without bank account (Margrabe’s formula?)

Question

I am considering two risky assets following the usual correlated GBM given by

$$\frac{\mathrm{d}S^{(i)}_t}{S^{(i)}_t}=\mu_i\mathrm{d}t+\sigma_i\mathrm{d}W^{(i)}_t,\quad i\in\{1,2\}$$

with

$$\mathrm{d}W^{(1)}_t\mathrm{d}W^{(2)}_t=\rho\mathrm{d}t.$$

Here's my current understanding: I know that in general, in a complete and no-arbitrage market setting, if I use $S^{(1)}_t$ as my numéraire, I can develop a measure $\mathbb{Q}_1$ such that $\tilde{S}^{(2)}_t=\frac{S^{(2)}_t}{S^{(1)}_t}$ is a martingale. By Girsanov's theorem, defining $\tilde{\sigma}=\sqrt{\sigma_1+\sigma_2-2\rho\sigma_1\sigma_2}$ and $\tilde{W}_t^{\mathbb{Q}_1}=\frac1{\tilde{\sigma}}\left(\sigma_2W^{(2)}_t-\sigma_1W^{(1)}_t\right)$, I obtain the driftless equation

$$\frac{\mathrm{d}\tilde{S}^{(2)}_t}{\tilde{S}^{(2)}_t}=\tilde{\sigma}\mathrm{d}\tilde{W}_t^{\mathbb{Q}_1}.$$

I realise Margrabe’s formula is the end goal, but some texts (one example p. 35) I have read include a risk-free asset or bank account. Notably, to obtain the above equations, they switch from the physical measure $\mathbb{P}$ to the risk-free measure $\mathbb{Q}$ first, then to the measure $\mathbb{Q}_1$. Some others claim that no bank account must be enforced. After reading these texts, I am now confused — if there is no such risk-free asset/account s.t. all the money must be invested in these risky assets, then:

1. What do I count as my risk-free interest rate (if any, given that I am working with two pure risky assets) under this numéraire?

2. Is it correct to claim that I cannot price all payoff types because the market is not complete due to lack of riskless asset, but I am I able to price, say, a spread option because it just so happens we can hedge it? If so, why are we even able to use this change of numéraire in the first place? How are we able to detect whether a particular type of payoff can be hedged or not? Is $|\rho|<1$ a sufficient condition for completeness of the market? What conditions are necessary and/or sufficient?

3. Is it possible to construct a riskless process $\mathrm{d}B_t=r_tB_t\mathrm{d}t$ via a self-financing portfolio replicated by the two assets? I have a feeling it is impossible in incomplete markets.

Answer 1

Under the risk-neutral measure both stocks follow the GBMs \begin{align} S^{(i)}_t=S^{(i)}_0\exp\left((r-q_i)t+\sigma_iW^{(i)}_t-\frac{\sigma_i^2t}{2}\right)\,,\quad i=1,2\,, \end{align} where the constant $r$ is the riskless interest rate and the constant $q_i$ is stock $S^{(i)}$'s dividend rate. The Margrabe formula says that the value of the option to exchange stock $S^{(2)}$ for stock $S^{(1)}$ at time $T$ is \begin{align}\tag{1} V&=e^{-q_1T}S^{(1)}_0N(d_1)-e^{-q_2T}S^{(2)}_0N(d_2)\,,\\ d_1&=\frac{\ln(S^{(1)}_0/S^{(2)}_0)+(q_1-q_2+\sigma^2/2)T}{\sigma\sqrt{T}}\,,\\ d_2&=d_1-\sigma\sqrt{T}\,,\\ \sigma&=\sqrt{\sigma_1^2+\sigma_2^2-2\sigma_1\sigma_2\rho}\,. \end{align} It is true (and was mentioned by William Margrabe in his original paper [1]) that in this formula the riskless rate $r$ does not occur. He wrote that "this may seem puzzling". In fact we rightfully expect the Black Scholes formula with strike $K$ to

1. contain the riskless rate,

2. be a special case of the Margrabe formula (1) when $\sigma_2=0$ and $q_2=0\,.$

In fact this is the case: When $\sigma_2=0$ and $q_2=0$ then $$ S^{(2)}_T=S^{(2)}_0e^{rT}\,. $$ When $S^{(2)}_T$ equals the Black-Scholes strike $K$ then

\begin{align} S_0^{(2)}=e^{-rT}K\,,\quad\quad d_1=\frac{\ln(S^{(1)}_0/K)+(q_1+r+\sigma^2/2)T}{\sigma\sqrt{T}}\,,\quad\quad\sigma=\sigma_1 \end{align} and (1) becomes $$ V=e^{-q_1T}S^{(1)}_0N(d_1)-e^{-rT}KN(d_2) $$ as expected.

Further remarks:

1. The model with two stocks, each following a GBM is complete. This means that every payoff that depends on the two stocks at maturity $T$ can be replicated with a self financing strategy in the two stocks. See [1].

2. The fact that we don't need the riskless asset $B_t=e^{rt}$ in the replication of the payoff $(S^{(1)}_T-S^{(2)}_T)^+$ does not mean that we are 'lacking' it. It is not needed. That's all.

3. An example of an incomplete model would be one where the stocks follow more complicated processes such as having stochastic volatility, or jumps with stochastic jump sizes.

[1] W. Margrabe, The value of an option to exchange one asset for another. Journal of Finance, Vol. 33, No. 1 (March 1978), 177-186.

Answer 2

Since you have asked in your title specifically about the case with no bank account, I would like to provide a slightly different answer than Kurt G. You have correctly inferred that then the model is incomplete. The $T$-claim $\mathcal{X} = 1$, i.e. a zero-coupon bond, cannot be replicated. Alternatively, note that there is no unique martingale measure. You can move the drift term from the normalized asset $\tilde{S}^{(2)}$ into either $W^{(1)}$ or $W^{(2)}$ to make $\tilde{S}^{(2)}$ a martingale under the corresponding changed measure $\mathbb{Q}$, in fact you have whole line $\{(\varphi_1, \varphi_2)\} \subset \mathbb{R}^2$ of possible such Girsanov kernels.

However, there is still a family of claims that can be priced (e.g. linear combinations of $S^{(1)}_T, S^{(2)}_T$). The spread option turns out to be one such claim. Namely, with $\mathcal{X} = \left(S^{(1)}_T - S^{(2)}_T\right)^+$, we have the discounted claim $$ \tilde{\mathcal{X}} := \frac{\mathcal{X}}{S^{(1)}_T} = \frac{\left(S^{(1)}_T - S^{(2)}_T\right)^+}{S^{(1)}_T} = \left(1 - \tilde{S}^{(2)}_T\right)^+. $$

Replicating $\mathcal{X}$ is equivalent to replicating $\tilde{\mathcal{X}}$ in the discounted economy $(\tilde{S}^{(1)}, \tilde{S}^{(2)}) = (1, \tilde{S}^{(2)})$. There, $\tilde{\mathcal{X}}$ is just a put option with strike $1$, so it can be replicated by the results for the Black-Scholes model. Its price is of course $$ \Pi_0\left(\tilde{\mathcal{X}}\right) = \mathbb{E}^{\mathbb{Q}_1}\left[\left(1 - \tilde{S}^{(2)}_T\right)^+\right], $$ and the price of $\mathcal{X}$ is obtained by multiplying by $S^{(1)}_0$.

The answer to question 2 is now straightforward. Any claim such that the discounted claim can be hedged in the discounted economy is replicable. Specifically, if the claim is simple and a function of $\tilde{S}^2_T$ and such that the corresponding Black-Scholes PDE with this function as the boundary condition emits a $C^{1, 2}$-solution, the hedging portfolio can be constructed as in the Black-Scholes framework.