Prove the Euro call option value has positive relationship with the risk-free rate under discrete time model (Binomial tree model)

Question

Could anyone show me how to prove that the European call option value has a positive relationship with the risk-free rate in a two-step binomial model with strike price K and different risk neutral probability q between each step? I know that in continuous model, the rho(call) is positive which shows the positive relationship but I am only familiar with the equations and don't really know how to prove that. Thank you.

Answer 1

Assume that the price of the next step are $u,d$, with probability $p,1-p$. The the discounted payoff is $e^{-r}(p(u - K)^+ + (1-p)(d-K)^+)$. Now suppose the interest rate $r$ is increased by $\Delta r$. Then the new discounted payoff would be $e^{-r - \Delta r}(p(e^{\Delta r}u - K)^+ + (1-p)(e^{\Delta r}d-K)^+) = e^{-r}(p(u - e^{-\Delta r}K)^+ + (1-p)(d-e^{-\Delta r}K)^+)$.

Since $e^{-\Delta r}K < K$, the new discounted payoff would be greater than the original one.

Answer 2

I was not able to reproduce @MainCom's example on a spreadsheet, hence I gave it a shot myself:

$$ \begin{align} C(r)&:=e^{-r\Delta t}\left(\frac{e^{r\Delta t}-D}{U-D}\left(U-K\right)^++\frac{U-e^{r\Delta t}}{U-D}\left(D-K\right)^+\right)\\ \Rightarrow C(r+\rho)&=e^{-r\Delta t-\rho\Delta t}\left(\frac{e^{r\Delta t + \rho\Delta t}-D}{U-D}\left(U-K\right)^++\frac{U-e^{r\Delta t + \rho\Delta t}}{U-D}\left(D-K\right)^+\right)\\ &=e^{-r\Delta t}\left(\frac{e^{r\Delta t}-De^{-\rho\Delta t}}{U-D}\left(U-K\right)^++\frac{Ue^{-\rho\Delta t}-e^{r\Delta t }}{U-D}\left(D-K\right)^+\right)\\ \end{align} $$ Let us now compare the first probability term $p(r):=\frac{e^{r\Delta t}-D}{U-D}$. As the denominator is independent of $r$, we simply compare

$$ \left(e^{r\Delta t}-De^{-\rho \Delta t}\right) \quad -\quad \left( e^{r\Delta t}-D\right) $$ Clearly,this is $D(1-e^{-\rho\Delta t})>0$, i.e. this term is increasing (and decreasing for $1-p$).