# Calculating the annual return on an option using a replicating porfolio

I am self-studying and encountered the following problem:

My idea was to calculate the price of the put using a replicating portfolio, then use the formula:

$$Pe^{\gamma h} = S\Delta e^{\alpha h} + \beta e^{rh}$$ to solve for $\gamma$, where $P$ is the put premium, $\alpha$ is the continuously compounded return on the stock, $\beta$ is the amount lent in the replicating portfolio, and $\gamma$ is the continuously compounded return on the option.

In this case $$\Delta = \frac{P_u - P_d}{S(u - d)}e^{-\delta h} = \frac{0 - 11.84485}{60.41285 - 33.15522}e^{0\cdot1} = -0.4345506$$ and

$$\beta = \frac{uP_d - dP_u}{u - d}e^{-rh} = \frac{1.40495(11.84485) - 0.77105(0)}{1.40495 - 0.77105}e^{-0.04\cdot1} = 26.22307,$$

giving a put premium of $$P = \Delta\cdot{}S + \beta = -0.4345506\cdot43 + 26.22307 = 7.53739.$$

Since I did not arrive at the same put premium as the textbook, I stopped here. I'm not sure where I am making my mistake at.

I know my formula for $\beta$ is correct because:

A successful replicating portfolio must satisfy: $P_d = \Delta S_d e^{\delta h} + \beta e^{rh}$ and $P_u = \Delta S_u e^{\delta h} + \beta e^{rh}$.

Then $\Delta = \frac{(P_d - \beta e^{rh})}{S_d}e^{-\delta h}$ and $\Delta = \frac{(P_u - \beta e^{rh})}{S_u}e^{-\delta h}$.

Therefore $(P_d - \beta e^{rh})e^{-\delta h} S_u = (P_u - \beta e^{rh})e^{-\delta h} S_d$.

Noting that $S_u = S_0\cdot u$ and $S_d = S_0 \cdot d$, we can eliminate $S_0$ and write

$P_d e^{-\delta h} u - \beta e^{rh}e^{-\delta h} u = P_u e^{-\delta h}d - \beta e^{rh - \delta h} d$

This implies that $P_d u e^{-\delta h} - P_u d e^{-\delta h} = \beta(e^{rh}e^{-\delta h}u - e^{rh}e^{-\delta h}d)$.

Hence $\beta = \frac{(P_d u - P_u d)e^{-\delta h}}{(u - d)e^{rh}e^{-\delta h}} = \frac{P_d u - P_u d}{u - d}e^{-rh}$

• What is $\delta$ in the formula for $\Delta$? Commented Sep 13, 2016 at 22:47
• $\delta$ is the continuously compounded dividend yield. Commented Sep 15, 2016 at 14:57

Your computation of $\Delta$ is correct. However, your computation of the cash amount is wrong. You choose the cash amount $\beta$ that you need to initially lend or borrow such that in the up state, the following holds

$$P_u = \Delta S_u e^{\delta h} + \beta e^{r h}.$$

We get

\begin{eqnarray} \beta & = & \left( P_u - \Delta S_u e^{\delta h} \right) e^{-r h}\\ & = & 0.4345506 \cdot 60.41285 \cdot e^{-0.04}\\ & = & 25.223067. \end{eqnarray}

Thus,

$$P = \Delta S + \beta = 6.537391$$

• I understand your solution and why it provides the correct $\beta$. One thing that I still don't understand is why my formula, $$\beta = \frac{uP_d - dP_u}{u - d}e^{-rh}$$ does not apply here and give the same $\beta$ that you calculated. That formula is what the textbook provided. Commented Sep 15, 2016 at 14:58
• I showed you how to obtain the correct expression for $\beta$ and your formula for it looks different. So the question is not really why your formula doesn't apply but rather how you obtained it since it seems wrong. Commented Sep 15, 2016 at 15:00
• Do you understand how the textbook derived it? Is it in a different context / sure it applies here? Don't just take "formulas from the textbook" for granted. Commented Sep 15, 2016 at 16:38
• The value of the replicating formula at time $h$, with stock price $S_h$, is $\Delta S_h + e^{rh}\beta$. At the prices $S_h = S_d$ and $S_h = S_u$, a replicating portfolio must satisfy: $\Delta \cdot S_d \cdot e^{\delta h} + (\beta \cdot e^{rh}) = P_d$ and $\Delta \cdot S_u \cdot e^{\delta h} + (\beta \cdot e^{rh}) = P_u$. The two unknowns $\Delta$ and $\beta$ allow us to solve for $$\beta = e^{-rh} \cdot \frac{u P_d - d P_u}{u - d}.$$ Commented Sep 16, 2016 at 0:47
• You are right - my mistake. But at least now I finally spotted what went wrong in the first place. You have the correct formula for $\beta$ but forget to apply discounting when computing the value - i.e. you don't take into account the $e^{-r h}$ term. Commented Sep 16, 2016 at 15:17