# Revisiting risk-neutral pricing of options

This is post #6 on the binomial option pricing model. The purpose of post #6:

Post #6: To revisit the notion of risk-neutral pricing. The idea of risk-neutral pricing is that the binomial option pricing formula can be interpreted as a discounted expected value. In risk-neutral pricing, the option value at a given node is a discounted expected payoff to the option calculated using risk-neutral probabilities and the discounting is done using the risk-free interest rate. Then the price of the option is calculated by working backward from the end of the binomial tree to the front. Even though the risk-neutral probabilities are not the true probabilities of the up and down moves of the stock, option pricing using risk-neutral probabilities is the simplest and easiest pricing procedure and more importantly produces the correct option price. In this post, we examine why this is the case.

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The binomial option pricing formula

In the post #1 on the binomial option pricing model, the following option pricing formula is derived (formula (4) in that post).

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$\displaystyle C=\Delta S + B=e^{-r h} \biggl(C_u \ \frac{e^{(r-\delta) h}-d}{u-d} +C_d \ \frac{u-e^{(r-\delta) h}}{u-d} \biggr) \ \ \ \ \ \ \ \ \ \ \ \ \ \ (1)$
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The formula has the appearance of a discounted expected value. The expected value refers to the result inside the parentheses, which is the expected value of the option value $C_u$ (when stock price goes up) and the option value $C_d$ (when stock price goes down). The calculation uses the probabilities $p^*$ and $1-p^*$:

$\displaystyle p^*=\frac{e^{(r-\delta) h}-d}{u-d} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (2)$

$\displaystyle 1-p^*=\frac{u-e^{(r-\delta) h}}{u-d}$

The values of $p^*$ and $1-p^*$ sum to 1 and are positive (discussed in the post #2 on the binomial option pricing model). Thus they can be interpreted as probabilities. The value inside the parentheses in (1) can thus be interpreted as the expected value of the option payoff in the next period that follows a given node. The formula (1) uses the risk-free rate to discount the expected value back to that given node. Using this formula, the price of the option is calculated by working backward from the end of the binomial tree to the front. Using formula (1) in this recursive fashion is called the risk-neutral pricing.

From a computational standpoint, formula (1) is clear. Something is peculiar about the expected value calculation and the discounting in formula (1). The expected value is calculated using $p^*$ and $1-p^*$. What is $p^*$? Is it really the probability that the stock will go up? There is no reason to believe that $p^*$ is the true probability of an up move in the stock price in one period in the binomial tree. Why is the true probability of stock price movement not used?

On the other hand, the expected value is counted from one period to the previous period using the risk-free rate. In the earlier posts on the binomial pricing model, we see that an option is equivalent to a leverage investment in the stock (e.g. a call is equivalent to borrowing the amount $B$ to partly finance the purchase of $\Delta$ shares). Thus an option is riskier than the stock. It is natural to think that discounting the value of an option should be done using the risk-free rate and instead using a rate of return equivalent to the option.

Our goal in this post is to show that the risk-neutral pricing approach produces the same option price as from using the more standard approach of using a true probability of a stock price up move and using a realistic discount rate. Even though using the more standard approach is possible, it is more cumbersome. Thus the risk-neutral pricing approach is easy to implement and produces the correct price. There is no reason to not use risk-neutral pricing.

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The risk-neutral world

Let’s look at the implication of investing in a risk-neutral world. Imagine a world where investors are indifferent between a sure thing and a risky investment as long as both investments have the same expected value. For example, one investment pays $25 with certainty. Another investment with equally likely payoff of$50 or 0. Both investments have the same expected value but the second one is much riskier. Normally a risk premium is needed in order to entice a risk-averse investor to hold the second investment. In a risk-neutral world, investors are indifferent between these two investment choices. We further assume that in a risk-neutral world investors are willing to hold risky assets without a risk premium, i.e. risky assets such as stock are expected to earn at the risk-free rate. Let’s see what happens when stock are expected to earn at the risk-free rate. Thus the end-of-period value of the stock is $e^{r h} S$ if $S$ is the initial stock price. Here $r$ is the annual risk-free rate and $h$ is the length of a period in years. Solving for $p^*$ in the following equation $\displaystyle p^* \ uS e^{\delta h}+(1-p^*) \ dS e^{\delta h}=e^{r h} S \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (3)$ produces the following answer: $\displaystyle p^*=\frac{e^{(r-\delta) h}-d}{u-d} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (4)$ which is exactly the risk-neutral probability of an up stock move in formula (2) above. Thus $p^*$ is the probability of an increase in the stock price in the scenario that the stock is expected to earn the risk-free rate. This is the reason that $p^*$ is called the risk-neutral probability of a up move in stock price. Thus the risk-neutral pricing procedure is the realistic method for pricing options in a risk-neutral world. But we do not live in a risk-neutral world. Most investors will demand a risk premium in order to bear risk. We show that risk-neutral pricing is also a realistic method of pricing in a world where investors are risk-averse. When we use risk-neutral pricing formula to price options, we are not saying that every investor is risk-neutral. Risk-neutral pricing is only an interpretation to formula (1). The best reason for using it is that it gives the correct result and is much easier to implement as compared to the more standard approach discussed below. ___________________________________________________________________________________ A more realistic investment world Suppose that investors do care about risk. As a result, we want to calculate an expected value of payoff using true probability of stock price movements and using the expected rate of return of the option to discount the expected value of payoff. To derive the true probability of an up stock move, suppose that the continuously compounded expected return on the stock is $\alpha$. Solve for $p$ in the following equation $\displaystyle p \ uS e^{\delta h}+(1-p) \ dS e^{\delta h}=e^{\alpha h} S \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (5)$ yields the following answer: $\displaystyle p=\frac{e^{(\alpha-\delta) h}-d}{u-d} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (6)$ For $p$ and $1-p$ to be between 0 and 1, the rate of return $\alpha$ must be compatible with the stock price movement factors $u$ and $d$. Specifically, we must have $\displaystyle d < e^{(\alpha-\delta) h} < u$. Given that we know $\alpha$, the expected return of the stock, we have just derived $p$, which is the probability of the stock going up. The following is then the expected payoff of the option in the next period: \displaystyle \begin{aligned} C&=p \ C_u + (1-p) \ C_d \\&=\frac{e^{(\alpha-\delta) h}-d}{u-d} \ C_u+\frac{u-e^{(\alpha-\delta) h}}{u-d} \ C_d \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (7) \end{aligned} The value of $C$ belongs to the next period. So we need to discount it back to the present period (or the present node in the tree). Let’s say the discount rate is $\gamma$. Then the following equation is satisfied: $\displaystyle e^{\gamma h}=\frac{S \Delta}{S \Delta+B} \ e^{\alpha h}+\frac{B}{S \Delta+B} \ e^{r h} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (8)$ Recall that an is equivalent to the portfolio of holding $\Delta$ shares of stock and the amount $B$ in lending (this is called the replicating portfolio). The make-up of the replicating portfolio is determined from the idea of replication: equating the option values and the values of the replicating portfolio, i.e. from solving the following equations. So the determination of $\Delta$ and $B$ has nothing to do with $p$ or $\alpha$. $\text{ }$ $\displaystyle \text{ } \left\{\begin{matrix} \displaystyle \Delta \times (dS \ e^{\delta h}) + B \ e^{r h}=C_d&\ \ \ \ \ \ \text{ }& \\ \text{ }&\text{ } \\ \Delta \times (uS \ e^{\delta h}) + B \ e^{r h}=C_u&\ \ \ \ \ \ \text{ } \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (9) \end{matrix}\right.$ $\text{ }$ The right hand side of (8) is the expected return of the replicating portfolio. The right hand side is simply the weighted average of the return of the $\Delta$ shares of stock and the amount $B$ in lending. Then $\gamma$ can be determined from solving equation (8) for $\gamma$. Once $\gamma$ is known, the option price $C$ is: $\displaystyle C=e^{-\gamma h} \ \biggl(\frac{e^{(\alpha-\delta) h}-d}{u-d} \ C_u + \frac{u-e^{(\alpha-\delta) h}}{u-d} \ C_d \biggr) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (10)$ Let’s recap the journey it takes to get the option price in (10). We assume an expected rate of return $\alpha$ for the stock in question, which leads to a probability $p$ for an up movement in the stock. The actual probability $p$ allows us to compute the expected option value in (7). To find the expected rate of return of the option, we take the weighted average of the returns of the stock and lending in the replcating portfolio. Then equation (1) gives the discounted value of the expected option value. One peculiar thing happens in the process of obtaining the answer in (10). We can obtain $\Delta$ and $B$ by solving the equations in (9). Then we would have obtained the option price $C=\Delta S+B$. Doing so does not require knowing $\alpha$, the expected return of the stock, or $p$, the real probability of an up move in stock price. If the goal is to obtain the option price, the steps for obtaining $p$ and $\gamma$ are redundant! The ultimate reason that $\alpha$, $p$ and $\gamma$ are not necessary is that the option price in (10) is the same as the option price obtained from using risk-neutral pricing, i.e. equation (1). We verify this fact in the next section. Then we examine some examples. ___________________________________________________________________________________ Option valuation using true probabilities Recall that the risk-neutral pricing formula (1) is identical to $C=\Delta S+B$. With a little bit of algebraic manipulation, we show that the option price in (10) is identical to $C=\Delta S+B$. First equation (10) is identical to the following: $\displaystyle e^{-\gamma h} \ \biggl(\frac{e^{(r-\delta) h}-d}{u-d} \ C_u + \frac{u-e^{(r-\delta) h}}{u-d} \ C_d + \frac{e^{(\alpha-\delta) h}-e^{(r-\delta) h}}{u-d} (C_u-C_d)\biggr) \ \ \ \ \ (11)$ where $\displaystyle e^{-\gamma h}=\frac{S \Delta+B}{S \Delta e^{\alpha h}+B e^{r h}}$. We show that the content within the big parentheses in (11) is the same as $S \Delta e^{\alpha h}+B e^{r h}$. Then (11) is identical to $C=\Delta S+B$. Based on the risk-neutral pricing formula (1), the first two terms inside the parentheses in (11) can be rewritten as: $\displaystyle e^{r h} (\Delta S+B)=\frac{e^{(r-\delta) h}-d}{u-d} \ C_u + \frac{u-e^{(r-\delta) h}}{u-d} \ C_d$ Denote the content inside the parentheses in (11) by $A$, we have the following derivation: \displaystyle \begin{aligned} A &=\frac{e^{(r-\delta) h}-d}{u-d} \ C_u + \frac{u-e^{(r-\delta) h}}{u-d} \ C_d + \frac{e^{(\alpha-\delta) h}-e^{(r-\delta) h}}{u-d} (C_u-C_d) \\&=e^{r h} (\Delta S+B) + \frac{e^{(\alpha-\delta) h}-e^{(r-\delta) h}}{u-d} (C_u-C_d) \\&=e^{r h} (\Delta S+B) + (e^{\alpha h}-e^{r h}) \ e^{-\delta h} \frac{C_u-C_d}{u-d} \\&=e^{r h} (\Delta S+B) + (e^{\alpha h}-e^{r h}) \ \Delta S \\&=S \Delta e^{\alpha h}+B e^{r h} \end{aligned} The above derivation shows that the content inside the big parentheses in (11) is identical to $S \Delta e^{\alpha h}+B e^{r h}$. This means that (10) is identical to $C=\Delta S + B$. Therefore it is not necessary to use real stock price probability and real discount rate to calculate the option price. When we do, we know that the result is the same as from using the risk-neutral pricing method. ___________________________________________________________________________________ Example We now examine examples to illustrate the point that risk-neutral pricing and valuation approach using true probabilities and true discount rate produce the same option price. We take two examples from previous posts and compare the two valuation approaches. Example 1 This is Example 1 in the post #4 on the binomial option pricing model. The example is to price a 1-year call stock option with strike price55. For the other details of this example, see Example 1 in the other post. Price this call option assuming that the expected annual rate of return of the stock is $\alpha=$ 12%. Compare this pricing with the risk-neutral pricing.

The following is the binomial tree obtained by using risk-neutral pricing.

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Example 1: option valuation using risk-neutral pricing
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$\displaystyle \begin{array}{lllll} \displaystyle \text{Initial Price} & \text{ } & \text{Period 1} & \text{ } & \text{Period 2} \\ \text{ } & \text{ } & \text{ } & \text{ } & \text{ } \\ \text{ } & \text{ } & \text{ } & \text{ } & S_{uu}=\ 95.45058 \\ \text{ } & \text{ } & \text{ } & \text{ } & C_{uu}=\ 40.45058 \\ \text{ } & \text{ } & \text{ } & \text{ } & \text{ } \\ \text{ } & \text{ } & S_u=\ 75.67718 & \text{ } & \text{ } \\ \text{ } & \text{ } & C_u=\ 21.76625 & \text{ } & \text{ } \\ \text{ } & \text{ } & \Delta=1.0 & \text{ } & \text{ } \\ \text{ } & \text{ } & B=- \ 53.91093 & \text{ } & \text{ } \\ \text{ } & \text{ } & \text{ } & \text{ } & \text{ } \\ S= \ 60 & \text{ } & \text{ } & \text{ } & S_{ud}=\ 62.44865 \\ C=\ 11.30954 & \text{ } & \text{ } & \text{ } & C_{ud}=\ 7.44865 \\ \Delta=0.70710 & \text{ } & \text{ } & \text{ } & \text{ } \\ B=- \ 31.11633 & \text{ } & \text{ } & \text{ } & \text{ } \\ \text{ } & \text{ } & \text{ } & \text{ } & \text{ } \\ \text{ } & \text{ } & S_d=\ 49.51187 & \text{ } & \text{ } \\ \text{ } & \text{ } & C_d=\ 3.26482 & \text{ } & \text{ } \\ \text{ } & \text{ } & \Delta=0.34498 & \text{ } & \text{ } \\ \text{ } & \text{ } & B=- \ 13.81577 & \text{ } & \text{ } \\ \text{ } & \text{ } & \text{ } & \text{ } & \text{ } \\ \text{ } & \text{ } & \text{ } & \text{ } & S_{dd}=\ 40.85710 \\ \text{ } & \text{ } & \text{ } & \text{ } & C_{dd}=\ 0 \end{array}$
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The following shows the calculation for the probability associated with the expected rate of stock return $\alpha=$ 0.12.

$u=e^{(r-\delta) h+\sigma \sqrt{h}}=e^{(0.04-0) 0.5+0.3 \sqrt{0.5}}=$ 1.261286251

$d=e^{(r-\delta) h-\sigma \sqrt{h}}=e^{(0.04-0) 0.5-0.3 \sqrt{0.5}}=$ 0.825197907

$\displaystyle p=\frac{e^{(\alpha-\delta) h}-d}{u-d}=\frac{e^{(0.12-0) 0.5}-d}{u-d}=$ 0.542639222

$\displaystyle 1-p=$ 0.457360778

Next find the rate of return of option at each node. Because the compositions of the replicating portfolio are different across the nodes, the option rate of return $\gamma$ is different.

\displaystyle \begin{aligned}e^{\gamma (0.5)}&=\frac{75.67718 (1)}{75.67718 (1)-53.91093} \ e^{0.12 (0.5)}+\frac{-53.91093}{75.67718 (1)-53.91093} \ e^{0.04 (0.5)} \\&=1.164959169 \end{aligned}

$\displaystyle \gamma=2 \ \text{ln}(1.164959169)=$ 0.305372077 (at the node for stock price $S_u$)

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\displaystyle \begin{aligned}e^{\gamma (0.5)}&=\frac{49.51187 (0.34498)}{49.51187 (0.34498)-13.81577} \ e^{0.12 (0.5)}+\frac{-13.81577}{49.51187 (0.34498)-13.81577} \ e^{0.04 (0.5)} \\&=1.238024651 \end{aligned}

$\displaystyle \gamma=2 \ \text{ln}(1.238024651)=$ 0.427034172 (at the node for stock price $S_d$)

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\displaystyle \begin{aligned}e^{\gamma (0.5)}&=\frac{60 (0.70710)}{60 (0.70710)-31.11633} \ e^{0.12 (0.5)}+\frac{-31.11633}{60 (0.70710)-31.11633} \ e^{0.04 (0.5)} \\&=1.176388903 \end{aligned}

$\displaystyle \gamma=2 \ \text{ln}(1.176388903)=$ 0.324898989 (at the initial node)

We are now ready to calculate the option value at each node.

\displaystyle \begin{aligned} C_u&=e^{-0.305372077 (0.5)} \ \biggl(0.542639222 \ (40.45058041) + 0.457360778 (7.448646452) \biggr) \\&=21.76624803 \end{aligned}

\displaystyle \begin{aligned} C_d&=e^{-0.427034172 (0.5)} \ \biggl(0.542639222 \ (7.448646452) + 0.457360778 (0) \biggr) \\&=21.76624803 \end{aligned}

\displaystyle \begin{aligned} C&=e^{-0.0.324898989 (0.5)} \ \biggl(0.542639222 \ (21.76624803) + 0.457360778 (3.264820059) \biggr) \\&=11.3095427 \end{aligned}

Note that the option price produced from the alternative approach is the same as from the risk-neutral approach. The following binomial tree shows all the results.

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Example 1: option valuation using true probabilities
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$\displaystyle \begin{array}{lllll} \displaystyle \text{Initial Price} & \text{ } & \text{Period 1} & \text{ } & \text{Period 2} \\ \text{ } & \text{ } & \text{ } & \text{ } & \text{ } \\ \text{ } & \text{ } & \text{ } & \text{ } & S_{uu}=\ 95.45058 \\ \text{ } & \text{ } & \text{ } & \text{ } & C_{uu}=\ 40.45058 \\ \text{ } & \text{ } & \text{ } & \text{ } & \text{ } \\ \text{ } & \text{ } & S_u=\ 75.67718 & \text{ } & \text{ } \\ \text{ } & \text{ } & C_u=\ 21.76625 & \text{ } & \text{ } \\ \text{ } & \text{ } & \gamma=0.305372077 & \text{ } & \text{ } \\ \text{ } & \text{ } & \text{ } & \text{ } & \text{ } \\ \text{ } & \text{ } & \text{ } & \text{ } & \text{ } \\ S= \ 60 & \text{ } & \text{ } & \text{ } & S_{ud}=\ 62.44865 \\ C=\ 11.30954 & \text{ } & \text{ } & \text{ } & C_{ud}=\ 7.44865 \\ \gamma=0.324898989 & \text{ } & \text{ } & \text{ } & \text{ } \\ \text{ } & \text{ } & \text{ } & \text{ } & \text{ } \\ \text{ } & \text{ } & \text{ } & \text{ } & \text{ } \\ \text{ } & \text{ } & S_d=\ 49.51187 & \text{ } & \text{ } \\ \text{ } & \text{ } & C_d=\ 3.26482 & \text{ } & \text{ } \\ \text{ } & \text{ } & \gamma=0.427034172 & \text{ } & \text{ } \\ \text{ } & \text{ } & \text{ } & \text{ } & \text{ } \\ \text{ } & \text{ } & \text{ } & \text{ } & \text{ } \\ \text{ } & \text{ } & \text{ } & \text{ } & S_{dd}=\ 40.85710 \\ \text{ } & \text{ } & \text{ } & \text{ } & C_{dd}=\ 0 \end{array}$
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Example 2
This is Example 1 in the post #5 on the binomial option pricing model. Example 1 in that post is to price a 6-month American put option in a 3-period binomial tree. The strike price of the option is $45. The following shows the specifics of the binomial trees. • The initial stock price is$40.
• The annual risk-free interest rate is $r=$ 0.05.
• The stock pays no dividends.
• The annual countinuously compounded standard deviation of stock return is $\sigma=$ 0.3.

Price this 6-month put option assuming that the expected annual rate of return of the stock is $\alpha=$ 15%. Compare with the results from Example 1 in the previous post.

The following is the binomial tree from the previous post showing option price based on risk-neutral pricing. Since this is for an American option, early exercise is permitted if it is optimal to do so. There are two nodes in the following tree where early exercise is optimal (the option value is in bold). For option valuation using true probabilities, the calculation at each node is also an either-or proposition, i.e. the option value is either the value from the discounted expected value using probabilities or the value from early exercise.

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Example 2 – Option valuation using risk-neutral pricing (from a previous post)
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$\displaystyle \begin{array}{llll} \displaystyle \text{Initial Price} & \text{Period 1} & \text{Period 2} & \text{Period 3} \\ \text{ } & \text{ } & \text{ } & \text{ } \\ \text{ } & \text{ } & \text{ } & S_{uuu}=\ 59.22258163 \\ \text{ } & \text{ } & \text{ } & C_{uuu}=\ 0 \\ \text{ } & \text{ } & S_{uu}=\ 51.96108614 & \text{ } \\ \text{ } & \text{ } & C_{uu}=\ 0 & \text{ } \\ \text{ } & \text{ } & \text{ } & S_{uud}=\ 46.3561487 \\ \text{ } & \text{ } & \text{ } & C_{uud}=\ 0 \\ \text{ } & S_u=\ 45.58994896 & \text{ } & \text{ } \\ \text{ } & C_u=\ 2.41285153 & \text{ } & \text{ } \\ S=\ 40 & \text{ } & S_{ud}=S_{du}=\ 40.67225322 & \text{ } \\ C=\ 6.024433917 & \text{ } & C_{ud}=\ 4.585624746 & \text{ } \\ \text{ } & S_d=\ 35.68528077 \text{ } & \text{ } \\ \text{ } & \mathbf{C_d=\ 9.314719233} \text{ } & \text{ } \\ \text{ } & \text{ } & \text{ } & S_{udd}=\ 36.28501939 \\ \text{ } & \text{ } & \text{ } & C_{udd}=\ 8.714980615 \\ \text{ } & \text{ } & S_{dd}=\ 31.83598158 & \text{ } \\ \text{ } & \text{ } & \mathbf{C_{dd}=\ 13.16401842} & \text{ } \\ \text{ } & \text{ } & \text{ } & S_{ddd}=\ 28.40189853 \\ \text{ } & \text{ } & \text{ } & C_{ddd}=\ 16.59810147 \\ \end{array}$

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The pricing results from using true probabilities will be identical to the risk-neutral pricing results. We will not show a diagram for the binomial tree. Instead, we show the calculation on some nodes.

The following shows the calculation for the probability associated with the expected rate of stock return $\alpha=$ 0.15.

$u=e^{(r-\delta) h+\sigma \sqrt{h}}=e^{(0.05-0) \frac{0.5}{3} +0.3 \sqrt{\frac{0.5}{3}}}=$ 1.139748724

$d=e^{(r-\delta) h-\sigma \sqrt{h}}=e^{(0.05-0) \frac{0.5}{3} -0.3 \sqrt{\frac{0.5}{3}}}=$ 0.892132019

$\displaystyle p=\frac{e^{(\alpha-\delta) h}-d}{u-d}=\frac{e^{(0.15-0) \frac{0.5}{3}}-d}{u-d}=$ 0.537859921

$\displaystyle 1-p=$ 0.462140079

The following is the calculation at the node where the stock price is $S_{ud}=$ 40.67225. \displaystyle \begin{aligned}e^{\gamma \frac{0.5}{3}}&=\frac{40.67225 (-0.86534)}{40.67225 (-0.86534)+39.78107} \ e^{0.15 \frac{0.5}{3}}+\frac{39.78107}{40.67225 (-0.86534)+39.78107} \ e^{0.05 \frac{0.5}{3}} \\&=0.878297298 \end{aligned} $\displaystyle \gamma=6 \ \text{ln}(0.878297298)=$ -0.778620804 \displaystyle \begin{aligned} C_{ud}&=e^{0.778620804 \frac{0.5}{3}} \ \biggl(0.537859921 \ (0) + 0.462140079 \ (8.714980615) \biggr) \\&=4.585624753 \end{aligned} At the node where the stock price is $S_{dd}$ = 31.83598, early exercise is optimal. There is no need to calculate the option value here using true probabilities. However, it is possible to calculate $\gamma$ if it is desirable to do so. This is done by solving for $\gamma$ in equation (10). $\displaystyle 13.16401842=e^{-\gamma \frac{0.5}{3}} \ \biggl(0.537859921 \ (8.714980615) + 0.462140079 \ (16.59810147) \biggr)$ $\gamma=$ -0.379059524 At the node where stock price is $S_d=$35.68528, early exercise is also optimal. So the option value is not obtained by a discounted expected value (risk-neutral probabilities or otherwise). We now look at the initial node.

\displaystyle \begin{aligned}e^{\gamma \frac{0.5}{3}}&=\frac{40 (-0.69683)}{40 (-0.69683)+33.89762} \ e^{0.15 \frac{0.5}{3}}+\frac{33.89762}{40 (-0.69683)+33.89762} \ e^{0.05 \frac{0.5}{3}} \\&=0.929959775 \end{aligned}

$\displaystyle \gamma=6 \ \text{ln}(0.929959775)=$ -0.435683676

\displaystyle \begin{aligned} C&=e^{0.435683676 \frac{0.5}{3}} \ \biggl(0.537859921 \ (2.41285153) + 0.462140079 \ (9.314719233) \biggr) \\&=6.024433917 \end{aligned}

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Remarks

The proof shown above and the calculation in the two examples show that option valuation using true probabilities based on the expected rate of return of the stock is not necessary. Risk-neutral pricing will produce the same results with much simpler and easier calculation.

One peculair point about option valuation using true probability and true discount rate that should be mentioned again. To discount the expected value of the option, we need to find the rate of return $\gamma$ of the option at each node. To find the rate of return $\gamma$ for the option at each node, we make use of the replicating portfolio $\Delta$ and $B$. By knowing $\Delta$ and $B$, we can derive the option value directly by $C=\Delta S + B$. Then we are done. Unless the goal is to find the expected return $\gamma$ of an option, the valuation approach of using real probability and actual rate of return of option is pointless.

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Practice problems

$\copyright \ \ 2015 \ \text{Dan Ma}$