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# The Binomial Models

After completing this reading, you should be able to:

1. Explain the concept of no-arbitrage and the risk-neutral approach to valuing derivative securities
• Understand the concept of no-arbitrage when comparing actual and synthetic calls or when comparing actual and synthetic puts.
• Understand the concepts underlying the risk-neutral approach to valuing derivative securities in the context of the Binomial Option Pricing Model.
1. Use the Binomial Option Pricing Model to calculate the value of European and American call and put options, along with the value of Asian and barrier options.
• Price options under a one-period binomial model on a stock with no dividends.
• Extend the binomial model to multi-period settings for pricing both European and American call and put options.
• Extend the binomial model to other underlying assets, including stock indices with continuous dividends, currencies, and futures contracts.

## No Arbitrage and the Risk Neutral Approach to Valuing Derivative Securities

### Arbitrage Assumption

Arbitrage describes a situation where an investor can make a risk-free trading profit. In a given market, an arbitrage opportunity exists if:

1. An investor can, at time 0, enter into a deal with zero initial costs, a strictly positive probability of future profit, and zero probability of future loss.
2. An investor enters into a deal that gives him/her an immediate profit with no risk of future loss.

In other words, arbitrage is exploiting a price imbalance in the same asset between two or more markets. For derivatives, this is taking advantage of the differences in prices of a unique asset to make a risk-free profit.

If such an opportunity exists, an investor will seek to multiply his portfolios to make an unlimited profit. However, this is mostly impossible since other active participants of the market would do the same. Thus, the prices in the market will change accordingly to eliminate the arbitrage opportunity.

Calculate the value of the two-period European call option, $$C_0$$ with an exercise price of 75. #### Solution Consider the following binomial tree: $$\begin{array} {} & {} & {} & \begin{matrix} S_{UU}=120 \\ C_{UU}=120-75=45 \end{matrix} \\ {} & \begin{matrix} { S }_{ U }={90} \\ C_U=? \end{matrix} & {\Huge \begin{matrix} \nearrow \\ \searrow \end{matrix} } & {} \\ \begin{matrix} S_0=60 \\ C_0=? \end{matrix} \begin{matrix} & {} & \\ &\Huge \nearrow & \\ &\Huge \searrow & \\ & { } & \end{matrix} & {} & \begin{matrix} {} \\ \\ { } \end{matrix} & \begin{matrix} { \begin{matrix} S_{UD}=75 \\ C_{UD}=\text{max}(0,75-75)=0 \end{matrix} } \\ \\ { \begin{matrix} S_{DU}=60 \\ C_{DU}=\text{max}(0,60-75)=0 \end{matrix} } \end{matrix} \\ {} & { \begin{matrix} S_D=45 \\ C_D=? \end{matrix} } & {\Huge \begin{matrix} \nearrow \\ \searrow \end{matrix} } & {} \\ {} & {} & {} & { \begin{matrix} S_{DD}=60 \\ C_{DD}=\text{max}(0,60-75)=0 \end{matrix} } \\ \end{array}$$ The movement factor for the second period is: $$U_1=\cfrac {120}{90}=\cfrac {4}{3}$$ And $$D_1=\cfrac {75}{90}=\cfrac {5}{6}$$ Now, the risk-neutral probability is: $$\pi_1=\cfrac {e^{-r}-U_2}{U_2-D_2}=\cfrac {e^{-0.05}-\frac {4}{3}}{\frac {4}{3}-\frac {5}{6}}=0.43588$$ Now, the possible derivative values at the second period if the share prices rise is 45 and 0 then the derivative payoff at the beginning of the second period is: \begin{align*} C_U & =e^{-r} [\pi_U c_{UU_1}+(1-\pi_1)c_{UD_1} ] \\ & =e^{-0.05} [0.43588×45+(1-0.43588)×0] \\ &=18.658 \\ \end{align*} The movement factors for the second branch (when the share price drops) are equal as above and so as the risk-neutral probability and hence: $$C_D =e^{-r} [\pi_2 c_{DU_2}+(1-\pi_2)c_{DD_2} ]$$ Where: \begin{align}U_2&=\frac{60}{45}=1.333\\ D_2&=\frac{37.5}{45}=0.8333\\ \pi_2&=\frac{e^{0.05}-0.833}{1.333-0.833}\end{align} Therefore, $$C_D =e^{-0.05} [0.43588×0+(1-0.43588)×0] =0 \\$$ And so, $$C_0 =e^{-r} [\pi_U c_U+(1-\pi_U)c_D$$ Where: $$U=\frac{90}{60}=1.5,\ \ D=\frac{45}{60}=0.75\ \text{and}\ \pi_U=\frac{e^{0.05}-0.75}{1.5-0.75}=\ 0.40169$$ Thus, \begin{align*} C_D &=e^{-0.05} [0.40169×18.658+(1-0.40169)×0] \\ & =7.1293 \\ \end{align*} As you must have realized by now, when the payoffs are zero, it makes the work easier. You may ignore these values to save you time. #### Example: Pricing European Put Option using the Two-Step Binomial Model A two-year European put has a strike price of60 on a stock with a current price of $54. A model with two steps has been constructed, and it is expected that the share price will move up or down by 20% at each step. The size of each step is one year. The continuously compounded risk-free rate of return applicable over the contract term is 4% per annum. Calculate the price of the put option. #### Solution Consider the following two-step Binomial-tree: $$\begin{array} {} & {} & {} & \begin{matrix} S_{UU}=77.76 \\ P_{UU}=\text{max}(60-77.76,0)=0\end{matrix} \\ {} & \begin{matrix} { S }_{ U }={64.8} \\ P_U=? \end{matrix} & {\Huge \begin{matrix} \nearrow \\ \searrow \end{matrix} } & {} \\ \begin{matrix} S_0=54 \\ P_0=? \end{matrix} \begin{matrix} & {} & \\ &\Huge \nearrow & \\ &\Huge \searrow & \\ & { } & \end{matrix} & {} & \begin{matrix} {} \\ \\ { } \end{matrix} & \begin{matrix} { \begin{matrix} P_{UD}=75 \\ P_{UD}=\text{max}(0,60-51.84)=8.16 \end{matrix} } \\ \\ { \begin{matrix} S_{DU}=51.84 \\ P_{DU}=\text{max}(60-51.84,0)=8.16\end{matrix} } \end{matrix} \\ {} & { \begin{matrix} S_D=43.2 \\ P_D=? \end{matrix} } & {\Huge \begin{matrix} \nearrow \\ \searrow \end{matrix} } & {} \\ {} & {} & {} & { \begin{matrix} S_{DD}=37.5 \\ P_{DD}=\text{max}(60-34.56,0)=25.44 \end{matrix} } \\ \end{array}$$ We apply the same principles as with the call option. However, we need to note that the put results in a payoff when the underlying asset price falls below the strike price. $$P_0=\pi P_1U+\left(1-\pi\right)P_1D$$ This can also be written as: $$P_0=e^{-2r\Delta t}\left[\pi^2P\left(U_U\right)+2\pi\left(1-\pi\right)P\left(U_D\right)+\left(1-\pi\right)^2P(D_D)\right]$$ In this case, the same probabilities of up and down movements apply at each step: $$\pi=\frac{e^{0.04}-0.8}{1.2-0.8}=0.602$$ Substituting the formula for values: $$P_0=e^{-2\times0.04\times1}\left[{0.602}^2\times0+2\times0.602\times0.398\times8.16+{0.398}^2\times25.44)\right]=\ 7.33$$ ### Generalization of the Two-Step Binomial Tree We can generalize the two-step binomial model using Figure 3. During each time step, it either moves up to $$U \times \text{Initial value}$$ or $$D \times \text{Initial value}$$. Let $$r$$ be the risk-free rate of interest and $$\Delta t$$ years the length of a period, then we know that: $$C_1 (U)=e^{-r\Delta t} [\pi_{U_1} C_2 {U_U}+(1-\pi_{U_1 } ) C_2 (U_D ) ]…………..(1)$$ And $$C_1 (D)=e^{-r\Delta t} [\pi_{U_2} C_2 (D_U )+(1-\pi_{U_2} ) C_2 (D_D ) ]…………..(2)$$ We know also that: $$C_0=e^{-r\Delta t} [\pi_{U_0} C_1 (U)+(1-\pi_{U_0} ) C_1 (D) ] ………(3)$$ Now, if we assume that $$\pi=\pi_{U_0}=\pi_{U_1}=\pi_{U_2}$$ then, substituting equations (1) and (2) in (3), we have: $$C_0=e^{-2r\Delta t} [\pi^2 C(U_U )+2\pi (1-\pi)C(U_D )+(1-\pi)^2 C(D_D)]$$ This is consistent with the principle of risk-neutral valuation, that is the variables are $$\pi^2$$, $$2\pi(1-\pi)$$ and $$(1-\pi)^2$$ are the probabilities of reaching upper, middle, and lower nodes. This principle will hold as more nodes are added. ### Increasing the Number of Steps in the Binomial Model Binomial models with one or two steps are unrealistically simple. Assuming only one or two steps would yield a very rough approximation of the option price. In practice, the life of an option is divided into 30 or more time steps. In each step, there is a binomial stock price movement. As the number of time steps is increased, the binomial tree model makes the same assumptions about stock price behavior as the Black–Scholes–Merton model, which we will see in detail in the next chapter. When the binomial tree is used to price a European option, the price converges to the Black–Scholes–Merton price as the number of time steps is increased. Evidently, it is easy to see that constructing a binomial tree is dependent on the calculation of the option payoff and the risk-neutral probability based on the information given. As stated before, American options can be exercised before the expiry date; therefore, appropriate binomial trees should be constructed. ### Binomial Model for American-Style Options American-style options can be exercised any time before the expiration date. The Binomial model allows us to examine the payoff from the option at the end of each node and determine which node the option will result in the maximum payoff and hence, early exercise. #### Calculating the price of an American Style Option Let us consider an American style call option and consider the following payoffs from a call option $$\begin{array} {} & {} & {} & { C_2(U_U)} \\ {} & { C }_{ 1 }(U) & \begin{matrix} & { \small {\pi_{U_1}} } & \\ &\Huge \nearrow & \\ &\Huge \searrow & \\ & {\small {1-\pi_{U_1}} } & \end{matrix} & {} \\ { C }_{ 0 } \begin{matrix} & { \small {\pi_{U_0}} } & \\ &\Huge \nearrow & \\ &\Huge \searrow & \\ & {\small {1-\pi_{U_0}} } & \end{matrix} & {} & \begin{matrix} {} \\ \\ { } \end{matrix} & \begin{matrix} { { C_2(U_D)}} \\ \\ { { C_2(D_U)}} \end{matrix} \\ {} & { C }_{ 1 }(D) & \begin{matrix} & { \small {\pi_{U_2}} } & \\ &\Huge \nearrow & \\ &\Huge \searrow & \\ & {\small {1-\pi_{U_2}} } & \end{matrix} & {} \\ {} & {} & {} & { { C_2(D_D)}} \\ \end{array}$$ American style options can be exercised before expiry. Thus, to compute the price, we check for the value of the option at each node. If the option’s value is greater if exercised, we assign the value to that node. Otherwise, the option is left unexercised, and we work backward on the binomial tree as usual. At any node, the value of the option is given by: $$C=\text{max}\left(S_T-K,e^{-r}\left[\pi_Uc_U+\left(1-\pi_U\right)c_D\right]\right)$$ The maximum of the payoff and the option’s value at the node are assigned to a given node. We then proceed backward on the binomial tree as before. The same principles will apply to the corresponding American style put options. Let us consider an example to make it clear. #### Example: American Style Put Option A two-year American style put has a strike price of$60 on a stock with a current price of $54. A model with two steps has been constructed, and it is expected that the share price will move up or down by 20% at each step. The size of each step is one year. The continuously compounded risk-free rate of return applicable over the contract term is 4% per annum. Calculate the price of the put option. #### Solution Consider the following binomial tree: $$\begin{array} {} & {} & {} & \begin{matrix} \text{Node D:}\\S_{UU}=77.76 \\ P_{UU}=\text{max}(60-77.76,0)=0\end{matrix} \\ {} & \begin{matrix}B\\ { S }_{ U }={64.8} \\ P_U=? \end{matrix} & {\Huge \begin{matrix} \nearrow \\ \searrow \end{matrix} } & {} \\ \begin{matrix} A\\ S_0=54 \\ P_0=? \end{matrix} \begin{matrix} & {} & \\ &\Huge \nearrow & \\ &\Huge \searrow & \\ & { } & \end{matrix} & {} & \begin{matrix} {} \\ \\ { } \end{matrix} & \begin{matrix} { \begin{matrix} \text{Node E:}\\S_{UD}=75 \\ P_{UD}=\text{max}(0,60-51.84)=8.16 \end{matrix} } \\ \\ { \begin{matrix} \text{Node F:}\\S_{DU}=51.84 \\ P_{DU}=\text{max}(60-51.84,0)=8.16\end{matrix} } \end{matrix} \\ {} & { \begin{matrix} C\\S_D=43.2 \\ P_D=? \end{matrix} } & {\Huge \begin{matrix} \nearrow \\ \searrow \end{matrix} } & {} \\ {} & {} & {} & { \begin{matrix} \text{Node G:}\\S_{DD}=37.5 \\ P_{DD}=\text{max}(60-34.56,0)=25.44 \end{matrix} } \\ \end{array}$$ We start at the final nodes and work backward as before. At node B, the value of the option $$P_U$$ is given as: $$P_U=e^{-r\Delta t}\left(\pi P_{UU}+(1-\pi)P_{UD}\right)$$ Where: $$\pi=\frac{e^{0.04}-0.8}{1.2-0.8}=0.602$$ Therefore, $$P_U=e^{-0.04\times1}\left(0.602\times0+0.398\times8.16\right)=3.1203$$ The price of the put at this node is$3.1203.

However, the put has zero payoff at this node, i.e.

$$\max{\left(0,60-64.8\right)}=0$$

Thus it would not be optimal to exercise at this node and the value at this node will be:

$$\max{\left(K-S_T,P_U\right)}=\max{\left(0,3.1203\right)}=3.1203$$

Hence, the value assigned to node B is $$P_U=3.1203$$.

At node C, the value of the option, $$P_D$$ is given as:

$$P_D=e^{-0.04}\left(0.602\times8.16+0.398\times25.44\right)=14.459$$

The option has a payoff from the put at this node is $16.8 calculated as: $$\max{\left(K-S_T,P_D\right)}=\max{\left(16.8,14.448\right)}=16.8$$ At the initial node, A, the value of the option, $$P_0$$ is: $$e^{-0.04}\left(0.602\times3.1203+0.398\times16.8\right)=8.229$$ The option has a payoff of$6 at this node, and thus early exercise is not optimal.

The option price, $$P_0$$, will thus be:

$$P_0=\max{\left(6,8.229\right)}=8.229$$

For American-style options, we check for the value of the option at each node. If the option’s value is greater if exercised, we assign the value to that node. Otherwise, the option is left unexercised, and we work backward on the binomial tree as usual.

### Binomial Model for Barrier and Asian Options

#### Barrier Options

A barrier option is an option whose existence depends upon the underlying asset’s price reaching a predetermined barrier level. It can be either:

1. A knock-out, implying it expires worthless if the underlying exceeds a certain specified price, effectively limiting profits for the holder and losses for the writer. In other words, the right to exercise the option fades if the underlying asset price intersects the barrier at any time in the option’s lifetime. That is to say; the option price acquires a fixed price $$F$$ the moment they intersect. They are classified into:
1. Up-and-out options

These types are exercised only if  the underlying asset price is below a predetermined value during the term of the option. Then, the payoffs are given by:

$$\text{Call up}-\text{and}- \text{out option}=\begin{cases} \text{max}⁡(0,S_{ T }-X), & \text{if max}⁡(S_{ 1 },S_{ 2 },…S_{ T })\ge B \\ F, & \text{if max}⁡(S_{ 1 },S_{ 2 },…S_{ T }) < B \end{cases}$$ $$\text{Put up}-\text{and}- \text{out option}=\begin{cases} \text{max}⁡(0,X-S_{ T }), & \text{if max}⁡(S_{ 1 },S_{ 2 },…S_{ T })\ge B \\ F, & \text{if max}⁡(S_{ 1 },S_{ 2 },…S_{ T }) < B \end{cases}$$

b. Down-and-out options

In this type, the option can only be exercised if the underlying asset price is higher than the barrier value. The payoffs are given by:

$$\text{Call down}-\text{and}- \text{out option}=\begin{cases} \text{max}⁡(0,S_{ T }-X), & \text{if max}⁡(S_{ 1 },S_{ 2 },…S_{ T }) > B \\ F, & \text{if max}⁡(S_{ 1 },S_{ 2 },…S_{ T })\le B \end{cases}$$ $$\text{Put down}-\text{and}- \text{out option}=\begin{cases} \text{max}⁡(0,X-S_{ T }), & \text{if max}⁡(S_{ 1 },S_{ 2 },…S_{ T }) > B \\ F, & \text{if max}⁡(S_{ 1 },S_{ 2 },…S_{ T }) \le B \end{cases}$$

II. A knock-in, implying it has no value until the underlying reaches a certain specified price (barrier level). It can be further be subdivided into:

a. Up-and-in options:

This type of barrier options is only exercised when the underlying asset price is above a certain level, say $B during the options life. Let the strike price of the option be X ,then the payoffs of this option at maturity are: $$\text{Call up}-\text{and}- \text{in option}=\begin{cases} \text{max}⁡(0,S_{ T }-X), & \text{if max}⁡(S_{ 1 },S_{ 2 },…S_{ T }) \ge B \\ 0, & \text{if max}⁡(S_{ 1 },S_{ 2 },…S_{ T }) < B \end{cases}$$ $$\text{Put up}-\text{and}- \text{in option}=\begin{cases} \text{max}⁡(0,X-S_{ T }), & \text{if max}⁡(S_{ 1 },S_{ 2 },…S_{ T }) \ge B \\ 0, & \text{if max}⁡(S_{ 1 },S_{ 2 },…S_{ T }) < B \end{cases}$$ b. Down-and-in options These are types of barrier options which are only exercised at maturity if the underlying asset price falls below a certain predetermined level barrier (B). The prices at the maturity are, therefore: $$\text{Call down}-\text{and}- \text{in option}=\begin{cases} \text{max}⁡(0,S_{ T }-X), & \text{if max}⁡(S_{ 1 },S_{ 2 },…S_{ T }) \le B \\ 0, & \text{if max}⁡(S_{ 1 },S_{ 2 },…S_{ T }) > B \end{cases}$$ $$\text{Put down}-\text{and}- \text{in option}=\begin{cases} \text{max}⁡(0,X-S_{ T }), & \text{if max}⁡(S_{ 1 },S_{ 2 },…S_{ T }) \le B \\ 0, & \text{if max}⁡(S_{ 1 },S_{ 2 },…S_{ T }) > B \end{cases}$$ The binomial pricing of the barrier is similar to that of the standard option, only that calculations of option payoffs at maturity are dependent on the barrier level. #### Example: Binomial Pricing Models on Barrier Options The current stock price of cooperation is$100. Over the next three months, the stock price could go up to $110 or go down to$90. A down-and-in barrier call option is set on this stock with a strike price of $80 and a barrier value of$100.

The risk free rate of interest is 5%

Calculate the option’s price assuming a one period binomial model

#### Solution

$$\begin{array} \hline {} & { } & \begin{matrix} { S }_{ 1 }=110 \\ C_1=\text{max}(0,S_T-X),\text{if max}(S_1,S_2,…,S_T)=B=0 \end{matrix} \\ \begin{matrix} { S }_{ 0 }=100 \\ C_0=? \end{matrix} & {\Huge \begin{matrix} \nearrow \\ \searrow \end{matrix} } & {} \\ {} & {} & \begin{matrix} { S }_{ 1 }=90 \\ C_1=\text{max}(0,S_T-X),\text{if max}(S_1,S_2,…,S_T) < B=90-80=10 \end{matrix} \\ \end{array}$$

Recall that a down-and-in barrier option is exercised at maturity if the underlying asset price falls below a certain predetermined level barrier (B) and the share price must be less than the barrier value and greater than the strike price; otherwise, it will be worthless.

Now we want to calculate the risk-neutral probability. We know that:

$$\pi=\cfrac {e^{r\Delta t}-D}{U-D}$$ $$U=\cfrac {110}{100}=1.1$$

$$D=\cfrac {90}{100}=0.9$$

$$\Rightarrow \pi=\cfrac {e^{r\Delta t}-D}{U-D}=\pi=\cfrac {e^{0.05×0.25}-0.9}{1.1-0.9}=0.563$$

We know also that:

$$C_0=e^{-r\Delta t} [\pi_{U_0} C_1 (U)+(1-\pi_{U_0} ) C_1 (D) ]=e^{-0.05×0.25} [0.56×0+0.437×10]=4.316$$

As you must have realized, you can easily extend the pricing to a two-period binomial model. So, basically, if you are able to calculate the payoffs at the terminal nodes and working backward the tree as usual.

The same principles also apply to put options.

#### Asian Options

In an Asian option, the payoff depends on the average price of the underlying asset over a period of time instead of standard options where the payoff is determined by the involvement of the underlying at a specific point in time.

The average can be calculated arithmetically as:

$$\text{Arithmetic average}=\cfrac {s_1+s_2+⋯+s_n}{n}$$

And geometrically as:

$$\sqrt [n]{s_1 s_2…s_n }$$

The payoff from the geometric Asian options is given by:

\begin{align*} \text{Call Option}& =\text{max}\left[ { \left( 0,\prod _{ i=1 }^{ n }{ { S }_{ i } } \right) }^{ \cfrac { 1 }{ n } }-K \right] \\ \text{Put Option}& =\text{max}\left[ {0,K- \left(0, \prod _{ i=1 }^{ n }{ { S }_{ i } } \right) }^{ \cfrac { 1 }{ n } }\right] \\ \end{align*}

While the payoff from the arithmetic Asian options is given by:

\begin{align*} \text{Call Option} & =\text{max}⁡\left[ 0,{ \left( \frac { 1 }{ n } \sum _{ i=1 }^{ n }{ { S }_{ i } } \right) }-K \right] \\ \text{Put Option} &= \text{max}\left[ 0,K-{ \left( \frac { 1 }{ n } \sum _{ i=1 }^{ n }{ { S }_{ i } } \right) } \right] \\ \end{align*}

We shall consider Asian options whose payoffs are dependent on the arithmetic average of the path of the price it takes. Moreover, we will assume that the underlying asset price is a discrete-time stochastic process with the increasing factors given by:

$$U=e^{(r-\delta)t+\sigma \sqrt{t}}$$

and

$$D=e^{(r-\delta)t-\sigma \sqrt{t}}$$

Where

$$\sigma$$ = the volatility observed from the data;

$$t$$ = the length of one period;

$$r$$ = the risk-free rate; and

$$\delta$$ = the rate of dividend payment.

The risk-neutral probability is then given by:

$$\pi=\cfrac {e^{(r-\delta)t}-D}{U-D}$$

#### Example: Pricing Asian Option using Binomial Models

The current price of a share is $100. An Asian option is set on this asset with a strike price of$100. The risk-free rate of interest is 8%, $$\sigma$$=0.30, $$\delta$$=0 and $$T$$=1.

Calculate the price of an arithmetic Asian call option using a two-period binomial

#### Solution

Using the forward tree formula, we have:

\begin{align*} u & =e^{ \frac {(0.08)}{2} + \frac {0.3}{\sqrt {2}} }=1.2868 \\ d & =e^{ \frac {0.08}{2}-\frac {0.3}{\sqrt {2}} } =0.84187 \\ \end{align*}

The risk-neutral probabilities are:

$$\pi_u=\frac{e^{0.08\times0.5}-0.84187}{1.2868-0.84187}=0.4471$$

and

$$1-\pi_u=1-0.4471=0.5529$$

We can now comfortably construct the period binomial tree:

$$\begin{array} \hline {} & {} & {} & {} & { S }_{ 2 }=165.59 \\ {} & {} & { S }_{ 1 }=100 \times 1.2868=128.68 & {\Huge \begin{matrix} \nearrow \\ \searrow \end{matrix} } & {} \\ \begin{matrix} { S }_{ 0 }=100 \\ V_0=? \end{matrix} & {\begin{matrix} \\ \begin{matrix} \begin{matrix} \Huge \nearrow \\ \end{matrix} \\ \Huge \searrow \end{matrix} \\ \end{matrix} } & {} & {} & { S }_{ 2 } =108.34 \\ {} & {} & { S }_{ 1 }=100 \times 0.8419=84.19 & {\Huge \begin{matrix} \nearrow \\ \searrow \end{matrix} } & {} \\ {} & {} & {} & {} & { S }_{ 2 }=70.84 \\ \end{array}$$

From the tree above, the two possible prices in the first year are 128.68 and 84.19 and 165.59,108.34 and 70.84 in the last year. The possible averages are:

$$\cfrac {128.68+165.59}{2}=147.135,\cfrac {128.68+108.34}{2}=118.51 ,\cfrac {84.19+70.84}{2}=77.515$$

and $$\cfrac {84.19+108.34}{2}=96.26$$

Now, using the strike price of X=100, the “Up” value is $$147.135-100=47.135$$, and the “Down” value is $$118.51-100=18.51$$.

Then the value of the option is:

$$V_0=e^{-0.08}\left({0.4471^2}^2\times 47.135+2\times 0.4471\times 0.5529\times 18.51\right)=17.14681$$

We can easily change from the arithmetic mean to geometric, which by now should be intuitive. Moreover, if we are not given the volatility, we could easily adjust the formula by omitting it.

## Options of Other Assets

Like options on share prices, we can construct binomial trees for other assets such as stock indices with continuous dividends, currencies, and futures contracts.

### Stock Indices with Continuous Dividends

Consider a stock paying a dividend yield at a rate of $$q$$ with a risk-free rate of interest $$r$$. This implies that the capital gain is $$r-q$$. If the stock price starts at $$S_0$$, then the expected value after a period length of $$\Delta t$$ must be $$S_0 e^{(r-q)\Delta t}$$.

This means that:

$$\pi S_0 U+(1-\pi) S_0 D=S_0 e^{(r-q)\Delta t}$$

Where:

$$\pi=\cfrac {e^{(r-q)\Delta t}-U}{U-D}$$

We simply now use $$e^{(r-q)\Delta t}$$ instead of $$e^{r\Delta t}$$.

Given the volatility of $$\sigma$$, the up and down factor movements are can be computed as: $$U=e^{(r-q)\Delta t+\sigma\sqrt{\Delta t}}$$ and $$D=e^{(r-q)\Delta t-\sigma\sqrt{\Delta t}}$$.

For a stock with continuously payable dividends, the up and down factor formulas are adjusted as follows:

$$U=e^{(r-q)\Delta t+\sigma\sqrt{\Delta t}}$$

and

$$D=e^{(r-q)\Delta t-\sigma\sqrt{\Delta t}}$$

#### Example: Valuing an Option using Binomial Models given Volatility

A stock currently has a price of $81 and has a dividend yield of 2%. A one-year European call option has been issued on this stock with a strike price of$80. The continuously compounded risk-free rate of return is 4%, and the stock has a volatility of 20%.

Calculate the price of the call using a two-period binomial tree.

#### Solution

We construct the Binomial tree as follows:

$$\begin{array} {} & {} & {} & \begin{matrix} \text{Node D:}\\S_{UU}=107.48 \\ C_{UU}=\text{max}(107.48-80,0)=27.48\end{matrix} \\ {} & \begin{matrix}B\\ { S }_{ U }=93.30\\ C_U=? \end{matrix} & {\Huge \begin{matrix} \nearrow \\ \searrow \end{matrix} } & {} \\ \begin{matrix} A\\ S_0=81 \\ C_0=? \end{matrix} \begin{matrix} & {} & \\ &\Huge \nearrow & \\ &\Huge \searrow & \\ & { } & \end{matrix} & {} & \begin{matrix} {} \\ \\ { } \end{matrix} & \begin{matrix} { \begin{matrix} \text{Node E:}\\S_{UD}=81 \\ C_{UD}=\text{max}(81-80, 0)=1 \end{matrix} } \\ \\ { \begin{matrix} \text{Node F:}\\S_{DU}=81 \\ C_{DU}=\text{max}(81-80)=1 \end{matrix} } \end{matrix} \\ {} & { \begin{matrix} C\\S_D=70.32 \\ C_D=? \end{matrix} } & {\Huge \begin{matrix} \nearrow \\ \searrow \end{matrix} } & {} \\ {} & {} & {} & { \begin{matrix} \text{Node G:}\\S_{DD}=61.04 \\ C_{DD}=\text{max}(61.04-80,0)=0 \end{matrix} } \\ \end{array}$$

We compute the value of U and D.

$$U=e^{(r-q)\Delta t+\sigma\sqrt{\Delta t}}=e^{(0.04-0.02)0.5+0.2\sqrt{0.5}} = 1.1635$$

and

$$D=e^{(r-q)\Delta t+\sigma\sqrt{\Delta t}}=e^{(0.04-0.02)\times0.5-0.2\sqrt{0.5}} = 0.8768$$

Next, we compute, $$\pi$$ the risk-neutral probability

$$\pi=\frac{e^{\left(r-q\right)\Delta t}-U}{U-D}=\frac{e^{\left(0.04-0.02\right)\times0.5}-0.8768}{1.1635-0.8768}=0.4648$$

The option price is computed using the two-step binomial formula as follows:

\begin{align}C_0&=e^{-2r\Delta t}\left[\pi^2C\left(U_U\right)+2\pi\left(1-\pi\right)C\left(U_D\right)+\left(1-\pi\right)^2C(D_D)\right]\\ & e^{-2\times0.04\times0.5}\left[{0.4648}^2\times27.48+2\times0.4648\times0.5352\times1+0\right]\\ &=\ 5.936\end{align}

### Options on Currencies

Foreign currency can be referred to as an asset providing a yield at the foreign risk-free rate of interest of $$r_f$$. Such options are characterized by two risk-free rates of return applicable in the jurisdictions to which the currency belongs.

For instance, if we are given that; the risk-free rate of return in Australia is 7%, the risk-free rate in the US is 5%, and one Australian dollar exchanges for 0.65 US Dollars on a US Dollar-denominated exchange, then we can compute the price of an option on this exchange rate as before but use the term $$e^{(r-r_f)\Delta t}$$ instead of $$e^{r \Delta t}$$  in the formula for estimating the risk-neutral probabilities.

In this case, $$r$$ will represent the risk-free rate in the US, $$r_f$$ will represent the risk-free rate in Australia, and $$\Delta t$$ the size of each step as before.

Given the volatility, the up and down factors are computed as:

$$U=e^{(r-r_f)\Delta t+\sigma\sqrt{\Delta t}}$$

$$D=e^{(r-r_f)\Delta t-\sigma\sqrt{\Delta t}}$$

Thus,

$$\pi=\frac{e^{(r-r_f)\Delta t}-D}{U-D}$$

The price of a call option using the two-step binomial is then computed as:

\begin{align}C_0&=\pi C_1\left(U\right)+\left(1-\pi\right)C_1(D)\\ &=e^{-2r\Delta t}\left[\pi^2C\left(U_U\right)+2\pi\left(1-\pi\right)C\left(U_D\right)+\left(1-\pi\right)^2C(D_D)\right]\end{align}

#### Example: Binomial Model on Currency Options

The US Dollar is currently worth 0.83 Euros. The exchange rate between the two currencies is subject to the volatility of 15%. The risk-free rate of return in the US is 6%, and the risk-free rate in Europe is 4%. A 6-month European call option has been written on this exchange with a strike price of 0.85 Euros.

Calculate the price of this option using the two-step binomial model.

#### Solution

Consider the following the binomial tree:

$$\begin{array} {} & {} & {} & \begin{matrix} \text{Node D:}\\S_{UU}=0.964 \\ C_{UU}=\text{max}(0.964-0.85,0)=0.114\end{matrix} \\ {} & \begin{matrix}B\\ { S }_{ U }=0.895\\ C_U=? \end{matrix} & {\Huge \begin{matrix} \nearrow \\ \searrow \end{matrix} } & {} \\ \begin{matrix} A\\ S_0=0.83 \\ C_0=? \end{matrix} \begin{matrix} & {} & \\ &\Huge \nearrow & \\ &\Huge \searrow & \\ & { } & \end{matrix} & {} & \begin{matrix} {} \\ \\ { } \end{matrix} & \begin{matrix} { \begin{matrix} \text{Node E:}\\S_{UD}=0.83 \\ C_{UD}=\text{max}(0.83-0.85, 0)=0 \end{matrix} } \\ \\ { \begin{matrix} \text{Node F:}\\S_{DU}=0.83 \\ C_{DU}=\text{max}(83-0.85,0)=0 \end{matrix} } \end{matrix} \\ {} & { \begin{matrix} C\\S_D=0.77 \\ C_D=? \end{matrix} } & {\Huge \begin{matrix} \nearrow \\ \searrow \end{matrix} } & {} \\ {} & {} & {} & { \begin{matrix} \text{Node G:}\\S_{DD}=0.714 \\ C_{DD}=\text{max}(0.714-0.85,0)=0 \end{matrix} } \\ \end{array}$$

We first compute the values of $$U$$ and $$D$$:

$$U=e^{(0.04-0.06)\times0.25+0.15\times\sqrt{0.25}}=1.0725$$

and

$$D=e^{(0.04-0.06)\times0.25-0.15\times\sqrt{0.25}}=0.9231$$

Next, we compute the risk-neutral probabilities

$$\pi=\frac{e^{\left(0.04-0.06\right)\times0.25}-0.9231}{1.0725-0.9231}=0.4813$$

The option price is computed using the two-step binomial formula as follows:

\begin{align}C_0&=e^{-2r\Delta t}\left[\pi^2C\left(U_U\right)+2\pi\left(1-\pi\right)C\left(U_D\right)+\left(1-\pi\right)^2C(D_D)\right]\\ &=e^{-2\times0.04\times0.25}\left[{0.4813}^2\times0.114+2\times0.4813\times\left(1-0.4813\right)\times0\right]\\&=\ 0.0259\end{align}

### Options on Future Contracts

There are no costs for an investor to take a long or a short position in a futures contract. It follows that in the risk-neutral world, a future price should have an expected growth rate of zero. Using the same notations in non-dividend shares as discussed above and let $$F_0$$ is the initial futures price, the expected future price at the end of one period of length $$\Delta t$$ should also be $$F_0$$ then it follows that:

$$\pi F_0 U+(1-\pi) F_0 D=F_0$$

Where

$$\pi=\cfrac {1-D}{U-D}$$

#### Example:  An Option on a Futures Contract

An investor has purchased a one-year call option on an underlying futures contract. The current price of the futures is $40. Over two 6-month steps, the futures price can either go up or down by 10%. The strike price for the call is$42, and the continuously compounded risk-free rate of return applicable over the contract term is 5% per annum.

Calculate the price of the call option.

#### Solution

Consider  the following binomial tree:

$$\begin{array} {} & {} & {} & \begin{matrix} \text{Node D:}\\F_{UU}=48.4 \\ C_{UU}=\text{max}(48.4-42,0)=6.40\end{matrix} \\ {} & \begin{matrix}B\\ { F }_{ U }=44\\ C_U=? \end{matrix} & {\Huge \begin{matrix} \nearrow \\ \searrow \end{matrix} } & {} \\ \begin{matrix} A\\ F_0=40 \\ C_0=? \end{matrix} \begin{matrix} & {} & \\ &\Huge \nearrow & \\ &\Huge \searrow & \\ & { } & \end{matrix} & {} & \begin{matrix} {} \\ \\ { } \end{matrix} & \begin{matrix} { \begin{matrix} \text{Node E:}\\F_{UD}=39.6 \\ C_{UD}=\text{max}(39.6-42, 0)=0 \end{matrix} } \\ \\ { \begin{matrix} \text{Node F:}\\F_{DU}=39.6 \\ C_{DU}=\text{max}(39.6-42,0)=0 \end{matrix} } \end{matrix} \\ {} & { \begin{matrix} C\\F_D=36 \\ C_D=? \end{matrix} } & {\Huge \begin{matrix} \nearrow \\ \searrow \end{matrix} } & {} \\ {} & {} & {} & { \begin{matrix} \text{Node G:}\\F_{DD}=32.4\\ C_{DD}=\text{max}(32.4–42,0)=0 \end{matrix} } \\ \end{array}$$

Given the percentage increase, the values of U and D are determined as:

\begin{align} U&=1.1\\ D&=0.9\end{align}

Next, we compute the risk-neutral probabilities

$$\pi=\frac{1-0.9}{1.1-0.9}=0.5$$

The price of the call option can then be computed using the two-step binomial formula as:

\begin{align} C_0&=e^{-2r\Delta t}\left[\pi^2C\left(U_U\right)+2\pi\left(1-\pi\right)C\left(U_D\right)+\left(1-\pi\right)^2C(D_D)\right]\\ &=e^{-2\times0.05\times0.5}\left[{0.5}^2\times6.4\right]\\ &=\1.52\end{align}

You will note that we have computed the price based on the increase in the forward price only over the two steps because all the other nodes lead to a payoff of zero.

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