The binomial option pricing model – part 1

This is post #1 on the binomial option pricing model. Even though this is post #1, there are two previous posts with examples to illustrate how to price options using the one-period binomial pricing model (example of call and example of put). The purpose of post #1:

    Post #1: Describe the option pricing formulas in the one-period binomial model.

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The one-period binomial option pricing model

We first consider the pricing of options on stock. The most important characteristic of the binomial option pricing model is that over a period of time, the stock price is assumed to follow a binomial distribution, i.e. the price of the stock can only take on one of two values – an upped value and a downed value. In this post, we describe how to price an option on a stock using this simplifying assumption of stock price movement.

Consider a stock with the following characteristics:

  • The current share price is S.
  • If the stock pays dividends, we assume the dividends are paid at an annual continuous rate at \delta.
  • At the end of a period of length h (in years), the share price is either S_h=uS or S_h=dS, where u is the up factor and d is the down factor. The factor u can be interpreted as one plus the rate of capital gain on the stock if the stock goes up. The factor d can be interpreted as one plus the rate of capital loss if the stock goes down.
  • If \delta>0, the end of period share price is S_h=uS e^{\delta h} or S_h=dS e^{\delta h}. This is to reflect the gains from reinvesting the dividends. Of course if \delta=0, the share prices revert back to the previous bullet point.

The end of period stock prices are shown in the following diagram, which is called a binomial tree since it depicts the 2-state stock price at the end of the option period.

    \text{ }
    Figure 1 – binomial tree
    binomial tree
    \text{ }

Now consider a European option (either call or put) on the stock described above. When the stock goes up, we use C_u to represent the value of the option. When the stock goes down, we use C_d to represent the value of the option. The following is the binomial tree for the value of the option.

    \text{ }
    Figure 2 – option value tree
    option values
    \text{ }

Replicating Portfolio
The key idea to price the option is to create a portfolio consisting of \Delta shares of the stock and the amount B in lending. At time 0, the value of this portfolio is C=\Delta S + B. At time h (the end of the option period), the value of the portfolio is

    \text{ }
    Time h value of the replicating portfolio

    \displaystyle \text{ } \left\{\begin{matrix} \displaystyle \Delta \times (dS \ e^{\delta h})  + B \ e^{r h}&\ \ \ \ \ \ \text{(when stock price goes down)}& \\ \text{ }&\text{ } \\ \Delta \times (uS \ e^{\delta h})  + B \ e^{r h}&\ \ \ \ \ \ \text{(when stock price goes up)}   \end{matrix}\right.

    \text{ }

This portfolio is supposed to replicate the same payoff as the value of the option. By equating the portfolio payoff with the option payoff, we obtain the following linear equations.

    \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{ }   \end{matrix}\right.

    \text{ }

There are two unknowns in the above two equations. All the other items – stock price S, dividend rate \delta, and risk-free interest rate r – are known. Solving for the two unknowns \Delta and B, we obtain:

    \text{ }
    \displaystyle \Delta=e^{-\delta h} \ \frac{C_u-C_d}{S(u-d)} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (1)
    \text{ }

    \displaystyle B=e^{-r h} \ \frac{u \ C_d-d \ C_u}{u-d} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (2)
    \text{ }

Once the replication portfolio of \Delta shares and B in lending is determined, the price of the option (the value at time 0) is:

    \text{ }
    C=\Delta S + B \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (3)
    \text{ }

After plugging in (1) and (2) into (3), the option price formula becomes:

    \text{ }
    \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) \ \ \ \ \ \ \ \ \ (4)
    \text{ }

The price of the option described above is C, either given by formula (3) or formula (4). One advantage of formula (4) is that it gives the direct calculation of the option price without knowing \Delta and B. Of course, if the goal is to create a synthetic option for the purpose of hedging or risk management, it will be necessary to know the make up of the replicating portfolio.

The \Delta calculated in (1) is also called the hedge ratio and is examined in greater details in in this subsequent post.

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Examples

Example 1
Let’s walk through a quick example to demonstrate how to apply the above formulas. Suppose that the future prices for a stock are modeled with a one-period binomial tree with u= 1.3 and d= 0.8 and having a period of 6 months. The current price of the stock is $50. The stock pays no dividends. The annual risk-free interest rate is r= 4%.

  • Determine the price of a European 55-strike call option on this stock that will expire in 6 months.
  • Determine the price of a European 45-strike put option on this stock that will expire in 6 months.

The two-state stock prices are $65 and $40. The two-state call option values at expiration are $10 and $0. Apply (1) and (2) to obtain the replicating portfolio and then the price of the call option.

    \text{ }
    \displaystyle \Delta=\frac{10-0}{65-40}=\frac{10}{25}= 0.4

    \displaystyle B=e^{-0.04(0.5)} \ \frac{1.3(0)-0.8(10)}{1.3-0.8}=-16 e^{-0.02}= -$15.68317877

    The replicating portfolio consists of holding 0.4 shares and borrowing $15.68317877.

    Call option price = 50 \Delta+B= $4.316821227

    \text{ }

The 2-state put option values at expiration are $0 and $5. Now apply (1) and (2) and obtain:

    \text{ }
    \displaystyle \Delta=\frac{0-5}{65-40}=\frac{-5}{25}=-0.2

    \displaystyle B=e^{-0.04(0.5)} \ \frac{1.3(5)-0.8(0)}{1.3-0.8}=13 e^{-0.02}= $12.74258275

    The replicating portfolio consists of shorting 0.2 shares and lending $12.74258275.

    Put option price = 50 \Delta+B= $2.742582753

    \text{ }

Example 1 is examined in greater details in this subsequent post.

More Examples
Two more examples are in these previous posts:

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What to do if options are mispriced

What if the observed price of an option is not the same as the theoretical price? In other words, what if the price of a European option is not given by the above formulas? Because we can always hold stock and lend to replicate the payoff of an option, we can participate in arbitrage when an option is mispriced by buying low and selling high. The idea is that if an option is underpriced, then we buy low (the underpriced option) and sell high (the corresponding synthetic option, i.e. the replicating portfolio). On the other hand, if an option is overpriced, then we buy low (the synthetic option) and sell high (the overpriced option). Either case presents risk-free profit. We demonstrate with the options in Example 1.

Example 2

  • Suppose that the price of the call option in Example 1 is observed to be $4.00. Describe the arbitrage.
  • Suppose that the price of the call option in Example 1 is observed to be $4.60. Describe the arbitrage.

For the first scenario, we buy low (the option at $4.00) and sell the synthetic option at the theoretical price of $4.316821227. Let’s analyze the cash flows in the following table.

    \text{ }

    Table 1 – Arbitrage opportunity when call option is underpriced

    \left[\begin{array}{llll}      \text{Expiration Cash Flows} & \text{ } & \text{Share Price = } \$ 40 & \text{Share Price = } \$ 65 \\      \text{ } & \text{ } \\      \text{Sell synthetic call} & \text{ } & \text{ } & \text{ } \\      \ \ \ \ \text{Short 0.4 shares}  & \text{ } & - \$ 16 & - \$ 26 \\      \ \ \ \ \text{Lend } \$ 15.683  & \text{ } & + \$ 16 & + \$ 16 \\      \text{ } & \text{ } \\      \text{Buy call }  & \text{ } & \ \ \$ 0 & \ \ \$ 10 \\      \text{ } & \text{ } \\            \text{Total payoff} & \text{ } & \text{ } \ \$ 0  & \ \ \$ 0    \end{array}\right]

    \text{ }

The above table shows that the buy low sell high strategy produces no loss at expiration of the option regardless of the share prices at the end of the option period. But the payoff at time 0 is certain: $4.316821227 – $4.00 = $0.316821227.

For the second scenario, we still buy low and sell high. This time, buy low (the synthetic call option at $4.316821227) and sell high (the call option at the observed price of $4.60). Let’s analyze the cash flows in the following table.

    \text{ }

    Table 2 – Arbitrage opportunity when call option is overpriced

    \left[\begin{array}{llll}      \text{Expiration Cash Flows} & \text{ } & \text{Share Price = } \$ 40 & \text{Share Price = } \$ 65 \\      \text{ } & \text{ } \\      \text{Buy synthetic call} & \text{ } & \text{ } & \text{ } \\      \ \ \ \ \text{Long 0.4 shares}  & \text{ } & + \$ 16 & + \$ 26 \\      \ \ \ \ \text{Borrow } \$ 15.683  & \text{ } & - \$ 16 & - \$ 16 \\      \text{ } & \text{ } \\      \text{Buy call }  & \text{ } & \ \ \$ 0 &  - \$ 10 \\      \text{ } & \text{ } \\            \text{Total payoff} & \text{ } & \text{ } \ \$ 0  & \ \ \$ 0    \end{array}\right]

    \text{ }

The above table shows that the buy low sell high strategy produces no loss at expiration of the option regardless of the share prices at the end of the option period. But the payoff at time 0 is certain: $4.60 – $4.316821227 = $0.283178773.

These two examples show that if the option price is anything other than the theoretical price, there are arbitrage opportunities and there is risk-free profit to be made.

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How to construct a binomial tree

In the binomial tree in Figure 1, we assume that the share price at expiration is obtained by multiplying the original share price by the movement factors of u and d. The binomial tree in Figure 1 may give the impression that the choice of the movement factors u and d is arbitrary as long as the up factor is greater than 1 and the down factor is below 1. In the next post, we show that u and d have to satisfy the following relation, else there will be arbitrage opportunities.

    \displaystyle d < e^{(r-\delta) h} < u \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (5)

Thus the choice of u and d cannot be entirely arbitrary. In particular the relation (5) shows that the future stock prices have to revolve around the forward price.

    \displaystyle dS < Se^{(r-\delta) h} < uS \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (6)

The purpose pf the factors u and d in the binomial tree is to incorporate uncertainty of the stock prices. In light of (6), we can set u and d by applying some volatility adjustment to e^{(r-\delta) h}. We can use the following choice of u and d to model the stock price evolution.

    \displaystyle u = e^{(r-\delta) h \ + \ \sigma \sqrt{h}}

    \displaystyle d = e^{(r-\delta) h \ - \ \sigma \sqrt{h}} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (7)

where

    \sigma is the annualized standard deviation of the continuously compounded stock return,

    \sigma \sqrt{h} is the standard deviation of the continuously compounded stock return over a period of length h.

The standard deviation \sigma measures how certain we are that the stock return will be close to the expected return. There will be a greater chance of a return far from the expected return if the stock has a higher \sigma. If \sigma=0, then there is no uncertainty about the future stock prices. The formula (7) shows that when \sigma=0, the future stock price is precisely the forward price on the stock. When the binomial tree is constructed using (7), the tree will be called a forward tree.

A note on calculation. If a problem does not specific u and d but assume a standard deviation of stock return \sigma, then assume that the binomial tree is the forward tree. We now use a quick example to demonstrate how to price an option using the forward tree.

Example 3
Everything is the same as Example 1 except that the up and down stock prices are constructed using the volatility \sigma= 30% (the standard deviation \sigma). The following calculates the stock prices at expiration of the option.

    \displaystyle uS = 50 \ e^{(0.04-0) 0.5 \ + \ 0.3 \sqrt{0.5}}= $63.06431255

    \displaystyle dS = 50 \ e^{(0.04-0) 0.5 \ - \ 0.3 \sqrt{0.5}}= $41.25989534

    \displaystyle u=\frac{63.06431255}{50}= 1.261286251

    \displaystyle d=\frac{41.25989534}{50}= 0.825197907

Using formulas (1), (2) and (3), the following shows the replicating portfolio and the call option price. Note that the binomial tree is based on a different assumption than that in Example 1. The option price is thus different than the one in Example 1.

    \text{ }

    \displaystyle \Delta=\frac{8.064312548-0}{63.06431255-41.25989534}= 0.369847654

    \displaystyle B=e^{-0.04(0.5)} \ \frac{1.261286251(0)-0.825197907(8.064312548)}{1.261286251-0.825197907}= –$14.95770971

    The replicating portfolio consists of holding 0.369847654 shares and borrowing $14.95770971.

    Call option price = 50 \Delta+B= $3.534672982

    \text{ }

The following shows the calculation for the put option.

    \text{ }
    \displaystyle \Delta=\frac{0-3.740104659}{63.06431255-41.25989534}= -0.171529678

    \displaystyle B=e^{-0.04(0.5)} \ \frac{1.261286251(3.740104659)-0.825197907(0)}{1.261286251-0.825197907}= $10.60320232

    The replicating portfolio consists of shorting 0.171529678 shares and lending $10.60320232.

    Put option price = 50 \Delta+B= $2.026718427

    \text{ }

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More examples

We present two more examples in illustrating the calculation in the one-period binomial option model where the stock prices are modeled by a forward tree.

Example 4
The stock price follows a 6-month binomial tree with initial stock price $60 and \sigma= 0.3. The stock is non-dividend paying. The annual risk free interest rate is r= 4%. What is the price of a 6-month 55-strike call option? Determine the replicating portfolio that has the same payoff as this call option.

We will use risk-neutral probabilities to price the option.

    \displaystyle uS = 60 \ e^{(0.04-0) 0.5 \ + \ 0.3 \sqrt{0.5}}= $75.67717506

    \displaystyle dS = 60 \ e^{(0.04-0) 0.5 \ - \ 0.3 \sqrt{0.5}}= $49.51187441

    \displaystyle C_u= 75.67717506 – 55 = 20.67717506

    \displaystyle C_d= 0

    \displaystyle u=\frac{75.67717506}{60}= 1.261286251

    \displaystyle d=\frac{49.51187441}{60}= 0.825197907

    \displaystyle p^*=\frac{e^{(0.04-0) 0.5} - 0.825197907}{1.261286251 - 0.825197907}= 0.447164974

    \displaystyle 1-p^*= 0.552835026

    \displaystyle C=(p^* \times C_u + (1-p^*) \times C_d) e^{-0.02}= 9.063023234

    \text{ }

    \displaystyle \Delta=\frac{20.67717506-0}{75.67717506-49.51187441}= 0.790251766

    \displaystyle B=e^{-0.04(0.5)} \ \frac{1.261286251(0)-0.825197907(20.67717506)}{1.261286251-0.825197907}= –$38.35208275

    The replicating portfolio consists of holding 0.79025 shares and borrowing $38.352.

    \text{ }

Example 5
The stock price follows a 3-month binomial tree with initial stock price $40 and \sigma= 0.3. The stock is non-dividend paying. The annual risk free interest rate is r= 5%. What is the price of a 3-month 45-strike put option on this stock? Determine the replicating portfolio that has the same payoff as this put option.

The calculation is calculated as in Example 3.

    \displaystyle uS = 40 \ e^{(0.05-0) 0.25 \ + \ 0.3 \sqrt{0.25}}= $47.05793274

    \displaystyle dS = 40 \ e^{(0.05-0) 0.25 \ - \ 0.3 \sqrt{0.25}}= $34.861374

    \displaystyle C_u= 0

    \displaystyle C_d= 45 – 34.861374 = $10.138626

    \displaystyle u=\frac{47.05793274}{40}= 1.176448318

    \displaystyle d=\frac{34.861374}{40}= 0.87153435

    \displaystyle p^*=\frac{e^{(0.05-0) 0.25} - 0.87153435}{1.176448318 - 0.87153435}= 0.462570155

    \displaystyle 1-p^*= 0.537429845

    \displaystyle C=(p^* \times C_u + (1-p^*) \times C_d) e^{-0.0125}= 5.381114117

    \text{ }
    \displaystyle \Delta=\frac{0-10.138626}{47.05793274-34.861374}= -0.831269395

    \displaystyle B=e^{-0.05(0.25)} \ \frac{1.176448318(10.138626)-0.87153435(0)}{1.176448318 - 0.87153435}= $38.63188995

    The replicating portfolio consists of shorting 0.831269395 shares and lending $38.63188995.

    \text{ }

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Remarks

The discussion in this post is only the beginning of the binomial pricing model. The concepts and the formulas for the one-period binomial option model are very important. The one-period model may seem overly simplistic (or even unrealistic). One way to make it more realistic is to break up the one-period into multiple smaller periods and thus produce a more accurate option price. The calculation for the multi-period binomial model is still based on the calculation for the one-period model. Before moving to the multi-period model, we discuss the one-period model in greater details to gain more understanding of the one-period model.

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Practice problems

Practice Problems
Practice problems can be found in the companion problem blog via the following links:

basic problem set 1

basic problem set 2

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

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