The binomial option pricing model – part 2

This is post #2 on the binomial option pricing model. In part 1, we derive the one-period binomial option pricing formulas. The purpose of post #2:

    Post #2: Discuss the underlying issues in the one-period model – e.g. arbitrage in the binomial tree and risk-neutral pricing.

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The one-period binomial option pricing formulas

For easier reference, we list out the option pricing formulas derived in part 1. The binomial tree models the stock price at expiration of the option.

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

The following is a tree showing the value of the option at expiration.

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

Replicating Portfolio

    \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{ }

Price of the Option

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

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Arbitrage in the binomial tree

In formulas (1), (2) and (4), it seems that we choose the up factor u and the down factor d arbitrarily. It turns out that the assumed stock price factors u and d should be set in such a way that arbitrage opportunities are not possible. The factors u and d must follow the following relationship.

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

Multiplying (5) by the initial stock price S yields the following:

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

The middle term in (6) is the forward price on the stock. The relationship (6) indicates that whatever the values of the up factor u and the down factor d are, the end of period upped stock price must be larger than the forward price and the downed stock price must be below the forward price. Violation of this requirement will yield arbitrage opportunities.

To see that arbitrage opportunities will arise if (5) is violated, suppose that e^{(r-\delta) h} > u. Multiply by the initial stock price produces Se^{(r-\delta) h} > Su. Since Su > Sd, we have the following:

    Se^{(r-\delta) h} > Su > Sd \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (a)

Based on the above inequality (a), the arbitrage opportunity: short e^{-\delta h} shares of stock (borrow that many shares and sell) and lend Se^{-\delta h} (the short sales proceeds). At time h, you need to buy back 1 share at price S_h. The value of the bond is Se^{-\delta h} e^{r h}=S e^{(r-\delta) h}. What occurs at time h is that you pay S_h to buy back 1 share and receive S e^{(r-\delta) h}. Based on (a), both Se^{(r-\delta) h} - Su > 0 and Se^{(r-\delta) h} - Sd > 0, which mean risk-free profit. So it must be the case that e^{(r-\delta) h} < u.

Suppose that e^{(r-\delta) h} < d. This also leads to arbitrage opportunities. Multiplying by the initial stock price produces the following:

    Se^{(r-\delta) h} < Sd < Su \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (b)

The arbitrage opportunity: borrow Se^{-\delta h} at the risk-free rate and use the borrowed fund to buy e^{-\delta h} shares of stock. The relationship (b) says that regardless of the stock price at time h (up or down), the stock price is always greater than the amount that has to be repaid. Thus there are risk-free profits in either case: 0 < Sd - Se^{(r-\delta) h} and 0 < Su - Se^{(r-\delta) h}.

Thus relationship (5) must hold for the stock price movement factors u and d. In fact, one way to set the factors u and d is to increase or decrease a volatility adjustment to the risk-free return factor e^{(r-\delta) h}. The resulting u and d are:

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

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

For more information about (7), see part 1.

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Risk-neutral pricing

At first glance, the pricing of an option on stock ought to require the use of a probability model. The price of the option depends on the price of the stock at expiration of the European option. The stock price at the end of the option period is uncertain. Thus to price the option, we need to find a way to characterize the uncertainty of the stock prices at expiration. Since the future stock prices are random, it is natural to think that we need a probability model to describe the uncertain stock prices. The above derivation of the binomial option pricing model shows that probabilities of the future stock prices are not necessary. All we use is the binomial assumption of stock prices. The trick is then to determine a replicating portfolio of holding \Delta shares and lending a dollar amount B. Because the replicating portfolio has the same payoff as the option, the movement of the stock prices (the up and the down prices) is irrelevant to the calculation of the price of the option.

However, there is a probabilistic interpretation of the option price in (4). Note that the terms \displaystyle \frac{e^{(r-\delta) h}-d}{u-d} and \displaystyle \frac{u-e^{(r-\delta) h}}{u-d} in formula (4) sum to 1.0. The two terms are also positive because of relationship (5). So they can be interpret as probabilities. So we have:

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

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

Then pricing formula (4) becomes:

    \text{ }
    \displaystyle C=\Delta S + B=e^{-r h} \biggl(p^* \ C_u +(1-p^*) \ C_d   \biggr) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (5)
    \text{ }

The formula p^* is called the risk-neutral probability. From a calculation standpoint, the risk-neutral probability is another way to calculate the price of an option in the one-period binomial model. Simply calculate the risk-neutral probabilities. Then use them to weight the option values C_u and C_d (and also discount to time 0).

If p^* and 1-p^* are interpreted as probabilities, then the pricing formula (5) says that the price of an option is the expected value of the end of period options values discounted at the risk-free rate. On the other hand, let’s use p^* and 1-p^* to compute the expected value of the stock prices.

    \text{ }
    \displaystyle  p^* uS+ (1-p^*) dS=\frac{e^{(r-\delta) h}-d}{u-d} uS+\frac{u-e^{(r-\delta) h}}{u-d} dS =S e^{(r-\delta) h}
    \text{ }

The last term in the above derivation is e^{(r-\delta) h}, which is the forward price on a stock that pays continuous dividends (derived in this previous post). Thus if we use p^* and 1-p^* to calculated the expected value of the stock prices, we get the forward price. This is why p^* and 1-p^* are called risk-neutral probabilities since they are the probabilities for which the expected value of the stock prices is the forward price. In particular, p^* is the risk-neutral probability of an increase in the stock price.

We conclude this post with an example on using risk-neutral probabilities to compute option prices. This example is Example 3 in part 1.

Example 1
Suppose that the future prices for a stock are modeled with a one-period binomial tree with volatility \sigma= 30% and having a period of 6 months. The current price of the stock is $60. The stock pays no dividends. The annual risk-free interest rate is r= 4%. Use risk-neutral probabilities to price the following options.

  • A European 60-strike call option on this stock that will expire in 6 months.
  • A European 60-strike put option on this stock that will expire in 6 months.

First calculate the u and d, and the stock prices at expiration:

    \displaystyle u = e^{(0.04-0) 0.5 \ + \ 0.3 \sqrt{0.5}}= 1.261286251

    \displaystyle d = e^{(0.04-0) 0.5 \ - \ 0.3 \sqrt{0.5}}= 0.825197907

    \displaystyle uS = 60 (1.261286251) = $75.67717506

    \displaystyle dS = 60 (0.825197907) = $49.51187441

Now the risk-neutral probabilities:

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

    \displaystyle 1-p^*=\frac{1.261286251-e^{(0.04-0) 0.5}}{1.261286251-0.825197907}= 0.552835026

Then the option prices are:

    C=e^{-0.04(0.5)} [0.447164974 (15.67717506) + 0.825197907(0)]= $6.871470666 (call)

    P=e^{-0.04(0.5)} [0.447164974 (0) + 0.825197907(10.48812559)]= $5.683391065 (put)

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Practice Problems

Practice problems can be found in in this blog post in a companion blog.

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

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