The binomial option pricing model – part 3

This is post #3 on the binomial option pricing model. The previous two posts (post #1 and post #2) discuss the calculation and issues for the one-period binomial option pricing model. The purpose of post #3:

    Post #3: Discuss the role of Delta (\Delta) in the replicating portfolio for an option. This number is also called the hedge ratio. In this post, the hedge ratio is discussed in the context of the one-period binomial option model.

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Hedging a short option position – two examples

Suppose that a market maker sells an option (on a stock). He is on the hook to sell (or buy) shares of the stock if the call (or put) buyer decides to exercise (i.e. when the share price of the underlying stock is above (or below) the strike price). He can hedge the risk of a short option position by creating a long synthetic option, i.e. creating a portfolio that replicates the same payoff of the option he sold. This replicating portfolio consists of \Delta shares of the stock and an appropriate amount of lending or borrowing. The \Delta is also called the hedge ratio and is the number of shares in the replicating portfolio to hedge away the risk from selling an option. Let’s discuss through two examples.

Example 1
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 following is the binomial tree shows the future state of the stock prices.

    \text{ }
    Figure 1 – Stock Price
    Future stock price
    \text{ }

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. What is the replicating portfolio for this call option.

This is Example 1 in the post #1 on binomial model. At the end of 6 months, the stock price is either $65 or $40 and the value of the option is either $10 (if stock price goes up) or $0 (if price goes down). According to the calculation in the previous post, the replicating portfolio consists of holding \Delta= 0.4 shares of the stock and $15.6832 in borrowing. The price of the call option is C=50 \Delta+B= 50(0.4) – 15.6832 = $4.3168.

The market maker makes $4.3168 per call option sold. But the market maker is also at risk of losing $10 (selling a share at $55 for a share that is worth $65) when the call buyer decides to exercise. To hedge this risk, the market maker can buy a synthetic call option that replicates exactly the call option he sold.

In this example, the hedge ratio is \Delta= 0.4, which is the ratio of the range of the values of the call to that of the stock across two possible outcomes. In this example, the calculation of \Delta is:

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

For every call option written by the market maker, 0.4 shares of stock must be held to hedge away risk. The reason is that the strategy of holding 0.4 shares and the borrowing of $15.6832 has the same payoff as the call option as indicated by the following two equations. Note that $16.00 is the end of period value of $15.6832.

    \text{ }

    \displaystyle \text{ } \left\{\begin{matrix} \displaystyle 0.4 \times 40  - 16=0&\ \ \ \ \ \ \text{ }& \\ \text{ }&\text{ } \\ 0.4 \times 65  - 16=10&\ \ \ \ \ \ \text{ }   \end{matrix}\right.

    \text{ }

The above two equations show the payoff of the replicating portfolio of holding 0.4 shares and the borrowing of $15.6832, which is exactly the same as the payoff of the call option in the example. By selling a call option in this example, the market maker is at risk of losing $10 when the stock price goes up. He can offset the loss by creating a replicating portfolio that gains $10. So a market maker can hedge away the risk from selling a call by buying a synthetic call (the replicating portfolio).

Example 2
In Example 1, we see that the hedge ratio \Delta is positive and is the number of stocks to hold to hedge away the risk of selling a call option. Now we consider \Delta for put options. We analyze the put option in the Example 1 of the post #1 on binomial model. The characteristics of the stock are as in Example 1. The stock prices are modeled with the same 6-month binomial tree as in Example 1, which is repeated here:

    \text{ }
    Figure 1 – Stock Price
    Future stock price
    \text{ }

The stock pays no dividends. The annual risk-free interest rate is r= 4%. Consider a European 45-strike put option on this stock that will expire in 6 months.

At the end of 6 months, the value of the option is either $0 (if stock price goes up) or $5 (if price goes down). According to the calculation in the previous post, the replicating portfolio consists of holding \Delta= -0.2 shares of the stock and $12.74258275 in lending. The price of the call option is C=50 \Delta+B= 50(-0.2) – 12.74258275 = $2.742582753.

The market maker makes $2.74258 per put option sold. But the market maker is also at risk of losing $5 (buying a share at $45 for a share that is worth only $40) when the put buyer decides to exercise. To hedge this risk, the market maker can buy a synthetic put option that replicates exactly the put option he sold.

In this example, the hedge ratio is \Delta= -0.2, which is the ratio of the range of the values of the put to that of the stock across two possible outcomes. In this example, the calculation of \Delta is:

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

The hedge ratio \Delta is negative. So instead of buying stock, like in Example 1, the market maker holds a short position in the stock, i.e. enter into a short sale for the stock. This means that the market maker borrows the shares and sell the borrowed shares for cash. A short position is a bearish position, i.e. investor enters into a short position in the hope that the price of the asset will fall. In this example, the market maker uses a short stock position because the payoff of a short stock position is exactly opposite of the payoff of a short put, i.e. the loss experienced by the market maker in the short put position is exactly offset by the gain in the short stock position.

Back to the example. For every put option written by the market maker, 0.2 shares of stock must be sold short to hedge away risk. The reason is that the strategy of shorting 0.2 shares and the lending of $12.74258 has the same payoff as the put option as indicated by the following two equations. Note that $13.00 is the end of period value of $12.74258.

    \text{ }

    \displaystyle \text{ } \left\{\begin{matrix} \displaystyle -0.2 \times 40  + 13=5&\ \ \ \ \ \ \text{ }& \\ \text{ }&\text{ } \\ -0.2 \times 65  + 13=0&\ \ \ \ \ \ \text{ }   \end{matrix}\right.

    \text{ }

The above two equations show the payoff of the replicating portfolio of shorting 0.2 shares and the lending of $12.74258, which is exactly the same as the payoff of the put option in the example. By selling a put option in this example, the market maker is at risk of losing $5 when the stock price goes down. He can offset the loss by creating a replicating portfolio that gains $5. So a market maker can hedge away the risk from selling a put by buying a synthetic put (the replicating portfolio).

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To further examine \Delta

Here’s the observation from the above two examples. From the perspective of a market maker, the hedge ration \Delta is the number of shares of stock required to hedge the price risk from selling an option. When selling a call option, the hedge ratio is positive, indicating that the marker maker is to hedge away the risk of a short call by going long on \Delta shares of stock with an appropriate amount in borrowing. When selling a put option, the hedge ratio is negative, indicating that the marker maker is to hedge away the risk of a short put by going short on \Delta shares of stock with an appropriate amount in lending.

The initial stock price in Example 1 and Example 2 is $50. In Example 1, the strike price of the call option is $55. Given the price position, there is no incentive for the call option buyer to exercise when the stock price is $50. The strike price of the put option in Example 2 is $45. So there is no incentive for the put option buyer to exercise when the stock price is $50. In other words, both options are out of the money. When can we say about the hedge ratio \Delta when the options are increasingly in the money? For the call option in Example 1, what if the initial stock price is not $50 but is higher, say $55, $60, $65, or $70? For these higher initial stock prices, the option will have an increasingly greater chance of being in the money. What can we say about \Delta? We examine these scenarios in Example 3. In Example 4, we will examine similar scenarios for the put option in Example 2.

Example 3
For the call option in Example 1, determine the replication portfolio and calculate the price of the call option as the initial stock price varies from $50, $55, $60, $65, to $70. The results are in the following table.

    \text{ }

    Table 1 – Call option hedge ratio when initial stock prices are increasing

    \left[\begin{array}{lllllllll}  \text{Call Option}     \\ \text{Strike Price} & \text{ } & \text{Initial Stock Price} & \text{ } & \text{Option Price}  & \text{ } & \text{Hedge Ratio } \Delta  & \text{ } & \text{Borrowing} \\      \text{ } & \text{ } \\      \$ 55 & \text{ } & \$ 50 & \text{ } & \$ 4.3168  & \text{ } & 0.4  & \text{ } & \$ 15.6832 \\       \$ 55 & \text{ } & \$ 55 & \text{ } & \$ 7.1228  & \text{ } & 0.6  & \text{ } & \$ 25.8772 \\      \$ 55 & \text{ } & \$ 60 & \text{ } & \$ 9.9287  & \text{ } & 0.7667  & \text{ } & \$ 36.0713 \\      \$ 55 & \text{ } & \$ 65 & \text{ } & \$ 12.7346  & \text{ } & 0.9077  & \text{ } & \$ 46.2654 \\       \$ 55 & \text{ } & \$ 70 & \text{ } & \$ 16.0891  & \text{ } & 1.000  & \text{ } & \$ 53.9109 \\               \end{array}\right]
    \text{ }

In Table 1, the initial stock prices are increasingly higher than the strike price. This means that the call option is increasingly in the money. As a result, the hedge ratio is increasingly becoming 1.0. To explain this phenomenon, let’s take the point of view of a market maker. Suppose that a market maker has sold a 55-strike call option. If the initial stock price is much higher than the strike price, it is much more likely that the option will finishes in the money. The market maker must then buy more shares initially in order to be able to cover the obligation of the short call position at expiration. Thus the hedge ratio \Delta increases as the initial stock price increases. When \Delta is 1, the option is all but certain to expire in the money that the market maker has to hedge by holding one share for one option.

Example 4
For the put option in Example 2, determine the replication portfolio and calculate the price of the put option as the initial stock price varies from $50, $45, $40, $35, to $30. The results are in the following table.

    \text{ }

    Table 2 – Put option hedge ratio when initial stock prices are decreasing

    \left[\begin{array}{lllllllll}  \text{Put Option}     \\ \text{Strike Price} & \text{ } & \text{Initial Stock Price} & \text{ } & \text{Option Price}  & \text{ } & \text{Hedge Ratio } \Delta  & \text{ } & \text{Lending} \\      \text{ } & \text{ } \\      \$ 45 & \text{ } & \$ 50 & \text{ } & \$ 2.7426  & \text{ } & -0.2  & \text{ } & \$ 12.7426 \\       \$ 45 & \text{ } & \$ 45 & \text{ } & \$ 4.9366  & \text{ } & -0.4  & \text{ } & \$ 22.9366 \\      \$ 45 & \text{ } & \$ 40 & \text{ } & \$ 7.1307  & \text{ } & -0.65  & \text{ } & \$ 33.1307 \\      \$ 45 & \text{ } & \$ 35 & \text{ } & \$ 9.3248  & \text{ } & -0.9714  & \text{ } & \$ 43.3248 \\       \$ 45 & \text{ } & \$ 30 & \text{ } & \$ 14.1089  & \text{ } & -1.0000  & \text{ } & \$ 44.1089 \\               \end{array}\right]
    \text{ }

In Table 2, the movement goes in the opposite direction. The initial stock prices are decreasingly lower than the strike price. This means that the put option is increasingly in the money. As a result, the hedge ratio is increasingly becoming -1.0. To explain this phenomenon, we again take the point of view of a market maker. Suppose that a market maker has sold a 45-strike put option. If the initial stock price is much lower than the strike price, it is much more likely that the put option will finish in the money. The market maker must then short more shares initially in order to be able to cover the obligation of the short put position at expiration. Thus the hedge ratio \Delta decreases as the initial stock price decreases. When \Delta is -1, the put option is all but certain to expire in the money that the market maker has to hedge by shorting one share for one option.

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Remarks

The discussion of \Delta in this post is from a market maker’s point of view. It is the number of shares a market maker needs to buy or short in order to cover the obligation of a short option position. When the initial price is sufficiently far from the strike price (when the option is extremely likely to expire in the money), the market maker must buy or short the stock on a one share to one option basis.

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Practice problems

Practice problems for this post are found in here.

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

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