Tag Archives: Synthetic Call

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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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}

Pricing a put option – an example

This post is a continuation of the example discussed in this previous post, which gives an example to illustrate the pricing of a call option using the binomial option pricing model. This post illustrates the pricing of a put option. Links to practice problems are found at the bottom of the post.

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The example

The following gives the information about the stock:

  • The stock of XYZ company is currently selling for $50 per share. The price per share 1 year from now is expected to increase to $65 or to decrease to $40. The stock pays no dividends.

Consider a put option with the following specifics:

  • The underlying asset of the put option is the XYZ stock.
  • The strike price is $55.
  • The option will expire in one year.
  • The option is assumed to be a European option, i.e. it can be exercised only at expiration.

The annual risk-free interest rate is 2%. There is a benefit to the buyer of the option described above. If the price of the stock goes down to $40 at the end of the 1-year period, the buyer of the put option has the right to sell a share of XYZ for $55 ($15 higher than the market price). If the price of the stock goes up to $65 at the end of the 1-year period, exercising the option would mean selling a share at $55 which is $10 below the market price, but the put option owner can simply walk away. The put option owner sells the stock only when he makes money. What would be the fair price of having this privilege? What is the fair price of this put option?

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Pricing the put option

In this example, the current stock price is $50 and the stock price can be only one of the two possible values at the end of the option contract period (either $65 or $40). The following diagram shows the future state of the stock prices.

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

The assumption of the 2-state stock prices in 1 year simplifies the analysis of the put option. The value of the put option at the end of 1 year is either zero or $15 (=55-40). Note that when the share price at the end of the 1-year contract period is higher than the strike price of $55, the put option expires worthless. The following diagram shows the value of the put option.

    \text{ }
    Figure 2 – Put Option Payoff
    put option payoff

    \text{ }

In the above diagram, the value of the put option at the end of 1-year is either $0 or $15. The value of the option at time 0 is C, which is the premium of the put option in this example. Our job here is to calculate C. The key to finding the value of the option is to compare the payoff of the put to that of a portfolio consisting of the following investments:

    Portfolio B

  • Short 0.6 shares of XYZ.
  • Lend $38.2277 at the risk-free rate.

The idea for setting up this portfolio is given below. For the time being, we take the 0.6 shares and the lending of $38.2277 as a given. Note that $38.2277 is the present value of $39 at the risk-free rate of 2%. Let’s calculate the value of Portfolio B at time 0 and at time 1 (1 year from now). The following diagram shows the calculation.

    \text{ }
    Figure 3 – Portfolio B Payoff
    replicating portfolio B payoff

    \text{ }

Note that the payoff of the put option is identical to the payoff of Portfolio B. Thus the put option in this example and Portfolio B must have the same cost. Since Portfolio B costs $8.2277, the price of the option must be $8.2277. The Portfolio B of 0.6 shares of stock in short sales and $15.683 in lending is a synthetic put since it mimics the put option described in the example. Portfolio B is called a replicating portfolio because it replicates the payoff of the put option in question.

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Arbitrage opportunities

In deriving the cost of the put option of $8.2277, we rely on the idea that if two investments have the same payoff, they must have the same cost. This idea is called the law of one price, which is a commonsensical idea and is also an important principle in derivative pricing. If the law of one price is violated, in particular if the price of the put option discussed here is not $8.2277, there would be arbitrage opportunities that can be exploited to gain risk-free profit.

What if the law of one price is violated? For example, what if the option were selling for a higher price (say $8.50)? If the price of the replicating portfolio is less than the price of the option, then we can “buy low and sell high” (i.e. buy the replicating portfolio and sell put option) and obtain a risk-free profit of $0.2723. The arbitrage is to buy the synthetic call (Portfolio B) at $8.2277 and sell the put option at $8.50. The following table shows the Year 1 cash flows of this arbitrage opportunity.

    \text{ }

    Table 1 – Arbitrage opportunity when put option is overpriced

    \left[\begin{array}{llll}      \text{Year 1 Cash Flows} & \text{ } & \text{Share Price = } \$ 40 & \text{Share Price = } \$ 65 \\      \text{ } & \text{ } \\      \text{Long synthetic put} & \text{ } & \text{ } & \text{ } \\      \ \ \ \ \text{Short 0.6 shares}  & \text{ } & - \$ 24 & - \$ 39 \\      \ \ \ \ \text{Receive the lending of } \$ 38.2277  & \text{ } & + \$ 39 & + \$ 39 \\      \text{ } & \text{ } \\      \text{Short put }  & \text{ } &  - \$ 15 & \ \ \$ 0 \\      \text{ } & \text{ } \\            \text{Total payoff} & \text{ } & \text{ } \ \$ 0  & \ \ \$ 0    \end{array}\right]

    \text{ }

The above table shows that buying a synthetic put (shorting 0.6 shares and lending $38.2277) and selling a put will have no loss at the end of 1 year. Yet, the time 0 cash flow is $0.2723 (=8.50 – 8.2277), and is thus a risk-less profit.

If the option is underpriced, then we can still buy low and sell high (in this case, buy put option and sell the replicating portfolio) and obtain risk-free arbitrage profit. For example, let’s say you observe a put option price of $8.00. Then the arbitrage opportunity is to buy the put option at $8.00 and sell a synthetic put (Portfolio B) at $8.2277. The time 0 payoff is $0.2723, which is a risk-less arbitrage profit. The following table shows the Year 1 cash flows.

    \text{ }

    Table 2 – Arbitrage opportunity when put option is underpriced

    \left[\begin{array}{llll}      \text{Year 1 Cash Flows} & \text{ } & \text{Share Price = } \$ 40 & \text{Share Price = } \$ 65 \\      \text{ } & \text{ } \\      \text{Short synthetic put} & \text{ } & \text{ } & \text{ } \\      \ \ \ \ \text{Long 0.6 shares}  & \text{ } & + \$ 24 & + \$ 39 \\      \ \ \ \ \text{Repay the borrowing of } \$38.2277   & \text{ } & - \$ 39 & - \$ 39 \\      \text{ } & \text{ } \\      \text{Long put }  & \text{ } & \ \ \$ 15 & + \$ 0 \\      \text{ } & \text{ } \\            \text{Total payoff} & \text{ } & \text{ } \ \$ 0  & \ \ \$ 0    \end{array}\right]

    \text{ }

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To complete the picture

The put option price of $8.2277 is derived by showing that the replicating portfolio has the same payoff as the put option. How do we know that the replicating portfolio consists of shorting 0.6 shares and lending of $38.2277?

In general, the replicating portfolio of a European option consists of \Delta shares of the stock and the amount B in lending at time 0 (borrowing if negative). By equating the payoff of the replicating portfolio and the payoff of the put option in this example, we have the following equations:

    \text{ }
    \displaystyle \begin{array}{ccc} \displaystyle 40 \ \Delta + B \ e^{0.02} & = & 15 \\ \displaystyle 65 \ \Delta + B \ e^{0.02} & = & 0  \end{array}
    \text{ }

Solving these two equations, we obtain \Delta=\frac{-15}{25}=-0.6 and B=39 \ e^{-0.02}=38.2277. Therefore, the replicating portfolio for the put option in this example consists of shorting 0.6 shares of the stock and $38.2277 in lending. The net investment for the replicating portfolio is $8.2277 (=-0.6(50)+38.2277). Because there are only two data points in the future stock prices, the option premium is a linear function of \Delta and B. The following is the premium of the call (or put) option using the one-period binomial tree

    C=\Delta \ S+B

where S is the stock price at expiration. The above formula gives the cost of the portfolio replicating the payoff of a given option. It works for call option as well as for put option. The above example shows that for put options, \Delta is negative and B is positive (i.e. shorting stock and lending replicate the payoff of a put). The number \Delta has a special interpretation that will be important in subsequent discussion of option pricing. It can be interpreted as the sensitivity of the option to a change in the stock price. For example, if the stock price changes by $1, then the option price, \Delta \ S + B, changes by the amount \Delta. In other words, \Delta is the change in the option price per unit increase in the stock price.

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Put-call parity

The put-call parity relates the price of a European call with a European put that has the same strike price and the same time to expiration. The following is a call on XYZ stock that is compatible to the put described above.

  • The underlying asset of the call option is the XYZ stock.
  • The strike price is $55.
  • The option will expire in one year.
  • The option is assumed to be a European option, i.e. it can be exercised only at expiration.

The previous post shows that the premium of this call option is $4.316821227. The put-call parity also derive the same cost for the put.

    \displaystyle \begin{aligned} P(55,1)&=C(55,1)-50+55 \ e^{-0.02} \\&=4.316821227-50+55 \ e^{-0.02} \\&=\$ 8.2277  \end{aligned}

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Remarks

The examples discussed in this post and in the previous post have value even though the examples may seem like an extreme simplification. These two examples are an excellent introduction to the subject of option pricing theory. The one-period example can be extended to a multi-period approach to describe far more realistic pricing scenarios. For example, we can break a year into many subintervals. We then use the 2-state method to describe above to work backward from the stock prices and option values of the last subinterval to derive the value of the replicating portfolio.

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Practice problems

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

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

Pricing a call option – an example

The example in this post illustrates how to price a call option using the one-period binomial option pricing model. The next post will present an example on pricing a put option. The two posts are designed to facilitate the discussion on the binomial option pricing (given in a series of subsequent posts). Links to practice problems are found at the bottom of the post.

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The example

The following gives the information about the stock:

  • The stock of XYZ company is currently selling for $50 per share. The price per share 1 year from now is expected to increase to $65 or to decrease to $40. The stock pays no dividends.

Consider a call option with the following specifics:

  • The underlying asset of the call option is the XYZ stock.
  • The strike price is $55.
  • The option will expire in one year.
  • The option is assumed to be a European option, i.e. it can be exercised only at expiration.

The annual risk-free interest rate is 2%. There is a benefit to the buyer of the option described above. If the price of the stock goes up to $65 at the end of the 1-year period, the owner of the option has the right to exercise the option, i.e., buying one share at the strike price of $55 and then selling it at the market price of $65, producing a payoff of $10. If the price of the stock goes down to $40 at the end of the 1-year period, the buyer of the option has the right to not exercise the option. The call option owner buys the stock only when he makes money. What would be the fair price of having this privilege? What is the fair price of this call option?

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Pricing the call option

In this example, the current stock price is $50 and the stock price can be only one of the two possible values at the end of the option contract period (either $65 or $40). The following diagram shows the future state of the stock prices.

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

The assumption of the 2-state stock prices in 1 year simplifies the analysis of the call option. The value of the call option at the end of 1 year is either $10 (=65-55) or zero. Note that when the share price at the end of the 1-year contract period is less than the strike price of $55, the call option expires worthless. The following diagram shows the value of the call option.

    \text{ }
    Figure 2 – Call Option Payoff
    call option payoff
    \text{ }

In the above diagram, the value of the call option at the end of 1-year is either $10 or $0. The value of the option at time 0 is C, which is the premium of the call option in this example. Our job here is to calculate C. The key to finding the value of the option is to compare the payoff of the call to that of a portfolio consisting of the following investments:

    Portfolio A

  • Buy 0.4 shares of XYZ.
  • Borrow $15.683 at the risk-free rate.

The idea for setting up this portfolio is given below. For the time being, we take the 0.4 shares and the borrowed amount of $15.683 as a given. Note that $15.683 is the present value of $16 at the risk-free rate of 2%. Let’s calculate the value of Portfolio A at time 0 and at time 1 (1 year from now). The following diagram shows the calculation.

    \text{ }
    Figure 3 – Portfolio A Payoff

    replicating portfolio payoff
    \text{ }

Note that the payoff of the call option is identical to the payoff of Portfolio A. Thus the call option in this example and Portfolio A must have the same cost. Since Portfolio A costs $4.317, the price of the option must be $4.317. The Portfolio A of 0.4 shares of stock and $15.683 in borrowing is a synthetic call since it mimics the call option described in the example. Portfolio A is called a replicating portfolio because it replicates the payoff of the call option in question.

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Arbitrage opportunities

In deriving the cost of the call option of $4.137, we rely on the idea that if two investments have the same payoff, they must have the same cost. This idea is called the law of one price, which is a commonsensical idea and is also an important principle in derivative pricing. If the law of one price is violated, in particular if the price of the call option discussed in this example is not $4.317, there would be arbitrage opportunities that can be exploited to gain risk-free profit.

What if the law of one price is violated? For example, what if the option were selling for a higher price (say $4.50)? If the price of the replicating portfolio is less than the price of the option, then we can “buy low and sell high” (i.e. buy the replicating portfolio and sell call option) and obtain a risk-free profit of $0.183. The arbitrage is to buy the synthetic call (Portfolio A) at $4.317 and sell the call option at $4.50. The following table shows the Year 1 cash flows of this arbitrage opportunity.

    \text{ }

    Table 1 – Arbitrage opportunity when call option is overpriced

    \left[\begin{array}{llll}      \text{Year 1 Cash Flows} & \text{ } & \text{Share Price = } \$ 40 & \text{Share Price = } \$ 65 \\      \text{ } & \text{ } \\      \text{Long synthetic call} & \text{ } & \text{ } & \text{ } \\      \ \ \ \ \text{Hold 0.4 shares}  & \text{ } & + \$ 16 & + \$ 26 \\      \ \ \ \ \text{Repay borrowed amount of } \$ 15.683  & \text{ } & - \$ 16 & - \$ 16 \\      \text{ } & \text{ } \\      \text{Short call }  & \text{ } & \ \ \$ 0 & - \$ 10 \\      \text{ } & \text{ } \\            \text{Total payoff} & \text{ } & \text{ } \ \$ 0  & \ \ \$ 0    \end{array}\right]

    \text{ }

The above table shows that buying a synthetic call (holding 0.4 shares and borrow $15.683) and selling a call will have no loss at the end of 1 year. Yet, the time 0 cash flow is $0.183 (=4.50 – 4.317), and is thus a risk-less profit.

If the option is underpriced, then we can still buy low and sell high (in this case, buy call option and sell the replicating portfolio) and obtain risk-free arbitrage profit. For example, let’s say you observe a call option price of $4.00. Then the arbitrage opportunity is to buy the call option at $4.00 and sell a synthetic call (Portfolio A) at $4.317. The time 0 payoff is $0.317, which is a risk-less arbitrage profit. The following table shows the Year 1 cash flows.

    \text{ }

    Table 2 – Arbitrage opportunity when call option is underpriced

    \left[\begin{array}{llll}      \text{Year 1 Cash Flows} & \text{ } & \text{Share Price = } \$ 40 & \text{Share Price = } \$ 65 \\      \text{ } & \text{ } \\      \text{Short synthetic call} & \text{ } & \text{ } & \text{ } \\      \ \ \ \ \text{Short 0.4 shares}  & \text{ } & - \$ 16 & - \$ 26 \\      \ \ \ \ \text{Receive the amount of } \$ 15.683  & \text{ } & + \$ 16 & + \$ 16 \\      \text{ } & \text{ } \\      \text{Long call }  & \text{ } & \ \ \$ 0 & + \$ 10 \\      \text{ } & \text{ } \\            \text{Total payoff} & \text{ } & \text{ } \ \$ 0  & \ \ \$ 0    \end{array}\right]

    \text{ }

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To complete the picture

The call option price of $4.317 is derived by showing that the replicating portfolio has the same payoff as the call option. How do we know that the replicating portfolio consists of holding 0.4 shares and the borrowing of $15.683?

In general, the replicating portfolio of a European call option consists of \Delta shares of the stock and the amount B in lending at time 0 (borrowing if negative). By equating the payoff of the replicating portfolio and the payoff of the call option in this example, we have the following equations:

    \text{ }
    \displaystyle \begin{array}{ccc} \displaystyle 40 \ \Delta + B \ e^{0.02} & = & 0 \\ \displaystyle 65 \ \Delta + B \ e^{0.02} & = & 40  \end{array}
    \text{ }

Solving these two equations, we obtain \Delta=\frac{10}{25}=0.4 and B=-16 \ e^{-0.02}=15.683. Therefore, the replicating portfolio for the call option in this example consists of 0.4 shares of the stock and $15.683 in borrowing. The net investment for the replicating portfolio is $4.317 (=0.4(50)-15.683). Because there are only two data points in the future stock prices, the option premium is a linear function of \Delta and B. The following is the premium of the call (or put) option using the one-period binomial tree

    C=\Delta \ S+B

where S is the stock price at expiration. The above formula gives the cost of the portfolio replicating the payoff of a given option. It works for call option as well as for put option. We will see that for put options, \Delta is negative and B is positive (i.e. shorting stock and lending replicate the payoff of a put). The number \Delta has a special interpretation that will be important in subsequent discussion of option pricing. It can be interpreted as the sensitivity of the option to a change in the stock price. For example, if the stock price changes by $1, then the option price, \Delta \ S + B, changes by the amount \Delta. In other words, \Delta is the change in the option price per unit change in the stock price.

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Put-call parity

The put-call parity relates the price of a European call with a European put that has the same strike price and the same time to expiration. The following is a put on XYZ stock that is compatible to the call described above.

  • The underlying asset of the put option is the XYZ stock.
  • The strike price is $55.
  • The option will expire in one year.
  • The option is assumed to be a European option, i.e. it can be exercised only at expiration.

By the put-call parity, the following gives the price of the put option.

    \displaystyle \begin{aligned} P(55,1)&=C(55,1)-50+55 \ e^{-0.02} \\&=4.316821227-50+55 \ e^{-0.02} \\&=\$ 8.2277  \end{aligned}

The next post will calculate the price of the same put using the binomial model.

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Remarks

We would like to comment that even though the example here may seem like an extreme simplification, the example has great value. First of all, this is an excellent introduction to the subject of option pricing theory. Secondly, the one-period example can be extended to a multi-period approach to describe far more realistic pricing scenarios. For example, we can break a year into many subintervals. We then use the 2-state method to describe above to work backward from the stock prices and option values of the last subinterval to derive the value of the replicating portfolio.

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Practice problems

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

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

Put-Call Parity, Part 2

Put-call parity is a key idea in option pricing theory. It provides a tool for constructing equivalent positions. The previous post gives a general discussion of the put-call parity. In this post, we discuss the put-call parity for various underlying assets, i.e. the parity relations in this post are asset specific. The following is one form of the general put-call parity. This is the version (0) discussed in the previous post.

    \text{ }
    Put-Call Parity
    \displaystyle PV(F_{0,T})=C(K,T)-P(K,T)+PV(K) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (0)
    \text{ }

The put-call parity has four components – the price of the call, the price of the put, the present value of the strike price and the present value of the forward price. In the general form of the put-call parity, the present value of the forward price completely take the dividends and time value of money into account. For a specific type of underlying asset, in order to make the put-call parity more informative, we may have to take all the interim payments such as dividends into account. Thus in the parity relations that follow, the general forward price is replaced with the specific forward price for that asset. Synthetic assets can then be created from the asset-specific put-call parity that is obtained.

The notations used here are the same as in the previous posts. The notation F_{0,T} is the forward price. All contracts – forward and options and other type of contracts – are set at time 0 (today) and are to end at time T. The strike price for the options is K. The letter r denotes the risk-free annual continuous interest rate. If the strike price K is paid for an asset at time T, its present value at time 0 is PV(K)=e^{-r T} K. All options discussed here are European options, i.e. they can be exercised only at expiration.

All the parity relations that follow will obviously involve a call and a put. To make this extra clear, the call and the put in these relations have the same strike price and the same time to expiration. Thus whenever we say buying a call and selling a put, we mean that they are compatible in this sense.

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Put-call parity for stocks

Forward prices for stocks are discussed here. For a non-dividend paying stock, the forward price is F_{0,T}=S_0 e^{r T}, i.e. the price to pay for the stock in the future is the future value of the time 0 stock price. The following is the put-call parity of a non-dividend paying stock.

    \text{ }
    Put-Call Parity – non-dividend paying stock
    \displaystyle S_0=C(K,T)-P(K,T)+e^{-r T} K \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (S1)
    \text{ }

The parity (S1) says that there are two ways to buy a non-dividend paying stock at time 0. One is the outright stock purchase (the left side). The other way (the right hand side) is to buy a call, sell a put and lend the present value of the strike price K. By buying a call and selling a put, it is certain that you will buy the stock by paying K, which is financed by the lending of PV(K)=e^{-r T} K at time 0. In both ways, you own the stock at time T. There is a crucial difference. In the outright stock purchase, you own the stock at time 0. In the “options” way, the stock ownership is deferred until time T. For the non-dividend paying stock, an investor is probably indifferent to the deferred ownership in the right hand side of (S1). For dividend paying stock, deferred ownership should be accounted for the parity equation.

    \text{ }
    Put-Call Parity – dividend paying stock (discrete dividend)
    \displaystyle S_0-PV(\text{Div})=C(K,T)-P(K,T)+e^{-r T} K  \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (S2)
    \text{ }

In (S2), \text{Div} refers to the dividends paid during the period from time 0 to time T and PV(\text{Div}) refers to the time 0 value of \text{Div}. The deferred stock ownership on the right hand side of (S2) does not have the dividend payments while the outright stock ownership has the benefit of the interim dividend payments. Thus the cost of deferred stock ownership must be reduced by the amount of the dividend payments. This is why the dividend payments are subtracted on the left hand side. The next parity relation is for a stock or stock index paying continuous dividend.

    \text{ }
    Put-Call Parity – dividend paying stock (continuous dividend)
    \displaystyle S_0 e^{-\delta T}=C(K,T)-P(K,T)+e^{-r T} K \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (S3)
    \text{ }

Continuous dividends are reinvested (as additional shares) where \delta is the annual continuous compounded dividend rate. The forward price is F_{0,T}=S_0 e^{(r-\delta) T}. The present value of the forward price is S_0 e^{-\delta T}, which is the left hand side of (S3). The left side of (S3) is saying that e^{-\delta T} shares at time 0 will accumulate to 1 share at time T. The right hand side is saying that buying a call, selling a put and lending out the present value of K at time 0 will lead to ownership of 1 share at time T.

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Synthetic stocks and other synthetic assets

In this section, we consider synthetic assets that can be created from the parity relations on stocks. These synthetic assets are parity relations. The left side of each of these relations is an asset that exists naturally in the financial market place. The right hand side is the synthetic asset – a portfolio that is an alternative asset that has the same cost and payoff, thus a portfolio that mimics the natural asset. For example, a synthetic stock is a combination of put and call and a certain amount of lending that will replicate the same payoff as owning a share of stock. In the next section, we will resume the discussion of put-call parity on underlying assets.

Each of the parity relation in this section is derived from an appropriate stock put-call parity by solving for the desired asset. For a synthetic stock, we put the stock on the left hand side by itself.

    \text{ }
    Synthetic stock – non-dividend paying
    \displaystyle S_0=C(K,T)-P(K,T)+e^{-r T} K \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (Syn1)
    \text{ }
    Synthetic stock – discrete dividend paying
    \displaystyle S_0=C(K,T)-P(K,T)+e^{-r T} K+PV(\text{Div})  \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (Syn2)
    \text{ }
    Synthetic stock – continuous dividend paying
    \displaystyle S_0 =(C(K,T)-P(K,T)+e^{-r T} K) \ e^{\delta T} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (Syn3)
    \text{ }

Note that (Syn1) is identical to (S1) since there is no dividend. The portfolio on the right hand side is the synthetic stock. For example, for (Syn2), the strategy of buying a call, selling a put, and lending out the present values of the strike price and the interim dividends is an alternative way to own a discrete dividend paying stock. There is a crucial difference between outright stock ownership on the left hand side and the deferred stock ownership on the right hand side. The synthetic stock pays no dividends. Thus the outright stock ownership is worth more than the synthetic stock. In other words, the cost of outright stock ownership exceeds the synthetic cost. By how much? By the present value of the interim dividends. This is why the present value of the dividend payments is added to the right hand side of (Syn2) and (Syn3).

Now we consider synthetic T-bills (or synthetic risk-free asset).

    \text{ }
    Synthetic T-bill – based on non-dividend paying stock
    \displaystyle e^{-r T} K=S_0-C(K,T)+P(K,T) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (T1)
    \text{ }
    Synthetic T-bill – based on discrete dividend paying stock
    \displaystyle e^{-r T} K+PV(\text{Div})=S_0-C(K,T)+P(K,T)  \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (T2)
    \text{ }
    Synthetic T-bill – based on continuous dividend paying stock
    \displaystyle e^{-r T} K=S_0 e^{-\delta T}-C(K,T)+P(K,T) \  \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (T3)
    \text{ }

In (T1), (T2) and (T3), the right hand side is the synthetic way of creating a T-bill. Let’s look at (T3).

    Relation (T3). In order to hold a synthetic T-bill, you buy e^{-\delta T} shares of stock, sell a call and buy a put at time 0. At time T, the e^{-\delta T} shares become 1 share, which will be used to meet the demand of either the call option or put option. If the stock price is more than K, the call buyer will want to exercise the call and you as a seller of the call will have to sell 1 share at the strike price K. If the stock price is less than K at time T, you as the put buyer will want to sell 1 share of stock at the strike price K. So in either case, you have the amount K at time T, precisely the outcome if you buy a T-bill with maturity value K.

Next we consider synthetic call options.

    \text{ }
    Synthetic call – based on non-dividend paying stock
    \displaystyle C(K,T)=S_0-e^{-r T} K+P(K,T) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (C1)
    \text{ }
    Synthetic call – based on discrete dividend paying stock
    \displaystyle C(K,T)=S_0-e^{-r T} K-PV(\text{Div})+P(K,T)  \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (C2)
    \text{ }
    Synthetic call – based on continuous dividend paying stock
    \displaystyle C(K,T)=S_0 e^{-\delta T}-e^{-r T} K+P(K,T) \  \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (C3)
    \text{ }

The right hand side of the above three equations are synthetic ways to buy a stock call option. They can be derived by solving for C(K,T) in the put-call parity relation in respective stock. It also pays to think through the cash flows on both sides. The right hand side of each of (C1) through (C3) consists of a leveraged position (stock purchase plus borrowing) and a long put to insure the leveraged position. For example, in the right hand side of (C1), borrow e^{-r T} K and buy one share of stock (the leveraged position). Then use a purchased put to insure this leveraged position.

Another way to look at synthetic call is that the right hand side consists of a protective put and borrowing. A protective put is the combination of a long asset and a long put. For example, the right hand side of (C1) consists of S_0+P(K,T) (a protective put) and the borrowing of e^{-r T} K, the present value of K.

Here’s the synthetic put options.

    \text{ }
    Synthetic put – based on non-dividend paying stock
    \displaystyle P(K,T)=C(K,T)-S_0+e^{-r T} K \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (P1)
    \text{ }
    Synthetic put – based on discrete dividend paying stock
    \displaystyle P(K,T)=C(K,T)-S_0+e^{-r T} K+PV(\text{Div})  \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (P2)
    \text{ }
    Synthetic put – based on continuous dividend paying stock
    \displaystyle P(K,T)=C(K,T)-S_0 e^{-\delta T}+e^{-r T} K \  \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (P3)
    \text{ }

The right hand side of each of (P1) through (P3) is a synthetic put, a portfolio that mimics the payoff of a put option. Note that the right hand side consists of a long call and a short stock position (this is a protective call) and the lending of the present value of K.

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Put-call parity for currencies

A previous post on forward prices shows that the currency forward price is F_{0,T}=x_0 \ e^{(r-r_f) T} where x_0 is the exchange rate (units of domestic currency per unit of foreign currency, e.g. dollars per euro), r is the domestic risk-free rate and r_f is the foreign currency risk-free rate. The present value of F_{0,T} is then e^{-r T} \ F_{0,T}=x_0 \ e^{-r_f T}, which is the number of units of the domestic currency (e.g. dollars) at time 0 in order to have one unit of foreign currency (e.g. euro) at time T. Substituting e^{-r T} \ F_{0,T}=x_0 \ e^{-r_f T} into the parity relation of (0), we have:

    \text{ }
    Put-Call Parity – Currencies
    \displaystyle x_0 \ e^{-r_f T}=C(K,T)-P(K,T)+e^{-r T} K \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (F1)
    \text{ }
    \displaystyle x_0 \ e^{-r_f T}-e^{-r T} K=C(K,T)-P(K,T) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (F2)
    \text{ }

In (F1) and (F2), we assume that the call and the put are denominated in dollars, i.e. both the strike price K and the put premium and call premium are denominated in dollars. For ease of discussion, let’s say the foreign currency is euro. The premium C(K,T) discussed here is in dollars and grants the right to pay K to get 1 euro. The premium P(K,T) discussed here is in dollars and grants the right to pay 1 euro to get K. Thus the strike price K is an exchange rate of USD per euro.

For example, let’s say K= 0.80 USD/Euro at time 0. If at time T the exchange rate is x_T= 0.9 USD/Euro, the call buyer would want to exercise the option by paying 0.8 USD for 1 euro. If at time T the exchange rate is x_T= 0.7 USD/Euro, then the long put position would want to exercise the put by paying 1 euro to get 0.8 USD.

The relation (F1) indicates that the difference in the call and put premiums plus lending the present value of the strike price is the same as lending the present value of the amount in dollars (the domestic currency) that is required to buy 1 euro at time T.

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Put-call parity for bonds

For a zero-coupon bond, the forward price is simply the future value of the bond price. For a coupon paying bond, the future price has to reflect the value of the coupon payments. In the following parity relations, B_0 is the bond price at time 0. The amount PV(\text{Coupons}) is the present value of the coupon payments made during the life of the options.

    \text{ }
    Put-Call Parity – zero-coupon bond
    \displaystyle B_0=C(K,T)-P(K,T)+e^{-r T} K \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (B1)
    \text{ }
    \displaystyle B_0-PV(\text{Coupons})=C(K,T)-P(K,T)+e^{-r T} K  \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (B2)
    \text{ }

Note that for the zero-coupon bond, the parity relation is similar to the one for non-dividend paying stock.

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Summary

The following is the list of all the asset specific put-call parity relations discussed in this post.

    \text{ }
    Forward/Futures
    \displaystyle e^{-r T} \ F_{0,T}=C(K,T)-P(K,T)+PV(K)
    \text{ }

    Non-dividend paying stock
    \displaystyle S_0=C(K,T)-P(K,T)+e^{-r T} K
    \text{ }

    Discrete dividend paying stock
    \displaystyle S_0-PV(\text{Div})=C(K,T)-P(K,T)+e^{-r T} K
    \text{ }

    Continuous dividend paying stock
    \displaystyle S_0 e^{-\delta T}=C(K,T)-P(K,T)+e^{-r T} K
    \text{ }

    Currency
    \displaystyle x_0 \ e^{-r_f T}=C(K,T)-P(K,T)+e^{-r T} K
    \text{ }

    Bond
    \displaystyle B_0=C(K,T)-P(K,T)+e^{-r T} K
    \text{ }

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

Basic insurance strategies – covered call and covered put

The use of options can be interpreted as buying or selling insurance. This post follows up on a previous post that focuses on two option strategies that can be interpreted as buying insurance – protective put and protective call. For every insurance buyer, there must be an insurance seller. In this post, we discuss two option strategies that are akin to selling insurance – covered call and covered put.

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Selling insurance against an asset position

The previous post discusses the strategies of protective put and protective call. Both of these are “buy insurance” strategies. A protective put consists of a long asset and a long put where the long put is purchased to protect against a fall in the prices of the long asset. A protective call consists of a short asset position and a long call where the long call option is purchased to protect against a rise in the prices of the asset being sold short. Both of these strategies are to buy an option to protect against the adverse price movement of the asset involved.

When an insurer sells an insurance policy, the insurer must have enough asset on hand to pay claims. Now we discuss two strategies where the investor or trader holds an asset position that can be used for paying claims on a sold option.

A covered call consists of a long asset and a short call. The insurance sold is in the form of a call option. The long asset gains in value when asset prices rise and the gains are used to cover the payments made by the call seller when the call buyer decides to exercise the call option. Therefore the covered call is to use the upside profit potential of the long asset to back up (or cover) the call option sold to the call buyer. The covered call strategy can be used by an investor or trader who believes that the long asset will appreciate further in the future but is willing to trade the long term upside potential for a short-term income (the call premium). This is especially true if the investor thinks that selling the long asset at the strike price of the call option will meet a substantial portion of his expected profit target.

A covered put consists of a short asset position and a short put. Here, the insurance sold is in the form of a put option. The short asset is used to back up (or cover) the put option sold to the put buyer. A short asset position is not something that is owned. How can a short asset position back up a put option? The short asset position gains in value when asset prices fall. A put option is exercised when the prices of the underlying asset fall. Thus a put option seller needs to pay claims exactly when the short asset position gains in value. Thus the gains in the short asset position are used to cover the payments made by the put seller when the put buyer decides the exercise the put option.

In this post, we examine covered call and covered put in greater details by examining their payoff diagrams and profit diagrams.

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Covered call

As mentioned above, a covered call is a position consisting of a long asset and a short call. Here the holder of the long asset sells a call against the long asset. Figure 1 is the payoff of the long asset. Figure 2 is the payoff of the short call. Figure 3 is the payoff of the covered call. Figure 4 is the profit of the covered call. The strike price in all the diagrams is K. We will see from Figure 4 that the covered call is a synthetic short put.

    \text{ }

    Figure 1 – Long Asset Payoff
    long asset position payoff

    \text{ }

Figure 1 is the payoff of the long asset position. When the asset prices are greater than the strike price K, the positive payoff is unlimited. The unlimited upside potential is used to pay claim when the seller of the call is required to pay claim to the call buyer.

    \text{ }

    Figure 2 – Short Call Payoff
    Short call position payoff

    \text{ }

Figure 2 is the payoff of the short call. This is the payoff of the call seller (i.e. the insurer). The call seller has negative payoff to the right of the strike price. The negative payoff occurs when the call buyer decides to exercise the call. The long asset payoff in Figure 1 is to cover this negative payoff.

    \text{ }
    Figure 3 – Long Asset + Short Call Payoff
    long asset position short call payoff
    \text{ }

Figure 3 is the payoff of the covered call, the result of combining Figure 1 and Figure 2. Unlike Figure 1, the long asset holder no longer has unlimited payoff to the right of the strike price. The payoff is now capped at the strike price K.

    \text{ }
    Figure 4 – Long Asset + Short Call Profit
    long asset position short call profit
    \text{ }

Figure 4 is the profit of the covered call. The profit is the payoff less the cost of acquiring the position. At time 0, the cost is S_0 (the purchase price of the asset, an amount that is paid out) less P (the option premium, an amount that is received). The future value of the cost of the covered call is then S_0 e^{r T}-P e^{r T}. The profit is then the payoff less this amount. The profit graph is in effect obtained by pressing down the payoff graph by the amount of S_0 e^{r T}-P e^{r T}. Because of the received option premium, S_0 e^{r T}-P e^{r T} is less than the strike price K. As a result, the flat part of the profit graph is above the x-axis.

Without selling insurance (Figure 1), the profit potential of the long asset is unlimited. With the insurance liability (Figure 4), the profit potential is now capped at essentially at the call option premium. In effect the holder of a covered call simply sells the right for the long asset upside potential for cash received today (the option premium).

The strategy of a covered call may make sense if selling at the strike price can achieve a significant part of the profit target expected by the investor. Then the payoff from the strike price plus the call option premium may represent profit close to the expected target. Let’s look at a hypothetical example. Suppose that the stock owned by an investor was purchased at $60 a share. The investor believes that the stock has upside potential and the share price will rise to $70 in a year. The investor can then sell a call option with the strike price of $65 with an expiration of 6 months and with a call premium of $5. In exchange for a short-term income of the call option premium, the investor gives up the profit potential of $70 a share. If in 6 months, the share price is more than $65, then the investor will sell at $65 a share, producing a profit of $10 a share ($5 in share price appreciation and $5 call premium). If the share price is below the strike price is 6 months, the investor then pockets the $5 premium.

Note the similarity between Figure 4 above and the Figure 11 in this previous post. Figure 11 in that previous post is the profit diagram of a short put. So the covered call (long asset + shot call) is also called a synthetic short put option since it has the same profit as a short put.

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Covered put

As indicated above, a covered put is to use the profit potential of a short asset position to cover the obligation of a sold put option. Figure 5 below is the profit of a short asset position. Figure 6 is the payoff of a short put option. Figure 7 is the payoff of the covered put. Figure 8 is the profit diagram of the covered put.

    \text{ }

    Figure 5 – Short Asset Payoff
    short asset payoff
    \text{ }

Figure 5 is the payoff of the short asset position. Holder of a short asset position is concerned about rising prices of the asset. The holder of the short borrows the asset in a short sales and sells the asset immediately for cash, which is then accumulated at the risk-free rate. The short position will have to buy the asset back in the spot market at a future date to repay the lender. If the spot price at expiration is greater than the original sale price, then the short position will lose money. In fact the potential loss is unlimited.

    \text{ }

    Figure 6 – Short Put Payoff
    Short put payoff
    \text{ }

Figure 6 is the payoff of a short put option. Recall that the short put payoff is from the perspective of the seller of the put option. When the price of the underlying asset is below the strike price, the seller has the obligation to sell at the strike price (thus experiencing a loss). When the asset price is above the strike price, the put option expires worthless.

    \text{ }

    Figure 7 – Short Asset + Short Put Payoff
    short asset short put payoff
    \text{ }

Figure 7 is the payoff of the covered call. With the covered call, the holder of the short asset can no longer profit by paying a price lower than the strike price for the asset to repay the lender. Instead he has to pay the strike price (this is the flat part of Figure 7). To the right of the strike price, the covered call continues to have the potential for unlimited loss.

    \text{ }

    Figure 8 – Short Asset + Short Put Profit
    short asset short put profit
    \text{ }

Figure 8 is the profit of the covered put, which indicates the profit is essentially the option premium received by selling the put option. Without selling the insurance (Figure 5), the short asset position has good profit potential when prices fall. With selling the insurance, the profit potential to the left of the strike price is limited to the option premium. The covered put is in effect to trade the profit potential (when prices are low) with a known put option premium.

Compare Figure 8 above with Figure 5 in this previous post. Both profit diagrams are of the same shape. Figure 5 in the previous post is the profit diagram of a short call. So the combined position of short asset + short put is called a synthetic short call.

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Synthetic put and call

Just a couple of more observations to make about synthetic put and synthetic call.

Note that Figure 3 (the payoff of long asset + short call) also resembles the payoff of a short put option, except that the level part of the payoff is not at the x-axis. So Figure 3 is the lifting up of the usual short put option payoff by a uniform amount. That uniform amount can be interpreted as the payoff of a long zero-coupon bond. Thus we have the following relationship.

    \text{ }
    payoff of “long asset + short call” = payoff of “short put + zero-coupon bond”
    \text{ }

Adding a bond lifts the payoff graph. However, adding a bond to a position does not change the profit. To see this, simply subtract the cost of acquiring the position from the payoff. You will see that for the bond, the same amount appears in both the cost and the payoff. Thus we have:

    \text{ }
    profit of “long asset + short call” = profit of “short put”
    \text{ }

As mentioned earlier, the above relationship indicates that the combined position of long asset + short call can be viewed as a synthetic short put. We now see that the covered call is identical to a short put.

Now similar thing is going on in a covered put. Note that Figure 7 resembles the payoff of a short call except that it is the pressing down of the payoff of a usual short call. We can think of this pressing down as a borrowing. Thus we have:

    \text{ }
    payoff of “short asset + short put” = payoff of “short call – zero-coupon bond”
    \text{ }

Adding a bond means lending and subtracting a bond means borrowing. As mentioned before, adding or subtracting a bond lift or depress the payoff graph but does not change the profit graph. We have:

    \text{ }
    profit of “short asset + short put” = profit of “short call”
    \text{ }

The above relationship is the basis for calling “short asset + short put” as a synthetic short call.

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

Basic insurance strategies – protective put and protective call

This post follows up on a previous post, which is an introductory discussion on options. In this post, we focus on the two basic strategies of using options as insurance – protective put and protective call. These two strategies are for investors or traders who want to buy insurance to protect profits that come from holding either a long or short position. For every insurance buyer, there must be an insurance seller. In the next post, we discuss covered call and covered put – basic strategies for investors or traders who want to sell insurance protection.

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Option strategies that are insurance protection

Suppose that an investor has been holding an asset that has gained in value. The long asset position held by this investor will suffer a loss if prices fall. A long put option (a purchased put) has positive payoff when prices are less than the strike price. Buying a put option will guarantee a minimum sale price (the strike price of the put option) should the investor wishes to sell the asset at a future date. Thus the risk management strategy of buying a put option to guard against the loss of a long position is called a protective put.

A protective call deals with an opposite situation. Suppose an investor or trader is holding a short position on an asset (e.g. the investor has short sold a stock). The short asset position held by this investor will suffer a loss if prices increase. A purchased call option has positive payoff when the asset prices are greater than the strike price. When the investor buys a call option with the same underlying asset, the strike price is in effect a minimum purchased price of the asset should there be a price increase, thus keeping the loss at a minimum. Thus the risk management strategy of buying a call option to guard against the loss of a short position is called a protective call.

Protective puts and protective calls are basic insurance strategies that can be used to protect profits from either holding a long asset position or a short position. Both of these strategies will minimize the loss in the event that the prices of the asset position move in the wrong direction. Of course, the insurance protection comes at a cost in the form of an option premium, which is a cash fee paid by the buyer to the seller at the time the option contract is made. In the remainder of the post, we examine the protective put and the protective call in greater details by examining the payoff and profit diagrams.

The put and call considered here are European options, i.e., they can be exercised only at expiration.

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Protective put

The protective put consists of a long asset position (e.g. owning a stock) and a long put option on the same asset. Our goal is to examine the payoff and profit of this combined position. We can then make some observations based on the profit diagram. Figure 1 is the payoff of the long asset. Figure 2 is the payoff of the long put option. Figure 3 is the payoff of the combined position. Figure 4 is the profit of long asset + long put. The strike price in all the diagrams is K. Instead of using a numerical example to anchor the diagrams, we believe that the following diagrams of payoff and the profit are clear. In fact, getting bogged down in a numerical example may make it harder to see the general idea. Asking questions such as – what happens when the asset is less than K, etc – will make the diagrams clear. In fact, reading the diagrams is a good concept check. An even better practice is to draw the payoff and profit diagrams on paper.

    \text{ }

    Figure 1 – Long Asset Payoff
    long asset payoff
    \text{ }

Figure 1 is the payoff of the long asset. The strike price K has no effect on the payoff of the long asset (Figure 1). The payoff of an asset is simply the value of the asset at a point in time. Thus the payoff is simply the asset price at a target date. The higher the asset price, the higher the payoff.

    \text{ }
    Figure 2 – Long Put Payoff
    long put payoff
    \text{ }

Figure 2 is the payoff of the long put option. For the long put, the payoff is K-S_T when it makes sense for the put option buyer to exercise. Thus the payoff is positive to the left of K. To the right of K, the put option expires worthless, thus the payoff is 0. The sum of Figure 1 and Figure 2 gives Figure 3.

    \text{ }
    Figure 3 – Long Asset Long Put Combined Payoff
    long asset long put payoff
    \text{ }

Figure 3 is the payoff of Long Asset + Long Put. The payoff to the right of the strike price K is flat, which is a sign of the insurance at work. The positive payoff of the long put neutralizes the effect of falling prices of the long asset, minimizing the loss from holding a long asset when the prices go south. This position of long asset + long put will enjoy the upside potential in the event that prices go up. Of course, such as good insurance product is not free. The next diagram will take cost into account. First, the following formula shows the payoff of long asset + long put.

    \text{ }

    \displaystyle \text{payoff of long asset + long put}=\left\{\begin{matrix} \displaystyle K&\ \ \ \ \ \ S_T \le K \\{\text{ }}& \\ S_T&\ \ \ \ \ \ S_T >K   \end{matrix}\right.

    \text{ }

    \text{ }
    Figure 4 – Long Asset Long Put Combined Profit
    long asset long put profit
    \text{ }

Figure 4 is the profit of long asset + long put. Recall that the profit of a position is the payoff less the cost of acquiring that position. What is the cost of acquiring a long asset and a long put? The cost of the long asset is S_0 e^{r T}, the future value of the price paid at time 0. The cost of the long put is P e^{r T}, where P is the put option premium paid by the buyer to the seller at time 0. Thus the cost of the long asset + long put is (S_0+P) e^{r T}. As a result, the profit graph is Figure 4 is obtained by pressing down the payoff in Figure 3 by the amount of the cost. The cost of the position is likely to be more than the strike price K. This is why K-\text{Cost} in Figure 4 is negative.

Without the insurance (Figure 1), the long asset position will suffer substantial loss in the event that the prices are low. With insurance (Figure 4), the potential loss for the long asset position is essentially the put option premium. But the long asset position still enjoys the upside profit potential (less the option premium).

Another observation that can be made about Figure 4 is that its shape is very similar to the profit of a long call option (compare Figure 4 above with the Figure 1 in this previous post). The profit diagram of long asset + long put does not merely resemble the profit of a long call (with the same strike price); it is identical.

Based on Figure 4, the investment strategy of long asset + long put mimics the profit of the long call position (with the same strike price). Thus the position of long asset + long call is called a synthetic call option.

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Protective call

The protective call consists of a short asset position (e.g. shorting a stock) and a long call option on the same asset. We now examine the payoff and profit of this combined position. Figure 5 is the payoff of the short asset. Figure 6 is the payoff of the long call option. Figure 7 is the payoff of short asset + long call. Figure 8 is the profit of short asset + long call. The strike price in all the diagrams is K. From Figure 8, we will see that the combined position is a synthetic put option.

    \text{ }

    Figure 5 – Short Asset Payoff
    short asset payoff
    \text{ }

Figure 5 is the payoff of the short asset position. Holder of a short asset position is concerned about rising prices of the asset. The holder of the short borrows the asset in a short sales and sells the asset immediately for cash, which is then accumulated at the risk-free rate. The short position will have to buy the asset back in the spot market at a future date to repay the lender. If the spot price at expiration is greater than the original sale price, then the short position will lose money. In fact the potential loss is unlimited.

    \text{ }
    Figure 6 – Long Call Payoff
    long call payoff
    \text{ }

Figure 6 is the payoff of the long call position. When the spot price at expiration is less than the strike price K, the call option expires worthless. When the spot price at expiration is greater than the strike price K, the payoff is S_T-K.

    \text{ }
    Figure 7 – Short Asset + Long Call Payoff
    short asset long call payoff
    \text{ }

Figure 7 is the payoff of the combined position of a short asset and a long call. The payoff of the combined position is flat to the right of the strike price. This is a sign of the insurance at work. The upside potential of the short call is limiting the loss of the short asset position. The positive payoff of the long call is S_T-K. The payoff of the short asset is -S_T when price is greater than the strike price. Then the combined payoff is -K when price is greater than the strike price. To further clarify, the following is the payoff of the combined position.

    \text{ }

    \displaystyle \text{payoff of short asset + long put}=\left\{\begin{matrix} \displaystyle -S_T&\ \ \ \ \ \ S_T \le K \\{\text{ }}& \\ -K&\ \ \ \ \ \ S_T >K   \end{matrix}\right.

    \text{ }

    \text{ }
    Figure 8 – Short Asset + Long Call Profit
    short asset long call profit
    \text{ }

Figure 8 is the profit of short asset + long call. To derive the profit, we need to subtract the cost of acquiring the combined position from the payoff. The profit graph is Figure 8 is the result of lifting up the payoff graph. That suggests that in this case the profit is the payoff plus a positive amount. This is indeed correct since the cost of acquiring the position is a negative number. Thus subtracting the cost from the payoff is in effect adding a positive number.

To see the above point, the cost of acquiring the initial position is a positive number if it is a cash outflow (you pay to buy an asset) and is a negative number if it is a cash inflow (you sell an asset). In a short position, you borrow the asset and sell it to get cash, which is -S_0 e^{r T} in this calculation. There is also the purchase of a call. Thus the total cost is -S_0 e^{r T}+P e^{r T}, which is likely a negative amount. So subtracting this negative cost from the payoff has the effect of lifting up the payoff graph.

Without the insurance of a long call (Figure 5), the short asset position has unlimited loss. With insurance (Figure 8), the loss of the short asset position is minimized, essentially the call option premium. The short asset position still enjoys the profit potential should asset prices fall (less the option premium).

Compare the above Figure 8 with the Figure 8 in this previous post, we see that they have the same shape. This is not coincidence. Both positions have the same profit. Thus the combined position of a short asset and a long call option is called a synthetic long put option since both have the same profit diagrams.

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Synthetic put and call

Just a couple of more observations to make about synthetic put and synthetic call.

Note that Figure 3 (the payoff of long asset + long put) also resembles the payoff of a long call option, except that the level part of the payoff is not at the x-axis. So Figure 3 is the lifting up of the usual long call option payoff by a uniform amount. That uniform amount can be interpreted as the payoff of a zero-coupon bond. Thus we have the following relationship.

    \text{ }
    payoff of “long asset + long put” = payoff of “long call + zero-coupon bond”
    \text{ }

Adding a bond lifts the payoff graph. However, adding a bond to a position does not change the profit. To see this, simply subtract the cost of acquiring the position from the payoff. You will see that for the bond, the same amount appears in both the cost and the payoff. Thus we have:

    \text{ }
    profit of “long asset + long put” = profit of “long call”
    \text{ }

As mentioned earlier, the above relationship indicates that the combined position of long asset + long put can be viewed as a synthetic long call. We now see that the protective put is identical to a long call.

Now similar thing is going on in a protective call. Note that Figure 7 resembles the payoff of a long put except that it is the pressing down of the payoff of a usual long put. We can think of this pressing down as a borrowing. Thus we have:

    \text{ }
    payoff of “short asset + long call” = payoff of “long put – zero-coupon bond”
    \text{ }

Adding a bond means lending and subtracting a bond means borrowing. As mentioned before, adding or subtracting a bond lift or depress the payoff graph but does not change the profit graph. We have:

    \text{ }
    profit of “short asset + long call” = profit of “long put”
    \text{ }

The above relationship is the basis for calling “short asset + long call” as a synthetic long put.

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