## 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? ___________________________________________________________________________________ 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.

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Figure 1 – Stock Price

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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.

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Figure 2 – Call Option Payoff

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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.

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Figure 3 – Portfolio A Payoff

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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. ___________________________________________________________________________________ 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 $\left[\begin{array}{llll} \text{Year 1 Cash Flows} & \text{ } & \text{Share Price = } \ 40 & \text{Share Price = } \ 65 \\ \text{ } & \text{ } \\ \text{Short synthetic call} & \text{ } & \text{ } & \text{ } \\ \ \ \ \ \text{Short 0.4 shares} & \text{ } & - \ 16 & - \ 26 \\ \ \ \ \ \text{Receive the amount of } \ 15.683 & \text{ } & + \ 16 & + \ 16 \\ \text{ } & \text{ } \\ \text{Long call } & \text{ } & \ \ \ 0 & + \ 10 \\ \text{ } & \text{ } \\ \text{Total payoff} & \text{ } & \text{ } \ \ 0 & \ \ \ 0 \end{array}\right]$ $\text{ }$ ___________________________________________________________________________________ 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. ___________________________________________________________________________________ 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. ___________________________________________________________________________________ Practice problems Practice problems can be found in the companion problem blog via the following links: ___________________________________________________________________________________ $\copyright \ \ 2015 \ \text{Dan Ma}$ Advertisements ## 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. ___________________________________________________________________________________ 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 $T$. ___________________________________________________________________________________ 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 $K$. ___________________________________________________________________________________ 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 $T$. ___________________________________________________________________________________ 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. ___________________________________________________________________________________ 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 $\displaystyle B_0=C(K,T)-P(K,T)+e^{-r T} K$ $\text{ }$ ___________________________________________________________________________________ $\copyright \ \ 2015 \ \text{Dan Ma}$ ## Put-Call Parity, Part 1 Put–call parity is a relationship between the price of a European call option and European put option with the same strike price and time to expiration. It is one of the most important relationships in option pricing. It provides a tool for constructing equivalent positions. This post is a general discussion of put-call parity. In the next post, we discuss put-call parity in greater details for various underlying assets – e.g. stocks, treasuries and currencies. ___________________________________________________________________________________ Synthetic forward – buying a call and selling a put Suppose you follow the strategy of buying a call and selling a put (at time 0) where both options have the same underlying asset, the same strike price $K$ and the same time $T$ to expiration. At time $T$, it is certain that you will buy the underlying asset by paying the strike price $K$. Too see this, if at expiration of the options, the asset price is more than $K$, then you, as a call buyer will want to exercise the call option and pay $K$ to buy the asset. If the asset price at expiration is less than $K$, then you as a call buyer will not want to exercise but the put buyer that bought from you will want to exercise the put option. As a result, you will also buy the asset by paying the strike price $K$. Thus by entering into a long call and a short put (on the same underlying asset, with the same strike and same time to expiration), you will end up buying the underlying asset at time $T$ at the strike price $K$. What is being described sounds very much like a forward contract – a contract in which you can lock in a price today to pay for an asset a time $T$ in the future. For this reason, the strategy of buying a call and selling a put is called a synthetic forward contract. ___________________________________________________________________________________ Put-call parity The above discussion on synthetic forward suggests that there are two ways to buy an underlying asset (e.g. a stock) at time $T$ in the future. They are: 1. Enter into a forward contract to buy the underlying asset by paying the forward price $F_{0,T}$ at time $T$. 2. Buy a call and sell a put today (on the same underlying asset, with the same strike price $K$ and the same time $T$ to expiration). The two different strategies generate the same payoff. Hence they must have the same cost. Otherwise there would be arbitrage opportunities. By the “no-arbitrage pricing” principle, the net cost of the two strategies must equal. The cost at time 0 of the “buy call sell put” strategy is $C(K,T)-P(K,T)$, plus the present value of the strike price $K$, where $C(K,T)$ and $P(K,T)$ represent the call option premium and put option premium, respectively. The cost at time $T$ of the forward contract strategy is the forward price $F_{0,T}$. Thus cost at time 0 of the forward contract strategy is the present value of $F_{0,T}$. We can now equate the costs of the two strategies. $\text{ }$ Put-Call Parity $\displaystyle PV(F_{0,T})=C(K,T)-P(K,T)+PV(K) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (0)$ $\text{ }$ The notation $PV(\cdot)$ denotes the time 0 value of an amount at the time $T$. Equation (0) is one form of the put-call parity, which is a statement that buying a call and selling a put is equivalent to a synthetic forward contract. It also tells us that buying a call and selling a put plus lending the present value of the strike price is equivalent to buying the underlying asset. Other versions can be derived by algebraically rearranging equation (0), some of which have interesting interpretations. The following is one of them. $\text{ }$ Put-Call Parity $\displaystyle C(K,T)-P(K,T)=PV(F_{0,T}-K) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (1)$ $\text{ }$ The left hand side of (1) is the net option premium – the premium paid for the call less the premium received for the put. When this amount is not zero, it is in effect the premium of the synthetic forward contract (this amount is the initial cash outlay for the synthetic forward contract). This is one difference between a synthetic forward and an actual forward. Note that an actual forward contract has zero premium (the initial cash outlay is zero). Another difference is that the “forward price” of the synthetic forward is the strike price $K$ of the options and while the forward price of the actual forward is $F_{0,T}$. Suppose that the strike price $K$ is chosen to be less than the actual forward price $F_{0,T}$. Then the holder of the synthetic forward contract can buy the asset at a price lower than the forward price. This is certainly a benefit. In order to get this benefit, the holder of the synthetic forward contract has to pay the net option premium, which is the result of the call being more expensive than the put. In this scenario, the net payment is a little higher at time 0. As a result, the payment at time $T$ is a little less. Suppose that the strike price $K$ is chosen to be more than the actual forward price $F_{0,T}$. Then the holder of the synthetic forward position is obliged to pay for the underlying asset at a price higher than the forward. It then makes sense for the holder of the synthetic forward position to be compensated by receiving a payment initially. This would occur if the put is more expensive than the call. In this scenario, the net payment is a little less at time 0, leading to a larger payment at time $T$. If the strike price is chosen to be the same as the forward price $F_{0,T}$, then equation (1) suggests that the synthetic forward mimic exactly the actual forward (both have zero premium). For this to happen, premiums for the put and the call must be equal. The right hand side of (1) is the value of the discount resulted from paying the strike price instead of the forward price. This version of the put-call parity says that the discount is identical to the net option premium. ___________________________________________________________________________________ Protective put and covered call The next two versions can be interpreted in terms of a protective put and a covered call. A protective put consists of a long asset position and a long put. It is the strategy of buying a put option to protect against the risk of falling prices of a long asset position. A covered call consists of a long asset position and a short call. The covered call uses the upside profit potential of the long asset to back up (or cover) the call option sold to the call buyer. First, the protective call version: $\text{ }$ Put-Call Parity $\displaystyle PV(F_{0,T})+P(K,T)=C(K,T)+PV(K) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (2)$ $\text{ }$ The left hand side of (2) is the time 0 cash outlay of buying the underlying asset and buying a put. The right hand side of (2) is time 0 cash outlay of buying a call option (with the same strike and time to expiration as the put) and buying a zero-coupon bond costing $PV(K)$. Thus equation (2) tells us that buying the underlying asset and buying a put on that asset (i.e. a protective put) have the same cost and generate the same payoff as the buying a call option and buying a zero-coupon bond. Adding a bond lifts the payoff graph but does not change the profit graph. Thus buying the asset and buying a put has the same profit as buying a call. Because of Equation (2), buying the underlying asset and buying a put is called a synthetic long call option. This point is also discussed in this previous post. Here’s the version of the put-call parity involving covered call. $\text{ }$ Put-Call Parity $\displaystyle PV(F_{0,T})-C(K,T)=PV(K)-P(K,T) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (3)$ $\text{ }$ The left hand side of (3) is the time 0 cash outlay of buying the underlying asset and selling a call on that asset (i.e. a covered call). The right hand side of (3) is the time 0 cash outlay of buying a zero-coupon bond costing $PV(K)$ and selling a put. Thus a covered call has the same cost and same payoff as buying a bond and selling a put. Once again, adding a bond does not change the profit. Thus a covered call has the same profit as selling a put. For this reason, a buying the underlying asset and selling a call is called a synthetic short put option. This point is also discussed in this previous post. ___________________________________________________________________________________ Summary As a summary, we gather the various versions of the put-call parity in one place along with their interpretations. $\text{ }$ Versions of Put-Call Parity $\text{ }$ $\displaystyle PV(F_{0,T})=C(K,T)-P(K,T)+PV(K) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (0)$ Interpretation: Time 0 cost of a long asset = Time 0 cost of (Long Call + Short Put + Long Bond). $\text{ }$ $\displaystyle C(K,T)-P(K,T)=PV(F_{0,T}-K) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (1)$ Interpretation: Net option premium (call option premium that is paid out less put option premium received) = the value of the discount as a result of paying the strike price instead of the forward price. $\text{ }$ $\displaystyle PV(F_{0,T})+P(K,T)=C(K,T)+PV(K) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (2)$ Interpretation: Time 0 cost of (Long Asset + Long Put) = Time 0 cost of (Long Call + Long Bond). The portfolio on the left (Long Asset + Long Put) is called a protective put. Because of (2), a protective put is considered a synthetic long call option. $\text{ }$ $\displaystyle PV(F_{0,T})-C(K,T)=PV(K)-P(K,T) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (3)$ Interpretation: Time 0 cost of (Long Asset + Short Call) = Time 0 cost of (Long Bond + Short Put). The portfolio on the left (Long Asset + Short Call) is called a covered call. Because of (3), a covered call is considered a synthetic short put option. $\text{ }$ In each of the above versions of parity, the portfolio of investments on the left side is equivalent to the portfolio of investment on the right side. More specifically, each version equates the costs of obtaining the portfolios at time 0. The bond indicated in the interpretations is a zero-coupon bond. A long position on a bond means lending. One comment about the four parity relations discussed here. We derive the first one, which is version (0) by comparing the cash flows of two equivalent investments. The other three versions are then derived by algebraically rearranging the first version. As a learning device, it is a good idea to think through the cash flows and payoff of versions (2) through (3) independently of version (0). Doing so is a great practice and will help solidify the understanding of put-call parity. Drawing payoff diagrams can make the comparison easier. It is also possible to just think through the cash flows of both sides of the equation. For example, let’s look at version (2). On the right side, you lend $PV(K)$ and buy a call at time 0. Then at time $T$, you get $K$ back. If the price of the underlying asset at that time is more than $K$, then you exercise the call – using the $K$ that you receive to buy the asset. So on the right hand, side, the payoff is $S_T-K$ if asset price is more than $K$ and the payoff is $K$ if asset price is less than $K$ (you would not exercise the call in this case). On the left hand side, you lend $PV(F_{0,T})$ and buy a put at time 0. At time $T$, you get $F_{0,T}$ back and you use it to pay for the asset. So you own the asset at time $T$. If the asset price at time $T$ is less than $K$, you exercise by selling the asset you own and receive $K$. Thus the payoff on the left hand side is $S_T-K$ if asset price is more than $K$ (in this case you don’t exercise the put and instead you profit from holding the asset). The payoff is $K$ if the asset price at time $T$ is less than $K$ (this is the case where you exercise the put option). The comparison shows that both sides of (2) have the same payoff at time $T$. Then it must be the case that they also have the same cost at time 0. Otherwise, there would be an arbitrage opportunity by buying the side that is low and sell the other side. The basic put-call parity relations discussed in this post can be used in a “cookbook” fashion to create synthetic assets. For example, version (0) indicates that buying a call, selling a put and lending the present value of the strike price $K$ has the same cost and payoff as buying a non-dividend paying stock. Thus version (0) is a basis for constructing a synthetic stock. In the next post, we discuss the put-call parity for different underlying assets. ___________________________________________________________________________________ $\copyright \ \ 2015 \ \text{Dan Ma}$ ## 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. ___________________________________________________________________________________ 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. ___________________________________________________________________________________ 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 $\text{ }$ 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 $\text{ }$ 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 $\text{ }$ 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 $\text{ }$ 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 at60 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.

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Figure 5 – Short Asset Payoff

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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.

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Figure 6 – Short Put Payoff

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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.

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Figure 7 – Short Asset + Short Put Payoff

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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.

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Figure 8 – Short Asset + Short Put Profit

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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.

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Figure 1 – Long Asset Payoff

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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.

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Figure 2 – Long Put Payoff

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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.

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Figure 3 – Long Asset Long Put Combined Payoff

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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.

$\text{ }$

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

$\text{ }$

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Figure 4 – Long Asset Long Put Combined Profit

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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.

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Figure 5 – Short Asset Payoff

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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.

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Figure 6 – Long Call Payoff

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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 $S_T-K$.

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Figure 7 – Short Asset + Long Call Payoff

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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.

$\text{ }$

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

$\text{ }$

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Figure 8 – Short Asset + Long Call Profit

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

## Introducing options

This post is an introductory discussion of options. After introducing the basic concepts and terminology, we use payoff and profit diagrams to highlight and summarize the risk and benefit characteristics of call and put options.

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Motivating options with forward contracts

Suppose you need to buy an asset at the future time $T$. The price of the asset right now is $S_0$. You are of course concerned that prices of the asset will rise at or before time $T$. You can buy the asset now by paying $S_0$. If for some reasons buying the asset now is not a practical approach, you can lock in a price $F_{0,T}$ now to pay for the asset at time $T$. In other words, you can purchase a forward contract today to buy the asset at time $T$. For a more detailed discussion on forward contracts, see the following three posts:

There are two parties in a forward contract, the long position (the buyer) and the short position (the seller). The forward contract obliges the buyer (holder of the long position) to pay the forward price $F_{0,T}$ at expiration for the asset (of course the other side is obliged to sell at the forward price). The buyer is obliged to buy even when it is not advantageous for him to do so (i.e. when the price of the asset tumbles below the forward price $F_{0,T}$). Could there be a contract in which the buyer can buy when it is advantageous to do so and the buyer can walk away when it is not advantageous to buy?

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An example of a call option

The answer to the above question is yes. This is a contract where the buyer (the long position) has all the upside potential but essentially has limited downside risk. This is a call option, which is a contract where the buyer has the right to buy as asset at a price set ahead of time but not the obligation to buy.

To make the idea of a call option clear, let’s look at an example. Suppose that an investor buys a call option contract by agreeing to pay $40 for a share of XYZ company in 3 months. After 3 months, if the XYZ stock price is$50, then the investor will pay $40 for a share of XYZ, reaping a payoff of$10 per share. If the price in 3 months is $30, then the investor will walk away because he does not want to pay$40 for a share of stock that is worth less.

A call option contract has two parties, the buyer (the long position) and the seller (the short position). In 3 months, if the XYZ stock price is $50 per share, then the seller will sell to the buyer a share of XYZ for$40 because the buyer will choose to buy. In this case, the seller is selling a stock that is worth $10 more than the selling price and thus has a loss of$10 a share. If in 3 months, the stock price is $30, then buyer will not buy so the seller is not obliged to sell. When the contract expires, the seller will have a payoff of$0 if the XYZ stock price is at or below $40 per share. On the other hand, the seller’s payoff is negative (i.e. has a loss) if the stock price at expiration is above$40 per share.

The call option buyer buys the asset only when he makes money. The buyer’s payoff is potentially unlimited (when the stock price is through the roof) and is at worse $0. On the other hand, the seller’s payoff is at best$0 and is potentially unlimited when prices rise. Who would want to enter into such a contract as a seller (the short position)? There is definitely an imbalance in the example just described. To remove this imbalance, the seller must be compensated for taking the disadvantageous position of being a seller of a call option. The compensation is in the form of a call option premium. For example, the buyer of the option will have to pay the seller $2.78 for the privilege of being in the long position. When the XYZ share price is above$40, the seller has the obligation to sell a share at a loss. But the seller will be able to keep $2.78 when the share price is below$40. How to price options will be a subject in many subsequent posts. For now we just take the option premium as a given.

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Call options and put options

The above example is for a call option. It is a contract where the buyer has the right but not the obligation to buy the underlying asset (this means that the counter party, the call option seller, has the obligation to sell). Another type of options is a put option. It is a contract where the buyer has the right but not the obligation to sell the underlying asset (this means that the counter party, the put option seller, has the obligation to buy). To make the discussion easier, let’s look at some standard terminology of options.

• Strike price (or exercise price). For a call option, it is the price at which the buyer will buy the underlying asset. For a put option, it is the price at which the buyer will sell the underlying asset. The strike price can be set at any value.
• Exercise. For a call option, this is the act of paying the strike price to receive the underlying asset. For a put option, this is the act of receiving the strike price to deliver the underlying asset. Regardless of the type (call or put), an option is exercised only when it is advantageous for the buyer to do so.
• Expiration. This is the date that is the deadline for the option being exercised by the buyer of the option. After the expiration date, the option will be worthless.
• Exercise style. The exercise style has to do with the timing of the exercise of the option. If the option can only be exercised on the expiration date, it is said to be a European-style option. If the option can be exercised at any time on or before the expiration date, it is said to be an American-style option. The distinction of “American” and “European” for options is not based on geography. Both types of options are traded worldwide.

In the above example, the call option has a strike price of $40. It has an expiration of 3 months. After 3 months, if the stock price is over the strike price of$40, the buyer can choose to exercise the option. Since the buyer can only exercise on the expiration date, it is a European-style option.

A call option is a contract in which the buyer has the right but not the obligation to buy the underlying asset at the strike price on or before the expiration (depending on the exercise style). This means that the counter party of the call option buyer (the call option seller) has the obligation to sell the underlying asset when the buyer decides to exercise. The call option seller is also called the call option writer.

A put option is a contract in which the buyer has the right but not the obligation to sell the underlying asset at the strike price on or before the expiration (depending on the exercise style). This means that the counter party of the put option buyer (the put option seller) has the obligation to buy the underlying asset when the buyer decides to exercise. The put option seller is also called the put option writer.

The assets that are transacted in option contracts can be financial assets (e.g. stock, stock index, currencies, and interest rates) or commodities (both physical commodities and commodity futures contracts).

In the remainder of the post, we focus on European-style options.

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Payoff and profit diagrams

The payoff of a derivative contract is the value of the contract at a point in time. Knowing the payoff is not the entire story. To know how profitable the contract is, we need to subtract the cost of acquiring the position from the payoff. Thus the profit of a position is the payoff less the future value of the original cost of acquiring the position.

A payoff diagram for a derivative contract is a graph in which the value of the derivative at one point in time is plotted against the price of the underlying asset. A profit diagram for a derivative contract is a graph in which the profit of the derivative at one point in time is plotted against the price of the underlying asset. Payoff diagrams and profit diagrams are important graphical tools to summarize the risk of the derivative contracts. We are only interested in the payoff value and profit at the time of expiration.

We present 12 diagrams below. There are 4 positions to consider:

1. Long call option
2. Short call option
3. Long put option
4. Short call option

We show three diagrams for each position – payoff of option, profit of option and the third one combining the option and a matching forward.

Comment
Going through the payoff and profit diagrams below will go a long way to solidify your understanding of the definitions and risk and benefit characteristics of four option positions listed above. It is very easy to recite the basic definition of call option and put option. For someone new (or fairly new) to options, it can be confusing in going through these 4 option position. For example, even though the position of the short put should be a short position, it can actually be regarded as a long position in the sense that it represents an obligation of the put seller to buy the underlying asset and that the holder of the short put benefit from rising prices. Therefore, a good way of sorting out these 4 positions is to draw these diagrams on paper. Then compare your diagrams to the ones given here.

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Payoff and profit diagrams of call options

_________________________________
Option Position 1 – Long Call Option
First, the payoff diagram of a long call option.

$\text{ }$

Figure 1

To the left of the strike price $K$, the payoff graph in Figure 1 coincides with the x-axis. To the right of the strike price $K$, the payoff graph is above the x-axis and increasing. The level part of the graph shows that this is an option – the right to walk away from the contract. The payoff is 0 on the left side of the strike price $K$ since the call buyer has no incentive to pay the strike price for an asset that is worth less than the strike price. To the right of the strike price $K$, the payoff is the spot price less the strike price. The following summarizes the payoff of a long call where $S_T$ is the spot price of the underlying asset at expiration.

$\text{ }$

$\displaystyle \text{payoff of a long call}=\left\{\begin{matrix} \displaystyle 0&\ \ \ \ \ \ S_T \le K \\{\text{ }}& \\ S_T - K&\ \ \ \ \ \ S_T >K \end{matrix}\right.$

$\text{ }$

The following is the profit diagram of a long call.

$\text{ }$

Figure 2

$\text{ }$

The profit graph of a long call option has the same shape of the long call payoff graph. The profit graph is the result of adjusting the payoff graph downward by the amount of the call premium. When considering profit, it is necessary to subtract the call option premium from the payoff. The call option premium is paid at time 0. The payoff is at time $T$. So the amount subtracted is the future value of the call option premium.

$\text{ }$

$\displaystyle \text{profit of a long call}=\left\{\begin{matrix} \displaystyle -\text{FV of option premium}&\ \ \ \ \ \ S_T \le K \\{\text{ }}& \\ S_T - K-\text{FV of option premium}&\ \ \ \ \ \ S_T >K \end{matrix}\right.$

$\text{ }$

Clearly the holder of the long call position benefit from rising prices. So does the holder of a long forward position. The following diagram compares a long call option with a long forward.

$\text{ }$

Figure 3

$\text{ }$

The dotted line in Figure 3 is the profit graph of a long forward. Note that it has no “flat” part. There is no optionality in a forward. Regardless of price movements, the forward contract holder (long or short) has the obligation to buy or sell. A long forward can be used to hedge against a price increase in the underlying asset. But the long forward position is exposed to risk if prices fall. When the asset price is below the strike place, the long forward position has the obligation to buy the underlying asset while the long call position has the right to walk away. The price of the right to walk away is the call option premium the long call buyer pays to call seller.

One more comment about the long forward. For a forward contract, the profit and the payoff are identical since there is no initial cost of acquiring the forward contract.

_________________________________
Option Position 2 – Short Call Option
The counter party to the call option buyer is the call option seller. The following shows the payoff diagram and the profit diagram of a short call.

$\text{ }$

Figure 4

Figure 5

$\text{ }$

Comparing Figure 4 and with Figure 1, it is clear that the call seller loses money where the call buyer makes money (to the right of the strike price $K$). In fact the loss for the short call is unlimited as the spot price increases as the gain is unlimited for the call buyer. To the left of the strike price, the call option buyer has no incentive to exercise. Thus the payoff to the short call is 0 to the left of the strike price. Figure 4 shows that the highest payoff for the short call option position is 0. As enticement, the payoff graph is lifted up by the amount of the call option premium to form the profit graph in Figure 5. The following further summarizes the payoff and profit of a short call option.

$\text{ }$

$\displaystyle \text{payoff of a short call}=\left\{\begin{matrix} \displaystyle 0&\ \ \ \ \ \ S_T \le K \\{\text{ }}& \\ K-S_T&\ \ \ \ \ \ S_T >K \end{matrix}\right.$

$\displaystyle \text{profit of a short call}=\left\{\begin{matrix} \displaystyle \text{FV of option premium}&\ \ \ \ \ \ S_T \le K \\{\text{ }}& \\ K-S_T+\text{FV of option premium}&\ \ \ \ \ \ S_T >K \end{matrix}\right.$

$\text{ }$

Both the short call position and the short forward position have similar risk characteristics. The following diagram compares a short call option with a short forward.

$\text{ }$
2

Figure 6

$\text{ }$

The dotted line in Figure 6 is the profit of the short forward contract. Both the short forward position and the short call position benefit from falling prices, except that the profit of the short call position is capped at the call option premium. When price rises, both positions are similar in that the losses are unlimited, except that the loss for the short call is reduced by the amount of the call option premium.

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Payoff and profit diagrams of put options

_________________________________
Option Position 3 – Long Put Option
The following shows the payoff diagram and the profit diagram of a long put option.

$\text{ }$

Figure 7

Figure 8

$\text{ }$

The dynamics of the put option are the opposite of the call option. Let’s compare Figure 7 with Figure 1. For a call option (Figure 1), the contract has positive payoff to the right of the strike price. For a put option (Figure 7), the contract has positive payoff to the left of the strike price. Thus a long call position makes money when prices go up and a long put position makes money when prices go down. Recall that a put option grants the right to its buyer to sell the underlying asset at the strike price. Thus when prices are low at expiration, the long put position can sell the asset at a price higher than what the asset is worth.

The profit graph in Figure 8 is obtained by lowering the payoff graph in Figure 7 by the amount of the put option premium. This is due to the fact that the put option buyer has to pay a premium to the seller to gain the privilege of walking away from the contract when the prices are higher than the strike price. The following further summarizes the payoff and profit of a long put option.

$\text{ }$

$\displaystyle \text{payoff of a long put}=\left\{\begin{matrix} \displaystyle K-S_T&\ \ \ \ \ \ S_T \le K \\{\text{ }}& \\ 0&\ \ \ \ \ \ S_T >K \end{matrix}\right.$

$\displaystyle \text{profit of a long put}=\left\{\begin{matrix} \displaystyle K-S_T-\text{FV of option premium}&\ \ \ \ \ \ S_T \le K \\{\text{ }}& \\ -\text{FV of option premium}&\ \ \ \ \ \ S_T >K \end{matrix}\right.$

$\text{ }$

Interestingly, the long put position and the short forward position have similar characteristics. The following diagram compares a long put option with a short forward.

$\text{ }$

Figure 9

$\text{ }$

The dotted line in Figure 9 is the profit graph of the short forward. While the long put position can walk away when prices rise, the short forward position is still obligated to sell at a loss. Of course, the long put has to pay a premium for the right to walk away when price rises.

One interesting observation is that the long put position can be regarded in some sense a short position despite the word long in its name. For example, the long put is a right to sell the underlying asset and the long put position benefits from falling prices. Note the similarity with the short forward position, which has an obligation to sell the underlying asset and which also benefits from falling prices.

_________________________________
Option Position 4 – Short Put Option

$\text{ }$

Figure 10

Figure 11

$\text{ }$

Recall that the short put option position (the seller of the put) has the obligation to buy the asset when the put buyer chooses to exercise. When the spot price at expiration is less than the strike price, the put seller will have to pay the strike price for an asset that is worth less than the strike price. This explains the downward slope to the left of the strike price in Figure 10 and Figure 11. The flat part of both Figure 10 and Figure 11 reflects the optionality – the right of the put buyer not to sell an asset that is worth more than the strike price. For this right to walk away, the put buyer will have to pay a premium to to the put seller. This explains that the profit graph in Figure 11 is the payoff graph lifted up by the amount of the put option premium.

The following further summarizes the payoff and profit of a short put option.

$\text{ }$

$\displaystyle \text{payoff of a short put}=\left\{\begin{matrix} \displaystyle S_T-K&\ \ \ \ \ \ S_T \le K \\{\text{ }}& \\ 0&\ \ \ \ \ \ S_T >K \end{matrix}\right.$

$\displaystyle \text{profit of a short put}=\left\{\begin{matrix} \displaystyle S_T-K+\text{FV of option premium}&\ \ \ \ \ \ S_T \le K \\{\text{ }}& \\ \text{FV of option premium}&\ \ \ \ \ \ S_T >K \end{matrix}\right.$

$\text{ }$

The short put position and the long forward position are similar in some sense. The following diagram compares a short put option with a long forward.

$\text{ }$

Figure 12

$\text{ }$

The dotted line in Figure 12 is the profit graph of the long forward. Figure 12 shows that both the long forward position and the short put position have substantial losses when price falls. The only difference is that the downside loss for the short put position is mitigated somewhat by the put option premium.

One interesting observation is that the short put option position can be regarded as a long position despite its name has the word short. It can be regarded as long in the sense that a short put position is obligated to buy asset from the put buyer and that a short put position benefit from rising prices. If prices rise, then the put buyer will not exercise and the put seller can pocket the put option premium. Note the similarity with a long forward. A long forward is an obligation to buy the underlying asset and also benefit from rising prices. The only difference is that the gain for a long forward is unlimited when prices rise, while the gain for a short put is only the option premium.

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Remarks

The risk and benefit characteristics discussed in all the above profit and payoff diagrams indicate that call options and put options can be used as insurance against loss in an investment position.

For example, put option is an insurance against a fall in the price of an asset. Let’s say you hold an asset and are planning on selling it at some point in the future. The risk is that the price may fall at the time of sales. To insure against such an adverse outcome, buy a put option with the same underlying asset and with strike price and expiration that match your need. At expiration of the put option, you will have a guarantee of a minimum sale price of your asset, which is the strike price. For this reason, the purchase of a put option is called a floor. A put option is a protection against falling prices of a long asset position.

On the other hand, a call option is insurance for an asset that we plan to own in the future. If we plan to buy shares of a stock at some future date, buying a call option on that stock now will guarantee a minimum purchased price. For this reason, buying a call option is called a cap. In general, a call option is a protection against rising prices for a short asset position.

This subsequent post discusses the insurance strategies of protective put and protective call.

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

## Creating synthetic forwards

When a customer buys a forward contract from a market maker, the market maker can create an offsetting position to protect against the risk of holding a short forward position. In this post, we explain how to create a synthetic forward contract to hedge a forward position. This post is a continuation of these two previous posts on forward contracts: An introduction to forward contracts and Putting a price on a forward contract.

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Synthetic forward contracts

Let’s say the market maker has sold a forward contract to a customer and the contract allows the customer to buy a share of stock at expiration. The customer has the long forward position and the market maker is holding the short forward position. To offset the risk of the short forward, the market maker can create a synthetic long forward position.

In this discussion, we assume that the stock in question pays annual continuous dividends at the rate of $\delta$. Thus the forward price is $F_{0,T}=S_0 \ e^{(r-\delta) T}$ (see equation (5) in this previous post). The following is the payoff of the long forward position:

$\text{ }$

$\text{Long forward payoff at expiration} = S_T-F_{0,T} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (1)$

$\text{ }$

The market maker that is in a short forward position will need to offset the long forward position in (1). To do that, the market maker can borrow the amount $S_0 \ e^{-\delta T}$ to buy $e^{-\delta T}$ shares of the stock at time 0. The stock purchase is financed by the borrowed amount. So there is no upfront cost to the market maker at time 0.

Now let’s look at what happens at time $T$. The $e^{-\delta T}$ shares will become 1 share at time $T$. The market maker can sell the 1 share to the customer at time $T$, thus receiving $S_T$. The market maker will also have to repay $S_0 \ e^{(r-\delta) T}$ to the lender, leaving the market maker with the amount $S_T-S_0 \ e^{(r-\delta) T}$. The following table summarizes the cash flows in these transactions.

$\text{ }$

Table 1 – Borrowing to buy shares replicates the payoff to a long forward

$\left[\begin{array}{llll} \text{Transaction} & \text{ } & \text{Time 0 Cash Flows} & \text{Time T Cash Flows} \\ \text{ } & \text{ } \\ \text{Buy } e^{-\delta T} \text{ shares of stock} & \text{ } & -S_0 \ e^{-\delta T} & +S_T \\ \text{ } & \text{ } \\ \text{Borrow } S_0 \ e^{-\delta T} & \text{ } & +S_0 \ e^{-\delta T} & -S_0 \ e^{(r-\delta) T} \\ \text{ } & \text{ } \\ \text{Total} & \text{ } & \text{ } \ \ 0 & \ \ S_T-S_0 \ e^{(r-\delta) T} \end{array}\right]$

$\text{ }$

In the above table, the payoff to the market maker is $S_T-S_0 \ e^{(r-\delta) T}$, which is exactly the long forward payoff indicated in equation (1). This means that the process of borrowing to buy shares of stock replicates the payoff to a long forward and thus is a synthetic forward contract. We have the following relationship.

$\text{ }$

$\text{Long forward} = \text{Long Stock} + \text{Short zero-coupon bond} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (2)$

$\text{ }$

If a market maker is holding a long forward position, then he can offset the risk of holding the long forward by creating a synthetic short forward contract. The cash flows in the synthetic short forward contract is simply the reverse of (2). Thus we have the following relationship.

$\text{ }$

$\text{Short forward} = \text{Short Stock} + \text{Long zero-coupon bond} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (3)$

$\text{ }$

Before we discuss how a market maker can use the strategies of (2) and (3) to hedge, we discuss other synthetic positions that can be obtained from relationship (2).

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Other synthetic positions

By manipulating the synthetic forward in the relationship (2), we can create a synthetic stock as well as a synthetic bond.

$\text{ }$

$\text{Long stock} = \text{Long Forward} + \text{Long zero-coupon bond} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (4)$

$\text{Long zero-coupon bond} = \text{Long Stock} + \text{Short Forward} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (5)$

$\text{ }$

If relationship (2) is understood, then (3), (4) and (5) are obtained by rearranging (2). For example, moving a long asset to the other side of the equation becomes a short. To further illustrate the idea of synthetically creating assets, we describe the cash flows for the transactions behind (4) and (5).

$\text{ }$

Table 2 – A long forward plus lending creates a synthetic share of stock

$\left[\begin{array}{llll} \text{Transaction} & \text{ } & \text{Time 0 Cash Flows} & \text{Time T Cash Flows} \\ \text{ } & \text{ } \\ \text{Long a forward} & \text{ } & \ \ 0 & \ \ S_T-F_{0,T} \\ \text{ } & \text{ } \\ \text{Lend } S_0 \ e^{-\delta T} & \text{ } & -S_0 \ e^{-\delta T} & \ \ S_0 \ e^{(r-\delta) T} \\ \text{ } & \text{ } \\ \text{Total} & \text{ } & -S_0 \ e^{-\delta T} & \ \ S_T \end{array}\right]$

$\text{ }$

Table 3 – Buying shares of stock and shorting a forward creates a synthetic bond

$\left[\begin{array}{llll} \text{Transaction} & \text{ } & \text{Time 0 Cash Flows} & \text{Time T Cash Flows} \\ \text{ } & \text{ } \\ \text{Buy } e^{-\delta T} \text{ shares of stock} & \text{ } & -S_0 \ e^{-\delta T} & \ \ S_T \\ \text{ } & \text{ } \\ \text{Short a forward } & \text{ } & \ \ 0 & \ \ F_{0,T}-S_T \\ \text{ } & \text{ } \\ \text{Total} & \text{ } & -S_0 \ e^{-\delta T} & \ \ F_{0,T} \end{array}\right]$
$\text{ }$

Looking at the Total row in table 2, the end result is that the market maker pays the time 0 price of $e^{-\delta T}$ shares and obtain the time $T$ value of one share. Thus the cash flows in Table 2 create a synthetic share of the stock.

The Total row of Table 3 tells us that the end result of Table 3 can be described in this way: the market maker lends out the amount $S_0 \ e^{-\delta T}$ at time 0. At time $T$, the market maker receives the future value of the loan, which is $F_{0,T}=S_0 \ e^{(r-\delta) T}$. Thus the cash flows in table 3 create a synthetic zero-coupon bond.

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How market makers use synthetic forwards

If the market maker is holding a short forward position, he can use relationship (2) to create a synthetic long forward to offset the short forward position. On the other hand, if the market maker is holding a long forward position, then the market maker can use relationship (3) to create a short forward to offset the long forward.

The following table displays the cash flows involved in hedging using the idea in (2). For easier reference, equation (2) is repeated below.

$\text{ }$

$\text{Long forward} = \text{Long Stock} + \text{Short zero-coupon bond} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (2)$

$\text{ }$

Table 4 – A market maker offsetting a short forward with a synthetic long forward

$\left[\begin{array}{lllll} \text{ } &\text{Transaction} & \text{ } & \text{Time 0 Cash Flows} & \text{Time T Cash Flows} \\ \text{ } & \text{ } \\ 1 & \text{Buy } e^{-\delta T} \text{ shares of stock} & \text{ } & -S_0 \ e^{-\delta T} & +S_T \\ \text{ } & \text{ } \\ 2 & \text{Borrow } S_0 \ e^{-\delta T} & \text{ } & +S_0 \ e^{-\delta T} & -S_0 \ e^{(r-\delta) T} \\ \text{ } & \text{ } \\ 3 & \text{Short forward} & \text{ } & \text{ } \ \ 0 & \ \ F_{0,T}-S_T \\ \text{ } & \text{ } \\ 4 & \text{Total} & \text{ } & \text{ } \ \ 0 & \ \ F_{0,T}-S_0 \ e^{(r-\delta) T} \end{array}\right]$

$\text{ }$

Row 2 in Table 4 is the short bond (borrowing cash has the effect of selling a bond). The borrowed cash is then used to buy stocks (the long stock in Row 1). Rows 1 and 2 form the synthetic long forward. Row 3 is the short forward position held by the market maker. Note that the total cash flow at time $T$ is $F_{0,T}-S_0 \ e^{(r-\delta) T}$, which is 0 assuming the no-arbitrage pricing principle. Thus the synthetic long forward neutralizes the actual short forward. All the ingredients of the last cash flow – forward price, spot price, risk-free interest rate and dividend yield – are known at time 0. Thus these transactions result in a risk-free position.

Table 4 illustrates a trading strategy that we want to highlight. A trading strategy in which an investor holds a long position in a security or commodity while simultaneously holding a short position in a forward contract on the same security or commodity is called a cash-and-carry. When using this strategy, the long position is held until the delivery date of the forward contract and is used to cover the obligation of the short position. Thus a cash-and-carry is risk-free.

Table 4 illustrates a cash-and-carry trade from the perspective of a market maker wishing to hedge the risk from a short position. When the cash-and-carry strategy is used by an arbitrageur, it is called a cash-and-carry arbitrage. The arbitrage strategy is to exploit the pricing inefficiencies for an asset in the cash (spot) market and the forward (or futures) market in order to make risk-less profits. The arbitrageur would try to carry the asset until the expiration date of the forward contract and the long asset position is used to cover the obligation of the short position. The strategy of cash-and-carry arbitrage is only profitable if the cash inflow from the short position exceeds the acquisition costs and carrying costs of the long asset position, i.e. $F_{0,T}>S_0 \ e^{(r-\delta) T}$, in which case an arbitrageur or market maker can use the strategy outlined in Table 4 to make a risk-free profit.

The following table displays the cash flows involved in hedging a long forward position using a synthetic short forward, i.e. using equation (3). For easier reference, equation (3) is repeated below.

$\text{ }$

$\text{Short forward} = \text{Short Stock} + \text{Long zero-coupon bond} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (3)$

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Table 5 – A market maker offsetting a long forward with a synthetic short forward

$\left[\begin{array}{lllll} \text{ } &\text{Transaction} & \text{ } & \text{Time 0 Cash Flows} & \text{Time T Cash Flows} \\ \text{ } & \text{ } \\ 1 & \text{Sell } e^{-\delta T} \text{ shares of stock} & \text{ } & +S_0 \ e^{-\delta T} & -S_T \\ \text{ } & \text{ } \\ 2 & \text{Lend } S_0 \ e^{-\delta T} & \text{ } & -S_0 \ e^{-\delta T} & +S_0 \ e^{(r-\delta) T} \\ \text{ } & \text{ } \\ 3 & \text{Long forward} & \text{ } & \text{ } \ \ 0 & \ \ S_T-F_{0,T} \\ \text{ } & \text{ } \\ 4 & \text{Total} & \text{ } & \text{ } \ \ 0 & \ \ S_0 \ e^{(r-\delta) T}-F_{0,T} \end{array}\right]$

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Row 1 in Table 5 is the short stock – borrowing the shares and sell them to receive cash. Then lend the cash from the sales of the borrowed stock (the long bond in Row 2). Rows 1 and 2 form the synthetic short forward. Row 3 in Table 5 is the long forward position held by the market maker. Note that the total cash flow at time $T$ is $S_0 \ e^{(r-\delta) T}-F_{0,T}$, which is 0 assuming the no-arbitrage pricing principle. Thus the synthetic short forward neutralizes the actual long forward. As in Table 4, all the ingredients of the last cash flow – forward price, spot price, risk-free interest rate and dividend yield – are known at time 0. Thus these transactions result in a risk-free position.

A reverse cash-and-carry is the reverse of cash-and-carry. Thus a reverse cash-and-carry is a trading strategy in which an investor holds a short position in a security or commodity while simultaneously holding a long position in a forward contract on the same security or commodity. Table 5 illustrates a reverse cash-and-carry from the perspective of a market maker. If the cash outflow from the long position is less than the selling proceeds and interest income of the short asset position, i.e. $F_{0,T}, then the market maker or an arbitrageur can use the strategy outlined in Table 5 to make a risk-free profit.

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