Tag Archives: Abitrage

The binomial option pricing model – part 2

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

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

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The one-period binomial option pricing formulas

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

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

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

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

Replicating Portfolio

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

    \text{ }

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

Price of the Option

    \text{ }
    C=\Delta S + B \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (3)
    \text{ }
    \text{ }
    \displaystyle C=\Delta S + B=e^{-r h} \biggl(C_u \ \frac{e^{(r-\delta) h}-d}{u-d} +C_d \ \frac{u-e^{(r-\delta) h}}{u-d}  \biggr) \ \ \ \ \ \ \ \ \ (4)
    \text{ }

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Arbitrage in the binomial tree

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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Risk-neutral pricing

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

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

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

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

Then pricing formula (4) becomes:

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

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

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

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

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

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

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

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

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

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

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

    \displaystyle uS = 60 (1.261286251) = $75.67717506

    \displaystyle dS = 60 (0.825197907) = $49.51187441

Now the risk-neutral probabilities:

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

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

Then the option prices are:

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

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

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Practice Problems

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

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

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The binomial option pricing model – part 1

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

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

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The one-period binomial option pricing model

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

Consider a stock with the following characteristics:

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

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

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

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

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

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

    \text{ }
    Time h value of the replicating portfolio

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

    \text{ }

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

    \text{ }

    \displaystyle \text{ } \left\{\begin{matrix} \displaystyle \Delta \times (dS \ e^{\delta h})  + B \ e^{r h}=C_d&\ \ \ \ \ \ \text{ }& \\ \text{ }&\text{ } \\ \Delta \times (uS \ e^{\delta h})  + B \ e^{r h}=C_u&\ \ \ \ \ \ \text{ }   \end{matrix}\right.

    \text{ }

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

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

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

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

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

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

    \text{ }
    \displaystyle C=\Delta S + B=e^{-r h} \biggl(C_u \ \frac{e^{(r-\delta) h}-d}{u-d} +C_d \ \frac{u-e^{(r-\delta) h}}{u-d}  \biggr) \ \ \ \ \ \ \ \ \ (4)
    \text{ }

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

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

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Examples

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

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

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

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

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

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

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

    \text{ }

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

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

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

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

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

    \text{ }

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

More Examples
Two more examples are in these previous posts:

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What to do if options are mispriced

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

Example 2

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

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

    \text{ }

    Table 1 – Arbitrage opportunity when call option is underpriced

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

    \text{ }

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

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

    \text{ }

    Table 2 – Arbitrage opportunity when call option is overpriced

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

    \text{ }

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

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

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How to construct a binomial tree

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

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

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

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

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

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

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

where

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

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

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

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

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

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

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

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

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

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

    \text{ }

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

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

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

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

    \text{ }

The following shows the calculation for the put option.

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

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

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

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

    \text{ }

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More examples

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

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

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

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

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

    \displaystyle C_u= 75.67717506 – 55 = 20.67717506

    \displaystyle C_d= 0

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

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

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

    \displaystyle 1-p^*= 0.552835026

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

    \text{ }

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

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

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

    \text{ }

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

The calculation is calculated as in Example 3.

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

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

    \displaystyle C_u= 0

    \displaystyle C_d= 45 – 34.861374 = $10.138626

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

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

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

    \displaystyle 1-p^*= 0.537429845

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

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

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

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

    \text{ }

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Remarks

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

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Practice problems

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

basic problem set 1

basic problem set 2

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

Pricing a put option – an example

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

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The example

The following gives the information about the stock:

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

Consider a put option with the following specifics:

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

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

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Pricing the put option

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

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

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

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

    \text{ }

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

    Portfolio B

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

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

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

    \text{ }

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

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Arbitrage opportunities

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

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

    \text{ }

    Table 1 – Arbitrage opportunity when put option is overpriced

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

    \text{ }

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

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

    \text{ }

    Table 2 – Arbitrage opportunity when put option is underpriced

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

    \text{ }

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To complete the picture

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

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

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

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

    C=\Delta \ S+B

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

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Put-call parity

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

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

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

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

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Remarks

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

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Practice problems

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

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

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.

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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.

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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.

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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.

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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.

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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)

    \text{ }

    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]

    \text{ }

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}<S_0 \ e^{(r-\delta) T}, then the market maker or an arbitrageur can use the strategy outlined in Table 5 to make a risk-free profit.

\text{ }

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

Putting a price on a forward contract

This post is a continuation of this previous post on forward contracts. The previous post discusses the basic features of forward contracts. How to price forward contracts is the subject of this post.

Suppose that you need to purchase a financial asset or commodity at time T in the future. The price of the asset is S_0 right now (at time 0). The price at time T is S_T, which is not known at time 0. You can wait until time T to buy the asset by paying S_T. Waiting could be risky since the price could increase substantially. So waiting would exposure you to the risk of price uncertainty and as a result profit uncertainty. You can then buy the asset at time 0 (now) and hold it to time T. Due to business reasons or other reasons, this may not always be practical. An alternative is to lock in a price F_{0,T} today to pay for the asset at time T.

The dynamics described above can apply to selling too. Suppose you have a financial asset or commodity that will be available for sales at time T. You can sell it at time T for the price S_T, which is unknown at time 0. Or you can lock in a price F_{0,T} today to sell the asset at time T.

The above scenario is in essence what a forward contract is. In this post, we discuss how to derive the forward price F_{0,T}. The focus here is on financial assets, in particular stocks, stock index and currencies.

As discussed in this previous post, a forward contract is a contract between two parties to buy or sell an asset at a specified price (called the forward price) on a future date. The forward price, the quantity F_{0,T} introduced above, is set today by the two parties in the contract for a transaction that will take place in a future date, at which time the buyer pays the seller the forward price and the seller delivers the asset to the buyer.

We also make the following simplifying assumptions:

  1. Trading costs and taxes are ignored.
  2. Individuals can always borrow or lend at a risk-free interest rate.
  3. Arbitrage opportunities do not exist.

In this post, we focus on the pricing of forward contracts on stock, stock indexes and currencies. Assume that the annual risk-free interest rate that is available for investors is r.

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Forward price on a stock – nondividend paying

The first case is that the stock pays no dividend. This simple case will help derive the case of paying dividends. Suppose that you want to own a share of a stock at time T in the future. Just like the scenarios described above, there are two ways to do this.

  • Buy a share at time 0 and hold it until time T.
  • Enter into a forward contract to buy one share of the stock at time T.

In the first way, you pay S_0 at time 0 to own the stock. In the second way, you pay F_{0,T} at time T to own the stock. In either way you own a share of the stock at time T. In the second way, in order to have the amount F_{0,T} available at time T, you can invest F_{0,T} \ e^{-rT} at time 0 at the risk-free interest rate compounded continuously. So at time 0, the cost outlay for the first way is S_0. At time 0, the cost outlay for the second way is F_{0,T} \ e^{-rT}. If there is to be no arbitrage, the two would have to be the same.

    F_{0,T} \ e^{-rT}=S_0 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (0)

    F_{0,T}=S_0 \ e^{rT} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (1)

Thus equation (1) gives the forward price on a stock that pays no dividend. The forward price in this case is simply the future value of the price of the asset at time 0.

If equation (1) is violated, then there would be arbitrage opportunities. Then traders can buy low and sell high to generate risk-free profit. To see how this work, suppose that F_{0,T}>S_0 \ e^{rT}. Then you can buy low and sell high. At time 0, borrow the amount S_0 and buy a share of the stock. At time 0, also sell a forward contract (i.e. enter into a short forward contract) at the forward price F_{0,T}. At time T, sell the share of the stock and obtain the forward price F_{0,T} and pay S_0 \ e^{rT} to the lender, producing a sure and positive profit F_{0,T}-S_0 \ e^{rT}.

On the other hand, suppose F_{0,T}<S_0 \ e^{rT}. This time the arbitrage strategy is still to buy low and sell high. You can buy a forward contract at the forward price F_{0,T} and simultaneously borrow a share and sell it at the price S_0. Invest the amount S_0 at the risk-free rate to obtain S_0 \ e^{rT} at time T. At time T, buy a share of the stock at the price F_{0,T} and then return it to the lender. The amount that remains is S_0 \ e^{rT}-F_{0,T}, which is a risk-free profit.

The above two arbitrage examples establish equation (1) as the correct forward price of a non-dividend paying stock.

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Forward price on a stock – discrete dividends

We now consider the case that the stock pays dividends in known amounts at known times during the life of the forward contract. In other words, this is the case that the frequency, the timing of the dividends and the amounts of the dividends are known ahead of time. To determine the forward price F_{0,T}, we still consider the two ways to own a share at time T.

  • Buy a share at time 0 and hold it until time T.
  • Enter into a forward contract to buy one share of the stock at time T.

However, there is now an important difference between these two ways. It is that the owner of the stock in the first way receives the dividends during the contract period while the owner of the forward contract is not entitled to receive dividends. By the time the forward contract owner receives the share at time T, she has missed out on all the dividend payments. So the forward contract owner must be compensated for the missed dividend payments. Consequently the forward contract owner should pay less than the price for an outright purchase at time 0. How much less? By the amount of the dividends. So we need to subtract the value of the dividends from the stock price.

The price of the first way (outright stock ownership at time 0) accumulated to time T is S_0 \ e^{rT}. Thus we need to subtract the cumulative value of the dividends from this price.

    \displaystyle F_{0,T}=S_0 \ e^{rT}-\text{CV of the dividends} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (3)

Here CV means cumulative value. To be more specific, suppose that during the contract period, there are n dividend payments d_1,d_2,\cdots,d_n received at times t_1,t_2,\cdots,t_n. Then the forward price should be S_0 \ e^{rT} subtracting the future values of the dividends at time T.

    \displaystyle F_{0,T}=S_0 \ e^{rT}-\sum \limits_{j=1}^n \ d_j \ e^{r (T-t_j)} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (4)

We illustrate the forward prices discussed thus far with an example.

Example 1
Suppose that the stock for XYZ company is $50 today. The annual continuously compounded risk-free rate is 3%. Calculate the following:

  • The price for a forward contract to deliver 500 shares of XYZ 6 months from now, assuming that the stock pays no dividends.
  • The price for a forward contract to deliver 500 shares of XYZ 6 months from now, assuming that the stock pays quarterly dividend of $1.50 with the first one occurring 3 months from now.

First consider the no dividend case. The forward price for one share is:

    F_{0,0.5}=50 \times e^{0.03(0.5)}=50 \times e^{0.015}=50.75565

Then the forward price for the contract is 500 F_{0,0.5}=25377.83.

Now consider the case with dividends. There are two dividend payments in the contract periods. The first one is accumulated forward for 3 months and the second one is assumed to be paid at expiration. The forward price for one share is:

    F_{0,0.5}=50 \times e^{0.03(0.5)}-1.50 e^{0.03(0.25)}-1.5=47.74436

Then the forward price for the contract is 500 F_{0,0.5}=23872.18.

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Forward price on a stock index – continuous dividends

We now consider an asset that pays dividends at an annual continuously compounded rate that is denoted by \delta. The dividends are paid continuously and reinvested back in the asset. So instead of receiving cash payment, the owner gets more shares. If the investor starts out with one share at time 0, she ends up with e^{\delta \ T} shares at time T.

For a stock index containing many stocks, assuming a continuous compounded dividend rate will simplify the discussion.

The forward contract owner wishes to pay F_{0,T} for one share of the stock index at time T. Again, there are two ways to do this.

  • Buy a e^{-\delta \ T} shares at time 0 and hold them until time T.
  • Enter into a forward contract to buy one share of the stock index at time T.

Recall that the dividends come in the form of additional shares. To get one share at time T, we need to start with e^{-\delta \ T} shares at time 0. So in the first way, we need to pay S_0 e^{-\delta \ T} at time 0. In the second way, the value at time 0 of the forward price is F_{0,T} \ e^{-rT}. Again, to avoid arbitrage, the two present values must equal. We have S_0 e^{-\delta \ T}=F_{0,T} \ e^{-rT}, producing the following:

    F_{0,T}=S_0 \ e^{(r- \delta) \ T} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (5)

In this case of continuous dividends, the dividends come in the form of additional shares. The forward contract owner misses out on the additional shares. So the forward contract owner must be compensated for not receiving the additional shares. Equation (5) indicates that the buyer of the forward contract is compensated by getting a smaller interest rate r-\delta. So in this sense, the dividend rate is like a negative interest rate.

We now illustrate the continuous case with an example.

Example 2
Suppose that a stock with current stock price of $50 pays a 10% continuous annual dividend. The annual continuously compounded risk-free rate is 4%. What is the price for a forward contract for the delivery of 100 shares of XYZ? The contract is to be expired 1 year from now. If you observe a forward price of $49 on a contract on the same stock with the same expiration date, what arbitrage strategy would you use?

The forward for price for one share is:

    F_{0,1}=50 \times e^{(0.04-0.10) \times 1}=50 \times e^{-0.06}=47.08822668

The forward price for 100 shares is 100 F_{0,1}=4708.82. If you observe a forward price of 49 instead of the true theoretical forward price of 47.088, do the following “buy low sell high” strategy.

    S_0=50, \delta=0.10, r=0.04 and T=1.

    Borrow S_0 e^{- \delta T} to buy e^{- \delta T} shares at time 0. Simultaneously sell a forward contract to buy one share at the forward price $49 one year from now.

    At the end of one year, e^{- \delta T} becomes 1 share. As the party holding the short forward position, sell the one share at $49. Then repay S_0 e^{- \delta T} e^{r T}=S_0 e^{(r- \delta) T}=47.088 to the lender. This produces a risk-free profit of $19.12 per share. The profit for 100 shares is $191.2.

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Currency forward price

We use a dollar/euro example to illustrate. Suppose that we want to obtain one euro at time T by paying dollars. There are two risk-free rates here as there are two currencies. Let r be the risk-free rate of the domestic currency (US dollars) and let r_f be the risk-free rate of the euro (here the subscript stands for foreign). Let x_0 be the exchange rate (dollar per euro) at time 0. Once again, there are two ways to obtain one euro at time T. The first way is to pay US dollars to buy euros now. Let’s work backward. To get one euro at time T, we need to have e^{-r_f T} euro at time 0. Thus we need to have x_0 e^{-r_f T} dollars at time 0. We have the following two ways.

  • Exchange x_0 \ e^{-r_f T} dollars into euros at time 0 and hold them until time T.
  • Enter into a forward contract to buy one euro at time T.

In the first way, we need to have x_0 \ e^{-r_f T} dollars ready at time 0. In the second way, we need to have F_{0,T} ready at time T or have F_{0,T} \ e^{-r T} ready at time 0. Equating the two, we have F_{0,T} \ e^{-r T}=x_0 \ e^{-r_f T}, leading to the following:

    F_{0,T}=x_0 \ e^{(r- r_f) \ T} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (6)

Note that equation (6) is just like equation (5). So the risk-free interest rate for the foreign currency plays the role of a continuous dividend rate. One way to interpret the rate difference r-r_f is that it is the cost of carry for a foreign currency. In this interpretation, we borrow at the domestic rate r and invest the borrowed funds in a foreign risk-free account paying at the rate r_f. The earnings from the foreign account will offset the cost of the domestic borrowing.

Example 3
Suppose that a dollar denominated forward contract calls for the delivery of 10 million yens at the end of 6 months. Suppose that the annual continuously compounded risk-free rate for yen is 3% and the annual continuously compounded risk-free rate for dollars is 1%. Currently the dollar/yen exchange rate is $0.008 per yen. Calculate the forward price in dollars for this contract.

Plugging in all the relevant inputs, the following dollar forward price per yen:

    F_{0,0.5}=0.008 \ e^{(0.01- 0.03) \ 0.5}=0.008 \ e^{-0.01}=0.007920399

Thus the dollar forward price per 10 million yens is 10000000 (0.007920399)=79203.9867.

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Remarks

One idea emerged in the derivation of the above forward prices is that dividends have the effect of a negative interest rate. The dividend payments are not received by the forward contract buyer since she only receives the stock at the expiration date. As a result, the forward contract buyer must be compensated for missing out on the dividends. In general, if the asset produces income before the forward contract buyer receives the asset, the effect of the missed income would be like a negative interest rate (to compensate for the missed income). This idea also applies to commodity forward pricing. If a commodity has income streams (e.g. it can be leased out), the forward price must reflect this negative interest rate. On the other hand, if a owning a commodity incurs expenses (e.g. storage costs), then the forward contract buyer will have to pay more for the commodity since the forward contract buyer has to compensate the commodity owner for the expenses.

The following table summarizes all the forward prices discussed above. The price F_{t,T} is the forward price of an asset set at time t to be purchased at a future time T. All the other variables are as discussed above.

\text{ }

    \left[\begin{array}{ll}      \text{Underlying asset} & \text{Forward price}  \\      \text{ } & \text{ } \\      \text{Non-dividend paying stock} & F_{t,T}=S_t \ e^{r(T-t)} \\      \text{ } & \text{ } \\      \text{Dividend paying stock (discrete)} & F_{t,T}=S_t \ e^{r(T-t)}-\text{CV of the dividends} \\      \text{ } & \text{ } \\      \text{Dividend paying stock (continuous)} & F_{t,T}=S_t \ e^{(r- \delta) \ (T-t)} \\      \text{ } & \text{ } \\       \text{Currency} & F_{t,T}=x_t \ e^{(r- r_f) \ (T-t)}     \end{array}\right]

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