Pricing Forwards and Swaps

Chapter 7

Pricing Forwards and Swaps

7.1 Forwards

Throughout this chapter, we will repeatedly use the following property of no-arbitrage:

P0(αxT + βyT) = αP0(xT) + βP0(yT).

Here, P0(wT) is the time zero market price of a security whose payoff at time T is wT, which could be a random variable if the security is risky.

Let St be the price of an asset at time t. Consider a forward contract on the asset which requires the buyer of the contract to buy the asset at time T with a fixed price K. The payoff of this forward contract for the buyer (who takes a long position in the contract) is then ST − K at time T . Note that having K dollar at T is equivalent to having K units of the zero-coupon bond that pays one dollar at T . Let Zt(T ) be the market price of this zero coupon bond at time t (t ≤ T ). Then, the payoff of the forward contract can be written as ST − KZ^T(T ) (note that ZT(T ) = 1) and therefore the market value of the contract at time zero is

P0(ST − KZ^T(T )) = P0(ST) − KP⁰(ZT(T )).

Let r be the risk-free, continuously compounding interest rate. Then, P0(ZT(T )) = Z0(T ) = e^−rT.

So the time zero value of the forward contract is P0(ST) − e^−rTK.

51

52 CHAPTER 7. PRICING FORWARDS AND SWAPS The forward price of the asset at time zero, F0, is the value of K such that the value of the forward contract is zero. That is, F0 is the solution to the following equation

P0(ST − F⁰ZT(T )) = P0(ST) − e^−rTF0 = 0, which implies that

F0 = e^rTP0(ST).

So, to determine the value of a forward contract and to determine the forward price of an asset, we need to figure out how to determine P0(ST). This is done again by no-arbitrage argument.

(1) S_T is the price of a financial asset that pays no income.

In this case, S_T is simply the payoff from holding the asset until time T . Since the cost of buying the asset at time zero is S0, no-arbitrage implies that

P0(ST) = S0

and therefore

P0(ST − KZ^T(T )) = S0− e^−rTK, and F0 = e^rTS0.

(2) ST is the price of a financial asset that pays a fixed income I at time T . In this case, the payoff of holding the asset until time T is ST + IZT(T ) instead of ST, so we have the following no-arbitrage condition:

P0(ST + IZT(T )) = S0. But

P₀(S_T + I) = P₀(S_T) + e^−rTI.

So, we have

P0(ST) = S0− e^−rTI, and F0 = e^rT(S0− e^−rTI) = e^rTS0− I.

(3) ST is the price of a financial asset that pays a fixed income I at time T1 < T .

Having I at T1 is the same as having IZT₁(T1)at T1. So, from (2), we have

S0 = P0(ST + IZT₁(T1))

= P0(ST) + IP0(ZT₁(T1))

= P0(ST) + e^−rT1I.

−e^−rT(e^{r(T −T1)}I) = S₀− e^−rT1I, and

F0 = e^rT(S0− e^−rT1I) = e^rTS0− e^{r(T −T1)}I.

Thus,

P0(ST) = S0− e^−rT1I and

F0 = e^rT(S0 − e^−rT1I) = e^rTS0 − e^{r(T −T1)}I.

(4) ST is the price of a stock that pays a fixed dividend yield d. That is, the payoff of owning the asset from 0 to T is STe^dT.

No-arbitrage condition implies that

S0 = P0(STe^dT) = e^dTP0(ST).

Thus,

P₀(S_T) = e^−dTS₀, and F₀ = e^rT(e^−dTS₀) = e^(r−d)TS₀.

(5) S_T is the price of a foreign currency (the exchange rate).

In this case, owning one unit of foreign currency at time zero will give you e^rfT units of foreign currency at time T , where rf is the risk-free interest rate in the foreign country. In domestic dollar units, the payoff is STe^rfT. This is equivalent to an asset that pays a fixed continuous dividend yield rf. From (4), then, we have

P0(ST) = e^−rfTS0, and F0 = e^(r−rf)TS0.

Note that F0 is the forward price of a foreign currency, or the forward ex- change rate.

(6) ST is the price of a commodity.

54 CHAPTER 7. PRICING FORWARDS AND SWAPS Holding commodities may be costly. Let U be the (time zero) present value of the storage cost, then, no-arbitrage condition says that

U + S₀ = P₀(S_T).

Thus,

F0 = e^rT(S0+ U).

If the holder of the commodity have to pay storage cost continuously and the cost is proportional to the value of the commodity, then, the present value of holding the commodity to time T is S0e^uT, where u is the rate of storage cost. In this case, the no-arbitrage condition is

S0e^uT = P0(ST), which implies that

F0 = e^(r+u)TS0.

Note that effect of the storage cost u on the forward price is like having a negative dividend −u.

Sometimes people hold commodities for reasons other than hedging or speculation. For example, a firm may hold inventories of a commodity to meet unexpected future demand. In this case, holding commodity may ac- tually generate some benefits to the holder. As a result, the actual economic cost of holding a commodity is less than S0e^uT. In this case, a firm is willing to hold the commodity even if P0(ST) < S0e^uT. This implies that

F0 = e^rTP0(ST) < e^(r+u)TS0.

That is, since the party who delivers the commodity at T enjoys some benefits from holding the commodity, she is willing to charge a forward price that is lower than implied by pure financial arbitrage. We measure the benefits by the so-called convenience yield y, which is the number such that

F0 = e^(r+u)TS0e^−yT = e^(r+u−y)TS0. When F0 < e^(r+u)TS0, the convenience yield y > 0.

Problem 1: Let Z_t(t₂) be the time t price of a zero-coupon bond that pays one dollar at time t2. Consider a forward contract that allows one to buy the zero-coupon bond at t1 (t1 < t2). Derive the expression for foward price the

zero-coupon bond, P (t1, t2). Suppose that R(0, t) is the zero-coupon interest rate for the period from 0 to t, (i.e., Z0(t) = e^−R(0,t)t), find the forward interest rate f (t₁, t₂) such that

P (t1, t2) = e^{−f (t1,t2)(t2−t1)}.

Problem 2: Consider a forward contract on a 5 year coupon bond that settles in one year. Suppose that the bond pays a coupon of C dollar semi-annually and that the zero-coupon interest rate is constant, r. Let B0 be the price of the 5 year coupon bond today. Express the forward price as a function of r, C and B0.

7.2 Swaps

Consider a standard interest rate swap. Let A and B denote the party that pays floating interest rates and the party that pays a fixed interest rate, respectively. Let T be the maturity of the swap, and assume that the frequence of the payment is m times per year. So the total number of payment dates are n = mT , and the time interval between two consequtive payments are ∆ = 1/m. Let ti = i/m be the ith payment date of the swap.

Then, we have ti−1= ti− ∆. Let X be the notional amount of the swap. For any t, let Rt(∆) be the annual LIBOR rate of maturity ∆ that is determined in the market at time t. By definition, this interest rate is the rate such that the time t value of a zero-coupon bond with maturity ∆ be

Zt(∆) = Pt($1 at t + ∆) = 1

(1 + _m¹R_t(∆)).

In the standard interest rate swap, the payments of party A and B on date ti are _m¹Rti−1(∆)X and _m¹RX, respectively. The value of party B’s payments is easy to determine. It is simply a porfolio of n zero-coupon bonds with maturity ti i = 1, , , , n and size _m¹RX. So,

VB =

" _n X

i=1

Z0(ti)

# 1 mRX.

The determiniation of the value of party A’s payments, however, is more complicated because, except for the first payment, all the future n − 1 pay- ments are uncertain at time 0. However, for the ith payment, _m¹Rti−1X we

56 CHAPTER 7. PRICING FORWARDS AND SWAPS know that

P_ti−1

1

mR_ti−1(∆)X



= XP_ti−1

 1

mR_ti−1(∆)



= X

 Pt_i−1

1

mRt_i−1(∆) + 1 − 1



= X

 Pt_i−1

1

mRt_i−1(∆) + 1



− Z^ti−1(∆)



= X

1

mR_ti−1(∆) + 1



P_ti−1($1 at t_i) − Zti−1(∆)



= X

1

mR_ti−1(∆) + 1

 1

(1 + _m¹Rt_i−1(∆)) − Zti−1(∆)



= X

1 − Zti−1(∆) .

Thus, the time zero value of the ith payment of party A is P0

 Pti−1

1

mRti−1(∆)X



= P0 X

1 − Z^ti−1(∆)

= XP0 $1 − Z^ti−1(∆) at ti−1

= X [Z0(ti−1) − Z⁰(ti)] .

Therefore, the total value of the floating side payments is VA =

Xn i=1

X [Z0(ti−1) − Z⁰(ti)]

= X [Z0(t0) − Z⁰(tn)] . Note that t0 = 0 and tn = T . So, we have

VA= X [1 − Z⁰(T )] .

When the two parties entered into the swap, the value of the payments on both sides must be the same. So, at the time when the swap is entered, the fixed interest rate R must be such that

VA= X [1 − Z⁰(T )] = VB=

" _n X

i=1

Z0(ti)

# 1 mRX,

which implies that

R = m1 − Z0(T ) Pn

i=1

Z0(ti) .