# Financial Derivatives/Basic Derivatives Contracts

## Forwards

Spot markets allow the purchase and sale of an asset today. By contrast a forward contract specifies the price at which an asset can be purchased or sold at some future date. Although a forward contract is classified as a derivative in many markets it is difficult to distinguish between the underlying and the forward contract. Large trading volumes in OTC forwards can in fact make them more significant than spot markets.

A forward contract does not require upfront payment. It is simply the purchase or sale of an asset at some future date at a fixed price (the forward price). Therefore the assumption is that the forward price reflects the value of this asset on this date. If this assumption is based on a market view, characterising a forward contract as a derivative is misleading.

The primary reason for the classification of a forward contract as a derivative is that in many cases its price can be derived through a no-arbitrage argument that relates the forward price of an asset to its spot price. For assets like oil this is not possible; given the spot price of a barrel of oil it is not possible to construct an arbitrage argument that relates it to the forward price. In the oil markets forwards or futures are effectively the underlying and cannot be understood as derivatives. In these markets the forward price of oil is similar in nature to the price of a stock: it reflects the current consensus of the market and has nothing to do with risk-neutral valuation.

In financial markets forwards can be determined through a no-arbitrage argument. Consider for example a forward on the USD vs EUR exchange rate. If today one euro can be exchanged for 1.3 dollars (${\displaystyle FX_{spot}}$) then in order to determine the forward exchange rate one year from now we can look at the following set of trades,

• We buy a one year forward that guarantees an exchange rate of ${\displaystyle FX_{oneyear}}$ dollars per euro.
• We borrow one dollar today.
• We exchange it for (1/1.3) euros and invest this amount in a deposit account.
• After one year we withdraw the principal and the interest earned and exchange them into dollars at ${\displaystyle FX_{oneyear}}$.

The net cashflow of this trade at expiry is,

${\displaystyle -(1+r_{USD})+{\frac {1}{FX_{spot}}}(1+r_{EUR})FX_{oneyear}}$

In the absence of arbitrage opportunities the net cashflow of this trade should be zero and therefore,

${\displaystyle FX_{oneyear}=FX_{spot}{\frac {(1+r_{USD})}{(1+r_{EUR})}}}$

Another example is a forward contract on a zero coupon one year bond , one year from now. Given the price of a one year bond ${\displaystyle P_{1year}}$ and a two year bond ${\displaystyle P_{2year}}$ we look at the following set of trades,

• Sell a one year zero coupon bond one year from now at forward price ${\displaystyle P_{1,1}}$.
• Buy a one year zero coupon bond today.
• Sell a two year zero coupon bond today.

Since ${\displaystyle P_{1year}>P_{2year}}$ we must borrow the difference. After one year we receive $1 from the one year bond and pay interest and principal on the amount borrowed. The two year bond has one year to maturity and we transfer it to the buyer of the forward in return for ${\displaystyle P_{1,1}}$. Therefore the net cashflow in one year is, ${\displaystyle -(P_{1year}-P_{2year})(1+r_{1year})+1+P_{1,1}}$ In the absence of arbitrage opportunities this cashflow must be zero. Since, ${\displaystyle P_{1year}={\frac {1}{(1+r_{1year})}}}$ we conclude that, ${\displaystyle P_{1,1}=P_{2year}/P_{1year}}$ It is interesting to note that the formula, ${\displaystyle P_{1year}={\frac {1}{(1+r_{1year})}}}$ is based on a "no-arbitrage" argument itself and the one year bond can be viewed as the "forward contract" for one dollar received in one year. Given the value of ${\displaystyle r_{1year}}$, if the price of the one year bond was different from ${\displaystyle 1/(1+r_{1year})}$ one could sell a one-year bond at a price ${\displaystyle P^{*}>P_{1year}}$. At expiry one dollar would be paid to the buyer of the bond but since the proceeds from the sale would have earned ${\displaystyle P^{*}(1+r_{1year})}$ they would cover this payment and leave a clear profit. Only if ${\displaystyle P^{*}=1/(1+r_{1year})=P_{1year}}$ the condition of no arbitrage holds. The main features of forward contract are : • It is an agreement between two parties to buy or sell assets at an agreed future point in time. • Buying or selling price of assets is determined in present time. • The transfer or delivery of assets is in future agreed period. • Any terms of contract can be negotiated (not standardized)between the parties involved in the forward contract • Transactions in forward contract are not transparent. • The difference between the spot and the forward price is the forward premium or forward contract. • If the price of stock is increased in future, investor(buyer) will gain and seller will lose. ## Futures Futures contracts, like forward contracts, specify the delivery of an asset at some future date. Futures contracts, unlike forward contracts, 1. Require the buyer or the seller of the futures contract to post margin. 2. Have minimum margin requirements; these requirements are achieved through a margin call. 3. Use the process of mark-to-market. There three requirements in practice are not unique to futures contracts. The best way to understand them is by looking at a specific futures contract. The corn futures contract trades at the Chicago Board of Trade (CBOT). The specifications of the contract are very strict and require the delivery of "no. 2 yellow" corn; if other grades of corn are delivered instead the price paid is adjusted [1]. The contract size can be in multiples of 5,000 bushels of corn. Futures can be purchased for delivery of corn in months December, March, May, July and September only. Trading this contract ceases on the business day nearest to the 15th calendar date of the delivery month. Delivery takes place two business days after the 15th calendar date of the delivery month. Assume that one lot (5,000 bushels) of the Jul-07 contract was bought at 418 cents/bushel on 24 January 2007. The exchange would require the buyer to post initial margin of$900. If the buyer does not post this amount of money in her account with the exchange, her order cannot be executed. For this contract the maintenance margin is the same; during the life of this futures contract the balance of the account cannot go below this level; if for any reason the balance of the account falls below the maintenance margin, the buyer of this contract will receive a margin call.

On the date on which the trade was executed the mark-to-market of the futures contract is zero. Assume that on the next trading date, the settlement price of the futures contract is 418 3/4 cents/bushel (settlement price is the price traded for a futures contract at the close of the trading session). The mark-to-market of the Jul-07 corn futures is,

${\displaystyle MtM=5,000*(4183/4-418)=\37.50}$

The balance on the buyer's account will now be $937.50. The account is like a normal deposit account and earns interest on its balance. If the market price of the Jul-07 corn contract drops in the following day, the mark-to-market could drop from$37.50 to $12.50. In this case$25 are withdrawn from the buyer's account and the balance is now $912.50. If on the last trading date of this contract (13 July 2007) the settlement price is 420 1/4 cents/bushel then the mark-to-market is$112.50. The final balance of the buyer's account is $1012.50 plus interest earned. Since the corn that will be delivered on the 17th July 2007 is worth$21,012.50, the buyer will pay this amount to the clearing house. The clearing house acts as counterparty in the transaction between the corn producer and the buyer and makes sure payments are made and corn is delivered to the warehouse nominated by the buyer.

Since the trader has earned $112.50 (plus interest) in effect the net payment for delivery of corn is$20,900. This is equivalent to paying 418 cents/bushel on the corn delivered. The futures contract has therefore enabled the buyer to purchase corn at the original price of 418 cents/bushel and hedge against price changes.

In order to compare the price of a forward contract ${\displaystyle F_{0}}$ and the price of a futures contract ${\displaystyle \Phi _{0}}$ we look at the following set of trades:

• We sell a forward contract to deliver a specific quantity of corn at some future date for price ${\displaystyle F_{0}}$.
• For dates ${\displaystyle i=0,1,2,...,N-1}$ we purchase a quantity ${\displaystyle q_{i}}$ of the futures contract so that the following conditions are satisfied:
• ${\displaystyle q_{0}(1+r_{1})^{N-1}=1}$
• ${\displaystyle (q_{0}+q_{1})(1+r_{2})^{N-2}=1}$
• ...
• ${\displaystyle (q_{0}+q_{1}+...+q_{N-1})(1+r_{N-1})=1}$

where ${\displaystyle r_{i}}$ is the daily interest rate applicable for period starting on date ${\displaystyle i}$ and ending on date ${\displaystyle N}$. This set of equations can be solved recursively. The value of the margin account on date ${\displaystyle N}$ will be,

${\displaystyle \sum _{i=1}^{N-1}\left(\sum _{j=0}^{i-1}q_{j}\right)(1+r_{i})^{N-i}[\Phi _{i}-\Phi _{i-1}]}$

To undestand the last equation, we know that on date ${\displaystyle i}$ the total quantity of futures purchased is ${\displaystyle q_{0}+q_{1}+...+q_{i-1}}$. By the end of date ${\displaystyle i}$ the mark-to-market change is equal to ${\displaystyle (q_{0}+q_{1}+...+q_{i-1})(\Phi _{i}-\Phi _{i-1})}$. Depending on the direction of the change ${\displaystyle \Phi _{i}-\Phi _{i-1}}$ this is a gain or a loss and earns or requires the payment of principal plus interest at the expiry date of the contract.

Given the conditions that give rise to the solutions for ${\displaystyle q_{i}}$, the last equation is equal to ${\displaystyle \Phi _{N}-\Phi _{0}}$. Since at expiry the price of the futures is equal to the spot price of the asset and therefore ${\displaystyle F_{N}=\Phi _{N}}$, if ${\displaystyle \Phi _{0}}$ is different from ${\displaystyle F_{0}}$ a risk-free profit can be generated. Note that there is no cost in entering into the series of futures contracts and depending on the sign of the difference ${\displaystyle F_{0}-\Phi _{0}}$ the strategy can be reversed. Therefore the forward price must be equal to the futures price.

The strategy used in this analysis assumes that when we purchase an additional quantity ${\displaystyle q_{i}}$ of the futures contract we know the interest rate for the period ${\displaystyle i+1}$ to ${\displaystyle N}$. Since in practise the actual value of the interest rate is not known assume that we can lock in a forward rate. However since we cannot predict the change in the mark-to-market of the futures contract in the period ${\displaystyle i}$ to ${\displaystyle i+1}$ we do not know the notional amount we must purchase.

Assume that on date ${\displaystyle i}$ we make the assumption that there will be no change in the mark-to-market of the futures and therefore there is no need to lock in a forward rate for the period ${\displaystyle i+1}$ to ${\displaystyle N}$. Since the most likely scenario is that we will be wrong, we will have to borrow or deposit the actual change in the mark-to-market at the spot rate for the period ${\displaystyle r_{i+1}}$.

As long as the error in our estimate of the mark-to-market change is independent of the spot rate we can expect that the costs/benefits will balance to zero. But if the mark-to-market change of the futures contract is a function of spot rate the costs/benefits will not balance to zero and the futures strategy described above will not be able to replicate the payoff of the forward. We conclude that when the futures contract is a function of the interest rate the futures price will not be equal to the forward price.

Another exception occurs when the futures price can change by large amounts from one date to the next. The term "large amounts" here means that a one day move accounts for a large percentage of the difference between ${\displaystyle \Phi _{0}}$ and ${\displaystyle \Phi _{N}}$. In this case on the date when this large price change occurs the error in the notional locked in at the forward rate is large enough to magnify the error in our estimate of the change in the mark-to-market. Furthermore, all subsequent mark-to-market changes are much smaller and cannot balance this cost/benefit. Fortunately, most exchanges limit the maximum change in the futures price that can occur from one date to the next. But if such large price moves are possible then, even if the futures price is not a function of the interest rate, the assumption that it is equal to the forward price is wrong.

In general, the relation between the futures and the forward price cannot be derived through a static arbitarge strategy unless interest rates have a deterministic term-structure. The derivation of the relation between the futures and the forward price of an asset is one of the first applications of dynamic hedging [Black 1976].

The main features of Future contract are:

• It is standardized contract made in terms of quantity, expiration date and settlement procedures etc.
• Transition in a future contract are fully transparant.
• It is traded in organized exchange and is "marked to market" daily.
• Physical delivery of underlying assets is virtually never taken.

## Swaps

A Swap is an agreement to exchange a sequence of cash flows over a period of time in the future in same or different currencies. Mainly used for hedging various interest rate exposures, they are very popular and highly liquid instruments. Some of the very popular swap types are

Fixed - Float (Same currency) Party P pays/receives fixed interest in currency A to receive/pay floating rate in currency A indexed to X on a notional N for a tenor T years. For example, you pay fixed 5.32% monthly to receive USD 1M Libor monthly on a notional USD 1 mio for 3 years. Fixed-Float swaps in same currency are used to convert a fixed/floating rate asset/liability to a floating/fixed rate asset/liability. For example, if a company has a fixed rate USD 10 mio loan at 5.3% paid monthly and a floating rate investment of USD 10 mio that returns USD 1M Libor +25 bps monthly, and wants to lockin the profit as they expect the USD 1M Libor to go down, then they may enter into a Fixed-Floating swap where the company pays floating USD 1M Libor+25 bps and receives 5.5% fixed rate, locking in 20bps profit.

Fixed - Float (Different currency) Party P pays/receives fixed interest in currency A to receive/pay floating rate in currency B indexed to X on a notional N at an initial exchange rate of FX for a tenor T years. For example, you pay fixed 5.32% on the USD notional 10 mio quarterly to receive JPY 3M Tibor monthly on a JPY notional 1.2 bio (at an initial exchange rate of USDJPY 120) for 3 years. For Nondeleverable swaps, USD equivalent of JPY interest will be paid/received (as per the Fx rate on the FX fixing date for the interest payment day). Note in this case no initial Exchange of notional takes place unless the Fx fixing date and the swap start date fall in the future. Fixed-Float swaps in different currency are used to convert a fixed/floating rate asset/liability in one currency to a floating/fixed rate asset/liability in a different currency. For example, if a company has a fixed rate USD 10 mio loan at 5.3% paid monthly and a floating rate investment of JPY 1.2 bio that returns JPY 1M Libor +50 bps monthly, and wants to lockin the profit in USD as they expect the JPY 1M Libor to go down or USDJPY to go up(JPY depreciate against USD), then they may enter into a Fixed-Floating swap in different currency where the company pays floating JPY 1M Libor+50 bps and receives 5.6% fixed rate, locking in 30bps profit against the interest rate and the fx exposure.

Float - Float (Same Currency, different index) Party P pays/receives floating interest in currency A Indexed to X to receive/pay floating rate in currency B indexed to Y on a notional N for a tenor T years. For example, you pay JPY 1M Libor monthly to receive JPY 1M Tibor monthly on a notional JPY 1 bio for 3 years.

In this case, company wants to lockin the cost from the spread widening or narrowing. For example, if a company has a floating rate loan at JPY 1M Libor and the company has an investment that returns JPY 1M Tibor+30 bps and currently the JPY 1M Tibor = JPY 1M Libor +10bps. At the moment, this company has a net profir of 40 bps. If the company thinks JPY 1M tibor is going to come down or JPY 1M Libor is going to increase in the future and wants to insulate from this risk, they can enter into a Float float swap in same currency where they pay JPY TIBOR +10 bps and receive JPY LIBOR+35 bps. with this, they have effectively locked in a 35 bps profit instead of running with a current 40 bps gain and index risk. The 5bps difference comes from the swap cost which includes the market expectations of the future rates in these two indices and the bid/offer spread which is the swap commission for the swap dealer.

Float - Float (Different Currency) Party P pays/receives floating interest in currency A Indexed to X to receive/pay floating rate in currency A indexed to Y on a notional N at an initial exchange rate of FX for a tenor T years. For example, you pay floating USD 1M libor on the USD notional 10 mio quarterly to receive JPY 3M Tibor monthly on a JPY notional 1.2 bio (at an initial exchange rate of USDJPY 120) for 4 years.

To explain the use of this type of swap, consider a US company operating in Japan. To fund their Japanese growth, they need JPY 10 bio. the easiest option for the company is to issue debt in Japan. As the company might be new in the Japanese market with out a well known reputation among the Japanese investors, this can be an expensive option. Added on top of this, the company might not have appropriate Debt issuance program in Japan and they might lack sophesticated treasury operation in Japan. To overcone the above problems, they can issue USD debt and convert to JPY in the FX market. Although this option solves the first problem, it introduces two new risks to the company. 1. Fx risk. If this if this USDJPY spot goes up at the maturity of the debt, then when the company converts the JPY to USD to pay back its matured debt, it receives less USD and suffers a loss 2. USD and JPY interest rate risk. If the JPY rates come down, the return on the investment in Japan might go down and this introduces a interest rate risk component.

First exposure in the above can be hedged using long dated FX forward contracts but this introduces a new risk where the implied rate from the Fx Spot and the Fx Frward is a fixed rate but the JPY invest ment returns a floating rate. Although there are several alternatives to hedge both the exposures effectively with out introducing new risks, the easiest and the most cost effective alternative would be to use a Float-Float swap in different currencies. In this, the company raises USD by issuing USD Debt and swaps it to JPY. It receives USD floating rate(so matching the interest payments on the USD Debt) and pays JPY floating rate matching the returns on the JPY investment.

Fixed - Fixed (Different Currency) Party P pays/receives fixed interest in currency A to receive/pay fixed rate in currency B for a tenor T years. For example, you pay JPY 1.6% on a JPy notional of 1.2 bio and receive USD 5.36% on the USD equivalent notional of 10 mio at an initial exchange rate of USDJPY 120.

Usage is similar to above but you receive USD fixed rate and pay JPY Fixed rate.

--192.147.54.3 05:07, 29 June 2007 (UTC)M G Naidu

Primarily used as hedging instruments, against varying interest payments. The base concept is quite easy to follow; you swap a fixed rate for a floating rate or vice-versa. In the case of companies that offer Variable Rate Bonds, they can enter into a swap agreement with a broker/dealer; where the company pays the broker a fixed rate as per agreement and the broker provides them with the floating rate, which can be used to make periodic coupon payments. In essence, the company has hedged it's risk against a sudden rate increase, as it is locked in a fixed rate over time. Swaps may be terminated with one party paying it's counterpart a certain fee, which may have been determined at time of initial agreement or may be based on future payments if interest rates were to remain constant.

## Options

An option is a financial instrument that gives the holder to purchase or sale the stated number of shares at pre determined price(exercise price) within or on certain future date. It can be defined as a contract between two investors( i.e. call writer and option buyer).

There are two types of stock options:

• Call Option: Call option gives the buyer a right to purchase the given stock at the strike price. Thus Call option is generally bought when the buyer is bullish about the underlying security.The value of call option can be calculated by following equation:

Vc= Max.(Vs</sub->- E,0)

Where,

Vc = value of call option

Max = Maximum

0 = Zero

Vs = Value of stock

E = Exercise price or strike price

Profit or Loss

Break- Even point

• Put Option: Similarly buying a put option gives you the right to sell the underlying stock at the strike price. Put option is bought when the buyer has bearish views about the underlying security.The value of Put option can be calculated by following equation:

Vp= Max(E - Vs</sub->,0)

Where,

Vp = value of Put option

Max = Maximum

0 = Zero

Vs = Value of stock

E = Exercise price or stike price

Profit or Loss

Break- Even point

Each option comes with an "Exercise Date". European options may only be exercised on the exercise date, whereas American options may be exercised at any time up till the exercise date.

Due to the put-call parity, it is possible to create artificial call or put options if the other is not available. Put options may also be used as a hedging instrument, against possible decline in value of the underlying stock.

While stocks with high volatility (modified duration) are high risk, options whose underlying stock have high volatility are actually better. They provide a possibility of a higher payout if the stock goes up in proportion to its volatility and the same amount of loss.

## Options on Forwards

In this case, the underlying asset on which the option is written is a forward contract. A market exists in which forward contracts are traded. We do not impose the martingale property on the s.d.e. for a forward price. Rather, given the current forward price $F(t,T)$,

${\displaystyle {\frac {dF(t,T)}{F(t,T)}}=\mu dt+\sigma dW(t)}$

In order to simplify the analysis we assume that ${\displaystyle \mu }$ and ${\displaystyle \sigma }$ are positive constants. The mark-to-market of a forward contract with arbitrary strike ${\displaystyle K}$ is,

${\displaystyle V(t,T)=B(t,T)[F(t,T)-K]}$

where ${\displaystyle B(t,T)=\exp[-r(T-t)]}$ and ${\displaystyle r}$ is the risk-free rate. An option on a forward gives the buyer of the option the right to purchase a forward contract with strike ${\displaystyle K}$ and expiry ${\displaystyle T^{*}}$ at some future date ${\displaystyle T. Lets price this option blindly the actuarial approach. This approach requires that the price of the option is given by taking the expectation of its payoff under the 'true' distribution of the forward price,

${\displaystyle C(t,T)=B(t,T)\mathbf {E} _{t}\left\{B(T,T^{*})[F(T,T^{*})-K]^{+}\right\}}$

where,

${\displaystyle F(T,T^{*})=F(t,T^{*})\exp \left[\left(\mu -{\frac {1}{2}}\sigma ^{2}\right)(T-t)+\sigma {\sqrt {T-t}}U\right]}$

and ${\displaystyle U}$ is a standard normal random variable. The expectation has the following simple solution,

${\displaystyle C(t,T^{*})=B(t,T^{*})[Fexp[\mu (T-t)]N(d_{1})-KN(d2)]}$

where,

${\displaystyle d1={\frac {\ln \left({\frac {F}{K}}\right)+\left(\mu +{\frac {1}{2}}\sigma ^{2}\right)(T-t)}{\sigma {\sqrt {T-t}}}}}$

${\displaystyle N(x)=Prob(U>x)}$ and ${\displaystyle d_{2}=d_{1}-\sigma {\sqrt {T-t}}}$. In the same way the price of a put is given by,

${\displaystyle P(t,T^{*})=B(t,T^{*})[-F\exp[\mu (T-t)]N(-d_{1})+KN(-d2)]}$

In the absence of arbitareg put-call parity requires the following equation to hold,

${\displaystyle C(t,T^{*})-P(t,T^{*})=V(t,T^{*})}$

This is equivalent to,

${\displaystyle F\exp[\mu (T-t)]-K=F-K}$

This is only possible if ${\displaystyle \mu =0}$. This transparent approach, first proposed by Emanuel Derman and Nassim Taleb [2], generates the arbitrage-free option price without the need for unrealistic assumptions about the viability of dynamic hedging. The only assumption we made was regarding the 'true' probability distribution function of the forward price. If we choose a more general approach, where ${\displaystyle F(T,T^{*})}$ has an arbitrary probability distribution function, then the value of a call option is given by,

${\displaystyle \mathbf {E} _{t}\left\{[F(T,T^{*})-K]^{+}\right\}=\mathbf {E} _{t}\left\{F(T,T^{*})\right\}{\tilde {P}}(F(T,T^{*})>K)-KP(F(T,T^{*})>K)}$

where ${\displaystyle P(\cdot )}$ is the 'true' probability distribution and ${\displaystyle {\tilde {P}}(\cdot )}$ is a probability distribution with the property,

${\displaystyle d{\tilde {P}}(F(T,T^{*}))={\frac {F(T,T^{*})}{\mathbf {E} _{t}\{F(T,T^{*})\}}}dP(F(T,T^{*}))}$

In the same way, the value of a put option is given by,

${\displaystyle \mathbf {E} _{t}\left\{[K-F(T,T^{*})]^{+}\right\}=-\mathbf {E} _{t}\left\{F(T,T^{*})\right\}{\tilde {P}}(F(T,T^{*})

By put-call parity,

${\displaystyle B(t,T^{*})[\mathbf {E} _{t}\left\{F(T,T^{*})\right\}-K]=B(t,T^{*})[F(t,T)-K]}$

Therefore,

${\displaystyle \mathbf {E} _{t}\left\{F(T,T^{*})\right\}=F(t,T)}$

i.e. under an arbitrary 'true' distribution the option on the forward is priced by using the martingale property for the forward price.

## Options on the Product of Two Asset Prices

The growth of the financial sector has resulted in products which are covered under the broad term "exotic derivatives". These derivatives are often written on indices which are derived from traded prices but which themselves are not traded. Depending on investor preferences an index can be a function of more than one asset prices and can be determined from the value of these asset prices from a single or a series of observations. Exotic derivatives can either be priced using analytic methods or numerical techniques. The framework used to price all exotic derivatives is based on the Black-Scholes option pricing theory, in which dynamic hedging is used to obtain an arbitrage-free equation for the option price. Although we can always obtain a p.d.e. for all exotic derivatives, an analytic solution cannot always be obtained. However, there exists a large range of exotics where an analytic solution is possible. An option on the product of two asset prices has an analytic solution.

Given two traded assets, an index can be created where the value of the index at some time ${\displaystyle t}$ is defined as,

${\displaystyle S(t)={\frac {P_{1}(t)P_{2}(t)}{P_{1}(0)P_{2}(0)}}}$

where ${\displaystyle t=0}$ is the time at which the index is created and ${\displaystyle S(0)=1}$. An option can be written on this index with payoff at expiry ${\displaystyle T}$,

${\displaystyle C(T)=\max[S(T)-1,0]}$

Since the option is only a function of ${\displaystyle P_{1}}$, ${\displaystyle P_{2}}$ and ${\displaystyle t}$, given the s.d.e.s for the prices of the two assets,

${\displaystyle {\frac {dP_{1}(t)}{P_{1}(t)}}=m_{1}dt+\sigma _{1}dW_{t}^{1}}$

${\displaystyle {\frac {dP_{2}(t)}{P_{2}(t)}}=m_{2}dt+\sigma _{2}dW_{t}^{2}}$

(where ${\displaystyle dW_{t}^{1}dW_{t}^{2}=\rho dt}$) Itô's lemma can be applied the price of the option to give,

${\displaystyle dC=\left[{\frac {\partial C}{\partial t}}+m_{1}P_{1}(t){\frac {\partial C}{\partial P_{1}}}+m_{2}P_{2}(t){\frac {\partial C}{\partial P_{2}}}+{\frac {1}{2}}\sigma _{1}^{2}P_{1}(t)^{2}{\frac {\partial ^{2}C}{\partial P_{1}^{2}}}+{\frac {1}{2}}\sigma _{2}^{2}P_{2}(t)^{2}{\frac {\partial ^{2}C}{\partial P_{2}^{2}}}+\sigma _{1}\sigma _{2}\rho P_{1}(t)P_{2}(t){\frac {\partial ^{2}C}{\partial P_{1}\partial P_{2}}}\right]dt+\sigma _{1}{\frac {\partial C}{\partial P_{1}}}P_{1}(t)dW_{t}^{1}+\sigma _{2}{\frac {\partial C}{\partial P_{2}}}P_{2}(t)dW_{t}^{2}}$

A portfolio consisting of \$1 of the option, ${\displaystyle -\partial C/\partial P_{1}}$ of asset 1 and ${\displaystyle -\partial C/\partial P_{2}}$ of asset 2 must therefore have an s.d.e. given by,

${\displaystyle d\left(C-{\frac {\partial C}{\partial P_{1}}}P_{1}(t)-{\frac {\partial C}{\partial P_{2}}}P_{2}(t)\right)=\left[{\frac {\partial C}{\partial t}}+{\frac {1}{2}}\sigma _{1}^{2}P_{1}(t)^{2}{\frac {\partial ^{2}C}{\partial P_{1}^{2}}}+{\frac {1}{2}}\sigma _{2}^{2}P_{2}(t)^{2}{\frac {\partial ^{2}C}{\partial P_{2}^{2}}}+\sigma _{1}\sigma _{2}\rho P_{1}(t)P_{2}(t){\frac {\partial ^{2}C}{\partial P_{1}\partial P_{2}}}\right]dt}$

Since this portfolio has no sources of risk, in the absence of arbitrage it must have an instantaneous return equal to the risk-free rate ${\displaystyle r}$. Therefore the last equation gives rise to the following p.d.e.:

${\displaystyle rC={\frac {\partial C}{\partial t}}+rP_{1}{\frac {\partial C}{\partial P_{1}}}+rP_{2}{\frac {\partial C}{\partial P_{2}}}+{\frac {1}{2}}\sigma _{1}^{2}P_{1}^{2}{\frac {\partial ^{2}C}{\partial P_{1}^{2}}}+{\frac {1}{2}}\sigma _{2}^{2}P_{2}^{2}{\frac {\partial ^{2}C}{\partial P_{2}^{2}}}+\sigma _{1}\sigma _{2}\rho P_{1}P_{2}{\frac {\partial ^{2}C}{\partial P_{1}\partial P_{2}}}}$

From the payoff function of this option we can deduce that the pricing equation can be transformed into a two-dimensional one with variables ${\displaystyle t}$ and ${\displaystyle P=P_{1}P_{2}}$. Note that,

${\displaystyle {\frac {\partial C}{\partial P_{1}}}=P_{2}{\frac {\partial C}{\partial P}}}$

${\displaystyle {\frac {\partial C}{\partial P_{2}}}=P_{1}{\frac {\partial C}{\partial P}}}$

${\displaystyle {\frac {\partial ^{2}C}{\partial P_{1}^{2}}}=P_{2}^{2}{\frac {\partial ^{2}C}{\partial P^{2}}}}$

${\displaystyle {\frac {\partial ^{2}C}{\partial P_{2}^{2}}}=P_{1}^{2}{\frac {\partial ^{2}C}{\partial P^{2}}}}$

${\displaystyle {\frac {\partial ^{2}C}{\partial P_{1}\partial P_{2}}}=P_{1}P_{2}{\frac {\partial ^{2}C}{\partial P^{2}}}+{\frac {\partial C}{\partial P}}}$

Therefore the p.d.e. can be simplified to,

${\displaystyle rC={\frac {\partial C}{\partial t}}+mP{\frac {\partial C}{\partial P}}+{\frac {1}{2}}\sigma ^{2}P^{2}{\frac {\partial ^{2}C}{\partial P^{2}}}}$

where,

${\displaystyle m=2r+\sigma _{1}\sigma _{2}\rho }$

and,

${\displaystyle \sigma ={\sqrt {\sigma _{1}^{2}+\sigma _{2}^{2}+2\sigma _{1}\sigma _{2}\rho }}}$

and boundary condition ${\displaystyle C(T)=\max[P(T)/P(0)-1]}$. This p.d.e. is the Black-Scholes p.d.e. for a call option and can be solved to give,

${\displaystyle C(0)=\exp[(m-r)T]N(h_{1})-\exp[-rT]N(h_{2})}$

where,

${\displaystyle h_{1}={\frac {\left(m+{\frac {1}{2}}\sigma ^{2}\right){\sqrt {T}}}{\sigma }}}$

${\displaystyle h_{2}={\frac {\left(m-{\frac {1}{2}}\sigma ^{2}\right){\sqrt {T}}}{\sigma }}}$

The same result can be obtained by starting with the risk-neutral processes for the two assets,

${\displaystyle {\frac {dP_{1}(t)}{P_{1}(t)}}=rdt+\sigma _{1}d{\tilde {W}}_{t}^{1}}$

${\displaystyle {\frac {dP_{2}(t)}{P_{2}(t)}}=rdt+\sigma _{2}d{\tilde {W}}_{t}^{2}}$

Using Itô's lemma, the process for the product of the two prices is,

${\displaystyle {\frac {dP(t)}{P(t)}}=mdt+\sigma dW_{t}}$

and the pricing equation derived using the p.d.e. follows.