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## A. PRICING OF CURRENCY FUTURES

## B. COST OF CARRY IN CURRENCY FUTURES PRICING

### How cost of carry gets into futures pricing in USDINR contracts?

### Futures price as an expected price

### Let us also understand what is Basis in case of currency contracts

## C. PRICING OF CURRENCY OPTIONS

*Source: NSE*
## D. FACTORS THAT IMPACT OPTIONS PRICING

### Key factors that drive the value of an option

### Premiums increase as options become further in-the-money.

### The longer the time until expiration, the higher the option price.

### The shorter the time until expiration, the lower the option price.

### The greater the expected volatility, the higher the option value.

## E. OPTIONS SPREADS IN CURRENCIES

### How does a spread actually play out in the currency market?

## F. BLACK & SCHOLES IN CURRENCY OPTIONS PRICING

Chapter 5
# Pricing Of Currency Futures And Options

It is by now clear that currency futures are a right and an obligation whereas currency options are a right without an obligation. That is why currency options are known as asymmetric products because the payoff for the buyer and the seller of the currency option are diametrically opposite. For example, the buyer of a currency option (call or put) has a limited risk to the extent of premium paid but unlimited returns once the premium cost is covered. On the other hand the seller of the option has obligation without the right. The trader who sells a call option has the obligation to sell whereas the seller of a put option has the obligation to buy. For taking on this obligation without the right, the option seller receives premium. This premium is the maximum return for the option seller.

When we talk of pricing of currency futures and options these are two different aspects altogether. Pricing of currency futures means (spot price + cost of carry). Pricing of currency options is the price of the right to buy or sell the currency at a particular strike price. In short, when you buy futures you pay for the future spot price today. When you buy options (call or put) you pay for the right without the obligation. This is the essential difference between the pricing of currency futures and currency options.

If you were to open the currency futures quotes for the most liquid USDINR contract you will find that the actual USDINR near month futures price is different from the spot price. Before we get into the nuances of pricing of currency futures, let us understand that the currency future is a derivative transaction. That means it derives its value from an underlying which is the USDINR exchange rate in this case. You must have also observed that the futures price of the currency instrument moves in sync with its underlying. If the underlying price falls, so would the futures price and if the price of the underlying increases then the futures price also goes up. However, the underlying price and the futures price differs and they are not really the same. They only converge on the date of expiry, which is the last business day of the month when the RBI reference rate is used to settle transactions. But during the month, the spot price and the futures price are different. Let us actually look at the live data of the USDINR futures contract in the Indian context based on NSE data.

Currency Pair | Spot Price | Futures Price | Basis | Cost of Carry |
---|---|---|---|---|

USDINR (Mar) | 69.3538 | 69.4325 | 0.0787 | 0.1135% |

USDINR (Apr) | 69.3538 | 69.7100 | 0.3562 | 0.5136% |

UDDINR (May) | 69.3538 | 69.9250 | 0.5712 | 0.8236% |

This is the live data that we have taken from NSE. But is there a theoretical model to price currency futures. In the above case you will see that the cost of carry is higher as the maturity goes farther. That is because the major carrying cost in currency futures is only the interest cost as unlike in commodity you do not have to worry about costs like storage, insurance, demurrage etc. Let us now look at a formula for pricing of futures.

*Currency Futures price = Spot price x {{1+Rf)n}*

Rf is the risk free rate

n is the time to expiry (expressed in days – 20 days is 20/360)

Of course, when we are pricing the futures of a stock or index, we deduct the dividend yield but in case of currencies there are not dividends involved and hence it is only the risk free rate that matters. This gives the theoretical price of the currency futures price and the actual currency futures price will vary based on a plethora of domestic and global factors.

When we talk of cost of carry and futures pricing, we need to understand about gross cost of carry, net cost of carry and basis. These 3 concepts are fundamental to the understanding of the pricing of futures. Let us start off with a basic lesson on futures.

Futures are derivative product, which means the value of the derivative is derived from the value of the underlying. In case of the USDINR futures, the underlying is the spot USDINR exchange rate. In other words, the futures price of the USDINR is nothing but the expected spot price after a particular tenure which could be 1 month, 2 months, 3 months or 6 months. Why is there a difference between the spot price and the futures price and why does the futures price keep differing as the expiry becomes longer? This difference can be understood through two simple pricing models for futures contracts viz. the Cost of Carry Model and the Expectancy Model.

However, remember that these models merely give you platform on which to base your understanding of futures prices. That said, being aware of these theories gives you a feel of what you can expect from the futures price of a stock or an index.

Cost of Carry Model assumes that any asset has a cost of carrying. For example, when you carry equities, there is demat charges and there is the opportunity cost of your investments locked up. In case you are holding commodities, then it gets a lot more complicated. For example, there is opportunity cost of funds locked plus other costs like storage, transportation, legal costs, taxes & stamp duty, demurrage, fines etc. All these will go into cost of carry. What happens in case of currency future? Since these currency futures are cash settled in India, there is no scope for any physical costs. Hence the only cost involved is the interest or opportunity cost and the transaction cost if any.

What does this mean for pricing of the futures contract? It means that the price of a futures contract (FP) will be equal to the spot price (SP) plus the net cost incurred in carrying the asset till the maturity date of the futures contract.

What about carry returns? We already understood carry cost; let us also look at carry return. Carry Return refers to any income derived from the asset while holding it like dividends, bonuses etc. While calculating the futures price of an index, the Carry Return is the average returns given by the index during the holding period. A net of these two is called the net cost of carry. However, currency futures contract does not pay dividends anyways and hence carry returns will not be applicable in case of currency futures contract on the USDINR. The formula can be represented as under:

*Futures Price = Spot Price + (Carry Cost – Carry Return)*

Since carry returns in case of currency futures is nil, the futures price in the above case will just be the sum of spot price and the cost of carry.

This is another approach called the Expectancy Model of futures pricing, which essentially states that the futures price of an asset is basically the expected spot price of the currency value at a future date. For example, if the dollar is likely to substantially strengthen in the coming months, then it will enhance the futures price of the USDINR contract. Conversely, if the rupee is likely to strengthen then the futures price could be at a very narrow premium to the spot price. There is a small flaw in the expected price approach. Unlike the Cost of Carry model, this model believes that there is no relationship between the present spot price of the asset and its futures price. That may not be exactly true. There is too much focus on expectations not appreciating the fact that expectations are by themselves driven by hard core fundamental data.

If you look at the futures price of USDINR and compare with spot price, you will observe that there is usually a difference between the futures price and the spot price. This difference is called the basis. If the futures price of an asset is trading above its spot price, then the basis for the asset is negative. This means, the markets are expected to rise in the future. On the other hand, if the spot price of the asset is higher than its futures price, the basis for the asset is positive. This is indicative of a bear run on the market in the future.

Unlike currency futures which represent the future spot price, the currency options are about the right without the obligation. Option is the price you pay for the asymmetric right that you get as a buyer without the obligation to delivery. But how are premiums decided for a currency option. Let us look at the option screen first.

Some of the key aspects of the above option screenshot have been highlighted. Let us see what they signify about this option.

- Symbol represents the specific pair contract where you are buying the option contract in currencies.
- Expiry date in any month for the currency options contract is 2 days prior to the last business day of the month.
- The call option represents that it is a right to buy the USDINR at the strike price of Rs.69.50/$. This is a right without the obligation.
- The price of Rs.0.2650 is the price of the right without the obligation to exercise the call option and buy USDINR at Rs.69.50/$. Here the view is dollar will strengthen.
- RBI reference rate is the previous day’s reference rate and that is different from the USDINR spot rate. RBI Reference rate is only used for final settlement.
- Open interest shows the numbers of USDINR contracts expiring in March 2019 for the particular call option to buy USDINR at 69.50 are currently open.

The question now is about options currency pricing. To cut a long story short, how is the price of Rs.0.2650 for the 69.50 USDINR call option is arrived at? What is also important is how and on what basis this price keeps changing since it is a dynamic price constant to frequent changes. Let us first understand the broad theoretical underpinning of pricing of currency options.

The theoretical option pricing models are used by option traders for calculating the fair value or the intrinsic value of an option on the basis of the influencing factors like market price, volatility, time to expiry, interest rates, strike price etc. The two most popular option pricing models are: Black Scholes Model which assumes that percentage change in the price of underlying follows a lognormal distribution. Then there is the Binomial Model which assumes that percentage change in price of the underlying follows a binomial distribution. In the real options world, the more commonly used approach to options pricing is the Black & Scholes model.

Just for the sake of understanding, the Black & Scholes model is only as old as the early 1970s. But this model formed the basis for pricing options risk and hence became extremely popular in pricing options. If the options volume globally today is running into trillions of dollars per day, then it owes to this extremely elegant but sophisticated options pricing model called the Black & Scholes model.

Unlike currency futures which represent the future spot price, the currency options are about the right to buy or sell the currency at a particular strike price. The theoretical value of the option (call or put) is determined based on the Black & Scholes model, which combines various factors and comes out with a value for the call and the put option.

There are various factors that influence the price of the call option. The key is to understand how options get priced and how to effectively use underpriced and overpriced options.

The table above captures the primary factors that influence the value of an option. Let us now look at each of these factors individually and how they influence the value of an option.

Six factors that influence option pricing are shown on the top row of the chart. As indicated, the underlying price and strike price determine the intrinsic value; the time until expiration and volatility determine the probability of a profitable move; the interest rates determine the cost of money; and dividends can cause an adjustment to share price. Although dividends are not applicable as you don’t earn dividends on currencies. However, the understanding of the role of dividend is important in understanding valuation of options.

Now let us look at each of the factors individually.

__Underlying Spot Price of the USDINR__

The most influential factor of the theoretical option premium is the current market price of the underlying asset or the spot price of the USDINR. In general, as the price of the underlying increases, call prices increase because the gap between the spot price and the strike price widen. But the put prices decrease because the gap between the spot price and strike price narrows or is negative. Conversely, as the spot price of the underlying USDINR decreases, call prices decrease in value while the put prices increase in value. The outcome can be summarized as under:

If USDINR spot prices ... | Call prices will ... | Put prices will ... |
---|---|---|

Increase | Increase | Decrease |

Decrease | Decrease | Increase |

__Strike Price of the Option (call or put)__

The strike price determines if the option has any intrinsic value. What do we understand by the intrinsic value in this case? Intrinsic value is the difference between the strike price of the option and the current price of the underlying asset. In other words, it is a measure of whether the particular option is in the money (ITM) or it is out of the money (OTM). The premium typically increases as the option becomes further in-the-money (where the strike price becomes more favourable in relation to the current underlying price). The premium generally decreases as the option becomes more out-of-the-money (when the strike price is less favourable in relation to the underlying security). The impact of the strike price on the option value can be quickly summarized as under:

__Time left to expiry of the option__

The time to expiry is the time left to expire. In case of USDINR currency contracts, they expire 2 days prior to the last business day of the month. The expiry is defined and communicated well in advance. The time to expiry influences the call and put option in the same direction. The longer an option has until expiration, the greater the chance it will end up in-the-money (profitable). As expiration approaches, the option's time value decreases. As a general rule, an option loses one-third of its time value during the first half of its life, and two-thirds of its value during the second half. Again this is a thumb rule and this can vary depending on the market conditions.

The underlying asset's volatility is a factor in time value: If the underlying is highly volatile, you can reasonably expect a greater degree of price movement before expiration. The opposite holds true where the underlying exhibits low volatility: The time value will be lower if the underlying price is not expected to move much. As stated earlier, in case of USDINR options, the time to expiry influences the call and put options positively. That means higher time to expiry means higher value of call and put while lower time to expiry means lower value of call and put. The outcome can be summarized as under:

__Estimated volatility as measured by standard deviation__

Volatility impacts the call and put options in a manner similar to the time to expiry. That is because, greater the time to expiry greater is the expected volatility. What do we understand by volatility here? Volatility is the degree to which price moves, whether it goes up or down. It is a measure of the speed and magnitude of the USDINR price changes. Historical volatility refers to the actual price changes that have been observed over a specified time period. Options traders can evaluate historical volatility to determine possible volatility in the future.

Implied volatility, on the other hand, is a forecast of future volatility and acts as an indicator of the current market sentiment. Here in the option valuation we shall focus only on the actual volatility and not on the implied volatility. While implied volatility can be difficult to quantify, option premiums are generally higher if the underlying exhibits higher volatility because it will have higher expected price fluctuations. To that extent higher volatility means higher value for calls and volatility and the reverse holds in case of lower volatility. This can be summarized as under:

__Why do risk free rates impact option value?__

The impact of interest may be small but it impacts nevertheless. Interest rate is used to calculate the present value. For example, the spot price refers to the current date but exercise price refers to the future expiry date. Hence to convert and compare we need to calculate the present value of the exercise price or the strike price. Interest rates and dividends have small, but measurable, effects on option prices. In general, as interest rates rise, call premiums increase and put premiums decrease. This is because of the costs associated with owning the underlying: The purchase incurs either interest expense (if the money is borrowed) or lost interest income (if existing funds are used to purchase the shares). In either case, the buyer will have interest costs. Looked at differently, higher interest rates reduces the present value of the exercise price and enhances the gap between the spot price and the strike price. This is positive for call options but negative for put options. We can summarize the findings as under:

If interest rates ... | Call prices will ... | Put prices will ... |
---|---|---|

Rise | Increase | Decrease |

Fall | Decrease | Increase |

__Dividends declared__

Why do dividends impact options values? When a company pays out dividends it is considered as partial liquidation of the company since the wealth of the company is reduced and paid out to that extent. That means the dividend paid out reduces the theoretical value of the stock and hence the stock price goes down. When the stock price goes down, the gap between the stock price and the strike price narrows and is negative for call options but positive for put options. When we talk of USDINR options the concept of dividends is not applicable as you don’t earn dividends on currencies. However, the understanding of the role of dividend is important in understanding valuation of options. The impact can be summarized as under:

If dividends ... | Call prices will ... | Put prices will ... |
---|---|---|

Rise | Decrease | Increase |

Fall | Increase | Decrease |

Spreads are a kind of a long/short that we have previously seen but they are a little more complicated and sophisticated. Let us look at two such examples. An option spread can be used in case of moderately bullish or moderately bearish situations. They are referred to as a bull call spread and bear put spread respectively. An option spread combines buying a lower strike call and selling a higher strike call if you are moderately bullish. Alternatively, you can also buy a higher strike put and sell a lower strike put if you are moderately bearish. How exactly does a spread work in these cases? The premium received on the option sold reduces the cost of the premium paid on the option bought and the risk is limited to the extent of the net cost.

The second type of spread is a calendar spread which involves the simultaneous purchase and sale of two different monthly expiries of futures of the same underlying futures. One can also effectively use calendar spreads wherein you buy in one contract and sell in the other contract. The idea is not take a view but to capitalize on mispricing. Both these strategies work effectively in lacklustre markets. When you bet on spreads, you don’t bet on any individual asset. Instead, you bet on the spread between two months either widening or narrowing.

Typically, institutions and traders, at times, do not want to bet on the direction of any currency or any of the macros. Then they resort to spreads. In a spread, you bet that one month currency will outperform the other month currency due to some macro changes expected. For example, if the US Fed is likely to increase the Fed rates next month, then you can expect the dollar to harden. That makes a case for buying the USDINR futures. Here you can create a calendar spread by selling the current month USDINR and buying the next month USDINR contract. We are not betting on the USD here but only on the fact that the spread between the current month and the next month will widen at which time we can unwind the transaction and take home the profits. Spreads are always relative. Here the bet is not on the USDINR but the bet is on the expectation that the next month USDINR will outperform the current month USDINR.

If you look at the table below, they represent the calendar spreads between combinations of various pairs of months.

Contract | Best Bid | Best Ask | Volume | LTP Difference | ||
---|---|---|---|---|---|---|

USDINR | ||||||

SPNOV18DEC18 | 1000 | 0.185 | 0.19 | 56 | 5,77,840 | 0.19 |

SPNOV18JAN19 | 1000 | 0.365 | 0.4225 | 300 | 14,396 | 0.4025 |

SPNOV18FEB19 | 500 | 0.485 | - | - | 1,038 | 0.575 |

SPNOV18MAR19 | 1000 | 0.71 | - | - | - | - |

SPNOV18APR19 | 1000 | 0.71 | - | - | - | - |

SPNOV18MAY19 | 1000 | 0.71 | - | - | - | - |

SPDEC18JAN19 | 1000 | 0.2125 | 0.2175 | 300 | 4,306 | 0.22 |

SPDEC18FEB19 | 100 | 0.42 | 0.4375 | 100 | - | - |

SPDEC18MAR19 | - | - | - | - | - | - |

SPDEC18APR19 | - | - | - | - | - | - |

SPDEC18MAY19 | - | - | - | - | - | - |

SPDEC18JUN19 | 1500 | 0.7125 | - | - | - | - |

SPJAN19FEB19 | 250 | 0.2025 | 0.215 | 1000 | 144 | 0.21 |

In the above if you expect the USDINR DEC-Jan spread to widen from 0.2 to 0.25, then you just buy this spread contract and then you can book the profits when your profit target is reached. It is as simple as that.

Black & Sholes formula calculates the fair value of the option and helps you to find out if call and put options are underpriced or overpriced. There is a very complex formula that Black & Scholes uses to value currency options. The formula for the option valuation is as under:

Most option trading terminals give you the automated calculation of call and put options using Black & Scholes. However, what is important is to understand the five factors underlying the model and that go into the above formula. There are 5 factors that impact the calculation of the fair valuation of an option…

First is the current market price (S) of the underlying USDINR contract. In case of call options, higher spot prices increase the value of the options. The reverse is true in case of put options.

Second is strike price (K) of the USDINR option. In case of a call option, a higher strike price reduces the fair value of the option. In case of a put option, the reverse is true.

Third is volatility (s or standard deviation) of the underlying. Higher the volatility higher is the value of the option. That is because; if the volatility is in your favour then the option is more valuable and if the volatility is against you then you just lost the premium. Higher volatility is positive for call options and for put options.

Fourth is time to expiry (t) of the option. Greater the time to expiry, more the probability of that you will make money on the option. Longer time to expiry is positive for call options and also for put options. Time to expiry is linked to volatility and expectations of volatility.

Lastly, the risk free interest in the market also matters. Why are interest rates relevant? Options strikes pertain to a future date and hence time value becomes material. Higher interest rates mean lower present value of the strike price and therefore higher option value in case of call options. The reverse is true in case of put options.

Having understood the broad contours of the factors that impact the Black and Scholes pricing let us also look at the underlying assumptions of the model. Following are some of the assumptions on which Black-Scholes model is based.

- The short selling of particular foreign currency (original model have word of securities) is permitted
- The trader has to incur no transaction costs and to pay taxes whenever they do the rebalancing of their portfolio
- In case of currency option, the foreign currency does not provide any type of regular income during particular time span of option
- The model behaves in such a way so that trader will not have any arbitrage opportunity i.e. gain or loss emerges out
- In case of currency option, the foreign currency trading is continuous
- The risk-free rate is ‘r’ and is consistent for all maturities period. The time period do not have any impact on the r
- The model is applied on the European option

It is often said that some of the assumptions of the Black & Scholes option pricing model may be a tad impractical. However, this model still gives the best approximation of the real market conditions. That is where the utility of the model comes. The theoretical value thrown up by the model has to be ratified by practical factors to get the real picture.