Based on [1] and [2].

This is about financial instruments. However, let's start with assets. An asset is anything, tangible (like real estate) or intangible (like a patent) of value that can be owned or controlled. Individuals, businesses and governments use assets to generate economic benefits.

A financial instrument is a specific type of asset. It represents a legal agreement involving monetary value of any kind. They can be traded or exchanged. Stocks, bonds, and derivatives are examples. As with many things, also financial instruments can be categorized based on various aspects.

• Structure and purpose (haven't come up with a better expression for this): cash | derivative
• Purpose and function: money market | capital market
• Asset class: fixed income | foreign exchange | equities | commodities | real estate | interest rates (I consider them equivalent to an asset class, whereas they may typically not be considered directly as such, but in any case a key factor influencing the other asset classes) | ...
• Trading venue: over the counter (OTC) | exchange traded
• ...

Cash instruments are financial assets whose value is directly influenced by market conditions. The primary purpose of cash instruments is to facilitate the efficient flow and transfer of capital, and they are usually used for straightforward transactions and investments. Examples of cash instruments are securities and loans. Securities, such as stocks and bonds, represent ownership and debt, respectively. Loans and Deposits are agreements between borrowers and lenders.

Derivative instruments derive their value from an underlying asset, such as stocks, bonds, commodities, currencies, interest rates, or market indexes. They are used for several purposes: hedging, speculation, and leverage. They are a corner stone of financial risk management. Examples are futures contracts, forwards, options, and swaps. Through financial engineering, derivatives allow for more complex financial strategies and can help manage risk more effectively.

Purpose of money markets is to facilitate short-term borrowing and lending, typically for periods of one year or less. They provide liquidity, allowing entities to manage their short-term cash needs efficiently. This helps maintain stability and ensures that businesses and governments can meet their operational expenses. Participants are governments, corporations, banks, and financial institutions. Examples of money market instruments are treasury bills, commercial papers, and certificates of deposit.

Capital markets enable long-term investment and financing, typically for periods longer than one year. They facilitate raising capital for growth and development projects. Companies issue stocks and bonds to fund expansion, while investors buy these securities to potentially earn higher returns over time. Participants are companies, governments, and individual investors. Examples of capital market instruments are stocks, bonds, and other long-term securities.

The introduced non-exhaustive views above overlap. Shares are equity assets, and they are also securities. Bonds are also securities, and they are fixed income assets.

The payoff function of derivatives is either linear or not, and hence the derivative either a linear or a non-linear instrument. A linear instrument can be hedged (statically) once, and the hedge will work until its maturity. The hedge of a non-linear instrument needs frequent adjustment to be effective.

Staying with derivatives...

The general situation with derivatives

We consider a derivative \(C\) on an underlying \(S_t\) which pays function \(f(S_T)\) at time \(t=T\). Further, we consider \(f\) piecewise smooth, meaning that except at a finite number of points it is an infinitely differentiable function. This latter consideration is convenient for pricing and risk management of the derivative. But the good point is that this convenient assumption is true of all market instruments.

A prerequisite

The pricing of all interest rate derivatives assumes and relies on a continuum of zero-coupon bonds which can be freely bought and sold, including short-selling as necessary. With this framework, these bonds provide discount factors for present-valuing of cashflows and hence pricing or valuing of the instruments. The price \(P(T)\) of a zero-coupon bond with a notional of one and maturity \(T\) is the discount factor for the period from now or spot to \(T\).

A simple derivative

One of the simple interest rate derivative instruments is a forward rate agreement (FRA). An interest rate swap is another example.

The relationship between zero-coupon rates, forward rates, and swap rates is something to consider, and the intuition of this relationship becomes apparent after understanding the building block instruments and their respective valuations.

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Container Option setting the reference rate, fixed or floating, above (cap) or below (floor) which the buyer / holder receives payments, calculated based on cap rate or floor rate and a notional.

Container, because, usually, many reference periods (for payments) are addressed, and each reference period is a caplet or floorlet and has, among other things, a payment date associated with it. And in case the reference rate is floating, usually 3 or 6 months, each caplet or floorlet also has a fixing date.

Understandably, each caplet is a European call option, and each floorlet is a European put option.

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Underlying is an interest rate paying instrument, and therefore it gives exposure to changes in interest rates.

The exposure is firstly in its price or value, and secondly in terms of the interest rate of the underlying.

Short- term interest rate (STIR) futures have (often) a 3-month interest rate security as underlying.

Long-term interest rate futures are bond futures.

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Two ways to quote a bond's price:

• Dollar value
• Yield

Primary market

Secondary market

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Based on [1], [2], and [3].

From a retailer's perspective, options can be considered directional instruments.

But pricing and hedging of options can be understood when considering the interbank perspective of market maker / dealer, in which options are instruments of volatility.

That is, for a retail investor owning a call option on an asset, a consistent upward move in the underlying may be perceived as good.

But a market maker owning the same call option may well prefer that the underlying price oscillates as much and as often as possible, and the more this happens, the more a long position (in a call or a put) will gain, and a short position (in a call or a put) will lose. This is so, because of the options gamma, the second derivative, and if it happens when the option is at-the-money, the gains of a long call or put position, and the losses of a short call or put, are maximized, given that gamma is maximized. The diagram below shows this.

Quote from an unknown source in [3]: "...the big potential profit from these trades is from gamma, in other words, large moves in the underlying rather than changes in implied vol."

Interest rate options

• Caps, floors, collars
• Swaptions
• Cancelable and extendible swaps
• Compound options
• Exotic options
• Embedded options

Exotic options

• Path-dependent: barrier, asian, lookback, cliquet, ladder
• Time-dependent: chooser, delayed or forward-starting
• Digital
• Multi-variate: rainbow, basket, spread, quanto

Option pricing models, such as Black-Scholes, assume that market prices of the underlying follow a random walk. However, because interest rates don't have this behavior of ever expanding, but rather periodically return and have a more confined range of values (for example, usually not exceeding, say, 10%), other models are better for pricing their options.

Note on pricing options vs linear derivatives and securities

For securities (such as bonds) and non-option derivatives (these are linear derivatives, such as FRA, forward, swap), the pricing is possible via construction of a static and riskless hedge to determine a fair price. This is so, because two things are given at maturity:

(a) a rigid relationship between the price of the derivative and prices of the underlying, and

(b) a definite procedure that takes place upon maturity.

For options, (a) is also given, but not (b).

That is, it's unclear whether the option will be exercised or not. This lack of certainty as to what will eventually happen on the maturity date makes options different compared to linear derivatives.

An option buyer has the right to decide later whether to exercise the option or not, and the time-value of the option is the value of this right.

Option pricing

Various approaches:

• Closed-form solution, such as the closed-form Black-Scholes formula (which is derived by applying a series of assumptions to the Black-Scholes PDE, which in turn is derived from the Black-Scholes model under consideration of arbitrage arguments)
• Binomial model
• Monte Carlo simulation
• Finite Difference method

No matter which, the same five inputs are needed:

1. Underlying asset price
2. Strike price
3. Time to option expiry
4. Carry (combination of interest rates and potential earnings on the underlying)
5. Volatility

Volatility is the only input that is not readily available. The others are either set in the contract, or observed in the market.

Option types

1. European: option holder can exercise at maturity only
2. Bermudan: at maturity, and at least one more date prior to that
3. American: any time up to maturity

How to view a European option

Based on [2, p29].