3.6 Financial derivatives and option pricing

Financial derivatives are contracts whose value comes from an underlying asset like a stock, bond, or commodity. They're central to actuarial work because they provide the tools for pricing risk, hedging liabilities, and building the quantitative models you'll encounter throughout financial mathematics.

This section covers derivative types, the major pricing models (Black-Scholes, binomial, Monte Carlo), option sensitivities (the "Greeks"), trading strategies, exotic options, credit derivatives, and the risk management and regulatory frameworks around them.

Types of financial derivatives

A financial derivative is a contract whose value derives from an underlying asset such as a stock, bond, commodity, currency, interest rate, or market index. Three main types show up repeatedly:

• Options give the holder the right but not the obligation to buy (call) or sell (put) an asset at a predetermined strike price by a specified date.
• Futures are standardized exchange-traded contracts that obligate both parties to transact an asset at a future date and price.
• Forwards work like futures but are traded over-the-counter (OTC), meaning they're privately negotiated and customizable.
• Swaps involve two parties exchanging cash flows based on a notional principal amount, such as swapping fixed-rate payments for floating-rate payments.

The key distinction between options and futures/forwards: options give you a choice, while futures and forwards lock you in.

Underlying assets of derivatives

Derivatives can be written on a wide variety of underlyings, and the risk-return profile of that underlying directly affects the derivative's value.

• Equity derivatives are based on stocks or stock indexes (e.g., stock options, index futures, total return swaps).
• Fixed income derivatives reference bonds, interest rates, or credit products (e.g., interest rate swaps, bond futures, credit default swaps).
• Commodity derivatives use physical commodities like crude oil or gold, or commodity indexes, as the underlying.
• Foreign exchange derivatives are based on currency exchange rates and include FX forwards, futures, options, and swaps.
• Other underlyings include volatility indexes (like the VIX), inflation rates, weather conditions, and cryptocurrency prices.

Derivative markets and exchanges

Derivatives trade in two main venues, and the distinction matters for counterparty risk analysis:

• Exchange-traded derivatives (futures, listed options) are standardized, centrally cleared, and offer price transparency. Central clearing greatly reduces counterparty risk because the clearinghouse stands between buyer and seller. Major exchanges include the Chicago Mercantile Exchange (CME), Intercontinental Exchange (ICE), Eurex, and the Japan Exchange Group (JPX).
• OTC derivatives are customized contracts negotiated bilaterally between counterparties, typically large financial institutions or corporations. The OTC market is significantly larger by notional value but carries additional counterparty risk and liquidity risk because there's no central clearinghouse guaranteeing performance (unless voluntarily cleared).

Pricing models for derivatives

Black-Scholes option pricing model

The Black-Scholes model is the foundational formula for pricing European-style options. It takes five inputs:

1. Current underlying asset price ($S$)
2. Strike price ($K$)
3. Time to expiration ($T$)
4. Risk-free interest rate ($r$)
5. Implied volatility ($\sigma$)

The Black-Scholes formula for a European call is:

$C = S \cdot N(d_1) - K e^{-rT} \cdot N(d_2)$

where:

$d_1 = \frac{\ln(S/K) + (r + \sigma^2/2)T}{\sigma\sqrt{T}}, \quad d_2 = d_1 - \sigma\sqrt{T}$

and $N(\cdot)$ is the standard normal cumulative distribution function.

Key assumptions to remember: the underlying follows geometric Brownian motion with constant drift and volatility, markets are frictionless (no transaction costs or taxes), and the option can only be exercised at expiration (European-style). Violations of these assumptions are why market prices can diverge from Black-Scholes values.

Binomial option pricing model

The binomial model uses a discrete-time lattice to model possible price paths. It's more flexible than Black-Scholes and can price American-style options (which allow early exercise).

How it works:

1. Divide the option's life into $n$ time steps.

2. At each step, the price moves up by factor $u$ or down by factor $d$.

3. Calculate the risk-neutral probability $p = \frac{e^{r\Delta t} - d}{u - d}$ of an up move.

4. Build the tree forward to get all possible prices at expiration.

5. Calculate the option payoff at each terminal node.

6. Work backwards through the tree, discounting expected payoffs at the risk-free rate at each node.

As the number of time steps increases, the binomial model converges to the Black-Scholes price for European options.

Monte Carlo simulation for option pricing

Monte Carlo methods use repeated random sampling to simulate thousands of possible price paths for the underlying asset, then estimate the option price as the average discounted payoff across all paths.

This approach is especially useful for:

• Path-dependent exotic options (like Asian or lookback options)
• Options with multiple underlying assets
• Complex payout structures where closed-form solutions don't exist

The tradeoff is computational cost. Accuracy improves with more simulated paths, but each additional path takes processing time. Variance reduction techniques (antithetic variates, control variates) help improve efficiency.

Option price sensitivities

The "Greeks" measure how an option's price responds to changes in different market variables. They're essential for risk management because they tell you exactly where your exposures are.

Delta and gamma

Delta ($\Delta$) measures the option's price sensitivity to a small change in the underlying price. Think of it as the slope of the option price curve.

• Call options: $0 < \Delta < 1$
• Put options: $-1 < \Delta < 0$
• An at-the-money call has a delta near 0.5, meaning a $1 move in the stock changes the option price by roughly $0.50.

Gamma ($\Gamma$) measures the rate of change of delta with respect to the underlying price. It's the curvature (convexity) of the option price curve. High gamma means delta is changing rapidly, which makes hedging more difficult. Gamma is highest for at-the-money options near expiration.

Vega and theta

Vega ($\nu$) measures sensitivity to changes in implied volatility. Long options have positive vega (they benefit from rising volatility), while short options have negative vega. Vega is highest for at-the-money options with longer times to expiration.

Theta ($\Theta$) measures time decay, the rate at which an option loses value as expiration approaches, all else equal. Long options have negative theta (they bleed value over time), while short options have positive theta (time decay works in your favor). Theta accelerates as expiration nears, especially for at-the-money options.

Rho sensitivity

Rho ($\rho$) measures sensitivity to changes in the risk-free interest rate.

• Call options have positive rho: higher rates reduce the present value of the strike price, making calls more valuable.
• Put options have negative rho: higher rates make puts less valuable for the same reason.

Rho is typically the smallest of the Greeks and only becomes significant for options with long times to expiration.

Option trading strategies

Covered calls vs protective puts

• Covered call: You own the underlying stock and sell a call option against it. You collect the premium, which provides income and a small downside cushion, but you cap your upside at the strike price. This strategy reflects a neutral-to-mildly-bullish view.
• Protective put: You own the underlying stock and buy a put option. The put establishes a floor on your losses (at the strike price minus the premium paid), acting like insurance. You keep full upside participation minus the cost of the put.

Bull spreads vs bear spreads

• Bull call spread: Buy a lower-strike call and sell a higher-strike call with the same expiration. You profit if the underlying rises moderately. Both your maximum gain and maximum loss are capped.
• Bear put spread: Buy a higher-strike put and sell a lower-strike put. You profit if the underlying falls moderately. Again, both risk and reward are limited.

Both spreads are cheaper than outright option positions because the sold option partially offsets the cost of the bought option.

Straddles vs strangles

• Long straddle: Buy a call and a put at the same strike and expiration. You profit from large price moves in either direction. The risk is that if the underlying stays near the strike, both options lose value to time decay and you lose the combined premium.
• Long strangle: Buy an out-of-the-money call and an out-of-the-money put. Cheaper than a straddle, but the underlying needs to move further before you profit.
• Short straddles/strangles are the opposite: you collect premium and profit from stable prices, but face potentially unlimited losses if the underlying moves sharply.

Butterfly spreads vs condor spreads

• Butterfly spread: Uses three strike prices. You buy one option at the lowest strike, sell two at the middle strike, and buy one at the highest strike (all equidistant). Maximum profit occurs if the underlying is exactly at the middle strike at expiration.
• Condor spread: Similar to a butterfly but uses four different strikes, creating a wider range of profitability. You sacrifice some maximum profit for a broader "sweet spot."

Both strategies profit from stable prices and have limited risk and limited reward.

Hedging with options

Delta hedging techniques

Delta hedging creates a position that is insensitive to small moves in the underlying price by making the portfolio delta-neutral (total delta = 0).

Steps for delta hedging:

1. Calculate the delta of your option position.
2. Take an offsetting position in the underlying asset. For example, if you're short a call with delta 0.6 on 100 shares, buy 60 shares to neutralize the delta.
3. Rebalance as the underlying price moves, because delta itself changes (that's gamma at work).

The main limitation: delta hedging only works for small price moves. For larger moves, gamma causes delta to shift, requiring continuous rebalancing. In practice, you rebalance at discrete intervals, which introduces hedging error.

Gamma hedging strategies

Gamma hedging aims to make a portfolio gamma-neutral so that delta stays more stable across larger price moves.

• You can't gamma-hedge with the underlying asset alone (the underlying has zero gamma). You need to use other options.
• Typically, you add an option position with opposite gamma to your existing position, then delta-hedge the combined portfolio.
• Gamma hedging reduces the frequency of rebalancing needed but adds complexity and cost.

Exotic options and pricing

Asian options vs barrier options

Asian options have payoffs based on the average price of the underlying over a specified period, rather than the price at a single point. Because averaging smooths out price spikes, Asian options are less susceptible to manipulation near expiration and are cheaper than equivalent vanilla options.

Barrier options have payoffs that depend on whether the underlying hits a specified price level (the barrier) during the option's life:

• Knock-in options only become active if the barrier is reached.
• Knock-out options become worthless if the barrier is reached.

Barrier options are cheaper than vanilla options because of the additional condition that must be met (or avoided).

Lookback options vs forward-start options

• Lookback options let the holder buy (call) or sell (put) at the most favorable price observed during the option's life. They're path-dependent and more expensive than standard options because they eliminate the risk of poor timing.
• Forward-start options have a strike price that will be set at a future date, typically based on the underlying price at that time. They allow investors to lock in future volatility exposure without committing to a specific strike now. These are common in employee stock option plans.

Quanto options and foreign exchange risk

Quanto options are cross-currency derivatives where the payoff is determined by a foreign asset but settled in the investor's domestic currency at a fixed exchange rate. This eliminates foreign exchange risk while maintaining exposure to the foreign asset's price movements.

Pricing quantos is more complex than standard options because it requires modeling the correlation between the underlying asset price and the exchange rate.

Credit derivatives and swaps

Credit default swaps (CDS)

A credit default swap is essentially insurance against a credit event on a reference entity (like a corporate bond issuer).

• The protection buyer makes periodic premium payments (the CDS spread) to the protection seller.
• If a credit event occurs (bankruptcy, failure to pay, or restructuring), the seller compensates the buyer, typically paying the difference between par value and the recovery value of the reference debt.
• The CDS spread reflects the market's perception of the reference entity's creditworthiness. A widening spread signals deteriorating credit quality.

CDS can be used to hedge actual credit exposure or to speculate on credit quality changes.

Total return swaps (TRS)

In a total return swap, one party pays a set rate (reference rate plus a spread) while the other pays the total return (income plus capital gains/losses) of an underlying asset such as an equity index, loan, or bond.

The TRS receiver gains economic exposure to the underlying asset without actually owning it. This makes TRS useful for leverage, financing, and gaining exposure to assets that may be difficult to purchase directly. Counterparty risk is a significant concern.

Credit-linked notes (CLNs)

A credit-linked note combines a credit derivative with a fixed income security into a single instrument. The CLN pays periodic coupons and principal at maturity, but these payments are contingent on the credit performance of a reference entity or portfolio.

If a credit event occurs, the investor may receive the defaulted debt or suffer a loss of principal. CLNs allow investors to take on credit risk exposure in a funded, tradeable format.

Risk management with derivatives

Value-at-Risk (VaR) for derivatives

Value-at-Risk (VaR) estimates the maximum potential loss on a portfolio over a specified time horizon at a given confidence level (e.g., "the 1-day 99% VaR is $5 million" means there's a 1% chance of losing more than $5 million in a day under normal conditions).

Three main calculation methods:

1. Historical simulation: Uses actual past returns to build a loss distribution.
2. Monte Carlo simulation: Generates random scenarios based on assumed distributions and correlations.
3. Parametric (variance-covariance): Assumes returns follow a normal distribution and uses portfolio variance to estimate VaR.

Limitations: VaR assumes normal market conditions, focuses on a single quantile, and tells you nothing about how bad losses could be beyond the VaR threshold.

Expected shortfall and tail risk

Expected shortfall (ES), also called Conditional VaR (CVaR), answers the question VaR leaves open: given that you've exceeded the VaR threshold, what's the average loss?

ES is a coherent risk measure, meaning it satisfies desirable mathematical properties including subadditivity (the risk of a combined portfolio is no greater than the sum of individual risks). VaR does not always satisfy this property.

Derivatives with non-linear payoffs and high gamma can generate significant tail risks that VaR alone may underestimate. ES captures these better.

Stress testing derivative portfolios

Stress testing subjects a portfolio to extreme hypothetical scenarios to assess potential losses beyond what VaR captures.

• Historical scenarios replicate past stress events (e.g., the 2008 financial crisis, the 2020 COVID crash).
• Hypothetical scenarios simulate plausible but unprecedented shocks (e.g., a sudden 200 basis point rate spike combined with a 30% equity drop).
• Reverse stress testing works backwards: it identifies which scenarios would cause a specified level of loss, helping reveal hidden vulnerabilities.

Stress tests should cover all major risk factors relevant to the derivatives held, including price, volatility, interest rates, credit spreads, and correlations.

Regulation of derivative markets

Dodd-Frank Act and derivatives

The Dodd-Frank Act (2010) was the U.S. legislative response to the 2008 financial crisis. Its key provisions for derivatives markets:

• Mandated central clearing for standardized OTC derivatives to reduce counterparty risk.
• Imposed margin requirements for non-cleared swaps.
• Required trade reporting to swap data repositories to increase transparency.
• Established the Volcker Rule, prohibiting banks from proprietary trading and restricting investments in hedge funds and private equity.
• Created the Financial Stability Oversight Council (FSOC) to monitor systemic risks and the Consumer Financial Protection Bureau (CFPB).

European Market Infrastructure Regulation (EMIR)

EMIR (2012) is the EU's equivalent regulatory framework for OTC derivatives. Its requirements closely parallel Dodd-Frank:

• Mandatory clearing for eligible OTC derivatives through central counterparties (CCPs).
• Margin and capital requirements for non-cleared derivatives.
• Reporting of all derivative contracts to trade repositories.
• Risk management, governance, and operational resilience standards for CCPs.

The overarching goals of both Dodd-Frank and EMIR are to reduce systemic risk, increase market transparency, and protect against the kind of cascading counterparty failures seen in 2008.