Financial Derivatives: A Quantitative Finance View

Explore how interest rates arise as market negotiated prices in the money and bond markets, influenced by risk, central bank policy, and the term structure, with the sofr rate.

Adopt an investor’s view of interest rates and learn how annual compounding drives future value through the formula fv equals (1 + r)^t times p0.

Explains how compounding frequency affects the future value of investments and derives the general formula for m times per year. Uses semiannual and monthly examples to illustrate.

Explore standard investment return measures, linking interest rates to gross and net returns, and compare annual versus monthly compounding to show how compounding affects the final value and return.

Relate interest rates with different compounding by equating future value of a dollar, deriving a conversion between compounding frequencies, and applying it to five percent annual to semiannual.

Master continuous compounding, where interest accrues instantly and future value follows the e^(rt) growth factor. Use logarithms to derive the continuously compounded rate equivalent to an annual rate with examples.

Explore time value of money, showing why a dollar today is worth more than a dollar in future due to opportunity cost, and how this concept underpins borrowing and interest.

Calculate the present value of future payments by applying time value of money and present value formulas, with annual, periodic, and continuous compounding examples.

Learn how discount factors express the time value of money and derive present and future values under annual and continuous compounding, with a focus on risk-free benchmarks.

Explore discounted cash flow analysis and the present value of a stream of cash flows using discount factors, with annual and continuous compounding examples.

Explore yield to maturity, the key bond yield measure that discounts coupons and principal to the market price. Learn about semiannual compounding and route finding to compute it.

Explore simple interest and day count conventions, including actual over 360 and actual over 365, with practical overnight loan calculations and how they affect total value.

Examine Libor as a benchmark for interbank borrowing and its trimmed-mean calculation. Learn why regulators are phasing out Libor and how Euribor Tibor and alternative reference rates will replace it.

Explore how the fed funds rate guides monetary policy under the Federal Reserve. The FOMC sets target levels and steers the rate with open market operations amid post-crisis evolution.

Explore Sonia, sterling overnight index average, risk-free rate replacing LIBOR in sterling markets. It's built from real transactions, has no term structure, and uses compounded in arrears for term rates.

Explore the secured overnight financing rate (SOFR) as the USD LIBOR replacement, derived from overnight treasury repo data, and its compounding options and credit-sensitive alternatives.

Explore yield curves as the central concept linking interest rates, term structure, and fixed income, and learn how discount curves relate to spot rate curves via discount factors.

Continue yield-curve construction by linearly interpolating Y(1) and Y(2) to derive D(1.5). Price the two-year bond using these discount factors with semiannual 7% coupons, yielding 150,224 dollars.

Adopt the default interest rate assumptions: risk-free rate, continuous compounding, and constant deterministic rates with a flat term structure. Bonds and swaps may relax these assumptions.

Explore how commodity prices move as products of supply and demand in agricultural, metals, and energy markets; cash and derivative markets shape behavior. Highlight inventory theory and convenience yields.

Model portfolios as asset collections and compute value as allocations times base-unit values. Include cash and debt via the risk-free rate and address negative portfolio value.

This lecture analyzes foreign currencies as a portfolio asset, defines exchange rates as prices, and shows how to value foreign cash using domestic and foreign risk-free rates.

Model dividends and convenience yields as continuously compounded returns, represent stocks and commodities as time-dependent allocations, and account for storage costs in pricing forwards and futures.

Learn long and short positions as portfolio exposures, including buying to go long and selling to go short, with cash market and derivatives across assets.

Define arbitrage as a riskless profit and explain its central role in derivative pricing. Show how the no-arbitrage axiom equates prices to maintain equilibrium.

Demonstrates a bond arbitrage under a non-trivial term structure: borrow to buy the underpriced bond and invest leftovers for riskless profit.

Explore extensions of the law of one price, including cash flow and inequality forms, and use replication arguments with portfolios and zero coupon bonds to price assets and identify arbitrage.

Explore how arbitrage justifies time value of money, present value, and discounted cash flow analysis by linking law of one price to replication of cash flows with zero coupon bonds.

Examine forward payoffs for long and short positions at expiration, and plot payoff curves S(T) minus K and K minus S(T) to understand forward value.

Apply the law of one price to value forwards. Build a replicating portfolio using the underlying asset and a debt to derive the forward value and forward price.

Show how cash and carry and reverse cash and carry arbitrage determine the forward price, ruling out inequalities and highlighting the cost of carry.

Calculate the forward price for a six-month forward on a zero-coupon bond, compare offers, and exploit arbitrage through long or short positions to secure risk-free profit.

Derives the fair forward price for a non dividend paying stock using the risk-free rate, and analyzes long and short forward positions as prices move to 130 or 170.

Derive the forward price for income-bearing assets using cash-and-carry arbitrage: F0 = (S0 − I) e^{rT}, with I the present value of the income.

Derive the forward contract value for income-generating assets using arbitrage and present value, showing how the fair forward price ensures zero initial value.

Derive and apply forward pricing for income-bearing assets with a known dividend yield, showing the forward price as e^{(r - y)T} S0 and a replicating portfolio.

Compute forward price for a dividend-paying stock by discounting dividends and applying forward formula, then analyze how the short forward's value changes as the stock moves to 220 or 270.

Understand fx forwards, which lock in a future exchange rate, and apply interest rate parity and cash-and-carry arbitrage to derive forward prices using domestic and foreign risk-free rates.

Explore FX forward pricing and arbitrage by calculating fair forward rates from spot rates and risk-free rates, and applying long and short hedging strategies on EURUSD and AUDUSD examples.

Compare forwards and futures, highlighting exchange trading and standardization. Explain how defined delivery details—underlying asset, contract size, delivery month, and locations—link futures to spot prices for risk management and speculation.

Explain how futures prices reflect market expectations and arbitrage, accounting for risk premium, and how they differ from forward prices, with daily marking to market and contract size effects.

Understand marking to market for futures with daily settlement for long and short positions, considering contract size and number of contracts, illustrated by a silver example.

Explore how futures margin accounts secure daily settlement by requiring initial and maintenance margins, triggering a margin call, and using collateral to cover gains and losses.

Explore how futures prices relate to spot prices, compare them with forward contracts, and how marking to market daily cash flows, arbitrage, and cost of carry support risk management.

As futures approach expiration, futures prices converge to the spot price, with equalization in the delivery month supported by straightforward arbitrage; robust delivery mechanics sustain this link.

The lecture shows that with zero cost of carry, futures deliver the same exposure to the cash price as an outright position, with marking to market reflecting price moves.

Hedge preexisting portfolio risks with futures by adding long or short futures to offset cash exposure, achieving a unitary hedge under zero cost of carry.

Futures hedging for an oil refiner uses Nimetz WTI futures to take long position and cover a short cash position, yielding a perfect hedge as spot moves 55 to 60.

Explore how non-zero cost of carry creates a basis between spot and futures prices, and how a unitary futures hedge converts price risk into basis risk.

Explore how a unitary futures hedge protects a 1 million euros cash position from euro-dollar moves, using eight December futures contracts and basis changes from -0.03 to -0.01 per euro.

A copper miner hedges price risk using December copper futures, shorting 900 contracts to cover 10,000 tonnes, showing how futures hedges dampen gains when cash prices rise.

Use futures to speculate on crude oil by gaining exposure with smaller margins and high liquidity, while avoiding storage, insurance, and transport costs of cash market exposure.

Compare cash versus futures exposure to a 25 million yen position, noting futures use low margins for high apparent returns yet entail significant risk of large losses.

Explore the LIBOR curve as a yield curve built from short term LIBOR rates and derivatives. Learn how to bootstrap discount factors from Eurodollar futures and LIBOR swaps through arbitrage.

Explore forward rates and their link to spot rates, using arbitrage with zero coupon bonds to derive forward prices and forward interest rates, including a worked LIBOR example.

Forward rate agreements are over-the-counter, cash-settled contracts that lock in a future interest rate and exchange the two interest payments with no actual borrowing.

Examine eurodollar futures, a hedge or speculation tool tied to three-month LIBOR, where price equals 100 minus the implied rate and every basis point moves $25 daily.

Learn how swaps exchange fixed and floating cash flows on notional value, focusing on interest rate swaps and variants, and note OTC to central clearing shifts and counterparty risk.

Valuate swaps by separately pricing the fixed and floating legs using discount factors and the swap rate. Learn replication with zero-coupon bonds and derive the fair swap rate formula.

Calculate the fair swap rate for a one-year, four-quarter interest rate swap with a 500,000 notional using the provided discount factors, then evaluate its value after a rate update.

Swaps Corp uses a one-year five-million notional swap to convert floating LIBOR to fixed payments, hedging against rising rates. The example compares swap versus no-swap and shows a saving.

Learn to bootstrap a LIBOR curve from money market instruments, using cash LIBOR, Eurodollar futures, and swaps to derive discount factors and spot rates across tenors.

Bootstrap the LIBOR curve for the short end by converting one-, two-, and three-month rates to decimals, computing five-digit discount factors, and deriving corresponding spot rates.

Build the LIBOR curve's middle range with eurodollar futures to extend discount factors from 3 to 18 months by substituting futures rates for forward rates.

Extend the LIBOR curve from 18 months to four years by deriving discount factors from two- and four-year swap rates, addressing gaps with interpolation, and completing the curve.

Explore how stochastic processes model the random evolution of asset prices, contrasting linear forwards and futures with nonlinear options, and develop intuition through random price paths.

Define a stochastic process as a family of random variables indexed by time. Differentiate discrete and continuous time, sample paths, and Brownian motion through joint distributions and independence.

Explore time series statistics by defining strict and weak stationarity, and learn how auto covariance function and autocorrelation function depend on lag, with practical sample estimates.

Compute daily and five-day continuously compounded returns, link them to logarithmic differences, and express K-period returns as a telescoping sum of one-period returns.

Explore the stylized facts of asset prices, including volatility regimes, near-zero autocorrelation of raw returns, and fat-tailed distributions across S&P 500, the pound sterling USD, and soybeans using log returns.

Study volatility clustering by linking price volatility to return volatility, using returns to identify distinct volatility regimes in real assets like the S&P 500, pound USD, and soybeans.

Explore the random walk as a discrete-time stochastic process and its use as a model for asset prices, with independent and identically distributed up or down steps.

Uncover the probability distribution of a random walk after n steps, showing S_n relates to a binomial distribution via a Bernoulli mapping and independence, with explicit even-step formulas.

Examine how random walks model asset price evolution and sample paths, compare discrete versus continuous time, and evaluate fixed-jump limitations; show how increasing time steps improves realism.

Explore criticisms of the random walk model for asset prices and present a case for a modified random walk that preserves independence and market efficiency while enabling sensible jump sizes.

Explore how scaling a random walk yields Brownian motion as a continuous-time stochastic process, guided by the central limit theorem, variance control, and independent increments for asset price modeling.

Explore how Brownian motion with drift adds a linear trend and volatility to model asset prices, demonstrated by the S&P 500 data and its comparison to a random walk.

Assess how Brownian motion improves asset price modeling with continuous time, independent increments, and normally distributed jumps, while acknowledging unresolved issues like negative values and level-insensitive volatility.

Model asset prices using the log normal model, i.e., geometric Brownian motion, where log prices follow Brownian motion, producing multiplicative updates and stationary, uncorrelated returns.

Explore the log-normal model’s treatment of asset prices, ensuring positivity and proportional jump sizes, while highlighting its failure to reproduce volatility clustering and fat tails.

Options give the right, not the obligation, to buy or sell an asset at a strike price by expiration, with calls and puts, premiums, and european versus american exercise.

Explore the payoffs of European options, including long and short call and put positions, and visualize the hockey-stick payoff functions and their relation to option value at expiration.

The lecture demonstrates how arbitrage and the law of one price bound call and put prices by comparing their payoffs to the underlying asset and to forward contracts.

Derive arbitrage-based bounds on option prices using the law of one price and payoff geometry, and justify them with direct arbitrage arguments for calls, puts, and forward positions.

Assume the call price exceeds the underlying price, short the call, buy the stock, and invest the premium; at expiration, profits arise in both outcomes, so C ≤ S.

Justify the third arbitrage inequality: the call price is at least the stock price minus the discounted strike. Use a no-arbitrage argument with short stock, long call, and cash investment.

Learn how nonnegative option premiums strengthen lower bounds for call and put prices, and how arbitrage opportunities arise when a call exceeds the underlying, with an example.

Analyze arbitrage bounds for American options on non dividend-paying stocks, show American calls equal European calls, and explain why early exercise is never optimal, with bounds for puts.

Explore how put-call parity links long call and short put payoffs with a long forward, using same strike to show arbitrage-free pricing and the forward payoff.

Derive put-call parity via direct arbitrage by linking a long put and short call to a forward with a common strike, using risk-free discounting and stock price S(T).

Explore the one-step binomial model for asset prices and derivative pricing, using arbitrage and a risk-free asset to determine a call's fair value today via delta hedging.

Use delta hedging in the one-step binomial model to price a derivative by forming a riskless delta stock position and short derivative, then discount the payoff.

Use risk neutral pricing to price derivatives in multistep binomial models, expressing time zero value as the risk neutral expectation of discounted time one values.

Explore risk neutral pricing in a one-step binomial model by valuing a call with strike 55 as the stock moves to 65 or 40 from 50, using p*=0.56 and q*=0.44.

Extend the one-step binomial model to a two-step framework with up and down factors, and price derivatives via risk-neutral arbitrage; the example yields 5.478 at time zero.

Derive the risk-neutral distribution of asset prices in a two-step binomial model by pricing derivatives as discounted risk-neutral expectations. Confirm results align with prior binomial pricing.

Extend the binomial model from two steps to three steps and beyond, showing stock prices and european derivatives follow up and down factors with risk-neutral pricing.

Derive the call premium in the binomial model via risk-neutral, discounted payoff expectation, and relate the result to the Black-Scholes framework.

Bridge discrete binomial models to continuous time log-normal asset prices, showing binomial option pricing converges to the Black-Scholes framework under risk-neutral dynamics.

Derive the Black-Scholes formula by linking a binomial model to a log normal asset process, pricing a call as a risk-neutral discounted payoff and evaluating the resulting integral.

Derive and apply the Black-Scholes formula for call and put prices, explore drift independence, implied volatility, and practical hedging using option Greeks.

Define the greeks as the sensitivities of option prices to risk factors, derived as partial derivatives, with delta, gamma, and vega computed from Black-Scholes for hedging and risk management.

See how theta, the derivative of option price with time, drives time decay for calls and puts in the Black-Scholes framework. Compare theta to re-pricing to observe time decay dynamics.

Explore dynamic hedging by constructing a delta-neutral portfolio using the option's delta and short stock, illustrated with a Black-Scholes based example that shows hedging adapts as prices move.

Explore how options provide exposure to volatility, using the Black-Scholes framework to show that call prices rise with volatility, enabling long or short volatility strategies with delta hedging.

Explore how Black-Scholes implied volatility is derived from market option prices, revealing the one-to-one relationship with price, and how implied volatility surfaces, smiles, and skews inform hedging and Greeks.