Finance
Bonds and options
Bond prices, yields, duration and convexity; option values, Greeks and implied volatility.
How fixed-income and derivative prices respond to rates, time and volatility.
Bonds
A bond’s price is its coupons and principal discounted at the market yield. Duration says how far the price falls when yields rise; convexity says how much that straight-line estimate misses.
—price
- Current yield
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- Macaulay duration
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- Modified duration
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- Convexity
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The straight line is what duration alone predicts. The gap to the curve is convexity: prices rise more when yields fall than they drop when yields rise.
- Priced on a coupon date with no accrued interest; between coupon dates, quoted (clean) prices exclude the interest built up since the last coupon.
- Duration and convexity assume every yield on the curve moves by the same amount.
Options
The Black–Scholes–Merton value of a European call or put, with the Greeks: how the value moves with the share price, time, volatility and interest rates. Add a market price to back out the volatility it implies.
—call value
- Delta
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- Gamma
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- Vega, per vol point
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- Theta, per day
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- Rho, per rate point
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- Black–Scholes assumes constant volatility and rates, and prices European options (exercised only at expiry). Real markets show a volatility “smile”, which is why implied volatility differs by strike.
- The chance of finishing in the money is the risk-neutral N(d₂), not a forecast of what will happen.
Sources
- Macaulay, F. R. (1938). Some Theoretical Problems Suggested by the Movements of Interest Rates, Bond Yields and Stock Prices in the United States since 1856. NBER. (Duration.)
- Black, F. & Scholes, M. (1973). The pricing of options and corporate liabilities. Journal of Political Economy, 81(3), 637–654; Merton, R. C. (1973). Theory of rational option pricing. Bell Journal of Economics, 4(1), 141–183. (Dividend yield.)