Key takeaways
– Parity price describes a price level at which two assets have the same economic value. It’s widely used in convertibles, options, commodities, and currency markets.
– For convertible bonds, parity reveals the break-even stock price at which converting into shares makes sense.
– In FX markets, parity concepts include purchasing-power parity (PPP) and interest-rate parity (IRP), which link exchange rates, prices, and interest rates.
– Put–call parity ties the fair values of calls, puts, the underlying stock, and the present value of the strike.
– “Percent of parity” and “parity price” are practical measures traders use to gauge premiums or discounts relative to conversion or intrinsic value.
1. What parity price means (conceptual overview)
Parity price is any price that makes two items equivalent in value. Examples:
– A convertible bond and the shares it can be converted into.
– A call option and its intrinsic value (when option price equals intrinsic value, it’s “at parity”).
– Two currencies at a one-to-one exchange rate (e.g., $1 = €1 would be parity).
– A commodity’s selling price compared to the historical or required price level for producers’ purchasing power.
2. Purchasing-power parity (PPP)
– Definition: PPP compares the price of a fixed “basket” of goods across countries after adjusting by the exchange rate. If prices differ, PPP suggests the exchange rate should adjust so goods cost the same across countries.
– Practical use: macroeconomic comparisons, long‑term exchange rate expectations, inflation-adjusted valuation of currencies.
– Limitations: ignores transport costs, tariffs, non-tradables, and short-term capital flows — useful more for long-run comparison than short-term trading.
3. Parity in commodities and agriculture
– Agricultural “parity price” historically referred to a price that preserves farmers’ purchasing power relative to some base period (e.g., 10-year average of earlier legislation).
– Modern use: commodity parity can be a benchmark for policy decisions or support purchases; for producers it helps assess whether market price covers costs (wages, interest, equipment).
4. Parity in foreign exchange markets
– Currency parity may mean a 1:1 exchange rate between two currencies.
– Related concept: purchasing-power parity (above) and interest-rate parity (next), both used to understand exchange-rate determination.
5. Interest-rate parity (IRP)
– Concept: Investors should not be able to get a riskless arbitrage profit by borrowing in one currency, converting and investing in another, and hedging the exchange risk with a forward contract.
– Covered interest-rate parity (CIRP) (no-arbitrage with forward cover):
Formula: F = S × (1 + i_dom) / (1 + i_for)
Where:
• F = forward exchange rate (quote: domestic currency per unit of foreign currency)
• S = spot exchange rate (domestic per foreign)
• i_dom = interest rate in domestic country
• i_for = interest rate in foreign country
(Careful with notation: define domestic/foreign clearly when applying.)
– Uncovered interest-rate parity (UIRP): replaces the forward rate with the expected future spot rate; it assumes expected returns equalize but does not use forward hedging — subject to risk premia and thus often fails in the short run.
– Practical arbitrage step (CIRP check):
1. Compute theoretical forward F_theory = S × (1 + i_dom)/(1 + i_for).
2. Compare with market forward F_market.
3. If significant deviation beyond transaction costs, construct covered arbitrage:
• Borrow in the currency with the lower after-cost borrowing rate.
• Convert at spot, invest in the other currency.
• Hedge by entering a forward contract to convert proceeds back at F_market.
• Lock in profit if F_market differs from F_theory enough to overcome costs.
6. Conversion parity for convertible bonds
– Definition: the implied stock price per share at which converting the bond into equity is break-even.
– Formula: Parity price per share = Market price of the convertible bond / Conversion ratio
• Conversion ratio = number of shares the bond converts into.
– Example:
• Convertible bond market price = $1,200.
• Conversion ratio = 20 shares.
• Parity price = 1,200 / 20 = $60 per share.
• If the underlying stock trades above $60, conversion is immediately favorable (ignoring other factors like time value, coupons, and liquidity).
– Practical steps when evaluating a convertible:
1. Compute conversion parity (above).
2. Compare parity to current stock price.
3. Consider other drivers: coupon yield, credit risk of issuer, time to maturity, call provisions, volatility (affects option component), and liquidity.
4. Calculate percent of parity (see section below).
7. Percent of parity (convertible bond metric)
– Common definition: Percent of parity measures how far the convertible is trading relative to its conversion value.
– Two equivalent ways to present:
• Percent of parity = (Market price of convertible) / (Conversion value) × 100
• Conversion value = conversion ratio × current stock price.
• If percent > 100%, convertible trades at a premium to its immediate conversion value.
• Alternate expression (implied stock parity): Percent of parity = (Stock price) / (Parity price) × 100
• Where Parity price = bond price / conversion ratio (so both formulas are algebraically consistent).
– Example:
• Bond price = $1,200; conversion ratio = 20; stock price = $55.
• Conversion value = 20 × 55 = $1,100.
• Percent of parity = 1,200 / 1,100 × 100 = 109.1% → bond trades ~9.1% premium to conversion value.
8. Parity in options trading
– Intrinsic parity: an option is “at parity” when its market price equals its intrinsic value (intrinsic value = max(0, S − K) for a call).
• Example: A call with strike $50 on stock trading at $60 has intrinsic value $10. If the option trades at $10, it’s at parity.
– Put–call parity (European options, no dividends assumed or include PV(dividends)/adjustments):
• Core equation: C + PV(K) = P + S
• C = price of a European call with strike K and maturity T
• P = price of the corresponding European put
• S = current spot price of the underlying
• PV(K) = present value of strike price K discounted at risk-free rate to maturity
• Rearranged: P = C + PV(K) − S (gives fair put price implied by call).
– Practical steps to use put–call parity:
1. Compute PV(K) using the appropriate risk-free rate and time to expiration.
2. Plug observed C, S, PV(K) to compute implied P.
3. Compare implied P to market P — if large divergence beyond costs, arbitrage strategies (buy underpriced, sell overpriced, hedge underlying, etc.) may exist. Note: dividends, early exercise (American options), transaction costs, and borrow constraints can break simple arbitrage in practice.
9. Risk parity (portfolio construction)
– Definition: A portfolio-construction approach that allocates risk budgets across asset classes (equities, credit, rates, commodities) rather than allocating capital by fixed weights.
– Basic idea: adjust leverages so each risk factor contributes equally to total portfolio volatility.
– Practical steps to implement a simplified risk-parity approach:
1. Choose asset classes and obtain historical or implied volatilities and correlations.
2. Estimate risk contributions for a candidate weight set (contribution = weight × asset volatility × correlation effects).
3. Solve for weights (and leverage) so each asset’s risk contribution is equal (or matches a target).
4. Adjust for constraints: liquidity, margin, risk limits, and rebalancing rules.
5. Monitor and rebalance periodically; consider dynamic vol-targeting or volatility scaling.
10. Covered vs. uncovered interest-rate parity — practical differences
– Covered IRP (CIRP): uses forward contracts to eliminate currency risk; typically holds in liquid markets due to arbitrage (ignoring costs).
– Uncovered IRP (UIRP): refers to expected future spot rates; because it involves risk (no hedge), it can fail in practice — investors demand risk premia.
– Practical implications:
• Use CIRP for no-arbitrage forward pricing and hedged cross‑currency strategies.
• Use UIRP only as a long-run expectation; for trading, account for risk premia and macro shocks.
11. Worked examples (quick calculations)
– Convertible parity example (revisited):
• Bond price = $1,200; conversion ratio = 20 ⇒ Parity = $1,200/20 = $60 per share.
• If stock = $65, conversion value = 20 × 65 = $1,300; converting gives immediate $100 gain vs bond price (ignoring bond coupons & other terms).
– Put–call parity example:
• Given: S = $100, strike K = $95, time to maturity T = 0.5 year, risk-free rate ≈ 2% (annual, continuous or discrete — use discrete for simplicity).
• PV(K) ≈ 95 / (1 + 0.02×0.5) ≈ 95 / 1.01 ≈ $94.06.
• Market call price C = $8.50. Implied P = C + PV(K) − S = 8.50 + 94.06 − 100 = $2.56. If market put differs materially, arbitrage may exist (adjust for dividends/early exercise if American).
– IRP forward-rate example:
• Spot USD/EUR S = 1.10 (USD per EUR), USD interest i_us = 1.0% (annual), EUR interest i_eur = 0.0%.
• Theoretical forward (1-year): F = 1.10 × (1 + 0.01) / (1 + 0.00) = 1.10 × 1.01 = 1.111.
• If market forward is 1.12, small arbitrage might exist after costs.
12. Practical steps for investors and traders (checklist)
– Convertible bond investor:
1. Compute parity price and percent of parity.
2. Compare to stock price and conversion value.
3. Assess bond features (coupon, maturity, call/put provisions).
4. Value the embedded option (use Black–Scholes or binomial if needed).
5. Decide whether to hold, convert, or trade the convertible based on yield vs. equity upside.
– Options trader:
1. Use intrinsic value to spot deep in- or out-of-money options.
2. Use put–call parity as a sanity check on European options.
3. Adjust for dividends and early-exercise risk with American options.
– FX/arbitrage trader:
1. Compute theoretical forward from IRP and compare to market forward.
2. Account for transaction costs, credit lines, and settlement constraints before executing covered arbitrage.
– Portfolio manager interested in risk parity:
1. Estimate volatilities/correlations.
2. Determine target risk contributions.
3. Solve for weights and manage leverage and rebalancing.
13. Important caveats and limitations
– Transaction costs, bid–ask spreads, financing constraints, and short-sale or borrowing constraints can eliminate apparent arbitrage.
– American-style options, dividends, and discrete cash flows require adjustments to simple put–call parity.
– IRP typically holds better for short maturities and in liquid currency pairs; UIRP is often violated in the short term due to risk premiums.
– Parity calculations ignore non-price considerations such as liquidity, credit risk, and taxation; always include these in real-world decisions.
14. The bottom line
Parity price is a versatile concept that helps investors compare equivalent economic values across instruments — convertibles vs. stock, options vs. stock and strikes, and currency returns across time. It provides a clear, often simple, computational framework for spotting conversions, relative mispricing, and potential arbitrage. However, practical trading must consider transaction costs, financing, taxes, and other market frictions that can prevent theoretical parity from being exploitable.
References and further reading
– Investopedia — “Parity Price” (source material):
– Hull, J. C., Options, Futures, and Other Derivatives — for put–call parity and option valuation
– BIS and academic literature on interest-rate parity and FX microstructure
Editor’s note: The following topics are reserved for upcoming updates and will be expanded with detailed examples and datasets.