Basic, lesson 1 of 3
Parity and no arbitrage bounds
A call, a put, the stock, and cash are tied together so tightly that knowing three prices gives you the fourth. The bounds that follow catch most mispriced quotes.
10 min, 3 checks
1Put call parity with interest
C - P = S - PV(K), with PV(K) = K x e^(-rT)
European options with the same strike and expiry, on a stock paying no dividend.
Why it holds: a long call plus a short put pays S - K at expiry whatever happens, exactly like owning the stock and owing K. Two things with the same payoff must cost the same today, so the difference between call and put equals the stock minus the present value of the strike.
Worked example
Stock 100, strike 100, rate 5%, one year
- 1PV(K) = 100 x e^(-0.05) = 95.12.
- 2C - P = 100 - 95.12 = 4.88.
The at the money call is worth 4.88 more than the put. With zero rates they would be equal.
Check 1
Stock 50, the present value of the 50 strike is 49, and the call is 4. What is the put worth?
2Bounds
- A call is worth at least S - PV(K) and at least 0. It is never worth more than the stock.
- A put is worth at least PV(K) - S and never more than PV(K).
- A call spread (long the lower strike, short the higher) is worth between 0 and the strike gap.
- An American call on a stock with no dividend is never worth exercising early: its value already exceeds S - K.
Check 2
A call on a stock trading at 40 is offered at 42. What do you do?
Check 3
Which combination has the same payoff as owning the stock with borrowed money (a forward at K)?
Terms in this lesson: Arbitrage.
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