Comparing Digital Put and Call Prices Across Strike Levels

Summary

The document considers the relationship between the prices of a digital put at strike K1 and a digital call at strike K2. Each pays one unit at expiry when the spot finishes on its designated side of the strike. Comparing their combined payoffs in different strike configurations gives a price relationship relative to a zero-coupon bond with the same maturity, which represents the present value of one unit paid at expiry.

When the strikes are equal, exactly one contract pays in the solution’s convention, so the combined value equals the bond value. When the put strike is below the call strike, at most one pays, making the combined payoff no greater than one and its price lower than the bond. When the put strike is above the call strike, at least one pays, so the combined payoff is no less than one and its price exceeds the bond. These conclusions assume the binary payoff definitions and strike comparisons stated in the exercise; boundary treatment at expiry is not discussed in the excerpt.

Key ideas

• A digital put pays one unit when expiry spot is below its strike, while a digital call pays one unit when spot is above its strike.
• A zero-coupon bond provides the benchmark value of one unit paid at expiry.
• With equal strikes, the solution says the combined digital payoff is one, so their prices sum to the bond price.
• If the put strike is lower than the call strike, their combined price is below the bond price.
• If the put strike is higher than the call strike, their combined price is above the bond price.

Tags

# What can we say about digital puts and calls with different strike prices?


# What can we say about digital puts and calls with different strike prices?












I am a noob to the field of quantitative finance. I am reading this book by Mark S. Joshi. Can you help me make sense of one of the exercise questions? Here is the question (from page 40 of the book):

> Exercise 2.3 Let $P$ be a digital put struck at $K_1$ and $C$ be a digital call struck at $K_2$. (A digital put pays 1 if spot is below the strike at expiry, and a digital call pays 1 if the spot is above the strike.) What can we say about the prices of $C$ and $P$ in each of the following cases? $K_1=K_2$; $K_1 < K_2$; $K_1 > K_2$.

Here are the solutions (from page 474):

> Exercise 2.3 Precisely one of the derivatives pays off so the value of the two together at expiry will be equal to 1. Therefore $$\text{DC}(K_1) + \text{DP}(K_1) = \text{ZCB}.$$ At most one of the derivatives pays off so the value of the two together at expiry will be 1 or 0. Therefore $$\text{DC}(K_1) + \text{DP}(K_1) < \text{ZCB}.$$ At least one of the derivatives pays off so the value of the two together at expiry will be 1 or 2. Therefore $$\text{DC}(K_1) + \text{DP}(K_1) > \text{ZCB}.$$

I have not seen this notation before. Do you know what the functions DC, DP and ZCB mean? I'm guessing DC denotes the pay-off for the digital call, DP denotes the pay-off for the digital put. What is ZCB? Is it 1, if so, why did the author not just use 1?

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.