Finance Derivative Securities Trimester 3a 2013 Assignment 2

Finance Derivative Securitiestrimester 3a 2013 Assignment 2 No

Let ( )C K denote a European vanilla Call option with strike price K. Assume that all options are identical except for strike price, and strike prices satisfy 1 2 3K K K

Question 1 [5 marks] What are the no-arbitrage lower bound, and the no-arbitrage upper bound, of the vertical spread ( ) ( )1 2C K C K− ?

Question 2 [10 marks] What is the functional relationship between the no-arbitrage values of the two vertical spreads, ( ) ( )1 2C K C K− and ( ) ( )2 3C K C K− ?

Paper For Above instruction

The problem involves understanding the bounds and relationships of European vanilla call options and their spreads. In the context of options pricing, the no-arbitrage bounds ensure that option prices do not allow riskless profit opportunities, which is fundamental in derivatives pricing theory. Here, I will explore both parts of the question concerning the bounds of a vertical spread and the relationship between two such spreads.

Part 1: No-Arbitrage Bounds of a Vertical Spread

A vertical spread involves purchasing a call option at strike K1 and selling a call at strike K2, with K1

Given zero interest rates, the no-arbitrage lower bound for a call option with strike K is its intrinsic value, which is max(S - K, 0). Since the question assumes all options are in a frictionless market with no arbitrage, the minimal value of a vertical spread ( ) ( )1 2C K C K− is the maximum of zero and the difference in intrinsic values, which simplifies to:

  • Lower bound: 0

because when the underlying price S is below K1, both options are worthless, making the spread zero, which respects the no-arbitrage condition.

As for the upper bound, the spread cannot exceed the difference between the strike prices, under the assumption of no arbitrage in the market, the maximum value of the spread occurs when the underlying price S exceeds K2. In such a case, both options are in-the-money, and the spread's maximum value is (K2 - K1).

  • Upper bound: (K2 - K1)

This is consistent with the fact that the value of the spread is capped by the intrinsic difference when the underlying is sufficiently high.

Part 2: Functional Relationship Between the Two Vertical Spreads

Next, consider the two vertical spreads: ( ) ( )1 2C K C K− and ( ) ( )2 3C K C K−. Their strike prices satisfy K1

The key insight is that these spreads are nested, and their no-arbitrage values are linked through the concept of sub- and super-hedging. Specifically, the spread ( ) ( )1 2C K C K− can be viewed as a subset of the larger spread ( ) ( )2 3C K C K−, given the ordering of strike prices.

From arbitrage principles, the spread ( ) ( )1 2C K C K− cannot exceed the spread ( ) ( )2 3C K C K−, because the latter encompasses a greater range of intrinsic value differences. Mathematically, this implies:

  • ( ) ( )1 2C K C K− ≤ ( ) ( )2 3C K C K−

Furthermore, the relationship can be expressed as a proportional or functional one, depending on the underlying's price S and strike structure. When considering the value functions, the spreads' prices are monotonic with respect to their strike prices, and the relation between them is linear under certain assumptions of no arbitrage and convexity of the call option price curve.

Hence, the functional relationship is approximately proportional, with the value of the smaller spread bounded above by the larger spread, reflecting their nested and Lipschitz-continuous nature in the no-arbitrage framework. Formally, one could express this as:

  • ( ) ( )1 2C K C K− = α · ( ) ( )2 3C K C K− where 0 ≤ α ≤ 1, depending on the specific underlying price relative to strikes.

This illustrates that the two spreads are linearly related within the no-arbitrage bounds, with their precise relationship governed by the underlying asset's value and the convexity of the call price function.

Conclusion

In conclusion, the no-arbitrage lower bound of a vertical spread ( ) ( )1 2C K C K− is zero, and the upper bound is the difference in strike prices (K2 - K1). The relationship between the two vertical spreads ( ) ( )1 2C K C K− and ( ) ( )2 3C K C K− is a monotonic, bounded proportionality, reflecting their nested nature and the convexity of option prices. These bounds and relationships are fundamental to ensuring consistency and absence of arbitrage opportunities in options markets, guided by the principles of derivatives pricing theory.

References

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