Arbitrage-Free Option Pricing by Convex Optimization

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1 Arbitrage-Free Option Pricing by Convex Optimization Alex Bain June 1, Description In this project we consider the problem of pricing an option on an underlying stock given a risk-free interest rate, the price of the stock and the prices of existing options on the stock. We define a series of convex programs that find upper and lower bounds on the option price that satisfy the no-arbitrage condition between the option and the given assets. We give methods to find interesting risk-neutral price distributions, such as those that are smooth, sparse or multimodal. Additionally, we describe a method to find the best arbitrage among the existing assets, if no such arbitrage-free price exists. We implemented our methods in a Python application and we describe our results using the application with real world prices. 2 Finance Background A distribution is risk-neutral if the expectation under the distribution of the discounted value of each asset at maturity is equal to the current price of the asset. An arbitrage is an investment opportunity where you are paid to accept the investment, and in all possible outcomes, the final value of the investment is non-negative. From Farkas lemma, a riskneutral distribution exists if and only if prices are arbitrage-free [BV04, page 263]. 3 Comparison to Previous Work The basic problem of computing upper and lower bounds on an option price given the prices of existing assets, assuming only the absence of arbitrage, has been studied in [BP02], [BV04] and [LW04]. Likewise, our approach makes no assumptions about the statistical distribution of the stock price at maturity. Although recent work has focused on pricing for index options [de06], we limit our model to a single underlying stock. Our method is similar to the pair of LP s described in [BV04, page 263], but is more robust for real world prices. Real world prices are often not arbitrage-free and one can typically find small arbitrages on the order of a fraction of a penny. Our model accounts for these small arbitrages and includes the bid-ask spread as in [PVZ10]. We implemented our model with cvxpy, a package for specifying and solving convex programs in Python [RB11]. 1

2 4 Pricing Method We consider an option on an underlying stock and n assets that include a risk-free bond, the stock and existing options on the stock. We assume that there are m possible scenarios for the value of the option and the existing assets at maturity. Our method is to price the option with a series of convex programs. First, we determine how much the prices of the assets need to be adjusted to be arbitragefree. The data consists of the m scenarios of the values of the n assets at maturity V R m n and the current bid and ask prices of the assets p bid, p ask R n. The variables are the riskneutral distribution y R m and the adjustments to the bid and ask prices l, u R n. Since we take the risk-free rate as given, we don t allow the price of the bond to be adjusted. This program has the form minimize l 1 + u 1 subject to l + p bid V T y p ask + u l 1 = 0, u 1 = 0 y 0 Then we solve a pair of LP s that find upper and lower bounds on the arbitrage-free price of the option, after adjusting the real world prices by l and u. This pair of LP s is similar to the pair from [BV04, page 263]. We augment the data with the adjustments l, u R n from (1) and the m scenarios of the value of the option at maturity v opt R m. The variables are the risk-neutral distribution y R m and the bound on the arbitrage-free option price p opt R. This pair of LP s has the form minimize/maximize subject to p opt l + p bid V T y p ask + u p opt = v T opty y 0 We find interesting risk-neutral distributions with a family of convex programs parameterized by the objective function f 0. The data is the same as in (2). The variable is the risk-neutral distribution y R m. This family has the form minimize subject to f 0 (y) l + p bid V T y p ask + u y 0 For example, if we let D R m m be the tridiagonal matrix with entries (1, 2, 1), i.e. the finite second derivative, we can find a maximally smooth risk-neutral distribution by choosing f 0 (y) = max( Dy ). A heuristic for finding a sparse distribution is to choose f 0 (y) = y 1. 5 Best Arbitrage Portfolio Method Since real world prices not always arbitrage-free, we describe a convex program that finds the portfolio of existing assets that achieves the best arbitrage. In this program, long positions 2 (1) (2) (3)

3 in the portfolio must be established at the ask price and short positions at the bid price. The data is the same as in (1). The variables are the long and short positions in each asset held by the portfolio x long, x short R n (with x short 0) and the cost of the portfolio and its value at maturity c, v R. The overall position of the portfolio is given by (x long + x short ). Since an arbitrage requires a guaranteed return over all possible outcomes, we take the value of the portfolio at maturity to be the elemementwise min of V (x long + x short ). This program has the form maximize subject to v c p T askx long + p T bidx short c min(v (x long + x short )) v x long + x short x long 0, x short 0 c 0, v 0 The optimal portfolio is an arbitrage since the constraints c 0, v 0 specify that you are paid to accept the portfolio and that it has a non-negative value at maturity. If the asset prices are arbitrage-free, the optimal value of this program will be zero. 6 Application We implemented our method as a standalone Python application. The application allows you to price a standard option such as a call or a put, or a custom option, which you specify by creating a payoff table that gives the value of the option at maturity based on the price of the stock. It includes tables for entering the risk-free interest rate, the stock price and prices of existing options. The application runs our method with cvxpy and displays the resulting price bounds and graphs of risk-neutral distributions using matplotlib [Hun07]. (4) 7 Results We downloaded market-on-close option prices for Apple (AAPL), Boeing (BA) and Hewlett- Packard (HPQ) August call options and Intel (INTC) May call options from Yahoo! Finance. We calculated a risk-free rate interest based on the 3-month U.S. Treasury bond yield. For each stock, we chose eight options at various strikes centered around the stock price. From our method (1) we found that option prices for INTC admit a small arbitrage. The objective value of (1) for INTC is We found that option prices for AAPL, BA and HPQ are arbitrage-free. We illustrate our results by comparing the bounds found by our method (2) to the real world bid-ask prices of options. The diagrams below show how our bounds converge toward the real bid-ask prices as we increase the number of assets. We started by pricing the option with just the stock and the bond. Then we added existing options one by one to the asset set and recalculated the bounds. The options were added in random order and our results 3

4 are averaged over 10 trials. The diagrams show that our bounds converge toward the real bid-ask prices, which are always within the bounds AAPL Aug $340 call option bounds and true bid ask spread Upper bound Lower bound True ask True bid BA Aug $85 call option bounds and true bid ask spread Upper bound Lower bound True ask True bid Price Stock price True bid ask Final bounds Price Stock price True bid ask Final bounds Number of existing options included in set of assets Number of existing options included in set of assets HPQ Aug $45 call option bounds and true bid ask spread Upper bound Lower bound True ask True bid Stock price True bid ask Final bounds INTC May $21 call option bounds and true bid ask spread Upper bound Lower bound True ask True bid Stock price True bid ask Final bounds Price 4 3 Price Number of existing options included in set of assets Number of existing options included in set of assets 8 Pricing Heuristic We observed that changing the objective of (2) to create a tradeoff objective between the bounds and the smoothness of the risk-neutral distribution results in bounds that are closer to the real bid-ask prices. The diagrams show how the bounds converge within the bid-ask prices as we increase the smoothness tradeoff parameter. We are excited by this result! BA Aug $77.50 call bound smoothness tradeoff converges to bid ask Upper bound Lower bound True ask True bid HPQ Aug $40 call bound smoothness tradeoff converges to bid ask Upper bound Lower bound True ask True bid Price 4.8 Price Log 10 of smoothness tradeoff parameter Log 10 of smoothness tradeoff parameter 4

5 9 Example As a concrete example, we used our Python application to price the AAPL Aug $350 call. Our assets consist of the stock (trading at $340.50), a risk-free bond and eight options at various strikes. The real bid-ask is $ Our program computed bounds of $ The diagrams below show the risk-neutral distributions for our lower and upper bounds. Lower bound distribution of stock price at maturity Upper bound distribution of stock price at maturity Probability 0.10 Probability Stock price Stock price Using our method (3), we found intriguing distributions, such as a smooth risk-neutral distribution and a multimodal distribution. The smooth distribution results in an option price of $15.67 (very close to the real bid-ask) and shows fat tails. The inability to account for fat tails is one of the major criticisms of the Black-Scholes model [HT09]. Smoothest distribution of stock price at maturity Lower bound-smoothness tradeoff dist. of stock price at maturity Probability Probability Stock price Stock price Since the AAPL option prices are arbitrage-free, the value of (4) is zero. However, we saw earlier that the INTC option prices are not arbitrage-free. The value of (4) for INTC is about 2/100 of a penny and has the arbitrage portfolio given in the following table. 5

6 Asset Position Asset Position Risk-free bond Stock $16 call 0.0 $20 call $17 call 0.0 $21 call 0.0 $18 call 0.0 $22 call 0.0 $19 call 0.0 $23 call 0.0 Table 1: Best arbitrage portfolio for INTC. The table shows the percentage each asset comprises in the portfolio. Short positions in the portfolio are indicated by negative numbers. The value of the arbitrage is $ , about 2/100 of a penny. 10 Conclusions and Future Work Our method finds reasonable option bounds using real prices. Our method robustly handles real prices by accounting for the bid-ask spread and for small arbitrages. We are able to find interesting price distributions, such as those that have fat tails or are multimodal. We observed that adding a smoothness tradeoff term to the objective of (2) is an effective heuristic for producing bounds that are close to the real bid-ask prices. An intriguing avenue of future work is to explore other smoothness constraints, such as limiting the KL distance of the risk-neutral distribution from a log-normal price distribution. Such a constraint would allow us to analyze how much real option prices differ from prices predicted by the Black-Scholes model, which assumes prices are log-normal. 6

7 References [BP02] [BV04] [de06] [HT09] D. Bertsimas and I. Popescu. On the relation between option and stock prices: A convex optimization approach. Operations Research, 50: , S. Boyd and L. Vandenberghe. Convex Optimization. Cambridge University Press, A. d Aspremont and L. El Ghaoui. Static arbitrage bounds on basket option prices. Math. Program., 106(3): , E. Haug and N. Taleb. Option traders use (very) sophisticated heuristics, never the Black-Scholes-Merton formula (7th version). SSRN elibrary, [Hun07] J. Hunter. Matplotlib: A 2d graphics environment. Computing In Science & Engineering, 9(3):90 95, May-Jun [LW04] P. Laurence and T. Wang. Sharp Upper and Lower Bounds for Basket Options. SSRN elibrary, [PVZ10] J. Peña, J. Vera, and L. Zuluaga. Computing arbitrage upper bounds on basket options in the presence of bid-ask spreads. Working paper available at http: // [RB11] T. Tinoco De Rubira and S. Boyd. cvxpy: A Python package for modeling optimization problems, version May

8 A Appendix: Stability of Price Bounds Since our method relies on discretizing the possible outcomes of the asset and option values into m scenarios, we would like to know how stable our bounds are as we change the discretization. The table below shows how our bounds change as we vary the number of scenarios and the range of the stock price at maturity. The table shows that our bounds are quite stable when the number of scenarios exceeds 100 and the price range exceeds % of the current stock price. Option Scenarios Price Range Option Bounds BA $77.50 call 100 $ $ BA $77.50 call 200 $ $ BA $77.50 call 200 $0-160 $ BA $77.50 call 400 $0-160 $ HPQ $40 call 100 $20-60 $ HPQ $40 call 200 $20-60 $ HPQ $40 call 200 $0-80 $ HPQ $40 call 400 $0-80 $ AAPL $350 call 100 $ $ AAPL $350 call 200 $ $ AAPL $350 call 200 $0-680 $ AAPL $350 call 400 $0-680 $ Table 2: Stability of calculated option price bounds. B Appendix: Application Screenshots The application was written in Python using the cross-platform library wxpython for the GUI. The screenshots are from a version of the application running on Windows. All of the graphs in the Example section of this report were produced by the application. 8

9 Figure 1: Tables for entering the prices of existing assets and calculating the arbitrage-free price adjustments. 9

10 Figure 2: Choosing to price a standard call or put option or a custom option and calculating the arbitrage-free price bounds. The Plot distribution buttons display the risk-neutral distributions associated with each bound. 10

11 Figure 3: Tools to find interesting risk-neutral distributions, including smooth distributions and distributions resulting from tradeoff objectives. 11

12 Figure 4: The best arbitrage and best investment portfolios. In this case, there is an arbitrage of 2/100 of a penny. The best guaranteed investment is to invest in the bond, which returns 17/100 of a penny (the risk-free rate). The difference is that you are paid to accept the arbitrage portfolio, where the investment portfolio requires you to commit funds to purchase the portfolio. 12