Bounding Option Prices Using Semidefinite Programming

SACHIN JAYASWAL

Department of Management SciencesUniversity of Waterloo, Canada

Project work for

MSCI 700 Fall 2007Semidefinite Programming: Models, Algorithms &

Computation

Course InstructorDR. M. F. ANJOS

Introduction

2

• Call Option: An agreement that gives the holder the right to buy the underlying by a certain date for a certain price.

• European vs. American call option

Definitions

3

• T: Specific time when the underlying can be purchased (Maturity)

• K: Specific price at which the underlying can be purchased (Strike Price)

• St: Price of the underlying (stock) at time t

• r: Risk-free interest rate

• α: Expected return on the underlying

• σ: Volatility in the price of the underlying

• C: Price of call option

Call Option Payoff

4

Call Option Pricing

5

• Call Option Pricing – An interesting and a challenging problem in finance

• Black-Scholes (1973) – Asset price follows geometric Brownian motion

• Stock prices observed in the market often do not satisfy this assumption

• Can we price the option without assuming any specific distribution for stock price?

Bounds on Option Price

6

Dual Problems

7

Propositions

8

SDP Formulation: Upper Bound

9

SDP Formulation: Lower Bound

10

Comparison with Black-Scholes

11

• Black-Scholes assume stock prices follow geometric Brownian motion

where N(·) is the cumulative distribution of a normally distributed random variable

Computational Experiments

12

Black-K UB LB UB LB UB LB Scholes

30 10.0652 10.0346 10.0652 10.0346 10.0349 10.0346 10.034635 5.1007 5.0404 5.1007 5.0404 5.0420 5.0404 5.040440 0.5784 0.0461 0.5783 0.0549 0.5771 0.3425 0.465845 0.0614 0.0000 0.0500 0.0000 0.0027 0.0000 0.000050 0.0309 0.0000 0.0212 0.0000 0.0003 0.0000 0.0000

2-moment 3-moment 4-moments = 0.2

Computational Experiments

13

Black-K UB LB UB LB UB LB Scholes

30 18.3218 10.8194 18.3218 10.8194 15.5444 10.8194 13.990135 15.0745 5.9559 15.0745 7.5681 12.7348 7.5681 11.244540 12.2832 1.0925 12.2832 4.9197 10.5476 4.9197 9.007345 9.9894 0.0000 9.9894 2.6033 0.8792 3.4252 7.205650 8.1766 0.0000 8.1766 0.8246 7.5836 2.6551 5.7649

s = 0.82-moment 3-moment 4-moment

Computational Experiments

14

Computational Experiments

15

Computational Experiments

16

Cutting Plane Method for Solving[UB_SDP]

17

Let (

1)

• Observations:

where represents the polyhedral set corresponding to the linear constraints of

• Adding constraint (1) tightens the bound.

• Relaxing SDP constraint on X, Z makes the problems LP

Cutting Plane Algorithm

18

Performance of the Cutting Plane Algorithm

19

K UB LB UB LB UB LB UB LB30 19 19 35 21 18 17 35 2435 17 15 39 37 17 15 37 2040 28 9 46 28 28 9 47 2945 29 5 41 10 25 4 38 750 24 6 36 10 24 5 35 8

K UB LB UB LB UB LB UB LB30 18 17 36 24 18 20 37 2635 19 15 38 18 21 18 39 2040 24 9 51 40 26 13 55 3945 25 5 35 8 22 3 32 550 25 4 34 7 22 3 31 6

s = 0.42-moment

s = 0.22-moment 3-moment

s = 0.62-moment 3-moment

3-moment

s = 0.82-moment 3-moment

Number of cuts required by the algorithm

Conclusions

20

• The SDPs produce good bounds on the option price in absence of the known distribution of the stock price.

• The approach may be used in pricing complex financial derivatives for which closed-form formula is not possible (Boyle and Lin, 1997).

21

Thank You!