_slides - lecture9

8/2/2019 _slides - Lecture9

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Financial Option Pricing

Lecture 9

Arbitrage Based Option Pricing

Sandra Nolte

[email protected]

KE 515Office Hours: TBA

School of Management, University of Leicester


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Outline

Non-Arbitrage Theorem Reminder

Arbitrage Option Pricing

Other kind of Options

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Non-Arbitrage Theorem

We have 3 assets and two states of nature. Assume that the

states of nature are: a good economy which is state 1 and abad economy which is state 2.

1. Asset 1 (riskless): bond with price now of1 and returns of

1+r in both states - where r is the risk free rate.

2. Asset 2 (risky): a stock with costs now S(t) and will cost in

the future, at time t + 1, either S1(t + 1) or S2(t + 1)

3. a call on the underlying stock which costs now c(t) and will

cost in the future, at time t + 1, either c1(t + 1) or c2(t + 1)

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The non-arbitrage theorem says that if we find positive

constants,1, 2 so that the system of equations:

1 = (1 + r)1 + (1 + r)2

S(t) = S1(t + 1)1 + S2(t + 1)2

c(t) = c1(t + 1)1 + c2(t + 1)2

is satisfied, then there are no arbitrage possibilities. And if there

are no arbitrage opportunities then positive constants, 1, 2

satisfying the above system of equations can be found.

Note that we can set: (1 + r)1 = p1, where p1 is some

probability and (1 + r)2 = p2, is also some probability. Both

probabilities add up to unity: p1 + p2 = 1.

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If we multiply S(t) with unity 1 = 1+r

1+r :1 + r

1 + rS(t) =

1 + r

1 + r(S1(t + 1)1 + S2(t + 1)2)

S(t) =

1

1 + r (S1(t + 1)1(1 + r) + S2(t + 1)(1 + r)2)

S(t) =1

1 + r(p1S1(t + 1) + p2S2(t + 1))

Because (1 + r)1 = p1 and since (1 + r)2 = p2

But p1S1(t + 1) + p2S2(t + 1) is the expected value of the stock

price, E[S(t + 1)] The above equation can be rewritten as:

S(t) = 11 + r E[S(t + 1)]

We can discount the risky stock value at the risk free rate,

using the probabilities p1 and p2

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From the third equation: for the call price of 25 2 = 0.5

From the second equation we get 1 = 0.25.

From the first equation you get: 1.1 0.2 5 + 1.1 0.5 = 0.825!

25 must not be the arbitrage free price.

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All concepts we have see so far are based on the assumption of noarbitrage

Put-Call Parity, volatility smile, binomial tree,..

Under the assumption of no-arbitrage, we can write the

Black-Scholes portfolio like d = rdt

But arbitrage opportunities exist in the real world!

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Arbitrage Option Pricing Model

Models dealing with arbitrage option pricing are the models by Ilinski

& Stepanenko (1999) or Fedotov & Panayides (2005), among others.

The idea here is to introduce an arbitrage return, sayx

, so thatthe return on the Black-Scholes portfolio, , becomes:

d = rdt + xdt;

where r is the risk free rate of interest.

x can follow a specific random process like the

Ornstein-Uhlenbeck process as in Ilinski & Stepanenko (1999)

Named after Leonard Ornstein and George Eugene Uhlenbeck.The process is stationary, Gaussian, and Markov. Over time, the

process tends to drift towards its long-term mean.

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This process is defined there as:

dx

dt= x(t) + (t)

where (t) is noise and is the parameter which reflects the

speed the market reacts on the arbitrage opportunity.

The derivative price obtained in the Ilinski models is an average

derivative price

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Fedotov & Panayides (2005) assume that the fluctuations aroundthe risk free rate of interest are not restricted to follow a

particular process, like the Ornstein-Uhlenbeck process. More

general approach.

2 sources of uncertainty:

1. fluctuation from the stock (usual geometric Brownian motion)

dS = Sdt + SdW,

where W is a Wiener process

2. uncertainty from a random arbitrage return from the bond

dB = rBdt + (t)Bdt;where r is the risk free interest rate, B the bond price and

(t) is some random process (which does NOT have to obey

any specific stochastic differential equation).

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They assume that the variations on the risk free return, which

mimic arbitrage returns, are on a different time scale.

The time scale is on a scale of hours and denoted by

As the authors remark, this time lies in between the time scale

for the stock return (infinitely Brownian motion fluctuations)

and the maturity time of the derivative (months): 0


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Using Itos Lemma, they obtain the classical Black-Scholesequation:

V

t+ rS

V

S+

1

2

2V

S22S2 = rV

But using both uncertainties they obtain:

V

t+rS

V

S+

1

2

2V

S22S2+r+(t)+(t)

V

SS V

+

= rV

Solving the equation, the authors find that the option price,

when taking into account arbitrage returns will be of the general

form: VBS(S) 2

U(S, )

where is defined as

T , U(S, ) is a function of (t) and VBS isthe Black-Scholes option price.

We get an option pricing interval!

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Other kind of Options

Employee Stock Options: call options on a companys stockgranted by the company to its employees, see chapter 157 in Hull.

Options on Currencies: large trades are possible, strike price,

expiry date and other features tailored to meet the needs of

corporate treasurers

Future Options: is the right not the obligation to enter into a

future contract at a specified time in the future, at a specified

future price , see chapter 17 in Hull.

Exotic Options: Packages, Nonstandard American Options

(Bermuda), Gap Options, Forward Start Options, Cliquet

Options, Compound Options, Chooser Options, BarrierOptions,Binary Options, Lookback options, Shout Options,

Asian Options, see chapter 25 in Hull.

Real Options: Options on real assets

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Readings

Hull J. C. (2000); Options, Futures and Other Derivatives ;Prentice Hall, Chapter 19.

Fedotov S., & S. Panayides (2005): Stochastic Arbitrage return

and its implication for option pricing, Physica A, 345, 207-217.

Ilinski K. & A. Stepanenko (1999): Derivative pricing with

virtual arbitrage, Quantitative Finance Papers Nr. 9902046

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