Finance and Derivatives: Theory and Practice SĂŠ bastien Bossu and Philippe Henrotte
Chapter 6 Binomial model
Chapter 6 Binomial model
1. Introduction
The valuation of derivatives is usually difficult and requires making economic assumptions. Important exceptions are forward and futures contracts, whose arbitrage price can be determined with minimal assumptions. The simplest and most didactic model for option valuation is the binomial model where we consider only two possible scenarios (up or down) over a given period. Despite its simplicity this approach leads to an efficient and robust pricing algorithm which is still very much in use on trading floors. It is also a beginner’s version of the Black-Scholes model.
Sébastien Bossu & Philippe Henrotte, Finance and Derivatives: Theory and Practice. Copyright © Dunod, Paris, 2002. English translation published by John Wiley & Sons Ltd, 2006.
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Chapter 6 Binomial model
2. Binomial trees 2.2. One-step model
Define:
Assume:
St the spot price of the underlying asset S Dt the value of a derivative on S (a function of t and St) T the maturity rf the risk-free rate for maturity T The initial price S0 of the underlying is known There are only two outcomes (u)p and (d)own for the final underlying price ST Absence of arbitrage, infinite liquidity
What is D0?
Sébastien Bossu & Philippe Henrotte, Finance and Derivatives: Theory and Practice. Copyright © Dunod, Paris, 2002. English translation published by John Wiley & Sons Ltd, 2006.
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Chapter 6 Binomial model
Figure p.88: One-step binomial tree S T(u ) [ DT(u ) ] S0 [D0]
ST( d ) [ DT( d ) ]
Sébastien Bossu & Philippe Henrotte, Finance and Derivatives: Theory and Practice. Copyright © Dunod, Paris, 2002. English translation published by John Wiley & Sons Ltd, 2006.
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Chapter 6 Binomial model
2.2. One-step model (2)
Consider a portfolio P which is long the derivative and short a quantity ∆ of the underlying. Mark-to-market value for all t: Pt = Dt – ∆St. Choose ∆ such that we get a risk-free portfolio, i.e. such that PT does not depend on ST:
PT(u ) = ST(u ) − ∆DT( u ) DT(u ) − DT( d ) → ∆ = = (u ) (d ) (d ) (d ) (d ) S − S PT = ST − ∆DT T T Sébastien Bossu & Philippe Henrotte, Finance and Derivatives: Theory and Practice. Copyright © Dunod, Paris, 2002. English translation published by John Wiley & Sons Ltd, 2006.
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Chapter 6 Binomial model
2.2. One-step model (3)
With this choice of ∆, the portfolio can be seen as a fixed cash flow whose present value must be, under penalty of arbitrage: PT P0 = (1 + rf )T But P0 = D0 – ∆S0 . Equating both results yields D0:
D0 = ∆ × S0 +
DT( g) − ∆ × ST( g)
(1+ r )
T
f
where DT(?) and ST(?) can be taken in outcome (u)p or (d)own.
Sébastien Bossu & Philippe Henrotte, Finance and Derivatives: Theory and Practice. Copyright © Dunod, Paris, 2002. English translation published by John Wiley & Sons Ltd, 2006.
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Chapter 6 Binomial model
2.2. One-step model (4)
∆ is the ratio of the change in value of the derivative to the change in the underlying price:
DT( u ) − DT( d ) VDT ∆ = (u ) = (d ) ST − ST VST
In general, this is different from the discrete differential:
DT − D0 ∆≠ ST − S0
But ST(u) and ST(d) could be chosen to make the two concepts convergent.
Sébastien Bossu & Philippe Henrotte, Finance and Derivatives: Theory and Practice. Copyright © Dunod, Paris, 2002. English translation published by John Wiley & Sons Ltd, 2006.
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Chapter 6 Binomial model
2.3. Multiple-step model
The binomial model is easily generalized to multiple steps. To illustrate how this is done, consider a put option to buy an ounce of gold at $450 in 2 months
Spot price: Strike price: Maturity:
S0 = $400 per troy ounce K = $450 per troy ounce T = 1/6 year
The expected evolution over 2 half-month periods is shown next slide Furthermore, analysts believe the risk-free rate will remain stable at 4% over each period
Sébastien Bossu & Philippe Henrotte, Finance and Derivatives: Theory and Practice. Copyright © Dunod, Paris, 2002. English translation published by John Wiley & Sons Ltd, 2006.
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Chapter 6 Binomial model
Figure p.89: Binomial tree for the price of gold $550 $500 $400 $400 $350
$390 $ $300
SÊbastien Bossu & Philippe Henrotte, Finance and Derivatives: Theory and Practice. Copyright Š Dunod, Paris, 2002. English translation published by John Wiley & Sons Ltd, 2006.
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Chapter 6 Binomial model
2.3. Multiple-step model (2)
With this information we can calculate the value p0 of the put. To do so we iterate the binomial model going backwards: Step t = 2 months: pT = max(0, K – ST) = max(0, 450 – ST)
Sébastien Bossu & Philippe Henrotte, Finance and Derivatives: Theory and Practice. Copyright © Dunod, Paris, 2002. English translation published by John Wiley & Sons Ltd, 2006.
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Chapter 6 Binomial model
Figure p.89: Value of the put at maturity $500
$550 [0]
$400 [$50]
$400 $350
$390 [$60] $300 [$150]
SÊbastien Bossu & Philippe Henrotte, Finance and Derivatives: Theory and Practice. Copyright Š Dunod, Paris, 2002. English translation published by John Wiley & Sons Ltd, 2006.
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Chapter 6 Binomial model
2.3. Multiple-step model (3)
Step t = 1 month: We use the one-step model twice for each of the ‘up’ and ‘down’ scenarios:
Up Scenario: 0 − 50 1 =− 550 − 400 3 0 − (− 13 ) × 550 1 = − × 500 + = $16.07 1/12 3 (1 + 4%)
∆ (u ) = p1( u )
Down Scenario: 60 − 150 ∆(d ) = = −1 390 − 300 150 − ( −1) × 300 (d ) p1 = −350 + = $98.53 1/12 (1 + 4%)
Sébastien Bossu & Philippe Henrotte, Finance and Derivatives: Theory and Practice. Copyright © Dunod, Paris, 2002. English translation published by John Wiley & Sons Ltd, 2006.
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Chapter 6 Binomial model
Figure p.90: Value of the put at steps t = 1 month and t = 2 months $500 [$16.07] $400 $350 [$98.53]
$550 [0]
$400 [$50] $390 [$60] $300 [$150]
SÊbastien Bossu & Philippe Henrotte, Finance and Derivatives: Theory and Practice. Copyright Š Dunod, Paris, 2002. English translation published by John Wiley & Sons Ltd, 2006.
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Chapter 6 Binomial model
2.3. Multiple-step model (4)
Step t = 0: We use the one-step model once again and obtain:
16.07 − 98.53 ∆= = −0.55 500 − 350 16.07 − (−0.55) × 500 p0 = −0.55 × 400 + = $70.12 1/12 (1 + 4%)
In comparison the price of a forward contract with the same characteristics is -$48.53, and $21.59 for a call.
Sébastien Bossu & Philippe Henrotte, Finance and Derivatives: Theory and Practice. Copyright © Dunod, Paris, 2002. English translation published by John Wiley & Sons Ltd, 2006.
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