🗊Презентация The binomial model for option pricing

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Option Pricing: The Multi Period Binomial Model Henrik Jönsson Mälardalen University Sweden
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Option Pricing: The Multi Period Binomial Model Henrik Jönsson Mälardalen University Sweden

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Contents European Call Option Geometric Brownian Motion Black-Scholes Formula Multi period Binomial Model GBM as a limit Black-Scholes Formula as a limit
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Contents European Call Option Geometric Brownian Motion Black-Scholes Formula Multi period Binomial Model GBM as a limit Black-Scholes Formula as a limit

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European Call Option C - Option Price K - Strike price T - Expiration day Exercise only at T Payoff function, e.g.
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European Call Option C - Option Price K - Strike price T - Expiration day Exercise only at T Payoff function, e.g.

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Geometric Brownian Motion S(y), 0y<t, follows a geometric Brownian motion if independent of all prices up to time y
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Geometric Brownian Motion S(y), 0y<t, follows a geometric Brownian motion if independent of all prices up to time y

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Black-Scholes Formula The price at time zero of a European call option (non-dividend-paying stock): where
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Black-Scholes Formula The price at time zero of a European call option (non-dividend-paying stock): where

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The Multi Period Binomial Model
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The Multi Period Binomial Model

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The Multi Period Binomial Model Let Let (X1, X2,…, Xn) be the vector describing the outcome after n steps. Find the set of probabilities P{X1=x1, X2 =x2,…, Xn =xn}, xi=0,1, i=1,…,n, such that there is no arbitrage opportunity.
Описание слайда:
The Multi Period Binomial Model Let Let (X1, X2,…, Xn) be the vector describing the outcome after n steps. Find the set of probabilities P{X1=x1, X2 =x2,…, Xn =xn}, xi=0,1, i=1,…,n, such that there is no arbitrage opportunity.

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The Multi Period Binomial Model Choose an arbitrary vector (1, 2, …, n-1) If A={X1= 1, X2= 2, …, Xn-1= n-1} is true buy one unit of stock and sell it back at moment n Probability that the stock is purchased qn-1=P{X1= 1, X2= 2, …, Xn-1= n-1} Probability that the stock goes up pn= P{Xn=1| X1= 1, …, Xn-1= n-1}
Описание слайда:
The Multi Period Binomial Model Choose an arbitrary vector (1, 2, …, n-1) If A={X1= 1, X2= 2, …, Xn-1= n-1} is true buy one unit of stock and sell it back at moment n Probability that the stock is purchased qn-1=P{X1= 1, X2= 2, …, Xn-1= n-1} Probability that the stock goes up pn= P{Xn=1| X1= 1, …, Xn-1= n-1}

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The Multi Period Binomial Model
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The Multi Period Binomial Model

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The Multi Period Binomial Model Expected gain = No arbitrage opportunity implies
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The Multi Period Binomial Model Expected gain = No arbitrage opportunity implies

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The Multi Period Binomial Model (1, 2, …, n-1) arbitrary vector No arbitrage opportunity 
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The Multi Period Binomial Model (1, 2, …, n-1) arbitrary vector No arbitrage opportunity 

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The Multi Period Binomial Model Limitations: Two outcomes only The same increase & decrease for all time periods The same probabilities
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The Multi Period Binomial Model Limitations: Two outcomes only The same increase & decrease for all time periods The same probabilities

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Geometric Brownian Motion as a Limit The Binomial process:
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Geometric Brownian Motion as a Limit The Binomial process:

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The binomial model for option pricing, слайд №14
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GBM as a limit Let and , Y ~ Bin(n,p)
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GBM as a limit Let and , Y ~ Bin(n,p)

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GBM as a Limit The stock price after n periods where
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GBM as a Limit The stock price after n periods where

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GBM as a Limit Taylor expansion gives
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GBM as a Limit Taylor expansion gives

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GBM as a limit Expected value of W
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GBM as a limit Expected value of W

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GBM as a limit By Central Limit Theorem
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GBM as a limit By Central Limit Theorem

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GBM as a limit The multi period Binomial model becomes geometric Brownian motion when n → ∞, since are independent
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GBM as a limit The multi period Binomial model becomes geometric Brownian motion when n → ∞, since are independent

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B-S Formula as a limit Let , Y ~ Bin(n,p) The value of the option after n periods = where S(t)= uY dn-Y S(0)
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B-S Formula as a limit Let , Y ~ Bin(n,p) The value of the option after n periods = where S(t)= uY dn-Y S(0)

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B-S formula as a limit The unique non-arbitrage option price As n → ∞
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B-S formula as a limit The unique non-arbitrage option price As n → ∞

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B-S formula as a limit where X~N(0,1) and
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B-S formula as a limit where X~N(0,1) and

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B-S formula as a limit
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B-S formula as a limit

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B-S formula as a limit
Описание слайда:
B-S formula as a limit

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B-S formula as a limit
Описание слайда:
B-S formula as a limit

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B-S formula as a limit
Описание слайда:
B-S formula as a limit

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