PowerPoint Presentation
Option Pricing
The Riskless Hedge Principle
Where business comes to life
The Binomial Model
Two most widely used:
Black and Scholes (B-S) Option Pricing Model (OPM), 1973.
Binomial option pricing model (BOPM), derived by Cox, Ross, and Rubinstein (1979), and Rendleman and Bartter (1979).
Both models are similar in a number of ways:
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The Binomial Model
Both models explain the equilibrium price of an option in terms of the same parameters: S, X, T, R, V.
Each model limited ( originally) to cases involving European options with no dividend payments. Both need to be adjusted.
Each determines the equilibrium price of an option in terms of arbitrage forces. Such arbitrage models are based on the law of one price.
Arbitrage will force the respective values to converge to one price.
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The Binomial Model
Both the B-S and the BOPM are based on valuing options in terms of 1) using a replicating portfolio( Synthetic option ) and 2) creating what is known as a riskless hedge (Delta hedge). Both approaches yield the same option price.
Major difference in the models emanates from the assumption each makes about the underlying stock price behaviour over time.( i.e its stochastic process )
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The Binomial Model
In the BOPM , time to expiry is partitioned into discrete, or finite periods of equal length.
In each period the stock is assumed to follow a binomial process in which it either increases or decreases. The equilibrium price is then found from an arbitrage strategy consisting of positions in the stock, option and a bond.
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The Binomial Model
The B-S model, assumes a continuous time process partitioned into infinitely small time periods.
The above arbitrage strategy is then implemented and revised continuously.
In fact, the B-S model is the limiting case of the BOPM as the time intervals get smaller and smaller.
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The Binomial Model
The two basic approaches (underlying principles):
Basic idea of a replicating portfolio ( synthetic option )
Basic idea of a riskless hedge ( delta hedged portfolio or synthetic riskless bond )
Further examples of both and some basic derivations
2) The general binomial model ( basic derivation and meaning of model parameters)
3) Option pricing examples using the general model.
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The Riskless Hedge
Using (S(o) =100, X= 100, r =5%.), we will construct a portfolio of the share and option that is riskless: It has the same value in both states of the world.
Since the portfolio has no risk, it must by definition earn the riskless rate of interest
( we do not require a risk premium as in the CAPM)
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The Riskless Hedge
Construct a portfolio of (Delta) long shares and short 1 call option, such that the portfolio is riskless in both states of the world.
If the shares tick up to £110, the value of the portfolio will be (10 = short an option with an intrinsic value of £10)
If the shares tick down to £95, the portfolio is worth
( the call = 0).
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The Riskless Hedge
The portfolio is riskless if delta is chosen so that its value is equal in both states of the world:
In other words, for the portfolio to be riskless we need to be long .6667 shares and short 1 call option.
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The Riskless Hedge
If the share ticks up the portfolio is worth:
(110 x .6667) – 10 = 63.337
If it ticks down, 95 x .667 = 63.337
Therefore it is riskless.
In the absence of arbitrage opportunities it must earn the riskless rate (5%).
Therefore its value today is 63.337/1.05 = 60.32
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The Riskless Hedge
What is the value of the call option today?
The stock is at £100 today, therefore the value of the portfolio today must be:
(100 x .6667) – C = 66.67-C.
This must equal 60.32
e.g 66.67-C = 60.32 = £6.35.
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The Riskless Hedge
Again this is only true in the absence of arbitrage opportunities.
Note: The riskless hedge is a synthetic riskless bond: The riskless portfolio must earn the riskless rate of interest, therefore it replicates a riskless bond.
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The Riskless Hedge
The concept of the riskless hedge is so central to option pricing that we will take a look at another slightly more extended example:
This time the stock dynamics are
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100
115
95
X=100
The Riskless Hedge
Giving the payouts to a call option:
Buying 1 share and selling 1 call would give payouts of:
115-15 = 100, if the share price rises
95-0 = 95, if the share price falls
It seems that we have bought too many shares
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c
15
0
The Riskless Hedge
If we buy 0.5 shares and sold 1 call:
0.5(115)-15 = 42.5, if the share rises
0.5(95)-0 = 47.5, if the share falls
We now appear to be buying too few shares
If we buy 0.75 shares and sell 1 call:
0.75(115)-15 = 71.25, if the share price rises
0.75(95)-0 = 71.25, if the share price falls
The payoffs are now equal so we have, in a roundabout manner, found the delta.
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The Riskless Hedge
As we saw earlier this is a risk-free portfolio. This portfolio then must pay us or give us a return equal to the risk free rate. So our return then is
The gross pay-off is the guaranteed sum of 71.25. The investment is the cost of the shares (0.75(100)) less the proceeds from the sale of the call.
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The Riskless Hedge
If we assume a per period risk free rate of 5%
Solving for c we obtain 7.143
Again the key thing to remember is that the only reason this works is because we have set up a fully hedged position which guarantees a payoff in the next period of time
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The Binomial Model
The above two dimensional examples are obviously very limited but serve to explain two of the extremely important principles that underpin all option pricing models.
The above principles can be used to derive a general binomial model which is what CRR etc did.
Before doing so, we need to examine a crucially important concept: risk neutral valuation.
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Risk Neutral Valuation
Any security dependent on other traded securities can be valued on the assumption that investors are risk neutral
Why can we make this assumption??
Nowhere in the above analysis have we used expected returns from the stock or option or mentioned the risk preferences of investors.
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Risk Neutral Valuation
In fact we do not mention risk anywhere. we are either replicating a call to price it or creating a riskless hedge to price it.
Even if investors are risk averse we could still price the option.
Assuming risk neutrality is an extremely powerful tool, because in this world all securities’ expected return is the risk free rate, and just by definition that must be the rate which we use to discount future cash flows.
This allows us to “create” risk neutral probabilities
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The Binomial Model & Risk-Neutral Valuation
The risk-neutral probabilities can be explained using something that is known as the “one-period forward equation”:
Suppose we have a one period binomial model for the stock: the stock can rise to S(u) or fall to S(d).
Assume p is the up risk-neutral probability and r is the risk free rate
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The Binomial Model & Risk-Neutral Valuation
Assume we are in a risk-neutral world: that is. assume the expected value of the stock is equal to its forward price one period later:
This is known as the one period forward equation.
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The Binomial Model & Risk-Neutral Valuation
In other words we are forcing the stock price in this binomial world (S(u) or S(d)) to grow at the risk free rate.
If we solve the above equation it reduces to:
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The Binomial Model & Risk-Neutral Valuation
Which is the same as the equation found for p earlier, except we have replaced R(f) the simple risk free rate with the continuously compounded riskless rate:
Using the continuously compounded rate is important when we have multiple binomial nodes to value. Using the simple riskless rate can lead to errors.
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The Binomial Model & Risk-Neutral Valuation
The fundamental reason we can set up a riskless hedge or form a replicating portfolio is because the option and the stock have the same source of underlying risk: the stock
This allows us to assume risk-neutrality and hence derive the risk- neutral probabilities, because we have either hedged the risk away or we have replicated the payoffs to the option in the next state of the world: it is riskless for the next discrete instant of time. Therefore we can assume that all securities in this world will grow at the risk free rate.
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An easy derivation of the binomial model
We will return to the riskless hedge as a starting point. Let h be the hedge ratio (delta), then if we buy h shares and sell 1 call, the payoffs will be:
hSU-C(u) if the share price rises
hSD-C(d) if the share price falls
However, we know that the two payoffs must be the same:
hSU-C(u) = hS(d)-C(d)
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An easy derivation of the binomial model
Solving for h we get the familiar:
Now, the return to this risk free investment, as we have seen, must be the risk free rate-
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An easy derivation of the binomial model
Choosing the payoff to the higher share price (which in the riskless hedge is the same as the payoff to the lower share price):
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Hence:
An easy derivation of the binomial model
Rearranging, we have:
Hence
The unknowns are h and c but we know the formula for h, so:
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An easy derivation of the binomial model
Substituting for h:
A bit of fiddling with the algebra gives:
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An easy derivation of the binomial model
Where p is the equation we came across earlier:
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The Binomial Model
Apart from the above risk neutral probabilities the BOPM is defined in terms of S, X, T, R and the upward and downward parameters u and d.
The first four are observable, u & d need to be estimated:
CRR obtain:
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The Binomial Model
d is equal to 1/u.
Intuitively d and u are determined from the stock price volatility, whether it be historical or implied (see later)
By inserting the appropriate values for u and d, which can be altered at different parts of the tree if desired, any progression of prices can be simulated.
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The Binomial Model
The above equations for p, u and d fully define any binomial option pricing tree.
So the value of a call option given one discrete time period is:
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