﻿ 2.4 Future Value

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2.4 CALL OPTION VALUATION:  A RISKLESS HEDGE APPROACH

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n the one-period binomial world, the stock either moves up or down from its current price.  Let u > 1 be the uptick, d < 1 be the downtick, and S be the current stock price.

If an uptick is realized, the end-of-period stock price is Su.  Otherwise, a downtick is realized, and the end-of-period stock price is Sd. You may recall from topics 2.2 and 2.3, the Riskless Hedge ExampleRHE_BIN and  Synthetic Option ExampleSOE_BIN, that in valuing the option you do not need to know the probability of the stock moving up or down.

Let us now consider how to formulate the general case for the one-period option pricing problem.  First, we will need some notation.

Notation

S   = current stock price;

Su = future high stock price (call this State H) ;

Sd = future low stock price (call this State L) ;

r    = 1 + the risk-free interest rate;

X   = strike price;

C  = current price of the call option, which is to be determined.

We will assume that u > r > d.  This is actually necessary to prevent arbitrage (if r > u, then you should sell the stock and invest the proceeds in the risk-free asset; if d > r, you should borrow at the risk-free rate and buy the stock).

We start with the call option.  The terminal values of the call are:

If both Cu and Cd are zero, then the call option has no value, so suppose that Cu  > 0 and you have a portfolio of +1 stock and -k calls.  The future payoffs from this portfolio can be depicted as follows in Figure 2.4:

Figure 2.4

Future Payoffs

The Hedge Ratio (k)

For a portfolio to be riskless, we have to choose k so that the payoff in both states is equal:

In this case we have a risk-free portfolio.  This requires

which is called the hedge ratio.

The Riskless Hedged Portfolio:  Call Options

The portfolio of one stock and k calls, where k is the hedge ratio, is called the riskless hedged portfolio.  The hedge ratio, k, tells you that  for every stock you hold, k call options must be sold.  The riskless (call option) portfolio is:

S - kC

The Cost of the Riskless Hedge

The cost of acquiring this portfolio today is S  - kC.  Since

the  end-of-period portfolio value is known with certainty.

Since the future value is riskless, the present value equals the future value discounted at the risk-free interest rate.  The end-of-period payoff can be defined from either the up- or downtick, because both are the same.  So let us fix this at the realized uptick value

By substituting for k, we can solve for the value of the call option C.

This gives us the price of the call option as a function of the current stock price, the future stock values, the strike price, and the risk-free interest rate.

Consider the example, where X = 20, S = 20, Su = 40, Sd = 10, and  one plus risk-free interest rate r = 1, so

which gives 2S - 3C = 20 so C = (2S-20)/3, just as before in topic 2.2 the Riskless Hedge Example.

The riskless hedge portfolio approach to pricing put options is described in the next topic titled Put Option Valuation:  A Riskless Hedge Approach.