![]() ![]() Suppose we sell 300 calls and buy 100 shares. The number of units of the underlying purchased for each option sold would be the hedge ratio: n = (c + - c -) / (S + - S -) = 0.3333. Investors would exploit this opportunity by selling the option and buying the underlying. Suppose the option of the previous example is selling for $3 - a clear case of price not equaling value. This is the same value we ended up with using the expectations approach. To guarantee the $25 outcome, investor would have to buy 1/3 share of the stock and sell 1 call option. The present value of the guaranteed $25 to be received in six months is: $25 / 1.06 0.5 = $24.28. How much should you pay for this risk-free position? Guaranteed outcome is $25 for this portfolio, regardless of the value of the stock at expiration. To see that this is true, consider the position of the portfolio at the expiration of the call if the investor writes 1 call: In the example, to form a perfectly hedged portfolio, an investor needs to buy 1/3 of a share of stock for each call that's written (sold), or buy 1 share of stock and sell (write) 3 calls. Regardless of which way the underlying moves, the portfolio value should be the same ( perfectly hedged). If stock price = S +, TV + = n S + - c +.If stock price = S -, TV - = n S - c.Let TV denote the terminal value of the portfolio at expiration. Use the hedge ratio to construct a portfolio of stocks and calls in which terminal payoff is state-independent. Calculate the hedge ratio (shares per call). Both American-style options and European-style options can be valued based on the no-arbitrage approach. This approach is used for option valuation and is built on the key concept of the law of one price, which says that if two investments have the same future cash flows regardless of what happens in the future, then these two investments should have the same current price. Therefore, today's value of the 1-period option is $2.385. Then compute the expected value of the call option.Ĭ 0 = / (1.06) 0.5 = $2.385 In general, European-style options can be valued based on this approach.Ĭompute risk-neutral probabilities of up and down states. The discount rate is the risk-free rate and the expectation is based on the risk-neutral probability measure. In this approach the option value is determined as the present value of the expected future option payouts. Risk-free rate of return = r = 6% (discrete and annual)īased on this information, tree diagrams for the stock value and call option payoffs (state-dependent) would be drawn as follows:.Time to expiration = T = 1/2 year (6 months).Exercise price of call option = $85 (X).Stock price at expiration = $90 (S +) or $75 (S -).We assume that the stock price will only take two possible values at the expiration date of the option. We start off by having one binomial period for a European call option. n = (c + - c -) / (S + - S -) (the hedge ratio: the number of shares of stock per option to hedge).c 0 = / (1 + r) (the price of the call option). ![]() π = (1 + r - d) / (u - d) (risk-neutral "up" probability). ![]() d = (S - / S 0) ("down-state" price relative).u = (S + / S 0) ("up state" price relative).c - = Max (0, S - X) (call price if the stock price goes down: "down-state").c + = Max (0, S + - X) (call price if the stock price goes up: "up state").r = risk-free rate, assume 1 compounding period.Progressive calculation of option value at each earlier node the value at the first node is the value of the option.Calculation of option value at each final node.Option valuation using this method is a three-step process: The valuation process is iterative, starting at each final node and working backwards through the tree to the first node (valuation date), where the calculated result is the value of the option. ![]() This price evolution forms the basis for the option valuation. Each node in the lattice represents a possible price of the underlying, at a particular point in time. It uses a discrete-time framework to trace the evolution of the option's key underlying variable via a binomial lattice (tree), for a given number of time steps between valuation date and option expiration. In finance, the binomial options model provides a generalisable numerical method for the valuation of options. ![]()
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