University:
Boston University
Course:
CAS EC 203 | Empirical Economics 1
Academic year:

2024
Views:

467

Pages:

16
Author:

eishuma5iuk

Option Pricing: Further developments 1 Topics for Today • Risk Neutral Probabilities • Binomial Option Pricing for an Arbitrary Term • The Black-Scholes Model 2 1. Risk Neutral Probabilities • We have seen that the actual probabilities of the good (g) and bad (b) events do not enter into the option pricing formula, because the value is based on the absence of arbitrage between a call option and a replicating portfolio • Yet, option pricing heavily relies on probability results to derive convenient expressions for option values. • We now will explore why this is so, using our example from last time. 3 Option pricing formula • Current option price depends on future option price, as follows R − b Vg g − R Vb = V + g −b R g −b R Vg Vb = q + (1 − q ) R R • This is a mathematical expression equivalent to the idea concept that the option’s current value is equal to the discounted, expected future value of the option with “probabilities” q and 1-q. 4 Application to two period model 5 Application to two period model = s 2 (g, = g) Vgg max(0, Pgg − E ) = s 1 ( g , b= ) Vgb max(0, Pgb − E ) = s 1 (b, g= ) Vbg max(0, Pbg − E ) = s 0 (b, b= ) Vbb max(0, Pbb − E ) 6 Application to two period model V [q 2 max(0, Pgg − E ) + 2q (1 − q ) max(0, Pgb − E ) + (1 − q ) 2 max(0, Pbb − E )] / ( R 2 ) 7 2. Valuation for many periods 1 V ( P, E , n) = Rn 1 = Rn n n! j n− j j n− j q q Pg b − E )] ( ) (1 ) max(0, − ∑ j = 0 j !( n − j )! n n! j n− j j n− j q q Pg b − E )] ( ) (1 ) [ − ∑ j = a j !( n − j )! where a is minimum number of upward moves if stock is to finish "in the money" 8 Dynamic hedging • Hedge ratio evolves through time, as term shortens and as stock price change h( P, E , n) = Vg −Vb P ( g −b ) from last time = V ( Pg , E ,n P−1)( g−−Vb()Pb , E ,n −1) 9 More generally • Risk neutral probability construct is a powerful one for mathematics of option valuation • Counterpart of portfolio replication concept • Backward induction can be used to value much more complicated options, with additional sources of time variation 10 3. Black-Scholes • The most famous option pricing formula • A limit of the binominal option pricing model when number of periods (n) per unit of time is made large σ t/n = g e= and b e −σ µ 1 p= [1 + t / n ]; σ 2 t/n µ= mean; σ = std dev 11 Black-Scholes formula V =Φ P (d1 ) − Ee rT Φ (d 2 ) log( P / E ) + (r + σ 2 / 2)T d1 = σ T log( P / E ) + (r − σ 2 / 2)T d2 = σ T 12 Hedge ratio • Delta hedging: if writing a call, you should buy enough stock to protect oneself against changes (delta) in the option value • Before it depended on Vu, Vb, P, g and b • Here it is Φ(d1), but this depends on similar factors 13 Limits to Black-Scholes • Need to put in volatility (σ): huge industry for estimating time-varying volatility • Based on normally distributed log returns • But stock returns have “fat tails”, particularly some types of stocks • BS is close to accurate for “at the money options”, but problematic for situations in which P is far below E 14 4. Other topics • Creating a synthetic put is just like creating a synthetic call, but with transactions reversed. • The linkage between puts is governed by a relationship called put call parity. If you buy a call (getting stock upside) and sell (write) a put (getting stock downside) then you are essentially holding the stock. 15 • Understanding puts is particularly important for understanding current developments. The owner of a home – in a nonrecourse state like California -- has (a) bought an asset; (b) created a bond, and (c) bought a put on the asset, with the right to sell it if the price declines a lot. Correspondingly, a mortgage issuer has bought a bond and written a put. 16