Primbs, MS&E Applications of the Linear Functional Form: Pricing Exotics

Primbs, MS&E Exotics Digitals Asians Barrier Lookbacks American Digitals Black Scholes Dividends Early cash flows

Primbs, MS&E The Black-Scholes formula: where under Q Basically, we just have to calculate the expectation by brute force! This time we use risk neutral pricing for a European call option:

Primbs, MS&E The Black-Scholes formula: under Q So: But so by symmetry where where N( ) is the distribution function for a standard Gaussian. Now, note that under Q we have:

Primbs, MS&E The Black-Scholes formula: Let’s also rewrite this in terms of z T. We actually just did that calculation: where corresponds to Brownian motion. This looks like a Gaussian with mean  T.

Primbs, MS&E The Black-Scholes formula: This looks like a Gaussian with mean  T. where z T is Gaussian with mean  T and variance T. is standard Gaussian where by symmetry

Primbs, MS&E The Black-Scholes formula: where: distribution function for a standard Normal (i.e. N(0,1))

Primbs, MS&E under Qor Question:Where does the Black-Scholes partial differential equation come from in this framework???? But we know that where under Q So by Ito’s lemma: Hence: The Black-Scholes pde! We know that all portfolios must earn the risk free rate under the risk neutral probabilities Q.

Primbs, MS&E Exotics Digitals Asians Barrier Lookbacks American Digitals Black Scholes Dividends Early cash flows

Primbs, MS&E Continuous dividends If the stock pays a continuous dividend at rate q then: So That is, the price of the stock has mean return (r-q). So, under Q, the stock price follows: If we purchase 1 share at time 0. Price = S 0 We have e qT shares at time T: Payoff = e qT S T Under reinvestment in the stock

Primbs, MS&E Continuous dividends Hence, we could compute the price of a European call option as: where under Q where: distribution function for a standard Normal (i.e. N(0,1))

Primbs, MS&E Exotics Digitals Asians Barrier Lookbacks American Digitals Black Scholes Dividends Early cash flows

Primbs, MS&E t = 0 t = 1 t = 2 t = 3 t = 4 t = 5 t = 6 What do we do with early cash flows: Reinvest in the bond! We can use our pricing formula at time T: at time t at time T Reinvest in the bond becomes We just discount from time t!

Primbs, MS&E Early cash flows are just discounted from the time they occur. This makes sense, because there was nothing particularly special about the time we called T. Furthermore, if we hold an asset, we could decide to sell it at any time of our choosing, and not wait until a specified expiration or maturity date. Our pricing formula must hold no matter what time we decide to sell, otherwise we could arbitrage by selling at smart times...

Primbs, MS&E Exotics Digitals Asians Barrier Lookbacks American Digitals Black Scholes Dividends Early cash flows

Primbs, MS&E European Digital (Binary) Option: where under Q K Payoff payoff = STST Risk Neutral Pricing: But: where (We did this calculation for the Black-Scholes Formula!) So: Cash or nothing call

Primbs, MS&E European Digital (Binary) Option: where under Q K payoff = STST Risk Neutral Pricing: So: Asset or nothing call K But: where (We did this calculation for the Black-Scholes Formula!)

Primbs, MS&E It should be clear that a standard European call option is a portfolio of 1 asset-or-nothing call and –K cash-or-nothing calls. One difficulty with digital options is that they have discontinuous payoffs. Theoretically, this is not a problem. However, in practice it can make hedging these options very difficult and risky. Notes:

Primbs, MS&E Exotics Digitals Asians Barrier Lookbacks American Digitals Black Scholes Dividends Early cash flows

Primbs, MS&E Asian Options: Note that under Q, S follows: or hence whereis Gaussian Recall that Asian options involve averages. Normally, we do not have closed form solutions for Asian options. The exception is when the option is on a geometric average: so I T is actually log-normally distributed! Geometric Average:

Primbs, MS&E If we can calculate the mean and variance of then we know everything about I T. Mean:

Primbs, MS&E If we can calculate the mean and variance of then we know everything about I T. Variance : T Tt s t > s by symmetry

Primbs, MS&E If we can calculate the mean and variance of then we know everything about I T. Variance : T Tt s t > s by symmetry 0 by independent increment.

Primbs, MS&E If we can calculate the mean and variance of then we know everything about I T. So, the geometric average I T is where Therefore, to price an average price call whose payoff is: we just need to compute: where You may check that this is the same as if I t follows the geometric Brownian Motion: in a risk neutral world. Which just looks like Black-Scholes on an asset paying a continuous dividend of and with volatility of

Primbs, MS&E Arithmetic Asian Options: The majority of Asian options involve arithmetic means: In this case, I T, is not log-normally distributed, and hence we cannot fit it into the Black-Scholes formula framework. However, it is common to compute the first two moments of I T and assume that its distribution is log-normal with the same first two moments. In this case, the Black-Scholes formula provides a quick and closed form approximation to the true price. This is sometimes referred to as the method of moments in pricing.

Primbs, MS&E Exotics Digitals Asians Barrier Lookbacks American Digitals Black Scholes Dividends Early cash flows

Primbs, MS&E Barrier Options x Knock-outs: The option is worthless if it hits the barrier. Knock-ins: The option is worthless until it hits the barrier. Barrier options are like normal European options, except that they are either activated, or become worthless when the underlying asset hits a pre-specified barrier. The basic types are: Also, the barrier can be hit on the way down (down-and-out, down-and-in) or it can be hit on the way up (up-and-out, up-and-in).

Primbs, MS&E Lookback Options max time 0 time T Lookback options depend on the maximum or minimum price achieved during the life of the option. A European lookback call option pays off: A European lookback put option pays off: To price these, we need to be able to compute the statistics of the maximum and minimum...

Primbs, MS&E American Digitals K time 0 time T American digital options payoff $1 the moment the strike price is hit. To evaluate this, we need to know the statistics of the hitting time... $1 where  represent the first hitting time of the strike price K.

Primbs, MS&E The statistics that I need to know for Barriers, Lookbacks, and American digitals are all related: (Densities can then be obtained by differentiation) The: max = Probability that the max is greater than B first hitting time . = Probability that the time to hit B is less that T B time 0 time T Probability that a barrier B is hit before time T

Primbs, MS&E The key trick here is the “reflection principle”. For every path that crosses the barrier but ends up below the barrier, there is a reflected path that ends up above the barrier. x C The Reflection Principle B time 0 time T Basic Idea: The only twist that we have to deal with is that the probability of the original path and reflected path may be different. However, if we start with Brownian motion, we don’t have this problem. 2B-C

Primbs, MS&E The reflection principle for Brownian motion: B x C time 0 time T B time 0 time T Assume that is a Brownian motion  Denote Then for C < B, let’s calculate: 2B-C 2B-dC dC x

Primbs, MS&E The reflection principle for Brownian motion with drift: Assume that Denote Then for C < B, let’s calculate: Use Girsanov to change the mean of z  to zero B x C time 0 time T 2B-C

Primbs, MS&E In a similar manner: Assume: (i.e. S follows a geometric Brownian motion) etc... where Then you can calculate: These provide the building blocks for closed form solutions...

Primbs, MS&E The calculations become quite messy, so I won’t go through them The main points are: There are closed form solutions for many barriers and lookbacks. American digitals almost have a closed form solution (you need to evaluate an integral by quadrature). These are a consequence of risk neutral pricing combined with the reflection principle. For some options, risk neutral pricing provides the most direct route to closed form solutions. Consult a good text (such as Hull) for the messy formulas...