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I want to value a special type of call option on a stock. It's like a regular european vanilla call, but with the added rule that if a certain event occurs (that is approx 10% probability) then they must sell the option for the whichever is lowest of (i) Current option price (ii) Option price it was bought for.

This rule makes the option less valuable. I wonder how much less?

I'm familiar with black scholes, GBM and monte carlo pricing of options.

Tools I have: Excel and R.

Edit 1: The event is independent of the stock price. The mean daily return is 2% and the yearly volatility is 29%.

Edit 2: The option matures in 3.25 years. Assume the event takes place with 10% probability on year 1 and year 2 and year 3.

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  • $\begingroup$ Hi, what exactly is your question? To tell you how much less valuable the option is we need a lot more input. Also this would strongly depend on the dependencies of this event and underlying stock price. $\endgroup$ – Cettt Feb 2 '18 at 10:20
  • $\begingroup$ I would like a formula for how much less the option is worth. Parameters for black scholes are: yearly vol = 29%, daily mean return = 2% $\endgroup$ – jacob Feb 2 '18 at 10:35
  • $\begingroup$ Such a formula depends very on the event! For example if the event is independent of the underlying Brownian Motion and can only happen at one certain point in time $U$ which is before the maturity of the option, then $$ E[Adj Option Price] = 0.9*C(0) - 0.1* E[(C(U) - C(0))^+], $$ where C(0) is the time 0 option price, and C(U) is the future time-U option price. If you do not specify anything more, then it will be difficult to answer. $\endgroup$ – Cettt Feb 2 '18 at 13:24
  • $\begingroup$ It is unclear whether the "(ii) option price bought on" is the price of the option including the added rule value. Can you clarify? This changes the complexity of the problem significantly in my view. $\endgroup$ – Daneel Olivaw Feb 2 '18 at 14:19
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    $\begingroup$ @DaneelOlivaw yes this is unclear. I changed the wording. You can think of it being 2 steps: firstly the price of a plain vanilla option is $P_0$. Secondly, the holder of this modified option gets $min[P_0, P_t]$ $\endgroup$ – jacob Feb 2 '18 at 14:34
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Pretty complex, but here's a way to simplify: this option is effectively a standard maturity $T$ European call option $C_T$ minus a compound call on that call option with strike $k_c = PV_{t_0}(C_T)$ that is exercised only if your event $E$ occurs at a time $\tau \leq T$ i.e. $CoC_\tau.1_{\tau \leq T}$ where $CoC_\tau=Max(PV_\tau(C_T)-k_c,0)$.

So "all" you need to do to value the PV adjustment $A_{PV}$ is dig up a compound call option approximation (e.g. in Haug's Complete Guide to Option Pricing Formulas) and plug that into a formula of the form (assuming independence between $E$ and $S$, and in discrete time formulation for practicality -you can do this at daily or weekly or monthly points in practice).

$A_{PV} = \Sigma_{i=1...N}PV_{t_0}(CoC_{t_i}).p(\tau = t_i)$

And your product should be worth $C_T - A_{PV}$

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Off the top of my head this seems like a solution to the equation:

E[Adj Option Price] = P(no event) * Option Price + P(event) * E[loss on event]

Then you need to calculate those items either numerically or analytically. if the probability of event is variable you also have to take the expectation over the RHS if it impacts the expected loss on event non-linearly. This might happen if you are modelling the event with, say a Poisson distribution.

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  • $\begingroup$ Does this approach really work? I think it leads to: E[loss] = E[Option price in 1 year] - Price of option today = 0 $\endgroup$ – jacob Feb 2 '18 at 12:12
  • $\begingroup$ if std option price is 100. 50% chance of discrete event midway through option period. On the event either the option price has fallen, no mtm hit, or it has gained and you forced to sell at 100, that is a loss. As example if the expected loss conditional on the event is 5 (parametrised by vol, vol of vol, and distribution) then you have 50% * 100 + 50% * -5 = 97.5. I am assuming and no doubt it matters that the event is independent of any pricing parameters. This is a fast answer and I admit I haven't though very deeply about it but I cannot see a reason why this would not work $\endgroup$ – Attack68♦ Feb 2 '18 at 12:38
  • $\begingroup$ The event is independent of the stock price $\endgroup$ – jacob Feb 2 '18 at 12:43
  • $\begingroup$ Didn't realise I couldn't edit comments, but above it should read 47.5 not 97.5. Another way of considering it is in the limits: if the probability of event is zero then you should expect the price of the adjusted option to be same as the standard option. If the probability of event is 100% then you are more than likely to face the scenario of consistently losing money (ignoring theta impacts to option price over time) therefore you would actually want to be paid to own this product, the price would be a negative number. $\endgroup$ – Attack68♦ Feb 2 '18 at 12:55
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Idea of pricing method: Take historical stock prices, estimate $\hat \mu$ and $\hat \sigma$. Specify strike and 3 years maturity for a plain vanilla call option to get the price $P_0$. Put these parameters in a Geometric Brownian Motion (GBM) to get $M$ different stock paths. For example M=1000. Maturity is 3 years. Put these price paths into the matrix S.mat which has M rows and T=250*3=750 columns.

Let's split up time intwo three steps: year 1, 2 and 3 which stops at T=250, T=500 and T=750 respectively.

There are M different paths and each one has their own list of stock prices. For example path $p=1$ has the following list: $(S^1_{t=0}, S^1_{t=1}, ..., S^1_{t=750})$ and it can be found in the first row of S.mat. Taking the stdev() of each path -- each row in the matrix -- gives a list of volatilities: $(\hat \sigma^1, \hat \sigma^2, ...., \hat \sigma^M)$.

After 1 year we have a list of stock prices that day: $(S^1_{t=250}, S^2_{t=250}, ..., S^M_{t=250})$. This is the 250th column of S.mat.

Now we can use black scholes:

  • We have a list of current stock prices (the 250th column of S.mat).
  • We have a list of volatilities (the stdev of each row in S.mat).
  • So we can get a list of M number of black scholes prices out of these two lists. All we need to do is to specify $K$ and time left (it is 2.25 years left). Each volatility $\hat \sigma^p$ in that list gives an option price at day 250: $P^1_{t=250}, P^2_{t=250}, ... P^p_{t=250}$. The mean of this list is $\bar P_{t=250}$.

The rule can be stated as: if an event U occurs, the owner of the option is forces to sell it for the least of $P_0$ and whatever the price of the option is that day, which in expectation is $\bar P_{t=250}$. This option has some real value and some time value.

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